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Lectures on QuantumMechanics forMathematics Students

STUDENT MATHEMATICAL LIBRARYVolume 47

Lectures on QuantumMechanics forMathematics StudentsL. D. FaddeevO. A. Yakubovskii

Translated byHarold McFaden

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Editorial BoardGerald B. Folland Brad G Osgood (Chair)Robin Forman :Michael Starbird

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2000 Mathematics Subject Classification. Primary 81-01, 8lQxx.

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Library of Congress Cataloging-in-Publication DataFaddeev, L D

[T,ektsii po kvantovoi mekhanike dlia studentov-tnatematikov English]Lectures on quantum mechanics for mathematical students / 1. D Facldeev,

0 A Yakubovskii [English ed ]p cin (Student mathematical library ; v 47)

ISBN 978-0-8218-4699-5 (alk paper)1 Quantum theory T Iakubovskii, Oleg Aleksandrovich TI Title

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Contents

Preface

Preface to the English Edition

ix

xi

§1. The algebra of observables in classical mechanics 1

§2. States 6

§3- Liouville's theorem, and two pictures of motion inclassical mechanics 13

§4. Physical bases of quantum mechanics 15

§5. A finite-diniensional model of quantum mechanics 27

§6. States in quantum mechanics 32

§7. Heisenberg uncertainty relations 36

§8 Physical meaning of the eigenvalues and eigenvectors ofobservables 39

§9. Two pictures of motion in quantum mechanics. TheSchrodinger equation Stationary states 44

§10. Quantum mechanics of real systems. The Heisenbergcommutation relations 49

§11. Coordinate and momentum representations 54

§12. "Eigenfunctions" of the operators Q and P 60

§13. The energy. the angular momentum, and other examplesof observables 63

v

Vi Contents

§ 14 The interconnection between quantum and classicalmechanics. Passage to the limit from quantumrriechanics to classical mechanics 69

15. One-dimensional problems of quantum mechanics. Afree one-dimensional particle 77

D6 The harmonic oscillator 83

§17 The problem of the oscillator in the coordinaterepresentation 87

18. Representation of the states of a one-dimensionalparticle in the sequence space 12 90

§19. Representation of the states for a one-dimensionalparticle in the space V of entire analytic functions 94

§20. The general case of one-dimensional motion 95

§21. Three-dimensional problems in quantum mechanics. Athree-dimensional free particle 103

§22 A three-dimensional particle in a potential field 104

§23 Angular momentum 106

§24. The rotation group 108

§25. Representations of the rotation group 111

§26. Spherically symmetric operators 114

§27. Representation of rotations by 2 x 2 unitary matrices 117

§28. ]Representation of the rotation group on a space of entireanalytic functions of two complex variables 120

§29. Uniqueness of the representations D 123

§30. Representations of the rotation group on the spaceL 2 (S2). Spherical functions 127

§31 The radial Schrodinger equation 130

§32. The hydrogen atom. The alkali metal atoms 136

§33. Perturbation theory 147

§34. The variational principle 154

§35. Scattering theory. Physical formulation of the problem 157

§36 Scattering of a one-dimensional particle by a potentialbarrier 1 59

Contents vii

§37. Physical meaning of the solutions V), and i'2 164

§38 Scattering by a rectangular barrier 167

§39. Scattering by a potential center 169

§40. Motion of wave packets in a central force field 175

§41. The integral equation of scattering theory 181

§42. Derivation of a formula for the cross-section 183

§43. Abstract scattering theory 188

§44. Properties of commuting operators 197

§45. Representation of the state space with respect to acomplete set of observahles 201

§46. Spin 203

§47. Spin of a system of two electrons 208

§48. Systems of many particles. The identity principle 212

§49. Symmetry of the coordinate wave functions of a systemof two electrons. The helium atom 215

§50. Multi-electron atoms. One-electron approximation 217

§51. The self-consistent field equations 223

§52. Mendeleev's periodic system of the elements 226

Appendix: Lagrangian Formulation of Classical Mechanics 231

Preface

This textbook is a detailed survey of a course of lectures given inthe Mathematics-Mechanics Department of Leningrad University formathematics students. The program of the course in quantum me-chanics was developed by the first author. who taught the course from1968 to 1973. Subsequently the course was taught by the second au-thor. It has certainly changed somewhat over these years, but its goalremains the same: to give an exposition of quantum mechanics froma point of view closer to that of a mathematics student than is com-mon in the physics literature. We take into account that the studentsdo not study general physics. In a course intended for mathemati-cians. we have naturally aimed for a more rigorous presentation thanusual of the mathematical questions in quantum mechanics, but notfor full matherriatical rigor, since a precise exposition of a number ofquestions would require a course of substantially greater scope.

In the literature available in Russian, there is only one bookpursuing the same goal, and that is the American mathematicianG. W. Mackey's book, Mathematical Foundations of Quantum Me-chanics. The present lectures differ essentially from Mackey's bookboth in the method of presentation of the bases of quantum mechan-ics and in the selection of material. Moreover, these lectures assumesomewhat less in the way of mathematical preparation of the stu-dents. Nevertheless, we have borrowed much both from Mackey's

ix

x Preface

book and from von Neumann's classical book, Mathematical Founda-tions of Quantum Mechanics.

The approach to the construction of quantum mechanics adoptedin these lectures is based on the assertion that quantum and classi-cal mechanics are different realizations of one and the same abstractmathematical structure. The feat ures of this structure are explainedin the first few sections, which are devoted to classical mechanicsThese sections are an integral part of the course and should not beskipped over, all the more so because there is hardly any overlap ofthe material in them with the material in a course of theoretical me-chanics. As a logical conclusion of our approach to the constructionof quantum rnecha nics, we have a section devoted to the interconnec-tion of quantum and classical mechanics and to the passage to thelimit from quantum mechanics to classical mechanics.

In the selection of the material in the sections devoted to appli-cations of quantum mechanics we have tried to single out questionsconnected with the formulation of interesting mathematical problems.Much attention here is given to problems connected with the theoryof group representations and to mathematical questions in the theoryof scattering. In other respects the selection of material correspondsto traditional textbooks on general questions in quwitum mechanics,for example, the books of V. A. Fok or P. A. M. Dirar.

The authors are grateful to V. M. Bahich, who read through themanuscript and made a number of valuable comments.

I, D. Faddeev and 0 A. Yakubovskii

Preface to the EnglishEdition

The history and the goals of this book are adequately described inthe original Preface (to the Russian edition) and I shall not repeatit here. The idea to translate the book into English came from thenumerous requests of my former students, who are now spread overthe world. For a long time I kept postponing the translation becauseI hoped to be able to modify the book making it more informativeHowever, the recent book by Leon Takhtajan, Quantum Mechanicsfor Mathematicians (Graduate Studies in Mathematics, Volume 95,American Mathematical Society, 2008), which contains most of thematerial I was planning to add, made such modifications unnecessaryand I decided that the English translation can now be published.

Just when the decision to translate the book was made, my coau-thor Oleg Yakubovskii died. He had taught this course for morethan 30 years and was quite devoted to it. lie felt compelled to addsome physical words to my more formal exposition The Russiantext, published in 1980, was prepared by him and can be viewed asa combination of my original notes for the course and his experienceof teaching it. It is a great regret that he will not see the Englishtranslation.

xi

xii Preface to the English Edition

Leon '17akhtajan prepared a short appendix about the formalismof classical mechanics. It should play the role of introduction for stu-dents who did riot lake an appropriate course. which was obligatoryat St. Petersburg University

I want to acid that the idea of introducing quantum mechanicsas a deformation of classical mechanics has become quite fashionablenowadays Of course. whereas the term "deformation'' is not usedexplicitly in the book. the idea, of deformation was a guiding principlein the original plan for the lectures.

L. D.FaddeevSt Petersburg. November 2008

§ 1. The algebra of observables in classicalmechanics

We consider the simplest problem in classical mechanics: the problemof the motion of a material point (a particle) with mass m in a forcefield V (x), where x(x2 . x2, x3) is the radius vector of the particle. Theforce acting on the particle is

F= -radV_--aVax

The basic physical characteristics of the particle are. its coordi-nates xr, x2, x3 and the projections of the velocity vector v(v1. V2, v3).All the remaining characteristics are functions of x and v; for exam-ple, the momentum p = mv, the angular momentum I = x x prnx x v, and the energy E = mv2 /2 + V (x) .

The equations of motion of a material point in the Newton formare

1dv aV dx

{ }mdt

ax ' dtIt will be convenient below to use the momentum p irr place of thevelocity v as a basic variable. In the new variables the equations ofsnot ion are written as follows-follows-

dpOV dxp( 2)

dt ax' dt m

Noting that = N and Lv = N where H = V (x) is theera C9p dx ft 2mI Iamilt onian function for a particle in a potential field. we arrive atthe equations in the Hamiltonian form3 dxOH dpOHH

do ap ' dt ax

It is known from a course in theoretical mechanics that a broadclass of mechanical systems, and conservative systems in particular,

I

2 L. D. Faddeev and 0. A. Yakubovskii

are described by the Hamiltoniann equationsall OH

(4) q1= a , pi=-- , i-1.2,.. npz qz

Here H =- 1I(g1, .. , q,1: pl ... . p,,) is the Hamiltonian function, qjand pi are the generalized coordinates arid momenta. and n is calledthe number of degrees of freedom of the system. We recall that, for aconservative system, the Hamiltonniaxn function H coincides with theexpression for the total energy of the system in the variables qi and pi.We write the Hamiltonian function for a system of N material pointsinteracting pairwise.

N 2 N N

(5) H = 1: "+ 1: Vi.AXi - Xj) + Vi(xi)

2mzi=l i<j z-1

Here the Cartesian coordinates of the particles are taken as the gener-alized coordinates q, the Yiumber of degrees of freedom of the systemis n = 3N, and Vi.? (xi -- x j) is the potential of the interaction of theU b and jth particles. The dependence of Vi j only on the differencexi - xj is ensured by Newton's third law. (Indeed, the force acting onthe ith particle due t o the jth particle is Fi j = -- =

J_ -F ji . )OX,

The pot exitials Vi (Xi) describe the interaction of the ith particle withthe external field The first t ernn in (5) is the kinetic energy of thesystem of particles

For any mechanical system all physical characteristics are func-tions of the generalized coordinates and momenta. We introduce theset % of real infinitely differentiable functions f (q1, ... , qn: pI, .... p,z),which will be called observables.1 The set % of observahles is obvi-ously a linear space and forms a real algebra with the usual additionand multiplication operat ions for functions. The real 2n-dixrnensionalspace with elements (q1, ... , q ; pl ..... p,,) is called the phase spaceand is denoted by M. Thus, the algebra of observahles in classicalmechanics is the algebra of real-valued smooth functions defined onthe phase space M.

We shall introduce in the algebra of observahles one more opera-tion. which is connected with the evolution of the mechanical system

1 we do not discuss the question of introducing a topology in the algebra of ob-serva,bles Fortunately, most physical questions do not depend on this topology

§ 1. The algebra of observables in classical mechanics 3

For simplicity the exposition to follow is conducted using the exampleof a system with one degree of freedom The liamiltoriian equationsin this case have the form

(6)OH OH

q = Op p =c9q

H = H(9,P)-

The Cauchy problem for the system (6) and the initial conditions

(7) qlt=o qo, pl t=o = pc

has a unique solution

(8) q == q(qo, po, t) p == p(qo, po, t ).

For brevity of notat ion a point (q. p) in phase space will sometimeshe denoted by p, and the Iiamiltonian equations will he written inthe form

(9) ft = v(p),

where v (p) is the vector field of these equations, which assigns to eachpoint p of phase space the vector v with components 211 r

OHOp aq

The Ilamiltonian equations generate a one-parameter commut,ative group of t ransforniations

Gt : M ---> M

of the phase. space into itrself,2 where Gt1z is the solution of the l{arnil-tonian equations with the initial condition GtpIt=o = p. We have theequalities

(10) Gt+s = GtG8 = GSGt, Gt C t.In turn, the transformations GL generate a family of transformations

Ut : 21 --a 21

of the algebra of observables into itself, where

(11) Ut f (p) = ft (p) = f (Gtp)

In coordinates, the fuiictiori ft (q, p) is defined as follows:

ft (qO, po) = f (q (qO. po. t), p ((10, po, t))-

2 V4'e assume, that the Hamiltonian equations with initial conditions (7) have aunique solution on the whole real axis It is easy to construct examples in whicha global solution and, correspondingly, a group of transformations C,, do not existThese cases arc' not interesting, arid we do not consider them

4 L. D. Faddeev and 0. A. YakubovskiT

We find a differential equation that the function ft (q. p) satisfies.To this end. we differentiate the identity f,-.-t(y) = f t (G sp) withrespect to the variable s and set s = 0

a fs-+ t (L) I aft (11)as

dfc(Gsµ)as

s=O at

= O ft (µ) ' vW =aft ax Oft 81I

=oaq

Op Op aq

Thus, the function ft (q, p) satisfies the differential equation

aft aH aft aH aft(13) at a p O p

and the initial condition

(14) ft (q p) It=() = f (q, p).

The equation (13) with the initial condition (14) has a unique solu-tion, which can be obtained by the formula (12): that is, to constructthe solutions of (13) it suffices to know the solutions of the Haniil-tonian equations.

We can rewrite (13) in the form

L(1`) dt

={H,ft}.

where {H, ft } is the Poisson bracket of the functions 11 and ft. Forarbitrary observables f and g the Poisson bracket is defined by

Of 0.9 of agIf, aq aOp q p

and in the case of a system with ra degrees of freedom

" of ag of agff,gl = Ei=1 dpc jqi aqi dpi

We list the basic properties of Poisson bracket s:

1) {f, g -i- Ah) = {f, g I -f- A { f j h I (linearity) :

2) {f,g} = - {g,f} (skew symmetry) :

3) {f, {g. h } } + {g, {h.f}} + {h. {f.g}} = 0 (Jacobi identity).

4) {f.gh} = g{ f. h.} + { f, g}h.

§ 1. The algebra of observables in classical mechanics 5

The. properties 1), 2), and 4) follow directly from the definitionof the Poisson brackets. The property 4) shows that the "Poissonbracket" operation is a derivation of the algebra of observables. In-deed, the Poisson bracket can be rewritten in the form

{f,g} =Xfg,

where X f = p -- aq dp is a first-order linear differential operator,and the property 4) has the form

X fgh = (X1g)h + gX fh.

The property 3) can be verified by differentiation, but it can be provedby the following argument. Each term of the double Poisson bracketcontains as a factor the second derivative of one of the functionswith respect to one of the variables; that is, the left-hand side of3) is a linear homogeneous function of the second derivatives. Onthe other hand, the second derivatives of h can appear only in thesump {f, {g, h J} + {g, {h, f } } = = (X1Xq -- X y X f) h, but a commutatorof first-order linear differential operators is a first-order differentialoperator, and hence the second derivatives of h do not appear in theleft-hand side of 3). By symmetry, the left-hand side of 3) does notcontain second derivatives at all: that is, it is equal to zero.

The Poisson bracket {f, g} provides the algebra of observableswith the structure of a real Lie algebra.3 Thus, the set of observableshas the following algebraic structure. The set 22 is:

1) a real linear space,

2) a commutative algebra with the operation f g;

3) a Lie algebra with the operation {f, g}.

The last two operations are connected by the relation

If, gh) = If, g1h + gj f, h}.

The algebra 2i of observables contains a distinguished element,namely, the Hamiltonian function H, whose role is to describe the

3 We recall that a linear space with a binary operation satisfying the conditions1)-3) is called a Lie algebra


variation of observables with time:dft - Hdt ~{ ,.ft}.

We show that the mapping Ut : % 2i preserves all the opera-tions in 2i:

h = f +g--+ht=ft+9t,h = fg-*ht=ftgt,h={f,g} -+ht ={ft,gt};

that is, it is an automorphism of the algebra of observables. Forexample, we verify the last assertion. For this it suffices to see thatthe equation and the initial condition for ht is a consequence of theequations and initial conditions for the functions ft and gt:

Oht oft agtat = , gt + 1f t ,

= {{H, ft}, gt} + Ift, {H, gt}} _ {H, Ift, gt}} _ {H, ht}.

Here we used the properties 2) and 4) of Poisson brackets. Further-more,

htlt=c = {ft,gt}It=o = {f,g}.Our assertion now follows from the uniqueness of the solution of theequation (13) with the initial condition (14).

§ 2. States

The concept of a state of a system can be connected directly with theconditions of an experiment. Every physical experiment reduces toa measurement of the numerical value of an observable for the sys-tem under definite conditions that can be called the conditions of theexperiment. It is assumed that these conditions can be reproducedmultiple times, but we do not assume in advance that the measure-Tnent will give the same value of the observable when the experimentis repeated. There are two possible answers to the question of how toexplain this uncertainty in the results of an experiment.

1) The number of conditions that are fixed in performing theexperiments is insufficient to uniquely determine the results of themeasurement of the observables. If the nonuniqueness arises only

§ 2. States 7

for this reason, then at least in principle these conditions can besupplemented by new conditions, that is, one can pose the experimentmore "cleanly", and then the results of all the measurements will beuniquely determined.

2) The properties of the system are such that in repeated exper-iments the observables can take different values independently of thenumber and choice of the conditions of the experiment.

Of course if 2) holds, then insufficiency of the conditions can onlyincrease the nonuniqueness of the experimental results. We discuss1) and 2) at length after we learn how to describe states in classicaland quantum mechanics.

We shall consider that the conditions of the experiment determinethe state of the system if conducting many repeated trials under theseconditions leads to probability distributions for all the observables.In this case we speak of the measurement of an observable f for asystem in the state w. More precisely, a state w on the algebra 2t ofobservables assigns to each observable f a probability distribution ofits possible values, that is, a measure on the real line R.

Let f be an observable and E a Borel set on the real line R. Thenthe definition of a state w can be written as

f7 E wt (E)

We recall the properties of a probability measure:

0 <, wf (E) <, 1 1 wf (0) = 0, wf (R) = 1,

and if E1 f1 E2 = 0, then w f(E1 U E2) = w f(E1) + w f(E2).

Among the observables there may be some that are functionallydependent, and hence it is necessary to impose a condition on theprobability distributions of such observables. If an observable cp isa function of an observable f, Sp = cp(f ), then this assertion meansthat a measurement of the numerical value of f yielding a value fo isat the same time a measurement of the observable ;p and gives for itthe numerical value cpo = p (f o) . Therefore, w f (E) and wp(f) (E) areconnected by the equality

(2) Ww(f ) (E) = wf (W-'(E)) 7


where cp-1 (E) is the inverse image of E under the mapping ;p-

A convex combination

(3) w1(E) = awl f(E) +0 - a)w2 f(E). 0 < a < 1,

of probability measures has the properties (1) for any observable fand corresponds to a state which we denote by

(4) w=awl +(1--a)w2.

Thus, the states form a convex set. A convex combination (4) ofstates wl and W2 will sometimes be called a mixture of these states.If for some state w it follows from (4) that w1 = w2 = w, then wesay that the state w is indecomposable into a convex combination ofdifferent states. Such states are called pure states, and all other statesare called mixed states.

It is convenient to take E to be an interval (-oo, A] of the realaxis. By definition, w f (A) = w f ((-oo, A] ), and this is the distributionfunction of the observable f in the state w. Numerically, w f (A) is theprobability of getting a value not exceeding A when measuring f inthe state w. It follows from (1) that the distribution function w f (A)is a nondecreasing function of A with w f (- oo) = 0 and w f (+ oo) = 1.

The mathematical expectation (mean value) of an observable fin a state w is defined by the formula4

(f I w) = I__00 A dwf (A).

We remaxk that knowledge of the mathematical expectations forall the observables is equivalent to knowledge of the probability dis-tributions. To see this, it suffices to consider the function 9(A - f) ofthe observables, where O (x) is the Heaviside function

fi, x > 0,

0, x<0.

It is not hard to see that

(5) wf (A) = (O(A - f) I w).

4The notation (f I w) for the mean value of an observable should not be confusedwith the Dirac notation often used in quantum mechanics for the scalar product (p, 0)of vectors

§ 2. States 9

For the mean values of observables we require the following con-ditions, which are natural from a physical point of view:

1) (CIw)=C,(6) 2) (f + Ag I uj) = (f I w) + A(g I uj),

3) (f2Jw)>0.

If these requirements are introduced, then the realization of thealgebra of observables itself determines a way of describing the states.Indeed, the mean value is a positive linear functional on the algebra% of observables. The general form of such a functional is

(7) (11w)= JAM .f(p,q)dluw(p,q),

where dpi, (p, q) is the differential of the measure on the phase space,and the integral is over the whole of phase space. It follows from thecondition 1) that

(a)JAM

dp,a (p, q) = pw (M) = 1.

We see that a state in classical mechanics is described by speci-fying a probability distribution on the phase space. The formula (7)can be rewritten in the form

(9) (1 I w) = fMA .f (p, q) pw (p, q) dp dq;

that is, we arrive at the usual description in statistical physics of astate of a system with the help of the distribution function p, (p, q),which in the general case is a positive generalized function. Thenormalization condition of the distribution function has the form

(10)IM

P(P, 4) dP d9' = 1.

In particular, it is easy to see that to a pure state there corre-sponds a distribution function

P(p, 9) = a(q - Po) a(p - Po),

where d(x) is the Dirac 6-function. The corresponding measure onthe phase space is concentrated at the point (qo, po), and a pure stateis defined by specifying this point of phase space. For this reason the

10 L. D. Faddeev and 0. A. Yakubovskil

phase space M is sometimes called the state space. The mean valueof an observable ,f in the pure state w is

(12) (f w) = f (go,po)

This formula follows immediately from the definition of the 6-function:

(13) f (qo, po) = fM f (q, p) 6(q - qo) b(p -- po) dq dp.

In mechanics courses one usually studies only pure states, whilein statistical physics one considers mixed states, with distributionfunction different from (11). But an introduction to the theory ofmixed states from the very staxt is warranted by the following cir-cumstances. The formulation of classical mechanics in the languageof states and observables is nearest to the formulation of quantummechanics and makes it possible to describe states in mechanics andstatistical physics in a uniform way. Such a formulation will enableus to follow closely the passage to the limit from quantum mechanicsto classical mechanics. We shall see that in quantum mechanics thereare also pure and mixed states, and in the passage to the limit, apure quantum state can be transformed into a mixed classical state,so that the passage to the limit is most simply described when pureand mixed states are treated in a uniform way.

We now explain the physical meaning of mixed and pure states inclassical mechanics, and we find out why experimental results are notnecessarily determined uniquely by the conditions of the experiment.

Let us consider a mixture

w=awl+(1-a)w2i 0<a<1,of the states wl and w2. The mean values obviously satisfy the formula

(14) (f I W) = Q(f I W1) + (1 - COU I W2)-

The formulas (14) and (3) admit the following interpretation. Theassertion that the system is in the state w is equivalent to the assertionthat the system is in the state wl with probability a and in the stateW2 with probability (1 - a). We remark that this interpretation ispossible but not necessary.

§ 2. States 11

The simplest mixed state is a convex combination of two purestates:

P(4, p) = aS(9 - 9i) bhp - Pi) + (1- a) 8(9 - 92) b(P - Pi)

Mixtures of n pure states are also possible:

p(q, p) =n

ai6(q -- qi) 6(p -- pi), ai > 0, > aj = 1.

Here ai can be interpreted as the probability of realization of the purestate given by the point qi, pi of phase space. Finally, in the generalcase the distribution function can be written as

p(q,p) = ZM p(qo, po) o(q _ qo) 6(p -- po) dqo dpo

Such an expression leads to the usual interpretation of the distribu-tion function in statistical physics: ff p(q, p) dq dp is the probabilityof observing the system in a pure state represented by a point in thedomain 11 of phase space. We emphasize once more that this interpre-tation is not necessary, since pure and mixed states can be describedin the framework of a unified formalism.

One of the most important characteristics of a probability distri-bution is the variance

(15) Owf = ((f- (fI.))2I) = (f2w) - (fIw)2.We show that mixing of states leads to an increase in the variance. Amore precise formulation of this assertion is contained in the inequal-ities

(16)

(17)

Aw f f + (1 - a)Aw2foWfAW9 > a&, f &1,,9 + (1 - a)L 2fO129,

with equalities only if the mean values of the observables in the statesw2 and W2 coincide:

(f I W1 ) = (f I W2), (9 1 W1) = (9 1 W2) -

In a weakened form, (16) and (17) can be written as

(18) 02 f %In1II(O2Wlf7 Ow2f)l

(19) O.a.fAm9' II11n(0,1.fOwi9, Owz.fAw29)

12 L. D. Faddeev and 0. A. Yakubovski

The proof uses the elementary inequality

(20) a + (1 - a)x2 - (a + (1 - a)X)2 U. -oc < x < 00.

with

(21) 0(1) = 0. p(X) > 0 for x 1.

Using (14) and (20); we get that,

olf = (I2 I W)- (I I w)2

= aq2 I Wi) + (1- a)(.f2 I L02)- [a(f lwi) + (1 - a)(.f I w2)]2

a(j' I wi) + (1 - a) (f2 I Wa) - a(f I wi)2 - (1- a)(.f l W2)2aA2 f + (1 _ (,k)A2

W1 W2

The inequality (16) is proved. The inequality (17) is a consequenceof (16). Indeed,

i . f I . g>(cil.f +(I -a)022f)(aA2,g+(1-x)022.9)[azf&1g + (1 - a)A"2fA&2g]2.

For pure states,

p(q, p) = 5(q - q0) a(p - po),

(f I w) = f (qo, po).

A2 f = f2(go,po) - f2(go,po) = 0;that is, for pure states in classical mechanics the variance is zero. Thismeans that for a system in a pure state, the result of a measurement ofany observable is uniquely determined. A state of a classical systemwill be pure if by the time of the measurement the conditions ofthe experiment fix the values of all the generalized coordinates andmomenta. It is clear that if a macroscopic body is regarded as amechanical system of N molecules, where N usually has order 1023,then no conditions in a real physical experiment can fix the values ofqo and po for all molecules, and the description of such a system withthe help of pure states is useless. Therefore, one studies mixed statesin statistical physics.

Let us summarize. In classical mechanics there is an infinite setof states of the system (pure states) in which all observables havecompletely determined values. In real experiments with systems of a

§ 3. Liouville's theorem, and two pictures of motion 13

huge number of particles. mixed states arise. Of course, such statesare possible also in experiments with simple mechanical systems. Inthis case the theory gives only probabilistic predictions.

§ 3. Liouville's theorem, and two pictures ofmotion in classical mechanics

We begin this section with a proof of an important theorem of Liou-ville. Let 11 be a domain in the phase space M. Denote by f(t) theimage of this domain under the action of a phase flow, that is, theset of points Gtp, /2 E Q. Let V(t) be the volume of 1(t). Liouville'stheorem asserts that

dV(t) o

dt

Proof.

v(t) = d1i _ r D(Geu)Jst D(µ)

dy, dit = dq dp.

Here D (Gt tt) / D (u) denotes the Jacobi determinant of the transfor-mation Gt. To prove the theorem, it suffices to show that

D(Gt p)(1) D (IL)

for all t. The equality (1) is obvious for t = 0. Let us now show that

2( ) dt D(µ)

For t = 0 the formula (2) can be verified directly:

d D(Gtµ)dt D(µ)

do

[D(,p) + D(Q, P)

D(9,p) D(4, P) t=oq a2x a2x(8a7Y

aq + ap) 1t=0 aqap-

c9paq- o.

d D(Gcµ) = 0.

For t # 0 we differentiate the identity

D(Gc+Bµ) D(Ge+Bµ) D(Geµ)D(µ) D(Giµ) D(µ)


with respect to s and set s = 0, getting

d D(Ctµ) _ Id D(G.Geµ) D(GcIj,) _ 0.dt D(p) Ldt D(Gtµ) c=u D(µ)

Thus, (2) holds for all t. The theorem is proved.

We now consider the evolution of a mechanical system. We areinterested in the time dependence of the mean values (f I w) of the ob-servables. There are two possible ways of describing this dependence,that is, two pictures of the motion. We begin with the formulationof the so-called Hamiltonian picture. In this picture the time depen-dence of the observables is determined by the equation (1.15),5 andthe states do not depend on time:

4!={Hft} = 0.dt

The mean value of an observable f in a state w depends on the timeaccording to the formula

(3) (ft 1w) = ft (i) p(p) dju = f(Gt u) p(i) d p,

or, in more detail,

(4) (ft I=fM

f(4(9o,Post),p(9o,po, t))P(9o,Po)dQodPo

We recall that q(qo, po, t) and p(qo, po, t) are solutions of the liamil-tonian equations with initial conditions qlt=v = Qo, plt=o = po

For a pure state p(q, p) = S(y - go)8(p - po), and it follows from(4) that

(ft I w) = f (q(qo, po, t), p(qo, po, t) ).

This is the usual classical mechanics formula for the time dependenceof an observable in a pure state.6 It is clear from the formula (4)that a state in the Hamiltonian picture determines the probabilitydistribution of the initial values of q and p.

5In referring to a formula in previous sections the number of the correspondingsection precedes the number of the formula

6In courses in mechanics it is usual to consider only pure states Furthermore, nodistinction is made between the dependence on time of an abstract observable in theHamiltonian picture and the variation of its mean value

§ 4. Physical bases of quantum mechanics 15

An alternative way of describing the motion is obtained if in (3)we make the change of variables Gtµ -> µ. Then

fM f (Gtµ) p(p) dp = JM f (µ)DD(A) (G-tp)

dµ

= f .f (y) pt (,u) dp = (11 wt)M

Here we have used the equality (1) and we have introduced the no-tation pt(p) = p(G_tµ). It is not hard to see that pt (p) satisfies theequation

(5) dot = -{H, pt},

which differs from (1.15) by the sign in front of the Poisson bracket.The derivation of the equation (5) repeats that word-for-word forthe equation (1.15), and the difference in sign arises because G_tµsatisfies the Hamiltonian equations with reversed time. The picture ofthe motion in which the time dependence of the states is determinedby (5), while the observables do not depend on time, is called theLiouville picture:

df =0 dpt = - H .dt dt { ' Pt}

The equation (5) is called Liouville's equation. It is obvious from theway the Liouville picture was introduced that

(ftI) = (1 IWt).This formula expresses the equivalence of the two pictures of motion.The mean values of the observables depend on time in the same way,and the difference between the pictures lies only in the way this de-pendence is described. We remark that it is common in statisticalphysics to use the Liouville picture.

§ 4. Physical bases of quantum mechanics

Quantum mechanics is the mechanics of the microworld. The phe-nomena it studies lie mainly beyond the limits of our perception, andtherefore we should not be surprised by the seemingly paradoxicalnature of the laws governing these phenomena.


It has not been possible. to formulate the basic laws of quantummechanics as a logical consequence of the results of some collectionof fundammiental physical experiments. In other words, there is so farno known formulation of quantum mechanics that is based on a sys-tem of axioms confirmed by experiment. Moreover, sortie of the basicstatements of quantum mechanics are in principle not amenable toexperimental verification. Our confidence in the validity of quantummechanics is based on the fact that all the physical results of thetheory agree with experiment. Thus, only consequences of the ba-sic tenets of quantum mechanics can be verified by experiment, andnot its basic laws. The main difficulties arising upon an initial studyof quantum mechanics are apparently connected with these circum-stances.

The creators of quantum mechanics were faced with difficultiesof the same nature, though certainly much more formidable. Exper-iments most definitely pointed to the existence of peculiar quantumlaws in the microworld, but gave no clue about the form of quantumtheory. This can explain the truly dramatic history of the creationof quantum mechanics and, in particular, the fact that its originalformulations bore a purely prescriptive character. They containedcertain rules making it possible to compute experimentally measur-able quantities, but a physical interpretation of the theory appearedonly after a mathematical formalism of it had largely been created.

In this course we do not follow the historical path in the con-struction of quantum mechanics. We very briefly describe certainphysical phenomena for which attempts to explain them on the basisof classical physics led to insurmountable difficulties. We then try toclarify what features of the scheme of classical mechanics describedin the preceding sections should be preserved in the mechanics of themicroworld and what can and must be rejected. We shall see thatthe rejection of only one assertion of classical mechanics, namely, theassertion that observables are functions on the phase space, makes itpossible to construct a scheme of mechanics describing systems withbehavior essentially different from the classical. Finally, in the follow-ing sections we shall see that the theory constructed is more generalthan classical mechanics, and contains the latter as a limiting case.


Historically, the first quantum hypothesis was proposed by Planckin 1900 in connection with the theory of equilibrium radiation. Hesucceeded in getting a formula in agreement with experiment for thespectral distribution of the energy of thermal radiation, conjecturingthat electromagnetic radiation is emitted and absorbed in discreteportions, or quanta, whose energy is proportional to the frequency ofthe radiation:

(1) E = hw,

where w = 2irv with v the frequency of oscillations in the light wave,and where h = 1.05 x 10-27 erg sec is Planck's constant.7

Planck's hypothesis about light quanta led Einstein to give anextraordinarily simple explanation for the photoelectric effect (1905).The photoelectric phenomenon consists in the fact that electrons areemitted from the surface of a metal under the action of a beam oflight. The basic problem in the theory of the photoelectric effectis to find the dependence of the energy of the emitted electrons onthe characteristics of the light beam. Let V be the work required toremove an electron from the metal (the work function) . Then the lawof conservation of energy leads to the relation

hw=V+T,where T is the kinetic energy of the emitted electron. We see thatthis energy is linearly dependent on the frequency and is independentof the intensity of the light beam. Moreover, for a frequency w < V/h(the red limit of the photoelectric effect), the photoelectric phenom-enon becomes impossible since T > 0. These conclusions, based onthe hypothesis of light quanta, are in complete agreement with ex-periment. At the same time, according to the classical theory, theenergy of the emitted electrons should depend on the intensity of thelight waves, which contradicts the experimental results.

Einstein supplemented the idea of light quanta by introducingthe momentum of a light quantum by the formula

(2) p = hk.

Tin the older literature this formula is often written in the form F. = hv, wherethe constant h in the latter formula obviously differs from the h in (1) by the factor2 ir


Here k is the so-called wave vector, which has the direction of prop-agation of the light waves. The length k of this vector is connectedwith the wavelength A, the frequency w, and the velocity c of light bythe relations

21r w(3) k

C

For light quanta we have the formula

E = pc,

which is a special case of the relativity theory formula

E = Vp2C2 + m2c4

for a particle with rest mass m 0.

We remark that the historically first quantum hypotheses in-volved the laws of emission and absorption of light waves, that is,electrodynamics, and not mechanics. However, it soon became clearthat discreteness of the values of a number of physical quantities wastypical not only for electromagnetic radiation but also for atomic sys-tems. The experiments of Franck and Hertz (1913) showed that whenelectrons collide with atoms, the energy of the electrons changes indiscrete portions. The results of these experiments can be explainedby the assumption that the energy of the atoms can have only defi-nite discrete values. Later experiments of Stern and Gerlach in 1922showed that the projection of the angular momentum of atomic sys-tems on a certain direction has an analogous property. It is nowwell known that the discreteness of the values of a number of ob-servables, though typical, is not a necessary feature of systems in themicroworld. For example, the energy of an electron in a hydrogenatom has discrete values, but the energy of a freely moving electroncan take arbitrary positive values. The mathematical apparatus ofquantum mechanics had to be adapted to the description of observ-ables taking both discrete and continuous values.

In 1911 Rutherford discovered the atomic nucleus and proposeda planetary model of the atom (his experiments on scattering of aparticles on samples of various elements showed that an atom has apositively charged nucleus with charge Ze, where Z is the number ofthe element in the Mendeleev periodic table and e is the charge of an


electron, and that the size of the nucleus does not exceed 10-12 cm,while the atom itself has linear size of order 10-8 cm). The plane-tary model contradicts the basic tenets of classical electrodynamics.Indeed, when moving around the nucleus in classical orbits, the elec-trons. like all charged particles that are accelerating, must radiateelectromagnetic waves. Thus, they must be losing their energy andmust eventually fall into the nucleus. Therefore, such an atom cannotbe stable, and this, of course, does not correspond to reality. One ofthe main problems of quantum mechanics is to account for the stabil-ity and to describe the structure of atoms and molecules as systemsconsisting of positively charged nuclei and electrons.

The phenomenon of diffraction of microparticles is completelysurprising from the point of view of classical mechanics. This phe-nomenon was predicted in 1924 by de Broglie, who suggested thatto a freely moving particle with momentum p and energy E therecorresponds (in some sense) a wave with wave vector k and frequencyw. where

p=hk, E=hw;

that is, the relations (1) and (2) are valid not only for light quantabut also for particles. A physical interpretation of de Broglie waveswas later given by Born, but we shall not discuss it for the present. Ifto a moving particle there corresponds a wave, then, regardless of theprecise meaning of these words, it is natural to expect that this impliesthe existence of diffraction phenomena for particles. Diffraction ofelectrons was first observed in experiments of Davisson and Germerin 1927 Diffraction phenomena were subsequently observed also forother particles.

We show that diffraction phenomena are incompatible with clas-sical ideas about the motion of particles along trajectories. It is mostconvenient to argue using the example of a thought experiment con-cerning the diffraction of a beam of electrons directed at two slits," thescheme of which is pictured in Figure 1. Suppose that the electrons

8Such an experiment is a thought experiment, so the wavelength of the electronsat energies convenient for diffraction experiments does not exceed 10-7 cm, and thedistance between the slits must be of the same order In real experiments diffractionis observed on crystals. which are like natural diffraction lattices


Ay

A

I

IU

C

Figure 1

from the source A move toward the screen B and, passing throughthe slits I and II, strike the screen C.

We are interested in the distribution of electrons hitting the screenC with respect to the coordinate y. The diffraction phenomena on oneand two slits have been thoroughly studied, and we can assert thatthe electron distribution p(y) has the form a pictured in Figure 2 ifonly the first slit is open, the form b (Figure 2) if only the second slitis open, and the form c occurs when both slits are open. If we assumethat each electron moves along a definite classical trajectory, then theelectrons hitting the screen C can be split into two groups, dependingon the slit through which they passed. For electrons in the first groupit is completely irrelevant whether the second slit was open or not,and thus their distribution on the screen should be represented by the

yb

y

P(y)

Figure 2


curve a; similarly, the electrons of the second group should have thedistribution b. Therefore, in the case when both slits are open, thescreen should show the distribution that is the sum of the distribu-tions a and b. Such a sum of distributions does not have anything incommon with the interference pattern in c. This contradiction indi-cates that under the conditions of the experiment described it is notpossible to divide the electrons into groups according to the test ofwhich slit they went through. Hence, we have to reject the conceptof trajectory.

The question arises at once as to whether one can set up an ex-periment to determine the slit through which an electron has passed.Of course, such a formulation of the experiment is possible; for thisit suffices to put a source of light between the screens B and C andobserve the scattering of the light quanta by the electrons. In order toat tain sufficient resolution, we have to use quanta with wavelength oforder not exceeding the distance between the slits, that is, with suffi-ciently large energy and momentum. Observing the quanta scatteredby the electrons, we are indeed able to determine the slit throughwhich an electron passed. However, the interaction of the quantawith the electrons causes an uncontrollable change in their momenta,and consequently the distribution of the electrons hitting the screenmust change. Thus, we arrive at the conclusion that we can answerthe question as to which slit the electron passed through only at thecost of changing both the conditions and the final result of the exper-iment.

In this example we encounter the following general peculiarity inthe behavior of quantum systems. The experimenter does not havethe possibility of following the course of the experiment, since to do sowould lead to a change in the final result. This peculiarity of quantumbehavior is closely related to peculiarities of measurements in the mi-croworld Every measurement is possible only through an interactionof the system with the measuring device. This interaction leads toa perturbation of the motion of the system. In classical physics onealways assumes that this perturbation can be made arbitrarily small,as is the case for the duration of the measurement. Therefore, the


simultaneous measurement of any number of observables is alwayspossible.

A detailed analysis (which can be found in many quantum me-chanics textbooks) of the process of measuring certain observablesfor microsystems shows that an increase in the precision of a nieasurexrient of observables leads to a greater effect on the system, andthe rrieasurenient introduces irncont rollable changes in the numericalvalues of some of the other observables. This leads to the fact, thata simultaneous precise measurement of certain observables becomesimpossible in principle. For example, if one uses the scattering of lightquanta to measure the coordinates of a particle, then the error of themeasurement has the order of the wavelength of the light: Ox ti A. Itis possible to increase the accuracy of the measurement by choosingquanta with a smaller wavelength, but then with a greater momeri-turn p = 27rh/A. Here an uncontrollable change Lip of the order ofthe momentum of the quantum is introduced in the numerical val-«es of the momentum of the particle. Therefore, the errors Ax andOp in the measurements of the coordinate and the momentum are,connected by the relation

Lax Op ^, 27Th.

A more precise argument shows that this relation connects only acoordinate and the momentum project ion with the same index. Therelations connecting the theoretically possible accuracy of a simul-taneous measurement of two observables are called the Heisenberguncertainty relations. They will be obtained in a precise formtrlat ionin the sections to follow. Observables on which the uncertainty rela-tions do not impose any restrictions are simultaneously measurable.We shall see that the Cartesian coordinates of a particle are shnulta-neously measurable. as are the projections of its momentum. but thisis not so for a coordinate and a momentumi projection with the. sameindex, nor for two Cartesian projections of the angular momentum.In the construction of quantum mechanics we must remember thepossibility of the existence of quantities that are not simultaneouslymeasurable


After our little physical preamble we now try to answer the ques-tion posed above. what features of classical mechanics should be kept,,and which ones is it natural to reject in the construction of a mechan-ics of the microworld? The main concepts of classical mechanics werethe concepts of an observable and a state. The problem of a phys-ical theory is to predict results of experiments, and an experimentis always a measurement of some characteristic of the system or anobservable under definite conditions which determine the state of thesystem. Therefore, the concepts of an observable and a state mustappear in any physical theory. From the point of view of the ex-perimenter, to determine an observable means to specify a way ofmeasuring it. W e denote observables b y the symbols a, b, c, ... , andfor the present we do not make any assumptions about their mathe-matical nature (we recall that in classical mechanics the observablesare functions on the phase space). As before, we denote the set ofobservables by 2t.

It is reasonable to assume that the conditions of the experiment,determine at least the probability distributions of the results of mea-surements of all the observables, and therefore it is reasonable to keepthe definition in § 2 of a state. The states will be denoted by w asbefore, the probability measure on the real axis corresponding to anobservable a by wa, (E) ; the distribution function of a in the state wby WQ(A), and finally, the mean value of a in the state w by (w I a).

The theory must contain a definition of a function of an observ-able. For the experimenter, the assertion that an observable b isa function of an observable a (b = f (a)) means that to measure bit suffices to measure a, and if the number a{ is obtained as a re-sult of measuring a, then the numerical value of the observable b isbo = f (ao). For the probability measures corresponding to a andf (a), we have

(4) Wf (a) (E) = Wa Y I (E))

for any states w.

We note that all possible functions of a single observable a aresimultaneously measurable, since to measure these observables it suf-fices to measure a. Below we shall see that in quantum mechanics this


example exhausts the cases of simultaneous measurement, of observ-ables; that is, if the observables bl, b2, ... are simultaneously measur-able, then there exist an observable a and functions fl, f2.... suchthat bz = fi(a).b2 = f2(a)J.. 0 .

The set of functions f (a) of an observable a obviously includesf (a) = Aa and f (a) = const, where A is a real number. The existenceof the first of these functions shows that observables can be multipliedby real numbers. The assertion that an observable is a constant meansthat its numerical value in any state coincides with this constant.

We now try to make clear what meaning can be assigned to asum a+ b and product ab of two observables. These operations wouldbe defined if we had a definition of a function f (a, b) of two variables.I iowevetr, there arise fundamental difficulties here connected with thepossibility of observables that are not simultaneously measurable. Ifa and b are simultaneously measurable, then the definition of f (a, b)is completely analogous to the definition of f (a). To measure the ob-servable f (a, b), it suffices to measure the observables a and b leadingto the numerical value f (ac, bo), where ao and bo are the numericalvalues of the observables a and b, respectively. For the case of ob-servables a and b that are not simultaneously measurable, there is noreasonable definition of the function f (a, b). This circumstance forcesus to reject the assumption that observables are functions f (q, p) onthe phase space, since we have a physical basis for regarding q andp as not simultaneously measurable, and we shall have to look forobservables among mathematical objects of a different nature.

We see that it is possible to define a sum a + b and a productab using the concept of a function of two observables only in thecase when they are simultaneously measurable. However, anotherapproach is possible for introducing a sum in the general case. Weknow that all information about states and observables is obtainedby measurements; therefore, it is reasonable to assume that there aresufficiently many states to distinguish observables, and similarly, thatthere are sufficiently many observables to distinguish states.

More precisely, we assume that if

(alw)=(biw)


for all states w, then the observables a and b coincide,9 and if

(aiwi) = (aIw2)

for all observables a. then the states w1 and W2 coincide.

The first of these assumptions rriakes it possible to define the suma + b as the observable such that

(a+ bow) _ (alw)+(bjw)

for any state w. We remark at once that this equality is an expressionof a well-known theorem in probability theory about the mean valueof a sum only in the case when the observables a and b have a commondistribution function. Such a common distribution function can exist,(and in quantum mechanics does exist) only for simultaneously mea-surable quantities In this case the definition (5) of a sum coincideswith the earlier definition. An analogous definition of a product isnot possible, since the mean of a product is not equal to the productof the means, not even for simultaneously measurable observables.

The definition (5) of the suns does riot contain any indicationof a way to measure the observable a + b involving known ways ofmeasuring the observables a and b, arid is in this sense implicit.

To give an idea of how much the concept of a sum of observablescan differ from the usual concept of a sum of random variables, wepresent an example of an observable that will be studied in detail inwhat follows. Let

P2 w2 Q2H= 2 + 2

The observable H (the energy of a one-dimensional harmonic oscilla-tor) is the sum of two observables proportional to the squares of themomentum and the coordinate. We shall see that these last observ-ables can take any nonnegative numerical values, while the values ofthe observable H must coincide with the numbers E74 = (n + 1/2)w,where n = 0, 1, 2, ...: that is, the observable H with discrete numer-ical values is a sum of observables with continuous values.

Thus, two operations are defined on the set Qt of observables:multiplication by real numbers and addition, and 21 thereby becomes

This assumption allows us to regard an observable as specified if a real numberis associated with each state


a linear space. Since real functions are defined on 2t, and in particularthe square of an observable, there arises a natural definition of aproduct of observables:

(a + b)2 - (a - b)2(6) aob= 4 .

We note that the product aob is commutative. but it is not associativein general. The introduction of the product a o b turns the set 2t ofobservables into a real commutative algebra.

Recall that the algebra of observables in classical mechanics alsocontained a Lie operation- the Poisson bracket {f, g}. This opera-tion appeared in connection with the dynamics of the system. Withthe introduction of such an operation each observable 11 generates afamily of autornorphisms of the algebra of observables:

Ut : 2t - 2t,

where Utf = ft: and ft satisfies the equationdft = Mdt AI


ft It--o = f.

We recall that the mapping Ut is an automorphism in view of thefact that the Poisson bracket has the properties of a Lie operation.The fact that the observables in classical mechanics are functions onphase space does not play a role here. We assume that the algebraof observables in quantum mechanics also has a Lie operation; thatis, associated with each pair of observables a, b is an observable {a, b}with the properties

{a,b} = -{b, a},{)ta+b,c} = A{a,c} + {b,c},

{a,boc}={a,b}oc+bo{a,c},{a,{b,c}} + {b,{c,a}}+ {c,{a,b}} =0.

Moreover, we assume that the connection of the Lie operation with thedynamics in quantum mechanics is the same as in classical mechanics.It is difficult to imagine a simpler and more beautiful way of describingthe dynamics. Moreover, the same type of description of the dynamics

§ 5. A finite-dimensional model of quantum mechanics 27

in classical and quantum mechanics allows us to hope that we canconstruct a theory that contains classical mechanics as a limitingcase.

In fact, all our assumptions reduce to saying that in the construc-tion of quantum mechanics it is reasonable to preserve the structureof the algebra of observables in classical mechanics, but to reject therealization of this algebra as functions on the phase space since weadmit the existence of observables that are not simultaneously mea-surable.

Our immediate problem is to see that, there is a realization of thealgebra of observables that is different from the realization in classi-cal mechanics. In the next section we present an example of such arealization, constructing a finite-dimensional model of quantum me-chanics. In this model the algebra 2l of obscrvables is the algebraof self-adjoint operators on the n-dimensional complex space C. Instudying this simplified model, we can follow the basic features ofquantum theory At the same. time, after giving a physical interpre-tation of the model constructed, we shall see that it is too poor tocorrespond to reality Therefore, the finite-dimensional model cannotbe regarded as a definitive variant of quantum mechanics. However,it will seem very natural to improve this model by replacing C'z by acomplex flilbert space.

5. A finite-dimensional model of quantummechanics

We show that the algebra 2i of observables can be realized as thealgebra of self-adjoint operators on the finite-dimensional complexspace C".

Vectors in C' will be denoted by the Greek letters ;p,, ... .Let us recall the basic properties of a scalar product:

1) 0) = (01 0 1

2) + Aq, *) = (C 7 VY) + A (777 0) 1

3) C) > 0 if =14 0.

Here A is a complex number.


Vectors e1, ... , e, form an orthonormall ° basis in C7t if

(2) (ei, e j) = bi J,

where b;J is the Kronecker delta symbol.

The expansion of an arbitrary vector in the vectors of the basisel, ... , e,z has the form

71

(3) -- i =

rl'he vect or is uniquely determined by the numbers 1it ...

H 1. .. , ').

If one chooses a basis, then one thereby chooses a concrete realizationfor vectors, or one specifies a representation. Let e1, ... , e,, be a basis.Then the vectors

n

-° 1] Uikek,k=1

i=1.2,...,n,

also form a basis if the matrix U = {Ujk} is invertible and

(5)Uik1 = Uki.

A matrix whose elements satisfy the equality (5) is said to be unitary.The passage from one basis to another is a unitary transformation.

If a representation has been chosen, then the scalar product isgiven by the formula

(6) inii=1

In the given basis, operators are represented by matrices:

(7) A H {A2k}, Aik = (Aek, ei).

Indeed, if 17 = thenn n n

rIi = e) = (Akekei) = > k (Aek, ei) = E AikG-k=1 k=1 k=1

1oIn what follows, the word orthonormal will often be omitted, since we do notconsider other bases


An operator A* is called the adjoint of A if

(8) 17) = (A*

, n)

for any pair of vectors and q. Obviously. A i = Aki. An operatoris said to be self-adjoint if A* = A. For a self-adjoint operator,Ails = Aki . It follows immediately from the definition of the adjointof an operator that

(AB)* = B*A*, (cA)* = aA*,

where rt is a complex number.

We now construct a realization of the algebra of observables. Let,21 be the set of self-acljoint operators on C7t. In what follows, self-adjoint operators will often be called observables. The operationsof addition and multiplication by a number are defined on the set,of operators in the usual way. If A E 2i, B E %, and A E R, then(A+B) E % and AA E %, because (A+B)* = A+B and (AA)* = AA.It is natural to regard t hese operations as operations of addition ofobservables and multiplication by a number.

Our next problem is to find out how to construct functions ofobservables. It can be assumed here that the usual definition of afunction of an operator works. We can justify this assumption afterwe have learned how to construct probability distributions for ob-servables in quantum mechanics. Then we shall be able to verify theformula w f(A) (E) = wA(f -1(E)), which is equivalent to the generaldefinition of a function of an observable as given in the precedingsection.

We recall that there are several equivalent definitions of a functionof an operator. If f (x) has a power series expansion

x(10) f (X) _ E Cnxn

n=0

valid on the whole real axis, then f (A) is defined by the formula

xf (A) c,A

n=0


A second definition uses the existence for self-adjoint operators of ancigenvector basis-ii

Ajpi = aiSpi, i = 1.... , n.

Here the Spi are cigenvectors with (pj, ;p3) = bi j, and the ai are eigen-values of the operator A. To define a linear operator f (A) it sufficesto define the result of the action of f (A) on the vectors of a basis Bydefinition,

(12) f (A) pj = f (ai) oj-

In an eigenvector basis the matrix A is diagonal with the eigenval-ues on the diagonal, that is, A13 = aibij. In the same representation,[1(A)]13 = f (ai) bid . We remark that a self-ad j oint operator corre-sponds to a real function, that is, f (A) E %.

The operation A o B is defined by the formula (4.0), which forself-adjoint operators has the form

(A + B)2 - (A - B)2 _ AB + BA(13) A o B =

4 2

The self-adjointness of the operator A o B is obvious.

It remains for us to construct a Lie operation. To this end, weconsider the commutator [A, B] = AB - BA of operators A and B.The operation [A, B] has the following properties:

(14)

1) [A, B) = - [Bj A],

2) [A+.AB,C] _ [A, C] + A[B, C'],

3) [A,BoC]=[A,B]oC+Bo[A,CJ,4) [A, [B, C]] + [B, [C, A]] + [C, [A. B]] = 0.

All these properties can be verified directly. We remark that theproperty 3) is valid also for the nonsynarietrized product. Indeed,

[A, BC] = ABC - BCA + BAC - BAC = [A, B]C + B [A, C].

We see that the commutator has the properties of a Lie operation, but[A, B] is not self-adjoint, that is, [A, B] V Z. However, the expression

"We re( all that the eigenvalues of a self-adjoi nt operator are real and that eigen-vectors corresponding to different eigenvalues are orthogonal if an eigenvalue hasmultiplicity r, then there are r linearly independent eigenvectors corresponding to it,and they can always be chosen to be orthonornial


(i/h) [A, B], which differs from the conirnutator only by the imagi-nary factor i/h, does satisfy all the requirements. We remark thatalgebras % constructed with different constants h are not isomorphic.The choice of h can be made only after comparing the theory withexperiment. Such a comparison shows that h coincides with Planck'sconstant. In what follows we use the notation

(15) JAI B}h=i-[A,

BI

and call {A, B } h the quantum Poisson bracket.

It is interesting to note that in classical mechanics we could havereplaced the definition

{f,g}-8fdg afagap dq aq Op

of the Poisson bracket by the definition

{fig} = o8f dg - Ofaap dq aq ap

where a is a real constant. It is not hard to see, however, that inclassical mechanics this new definition of the Poisson bracket leadsto an algebra that is isomorphic to the original algebra. Indeed, thechange of variables p = a j p', q = sgn a f a j q' returns us to the olddefinition.

An important role in quantum mechanics is played by the traceTr A of an operator. By definition,

TrA = >Ajj -- (Ae,e).i=1 i=1

We recall the basic properties of this operation. The trace does notdepend on the choice of basis. In particular, if we take an eigenvectorbasis for an operator A, then

nIA'A=Eai;

i=x

that is, the trace is the sum of the eigenvalues. If aZ is a multipleeigenvalue, then it appears as a term in the sure as many times as itsmultiplicity.


The trace of the product of two operators does not depend on theorder of the factors:

Tr A B = Tr BA.

In the case of several fact ors, cyclic permutations can be taken underthe sign of Tr.

Tr ABC = '1'r BCA.

The trace of a self-adjoins matrix is a real number, since its eigenval-ues are real.

§ 6. States in quantum mechanics

In this section we show how states are given in quantum mechanics.We recall that we kept the definition in § 2 of a state. There we alsoshowed that specifying the probability distributions is equivalent tospecifying the mean values for all observables. The arguments thatled us to this result remain in force also in quantum mechanics sincet hey did not use a concrete realization of the algebra of observables ofclassical mechanics. To specify a state means to specify on the algebra% of observables a functional (w I A) with the following properties:

rii

1) (1\A+Bjw)=1\(Ajw)+(Bjw),

2) (421w)>O ,

(CIw)=C,3)

4) (AIw)=(AIw).

We have already discussed the property 1) . The property 2) expressesthe fact that for the observable A2, which has the sense of being non-negative, the mean value cannot be a negative number. The property3) asserts that the mean value of the constant observable C in anystate coincides with this constant. Finally, the property 4) says thatthe mean values are real. Thus, a state in quantum mechanics is apositive linear functional on the algebra 2 of self-adjoint operators.The general form of such a functional is

(2) (A I w) = Tr MA,

§ 6. States in quantum mechanics 33

where M is an operator on C' satisfying the conditions

1) M*=M,(3) 2)

3) Tr M = 1.

The operator M is called the density matrix, and in quantum me-chanics it plays the same role as the distribution function p(q, p) doesin classical mechanics.

We show that the properties of the density matrix follow fromthe properties formulated above of the averaging functional (A I U)).Indeed, since the functional (A I w) is real,

n n

Tr AM = Ails Mki = A*i Miki,k=1 i,k=1

n

_ 1: Aki *k = Tr AM* = Tr AM.i,k=1

Setting X = A, + i A2 , where A 1 and A2 are self-adj oint operators,we get from the last equality that

TXM=TrXM*,

where X is an arbitrary operator on Cn . The property 1) follows atonce from the arbitrariness of the operator X.

We now use the positivity of the functional (A I u)):

TrA2M0.

Let A = P,7, where 1-11 is the operator of projection on a normalized

vector r, (I I q I I = 1) :

Pn =(C'9)To compute the trace it is convenient to use any basis in which thefirst basis vector is,-r/ (e l = Ti, e2, ... , (2,L) , so that

Tr Pn M = Tr P,,M = (Mrf, 77) > 0.

We see that the positivity of the density matrix is a necessary con-dition for the positivity of the functional (A I w). The sufficiency is


verified as follows,.n n

Tr A2M = Tr AMA = E(AMAei, ei) = E(MAei, Aei) > 0,i=1 i=1

since each term under the summation sign is nonnegative

Finally, the normalization condition Tr M = 1 of the densitymatrix follows at once from the property 3) of the functional.

Thus, we have showed that states in quantum mechanics are de-scribed by positive self-adjoint operators with trace 1. (Recall that astate in classical mechanics is given by a nonnegative function p(q, p)normalized by the condition f p(q, p) dq dp = 1.)

Any operator M with the properties (3) describes some state ofthe system; that is, there is a one-to-one correspondence between theset of states and the set of density matrices M.

wHM.If Ml and M2 are operators with the properties (3), then it is obviousthat a convex combination

M=aM1+(1-a)M2, 0<a<1,of them also has these properties and hence corresponds to some state

w=awl+(1--a)w2.We see that the set of states in quantum mechanics is convex.

All the requirements of the density matrix are satisfied by a one-dimensional projection P (11011 = 1). Indeed,

1) (P1 ,rl)_(i1, P7l)2) 0,

=(b, )=1.3) TrPO

At the end of the section we show that the state P. cannot bedecomposed into a convex combination of different states; that is, itis a pure state (Recall that in classical mechanics a pure state hasa distribution function of the form p(q. p) = b(q - go)S(p - pc).) Wenote that a pure state is determined by the specification of a vector', but there is not a one-to-one correspondence between pure states

and vectors, because if 0' differs from 0 by a numerical factor with

§ 6. States in quantum mechanics 35

modulus 1, then P,,' = PW Thus. corresponding to a pure state is aclass of vectors with norm 1 that differs from each other by a factor(-iQwith aER.

The vector i' is usually called a state vector, and the space inwhich the self-adjoint operators (observables) act is called the statespace. For the present we take the space C" as the state space.

We show that any st ate in quantum mechanics is a convex com-bination of pure states. To this end, we observe that M, like any

operator, has an eigenvector basis:self-elf

Mcpi =

Let us expand an arbitrary vector in this basis and act on it by theoperator M:

n

- (i=1

n n

-- Pi v, Vii) Oi = > pi Poi .i-I i=1

Since is arbitrary,

(4) M =EIuij SOi

i=1

All the numbers lui are nonnegative since they are eigenvalues of apositive operator. Moreover. Enj=j /,ti = Tr M = 1, and hence thestate M is indeed a convex combination of the pure states P,,i . Forthe mean value of an observable in a pure state w P, , the forrrlula(2) takes the form

(5) (AIw) = (A), I1'1I = 1.

For the case of a mixed state w M pi Pit , we haven

r /(6) (A I w) = 1:,ui

i-1

This formula shows that. as in classical mechanics, the assertion thata system is in a mixed state M = E i r 1-i PS,, is equivalent to theassertion that the system is in the pure state P., with probabilityifori=1,...,n.


In concluding this section, we show that the state described by thedensity matrix M = P is pure. We have to show that the equality

(7) 'V =aM1+(1-o)M2, 0<a <1,implies that M1 = M2 = Pv.

For the proof we use the fact that for a positive operator A andany vectors ;p and

(8) I(A)(Ap).

It follows from (8) that AW = 0 if (Acp, gyp) = 0. (The inequality (8) isa condition for the positivity of a Hermitian form: (A(ap + 60). app +bib) = (Ap,p)a+ (Acp, V))ab + (Aip.p)ab+ (AiP,ib)bb > 0,where aand b are complex numbers.)

Let RI be the subspace orthogonal to the vector ib. Then P.0 = 0if V E f l . Using (7) and the positivity of the operators Ml and M2,we have

0 < a (Ml p, P) < a (Ml 0, p) + (1 -- a) (M2 gyp, Sp) = 0:

that is, M1 p = 0. Since M1 is self-adjoint, we get for any vectorthat

OERJJ

hence, M1 = Q V), and in particular, Mxib = C,, Here C#) is aconstant depending on the vector An arbitrary vector can berepresented in the form

ipExz,hence, Ml = CJP that is, All = C,,, Pg,. Finally, we get from thecondition Tr M1 = 1 that C. b = 1, and thus M1 = Pp and M2 = P,.

The converse can also he proved If a state is pure, then thereexists a vector 0 E Cn with I I V 1 I = 1 such that M = Per,P.

§ 7. Heisenberg uncertainty relations

In this section we show how the uncertainty relations mentioned in4 follow from the mathematical apparatus of quantum mechanics,

and we give them a precise formulation.

§ 7. Heisenberg uncertainty relations 37

Let us consider the variance of two observables A and B in a statew. We recall that the variance of an observable is defined by

A2 A = (w I (A - Aavg) 2) _ (w I A2) - (w I A)2'

where Aavg = (w I A) . Let Ow A be denoted by Ow A .

The uncertainty relations assert that for any state w

1 A, A B >w w IIt

suffices to prove the uncertainty relations for pure states. Formixed states they will then follow from the inequality (2.17), whichholds also in quantum Ynechanics, since its derivation does not dependon the realization of the algebra of observables.

We begin with the. obvious inequality

((A + jaB),O, (A+iaB)iP) > 0, 11,011=17 a E R.

Expanding the left-hand side, we get that

(A2b,i') + a2(B2 b, s) + ia((AB --- BA) ,') > 0.Using the definition (5.15) of the quantum Poisson bracket and theformula (6.5) for the mean value of an observable in a pure state

we rewrite. the last inequality in the form

(A21w)+a2(B21w)-ah({A,B}hlw) > 0

for any a E R, and hence

(A21w)(B21w)>h ({A,B}hIw)2.

The uncertainty relations (1) follow immediately from this inequalityif we make the substitutions A A - Aavg, B B - and takeinto account that {A, B}h = ((A - Aavg), (B - Bg) }h.

The uncertainty relations show that the variances of two observ-ables can be simultaneously zero only if the mean value of the Poissonbracket is zero. For commuting observables, { A, B}h = 0, and theright-hand side is zero for all states. In this case the uncertainty rela-tions do not impose any restrictions, and such observables are simul-twieously measurable. Below we shall see that there really is a theo-retically possible way of simultaneously measuring such observablesThe strongest restrictions are imposed on the pairs of observables


such that ({A, B } h I w) 0 for all w, for example, if {A. B } h = const.In this case there do not exist any states in which the variances ofboth observables are equal to zero. We shall see that for a coordinateand momentum projection with the same index,

fP-Qlh = Il

where I is the identity operator. The uncertainty relations for theseobservables take the form

&WPOWQ > h/2;

that is, in nature there are no states in which a coordinate and mo-mentum projection with the same index have simultaneously com-pletely determined values. This assertion is valid for both mixed andpure states, and therefore the pure states in quantum mechanics, un-like those in classical mechanics, are not states with zero variance ofall observables.

We can now go back to the question posed at the beginning of § 2:what can explain the nonuniqueness of the results of experiments? Wehave seen t hat in classical mechanics such nonuniqueness is connectedsolely with the conditions of the experiment. If in the conditions of theexperiment we include sufficiently many preliminary measurements,then we can be certain that the system is in a pure state, and theresults of any experiment are completely determined. In quantummechanics the answer to this question turns out to be quite different.The uncertainty relations show that there is not even a theoreticalpossibility of setting up the experiment in such a way that the resultsof all measurements are uniquely determined by the conditions of theexperiment We are therefore forced to assume that the probabilisticnature of predictions in the xnicroworld is connected with the physicalproperties of quantum systems.

These conclusions seem so unexpected that one might ask aboutthe possibility of introducing so-called "hidden parameters" into thetheory. One might suppose that the description of a state of thesystem with the help of the density matrix M = Pv is not complete;

§ 8. Physical meaning of eigenvalues and eigenvectors 39

that is, in addition to PV, we should specify the values of some param-eters x ("hidden parameters" ),12 and then the description becomessufficient for the unique prediction of the results of any measurement.The probabilistic character of the predictions in a state w P canthen be explained by the fact that we do not know the values of thehidden parameters, and there is some probability distribution withrespect to them. If the state determined by the pair (P,1,, x) is de-Ynoted by wx, then our assumption reduces to the state w being aconvex combination of the states w, We know that P,, does not de-compose into a convex combination of operators with the propertiesof a density matrix; that is, there are no operators M correspond-ing to the states wx. The way of describing the states in quantummechanics is determined by the choice of the algebra of observables.The assiunption that there are states that do not correspond to anydensity matrices forces us to reject the assertion that the observablesare self-ad j oint operators, that is, to reject the basic assumption ofquantum mechanics. Thus, we see that the introduction of hiddenparameters into quantum mechanics is impossible without a radicalrestructuring of its foundations.

§ 8. Physical meaning of the eigenvalues andeigenvectors of observables

In this section we consider questions involving the physical interpre-tation of the theory. First of all we must learn how to constructdistribution functions for observables in a given state. We know thegeneral formula

(L) WA (A) = (O(A - A) I w)l

where O (x) is the Heaviside function. To construct a function of theobservable O (A - A), we consider the equation

A'i = aicpi.

12Tlic "hidden parameters" x can be regarded as elements of some set X We donot rriake any assumptions about the physical nature of these parameters, since it isirrelevant for the arguments to follow


Just for simplicity of the arguments we assume for the present that, theeigenvalues are distinct: that is, the spectrum of the operator A is sim-ple, and we number the eigenvalues in increasing order: al < < an.Denote by P,,; the operators of projection on the eigenvectors Weintroduce the operator

(2) PA (A) = 1: PYi

a,<a

(the subscript i under the summation sign takes the values such thatai < A) . We show that

(3) PA(A) = a(a - A).

To prove this equality it, suffices to show that the operators PA (A)and ©(A - A) act in the same way on the basis vectors cpi. Using the.definition of a function of an operator, we have.

8(A-A)cps =O(A-aj),oj0, A < a..

On the other hand,

PA (A) Pj = 1: PP, oj =01 > aj.

ai<a 107 < aj.

The last equality was written taking into account that P,0i P3 = bij cpjand that the operator Pp, appears under the summation sign only ifA > a3 . It is easy to see that PA (A) is a projection, that is, PA (A) =PA (A) and PA (A) = PA(A). Obviously, PA (A) = 0 for A < a1 andPA (A) = 1 for A > an. The operator PA (A) is called the spectralfunction of the operator A.

Now it is not hard to get an explicit form for the distributionfunction WA (A), w M:

WA(A) = (PA(A)I) = Tr MPA(A) = Tr M 1: Phi = > Tr MP', ,

ai <A ai <,\

and finally,

(4) WA (A) = (Ma,').,

§ 8. Physical meaning of eigenvalues and eigenvectors 41

1

30,

a1 a2 a3 a1

Figure 3

an-1 an X

Recall that (Mp, cPi } > 0. We see that WA (A) is a piecewise const antfunction having jumps at the. values of A coinciding with eigenval-ues. at which points it is right-continuous. Its graph is pictured inFigure 3

From the, form of the distribution function it follows that thepi obability of obtaining a value of A that coincides only with one ofthe eigenvalues is nonzero. The probability Wi of getting the valueai in a measurement is equal to the size of the jump of the functionwA (A) at this point:

(5) Wi = (Mp,cp).The sum of the Wi is 1, as should be expected, since

n

>W3 = Tr M = 1.

(6)

For a pure. state w H P. the formula (5) takes the form

Wi = I

Finally, if the system is in the state P,,,, that is, in the pure statedetermined by one of the eigenvectors of the observable A, then Wi =aiJ in view of the formula (6). In such a pure state the number ai isobtained with certainty in a measurement of A.

Thus far we have assumed that the operator A has a simple spec-trum. Generalization of the results obtained to the case of a multiple

r) correspond tospectrum when several eigenvectors c0i 1)' ;pi 2), . . . , (Pi

an eigenvalue ai is not difficult. It suffices to replace the projections

42 L. D. Faddeev and 0. A. YakubovskiY

Pp, by operators Pi projecting on the eigenspaces corresponding tothe eigenvalues a i

r

k=1

Then the probability of getting the value ai in a measurement of theobservable A in the general case is given by the formula

r(7)i - Ek-1

and in the case of a pure state

(8)

r

k=1

We can now show that the usual definition of a function of anoperator agrees with the concept in § 4 of a function of an observ-able. Indeed, the operators A and f (A) have a common systemn ofeigenvectors

AcOZk) = aicO{k),

f (A)Oik)

_ f (ai)Soik)I

and the number cpik)) is the probability of getting thevalue ai for the observable A in a measurement and at the same theprobability of getting the value f (ai) for the observable f(A). Fromthis it follows at once that

Wf(A)(E) = WA (f -'(E)).

We remark also that in the states determined by the eigenvectorsof A, the observables A and f (A) simultaneously have completelydetermined values ai and f (ai ), respectively.

We can summarize the results connecting the mathematical ap-paratus of the theory with its physical interpretation as follows

1) An observable A in a state w M has the mean value

(A I w) = Tr MA

acid the distribution function

WA (A) = TY MPA (A) -

§ S. Physical meaning of eigenvalues and eigenvectors 43

For a pure state M = P,b, jI 11 = 1, we have

(A = (AiPj v),

WA (/\) PA (/\) 01 0) -

2) The set, of eigenvalues of an observable A coincides with theset of possible results of a measurement of this observable.

3) In measuring an observable A, the probability WW of obtaininga number coinciding with one of its eigenvalues is computed by theforrriula (7) in the general case and by the formula (8) for pure states.

4) The eigenvectors of an observable A determine the pure states,in which the observable with certainty takes a value equal to thecorresponding eigenvalue.

We see that the model constructed satisfies many of the physicalrequirements of the mechanics of the microworld that were formulatedin § 4. It admits the existence of observables that are not simultane-ously measurable, and it explains the uncertainty relations. Observ-ables having a discrete set of values can be described in a natural wayin the framework of this model. On the other hand, we also see thelimitations of such a model. Any observable cannot have more than nvalues, where n is the dimension of the state space C', and hence thismodel does not let us describe observables with a continuous set ofvalues. Therefore, it is hard to believe that such a model can describereal physical systems. However, these difficulties do not arise if wepass to n = oo and take the state space to be a complex Hilbert spaceand the observables to be self-adjoint operators acting in this space.We shall discuss the choice of a state space for concrete physical sys-tems and the rules for constructing observables for them in § 11, butfor the time being we continue to study our finite-dimensional model.A finite-dimensional model is convenient for us also in that we do nothave to deal with the purely mathematical difficulties encounteredin the spectral theory of unbounded self-adjoint operators acting inFlilbert space. We remark that the basic aspects of this theory wereworked out by von Neumann in connection with the needs of quantummechanics


§ 9. Two pictures of motion in quantummechanics. The Schrodinger equation.Stationary states

In this section we discuss questions of the dynamics of quantum sys-tems. In essence, the basic assumption was made already in § 4 whenwe considered the necessity of introducing a Lie operation in the al-gebra of observables in quantum mechanics and the connection ofthis with the dynamics. Therefore, we begin directly by postulatingthe so-called Heisenberg picture, which is an analogue of the classicalHamili onian picture. Just as in classical mechanics, the observablesA depend on time while the states w H M do not:

(1)

dt

dMdt

0,

=

Here H is the total energy operator of the system and is an analogueof the Ilarniltonian function of classical mechanics. The operator 11 issometimes called the Schrodinger operator. In form, the Heisenbergpicture is exactly the same as the Flamiltonian picture, except thatthe right-hand side contains the quantum Poisson bracket instead ofthe classical one.

As in classical mechanics, the equation

(2) = H A t) dt { )}htogether with the initial condition

(3) A(t)I c=n = A

gives a one-parameter family of aut onmorphisms of the algebra 2t ofobservables.

The dependence of the rriean value on the time is determined bythe formula

(A(t) I= TrA(t) M.The equation (2) with the initial condition (3) has a unique solution,which can be written in the form

(4) A(t) = e,HtAe- ,z jet.

§ 9. Two pictures of motion in quantum mechanics 45

Indeed,dA(t) i

= - --[H,A(t)] = {H,A(t)}h.

It is clear that the initial condition is satisfied.

The operator U(t) = e- Ht appearing in the solution (4) is calledthe evolution operator. The evolution operator is a unitary operatorin view of the self-adjointness of H:

U*(t) = e h Ht = U-1(t).

The set of operators U(t) forms a one-parameter group*

U(t2) U(t1) = U(tl + t2), U-1(t) = U(-t)-

If the observables H and A commute, then {H, A } h = 0, and themean value of A does not depend on the time. Such observables arecalled quantum integrals of motion.

In quantum mechanics there is a second equivalent picture of themotion which is an analogue of the classical picture of Liouville. Toformulate this picture, we transform the formula for the mean value:

(A(l) Tr U* (t) AU(t) M

= 'Ir AU(t) MU*(t) = 'ir AM (t) = (A I w(t)),

where

(s) w(t) H M(t) = U(t) MU'(l)

and MW is the unique solution of the equation

dt

with the initial condition

= -try, tvl kc)jla

(7) M(t)It=o = M.

We have arrived at the so-called Schr" inger picture. In this pictureit turns out that the states depend on the time:

dM (t)= -j11tM(0jhidtdAT 0.


From the very way it was introduced, the Schrddinger picture is equiv-alent to the Heisenberg picture, since the dependence of the meanvalues of the observables is the same in these pictures.

Let us now consider the dependence on time of the pure states inthe Schrddinger picture. According to the general formula (5).

P (t) = U(t)1 VU* (t).

We choose the dependence of the state vector 1(t) on time in such away that

Pi(t) = 1 3(E).

We verify that this condition is satisfied by

(9) fi(t) = U(t)

For an arbitrary vector ,

11P {t) _ ( p(t)) V) (t) U(t) /) U(t)= U(t)(U* (t) , ) = U(t)P0U* (t) = PO (t)

Thus, in view of the formula (9), the dependence on time of thevector (t) guarantees the correct dependence on time of the purestate. We note that (9) does not and cannot necessarily follow from(5), since a state vector is determined by the state only up to a factor of modulus 1. In spite of this, it is always assumed in quantummechanics that the dependence of state vectors on the time is deter-mined by the formula (9). We remark that the evolution does notchange the normalization of the state vector; that is,

II'(t)II =1IV)1I1

since U (t) is a unitary operator.

The vector p' (t) satisfies the equation

(io)


ihd dtt) -11*(t)

IP(t)1t=O _

The equation (10) is called the Schrddinger equation and is the basicequation in quantum mechanics.

§ 9. Two pictures of motion in quantum mechanics 47

We now consider states that do not depend on the time in theSchrodinger picture. Such states are called stationary states Obvi-ously, the mean value of any observable and the probabilities of itsdefinite values in stationary states do not depend on the time Thecondition of stationarity follows at once from (6) and has the form

(11) {H,M}h=O.

Let us consider pure stationary states. From (11) we have13

HP,(t) = Pi(t)H.

We apply the left- and right-hand sides of this equality to the vectorp(t)

F1 fi(t) = (Hip(t),1Ii(t))?/,'(t).

The number E = (U1V)(t), i(t)) = Tr 1",p(t) H does not depend on thetime, and we see that for any t the vector O (t) is an eigenvector of11 with eigenvalue E. Therefore, the Schrodinger equation for thevector b(t) takes the form

ihdo(t) - E .d '(t)t

The solution of this equation is

(12) b(t) = zp(O) e-hEt.

Thus, the pure stationary states are the states with definite en-ergy, and a vector determining such a state depends on the timeaccording to the formula (12).

The equationThi

is sometimes called the stationary (or time-independent) Schrodingerequation. The basic problems in quantum mechanics reduce to thesolution of this equation. According to the general physical inter-pretation, the numbers Ea are the possible values of the energy (theenergy levels) of the system. The state corresponding to the smallestvalue El of the energy is called the ground state of the system, andthe other states are called excited states. If the eigenvectors cpi of

3 We note that the independence of P(t) from the time does not in any wayimply that the vector p(t), which satisfies the Schrodinger equation, does not dependon the time

48 L. D. Faddeev and 0. A. YakubovskiI

H are known, then it is easy to construct a solution of the Cauchyproblem

ihdo(t) = H

dtV(t)}

V/ (t) I t=O = 0,

for the Schrodinger equation. For this it suffices to expand the vectorzG in the eigenvector basis of the operator H,

n

E ciWoi C2 ... (tb, 7'Z)

i-1

and to use the formula (9). Then we get the solution of the Cauchyproblem in the form

7z(13)(t} --- hit2=1

This formula is usually called the formula for the expansion of a so-lution of the Schrodinger equation in stationary states.

In the conclusion of this section we obtain the time-energy un-certainty relation. Setting B = H in (7.1), we have

w .1({ AH}h(14) w w2

Recalling that t = {H, A } h in the Heisenberg picture and using theobvious equality dt = dc I w), we rewrite the relation (14) as

A, A H>h` W2

dAavg

dt

We introduce the time interval &, t A = Ow A / I dALvg. In the time

&,t A the mean value of the observable Aa,,,g is displaced by the"width" &,A of the distribution. Therefore, for the state w andthe observable A, OW t A is the characteristic time during which thedistribution function WAN can change noticeably. It follows fromthe inequality

&WtA&H > h/2

that the set of values of A,1 A for all possible observables A in thestate w is bounded from below.

§ 10. Quantum mechanics of real systems 49

Setting At = infA OWtA arid denoting &,H by A E, we get thatfor any state w.

(15) AE At > h/2.

This is the time-energy uncertainty relation. The physical meaningof this relation differs essentially from the meaning of the uncertaintyrelations (7.1). In (7.1), &,A and A , B are the uncertainties in thevalues of the observables A and B in the state w at, one and the sameniorrient of time. In (15), Al is the uncertainty of the energy, andthis does not depend on the time, and At characterizes the time overwhich the distribution of at least one of the observables can changenoticeably. The smaller the time At, the more the state w differs froma stationary state. For strongly nonstationary states, At is small andthe uncertainty DE of the energy must be sufficiently large. On theother hand, if of = 0, then At = oo. In this case the state isstationary.

§ 10. Quantum mechanics of real systems. TheHeisenberg commutation relations

We have already mentioned that the finite-dimensional model con-structed is too poor to correspond to reality and that it can be im-proved by taking the state space to be a complex Hilbert space fland the observables to be self-adjoint operators acting in this space.

One can show that the basic formula for the mean value of anobservable A in a state w keeps its form

(1) (A I w) = Tr AM,

where M is a positive self-adjoins operator acting in fl with trace 1.

For any self-adjoint operator there is a spectral function PA (A),that is, a family of projections with the following properties:

1) PA (A) < PA (li) for A < p, that is, PA (A) PA (li) = PA (A) ;

2) PA (A) is right-continuous, that is, lim;,_.., A+c PA (1u) = PA (A) ;

3) PA(-oo) = lima-,_0CPA (A) = 0 and PA (oo) = I;

4) PA (A) B = B PA (A) if B is any bounded operator commutingwith A.


ifA vector p belongs to the domain of the operator A (cp E D(A))

and then

(2)

jA2d(PA(A),) < oc,

00

AdPA(A) o.x

A function f (A) of the operator A is defined by the formula

(3) f (A) p = f-00

f (A) dPA(A) p-

The domain D (f (A)) of this operator is the set of elements :p suchthat

oc.

The spectrum of a self-adjoint operator is a closed subset of thereal axis and consists of all the points of growth of the spectral func-tion PA (A) . The jumps of this function correspond to the eigenvaluesof the operator A, and PA (A + 0) - PA (A -- 0) is the projection on theeigenspace corresponding to the eigenvalue A. The eigenvalues formthe point spectrum. If the eigenvectors form a complete system, thenthe operator has a pure point spectrum. In the general case the spacecan be split into a direct sum of orthogonal and A-invariant subspaces7x and H2 such that A acting in the first has a pure point spectrumand acting in the second does not have eigenvectors. The spectrumof the operator acting in the subspace f2 is said to be continuous.

It follows from the basic formula (1) that in the general case thedistribution function of A in the state w H M has the form

(4) WA (A) = Tr PA (A) M,

while for pure states M = Pp, 11,011 = 1, we have

(5) WA(A) = (PA(A),O, ).

Unlike in the finite-dimensional model, the function WA (A) is not nec-essarily a step function. The set of admissible values of an observableA coincides with the set of points of growth of the function WA (A) forall possible states w. Therefore, we can assert that the set of possibleresults of measurements of A coincides with its spectrum. We see


that the theory makes it possible to describe observables with both adiscrete and a continuous set of values.

Our problem now is to give rules for choosing the state spacesand to learn how to construct the basic observables for real physicalsystems. Here we shall describe quantum systems having a classi-cal analogue. The problem is posed as follows. Suppose that wehave a classical system, that is, we are given its phase space andllamiltonian function. We must find a quantum system, that is, con-struct a state space and a Schrodinger operator, in such a way thata one-to-one relation f A f is established between the classicalobservables (functions on the phase space) and quantum observables(operators acting in the state space). Furthermore, the Hamiltonianfunction must correspond to the Schr"finger operator. This one-to-one relation certainly cannot be an isomorphism fg F-- / -+ A f o A9,f f, g } +- / -* {A, A9 J h (it is therefore that quantum mechanics dif-fers from classical mechanics), but it must become an isomorphism ash -+ 0 (this ensures that quantum mechanics approaches classical me-chanics in the limit). The quantum observables A f usually have thesame names as the classical observables f. We remark that we mustnot exclude the possibility of the existence of quantum systems thatdo not have simple classical analogues. For such systems there can beobservables that do not correspond to any function of the generalizedcoordinates and momenta.

The correspondence rules and the approach to the classical me-chanics limit will be described at length in § 14. For the present weestablish a correspondence only between the most important observ-ables, and we show how to construct the state space for the simplestsystems.

Let us first consider a material point. Its phase space is six-dimensional, and a point in it is determined by specifying three Caxte-sian coordinates q1, q2, q3 and three momentum projections p1, p2, p3.It is not hard to compute the classical Poisson brackets for any pairsof these observables:

{q,q}=0, {pi,pj} =0, {pi,gj}--Sip, i,j=1,2,3.


For a particle in quantum mechanics we introduce six observables Q,Q2- Q3- P1, P2. PP.3 for which the quantum Poisson brackets have thesame values:

(6) {Q1.Qj}h = 0, {P, P3 }h = o, {IQj}h =- Ihj3.

where I is the identity operator. These observables will be called thecoordinates and momentum projections.

Below we shall justify this association of operators with coordi-nates and momenta by the following circumstances. After studyingthe observables determined by the relations (6), we shall see that theyhave many of the properties of classical coordinates and xrionienta.For example, for free particles the projections of the momentum areintegrals of motion, and the mean values of the coordinates dependlinearly on the time (uniform rectilinear motion).

Further. the correspondence qi H "L}i, pi E--3 PZ, i = 1, 2, 3, isfundamental for us for the construction of a general correspondencef (q, p) H Af . The quantum observables defined in this way, includingthe coordinates and momenta, turn into the corresponding classicalobservables in the limit.14 Finally, the correspondence rules f ; Almake. the theory completely concrete, and its results can be verifiedin experiments. It is the agreement of theory with experiment thatshould be regarded as the definitive justification of the assumption(6) and of all of quantum mechanics.

The conditions (6) are called the Heisenberg quantization condi-tions. Together with (7.1) they at once imply the Heisenberg uncer-tainty relations for the coordinates and momenta. We have alreadydiscussed these relations.

The formulas (6) are often written for the commutators:

(7) [Q7, QkJ = O, [PjiPk1=O, [Q3, Pk]= ihSjk.

These relations are called the Heisenberg commutation relations.

14The exact meaning of this assertion is discussed in § 14


We now state without proof the remarkable von Neumanii-Stonetheorer1m.15

For the system of relations (7) there exists a unique irreduciblerepresentation by operators actin( in a Hilbert space (unique tip to aunitary transformation).

We recall what an irreducible representation is. One usually usestwo equivalent definitions:

1) A representation of the relations (7) is said to be irreducibleif there does not, exist an operator other than a multiple C1 of theidentity that commutes with all the operators Qi and Pi.

2) A representation is said to be irreducible if there does not exista nontrivial subspace Ho of x that is invariant under all the operatorsQ1 and P.

We verify the equivalence of these definitions. If there is a non-trivial subspace Ho invariant under Qi and I i, then the projectionPI-j0 onto this subspace commutes with Qi and Pi, and obviouslyPH() CL

If there is an operator A CI commuting with all the Qi and Pi,then A* and hence also A + A* commute with Qi and Pi. Therefore,we can assume from the very beginning that A is self-adjoint. Theself-adjoint operator A has a spectral function PA (A) which for some Ais different from zero and CI. The operator PA (A) commutes with Qiand Pi, and thus the subspace Ho onto which it projects is invariantwith respect to Qi and Pi.

It follows from the von Neumann-Stone theorem that if we findsome representation of the Heisenberg commutation relations andprove that it is irreducible, then all other irreducible representationswill differ from it by a unitary transformation.

We know that the physical interpretation of the theory is basedon the formula (A I w) = RAM for the mean value. The right-handside of this formula does not change under a unitary transformation

15The formulation given is not precise, since there are subtleties connected withthe unboundedness of the operators P and Q that we are not going to discuss Sim-ilar difficulties do not arise if instead of the operators P and Q themselves we con-sider bounded functions of these operators We present a precise formulation of thevon Neumann-Stone theorem in § 13


of all the operators. Therefore, the physical results of the theory donot depend on which of the unitarily equivalent representations wechoose.

§ 11. Coordinate and momentumrepresentations

We describe the so-called coordinate representation most often usedin quantum mechanics. It is sometimes called the Schrodinger repre-sentation. As the state space for a material point in this representa-tion, we choose the space L2 (R3) of square-integrable complex-valuedfunctions p (x) = Sp(x 1 , X2, X3). The scalar product is defined by

(1) ((P1, 02) = 1(X) So2 (x) dX.R3

The operators Qj and Pj, = 1, 2, 3, are introduced by the for-rnulas

Qj c0 (X) = xj fi(x),

(2)h (9

Pj O(x) i O(x)8xj

These are unbounded operators, and this corresponds to the phys-ical meaning of the observables, since the numerical values of thecoordinates and momenta are unbounded. The operators Q., and Pthemselves and all their positive integer powers are defined on the do-main D formed by the infinitely differentiable functions that decreasefaster than any power. Obviously, D is dense in W. It can be shownthat the operators Qj and P3 have unique self-adjoint extensions uponclosure. It is easy to verify that these operators are symmetric. Forexample, for P1 and any functions b, ;p E D we get by integrating byparts that

(Pi 5P, } = dx V)(x) --0C 11)0

dx2 dx3.Ax)'P(X)R.3 i ox, «, f0C z 00

h 8ip(x) h ft (x)3

dx so(x)i dxi

=3

dx So(x) i ax, - p, Pi h)JR R

§ 11. Coordinate and momentum representations 55

Let us verify the commutation relations (10.7). The first two ofthem are obvious, and the third follows from the equalities

ha h haPkQ3Se(x) i vex (xjfi(x)) bjk'P(x) + xj- SP(X),

2 i(Xkf,

SP(X) = xja

Q' 3 i x S O

Thus, we have indeed constructed a representation of the relations(10.7) by linear operators. The irreducibility of this representationwill be proved later

We present one more example of a representation (the momentumrepresentation). In this representation the state space 7-1 is also thecomplex space L`2 (R3) with elements co(p), and the operators Q.7 andPJ , j = 1, 2, 3, are defined as follows:

aQj P(p) = i 5P-jh

(3)SP(P),

P(p) _ pj SP(P)-

The commutation relations (10.7) are verified in the same wayas for the coordinate representation, and the proof of irreducibility ist he same. By the von Neumann Stone theorem, these representationsare unitarily equivalent; that is, there is a unitary transformation 16

Se(x) - (p)such that the operators defined by (3) are transformed into the oper-ators (2). It is not hard to see that the Fourier transformation

3

(4) x =SP() (27rh cP(P) PR3

3

2

(5) SP(p) = 2i he- hp"cp(x) dxj s

is such a transformation

The unitarity of the Fourier transformation follows from the Par-seval equality

fR3 Isa(X)I2 ax = 1R3 dp.

16We intentionally use one and the same symbol to denote the different functions.P(x) and cp(p), since both these functions represent one and the same state p


to the expression (4), we get thatApplying the operator h OX jJ

h a 1

i a:r.j 27rh

23

pX -1 f' h pX. 3

(P) dP =27Th

ppp(P) dPR JR

We see that tinder the unitary transformation W, the operatortt Q is transformed into the operator of multiplication by the variablez {`fix J

Pi. It can he shown similarly that

axjp(x) W----p ih SA(P)

p3

Thus. we have proved the unitary equivalence of the coordinate andthe momentum representations.

We write the formulas for the transformation of integral operatorsin passing from the coordinate to the momentum representation:

A(X,Y) P(Y) dY,A4p(X) =fR

AiP(P) =

where

3

1 2

(27rh iRS e h P dx

()3fR9 ee' 1iP(9) dxdYdq

P q) F(9) d4,fR3 A(

3 ,(6) A(p, q) = 27Th

_ pXA(x, y) ,dxdy.fR6

If an operator in the coordinate representation is the operator ofmultiplication by the function f (x), then it can be regarded as theintegral operator with kernel

A(x, y) = f (x) J(x - y).

In the momentum representation the kernel of such an operator de-pends only on the difference of the variables and has the form

(7) A(P,q)= ()3ff(x)e1_cix.


The transformation formulas from the momentum representationto the coordinate representation differ by the sign in the exponent:

31

(8) A(X,Y)= (27rh)f1R;6 e

hpA(Aq)e

"`"'dPdq,

and if A(p, q) = F(p)8(p - q), then

1 3

(9) A(x, y) = (27rh ) fR3F(p) eT p(X-r) dp.

We now explain the physical meaning of the functions (x) and,o(p), which are usually called wave functions. We begin with thewave function in the coordinate representation yp(x). In this repre-sentation the operators Q (i = 1, 2; 3) are the operators of multipli-cation by the variables x1; that is, the coordinate representation is aneigenrepresentation for the operators Q j. To see this we make a littlemathematical digression.

To give a function representation of the Hilbert space 7-1 meansto specify a one-to-one correspondence between vectors in this spaceand functions of a real variable, p ;p(x), and to specify a measuredm on the real axis such that

(Sp, *) = f(x)Ydrn(x).The values of ;p(x) can be complex numbers or functions of othervariables, that is, they are elements of some additional space, and

(x)'(x) then donotes the scalar product in this additional spaceIn the first case the representation is said to be simple, and in thesecond to be multiple. Two functions p(x) are regarded as the sameif they differ only on a set of m-measure zero.

A representation is called a spectral representation for an opera-tor A if the action of this operator reduces to multiplication by somefunction f (x).

A representation is called a direct representation or eigenrepre-sentation for an operator A if f (x) = x:

H xo(x)


The spectrum of the operator A can be described as the support of thedistribution function of the measure. The jumps of the measure cor-respond to the eigenvalues. If the measure is absolutely continuous,then the spectrum is continuous.

We recall that in the finite-dimensional space Cn we defined aneigenrepresentation for an operator A to be a representation in whichthe vectors SP were given by the coefficients Spj of the expansion in aneigenvector basis of A Suppose that the spectrum of A is simple:

Aii = ai V)i

>P = Oi lPi , SPi = (W,lii).i=1

With a vector Sp one can associate a function w(x) such that Sp(ai) =Spi (the values of SP(x) for x ai can be arbitrary). The distributionfunction of the measure is chosen to be piecewise constant with unitjumps at x = ai. The operator A can then be given by the formula

ASP H xSP(x),

since this expression is equivalent to the usual one:

Oi ai;PlA _

SPn a'.t 'p

Corresponding to a multiple eigenvalue ai are several eigenvec-tors. We can introduce the function Sp(x) whose value at the point aiis the vector with components 0ik, k = 1, 2, ... , ri. In this case therepresentation is multiple. and its iriultiplicity is equal to the multi-plicity ri of the eigenvalue ai.

We see that in an eigenrepresentation an operator is an analogueof a diagonal rrmatrix on C".

In an eigenrepresentation for an operator A it is easy to describea function f (A)

(10) I (A) p H .f (x) p(x).


Iri particular, for the spectral function PA (A) = ©(a -- A) the formula(10) gives

PA (A) O +-+ O(A - X) V(X) =fcp(x), x < A,

1.0, x > A.

The spectrum of A will be completely described if for A we can con-struct a spectral representation of it. A transformation carrying agiven representation into a spectral representation is called a spectralt rarisformation.

Let us return to the coordinate representation. We see thatit really is a multiple eigenrepresentation for the three operatorsQ1, Q2, Q3:

Ql -* xi So(xl ).

The value of cp(x1) is a vector in an additional space, in this case afunction of the variables x2 and x3. The scalar product is defined bythe formula

0) = [(xi)(dx,R

that is. in the coordinate representation the measure m(x) is Lebesguemeasure, and the support of its distribution function is the whole realaxis, arid hence the spectrum of the coordinate is continuous and fillsthe whole axis.

The distribution function of the coordinate Q1 in a pure statew +-+ 1-11 has the forni

WRY (A) = (PQ1(A)'.p,) = o(xj) o(xj dxl,A

from which it follows that

P(xl , x2, x3) (x1,x2.x3)dx2 dx3o(x1) (Xi) =R2

is the density of the distribution function of the coordinate Q1. Theexpressions for the densities of the distribution functions of Q2 andQ3 are written similarly. It is natural to expect that I (x)12 is thedensity of the distribution function common for all the coordinates;that is, the probability of finding a particle in a region f of three-dimensional space is determined by the expression .f 1 dx. We


prove this assertion in the section devoted to systems of commutingobservables.

The momentum representation is an eigenrepresentation for thethree operators P1, P2 , P3, and I W (p) 2 is the density of the distribu-tion function common for the three projections of the momentum.

We can now look at the uncertainty relations for the coordinatesand momenta from a new point of view. We see that these relationsare explained by a well-known property of the Fourier transforma-tion. The more strongly the function ;p (x) is concentrated, and thusthe smaller the uncertainties A;,, Qi of the coordinates are, the morethe Fourier transform ap(p) is spread out, and thus the greater theuncertainties 0, Pi of the momenta.

§ 12. "Eigenfunctions" of the operators Q and P

We now consider the equations for the eigenvectors of the operatorsQ and P For simplicity we consider a particle with one degree offreedom. In the coordinate representation these equations have theform

(1) (X) = X0 Oxo W -

(2) It =wnW

Solving them, we gel that

(3) 'OX" (X) = 6(X - XO).

pXen

(The first formula follows at once from the property x6(x) = 0 of the6-function, and the second is obvious. The choice of the normalizationconstant will be clear from what follows.)

Although an "eigenfunction" x,, (x) of the coordinate operator isa generalized function while app (x) is an ordinary function, they havein common that they both fail to be square integrable; that is, they donot belong to L2 The operators Q and P do not have eigenfunctionsin the usual sense of the word.

§ 12. "Eigenfunctions" of the operators Q and P 61

To understand how the functions cpx (x) and ypp (x) are connectedwith the spectrum of the operators Q and P, we recall the meaning ofeigenvectors in C'. The problem of finding the eigenvalues and eigen-vectors of a matrix A is connected with the problem of diagonalizingthis matrix by a similarity transformation, or in other words, withthe transformation of an operator from some initial representation toan eigenrepresentation. A self-adjoint operator A on C' always hasa basis consisting of eigenvectors:

A i = (pi, 0k) = Sik

We look for the eigenvectors in the initial representation, in which

I ... I 10 I

i Gl oMrn(i) )-O

The components of the vector in an eigenrepresentation of A arecomputed from the formula

(5)

n

kP(i)k=1

We see that the matrix consisting of the numbers Uik = ;pit) con-jugate to the, components of the eigenvectors implements a spectraltransformation:

it

; = EU is unitairy, since

n n

GoiUik UUj = k2)'p .7) , Vii) = Sajk=1 k=-1 k-1

If in (5) we formally replace p by and by gy(p). then we getthat

_ ( p)P

oror

z()fCe(x)dx.(6)


This formula is already familiar to us. We see that the functionU(p, x) = cpp(x) is the kernel of a unitary operator carrying the coor-dinate representation into the momentum representation (the eigen-representation for the operator P). An "eigenfunction ' of the coordi-nate operator Q can be interpreted in the same way. The coordinaterepresentation is an eigenrepresentation for Q; therefore, the operatorimplementing the spectral transformation must be the identity, and

,o (x) = 6(x - x0) is the kernel of the identity operator.

We see from these examples that even though the "eigenfunc-tions" of the continuousspectrum are not eigenelements in the usualsense of the word, their connection with the spectral transformationremains the same as for eigenvectors in a finite-dimensional space.

We remark that there is a construction enabling one to give so-lutions of the equation

Acpa = A(pa

the precise meaning of eigenvectors even for the case when ;Pa doesnot belong to R. To this end we introduce a broader space V D Nwhose elements are linear functionals defined on some space dD C fl.The pair of spaces V can be constructed in such a way that eachself-adjoint operator acting in f has in V a complete system ofeigenvectors. The eigenelements of A in and not in fl are. calledgeneralized eigenelements. If A has a simple spectrum. then anyV' E has an eigenelement decomposition

c(A) p,\ dA.JR

The Fourier transformation can be interpreted as an eigenfunc-tion decomposition of the momentum operator :pp (x) :

(8)1

') _ (27rh) (p) ez'sI dpR

The normalization constant (1/2irh)1/2 in the expression for an eigen-function of the momentum corresponds to the normalization

JR

§ 13. The energy, the angular momentum, and others 63

This formula is a consequence of the well-known integral representa-tion for the 6-function:

(10)At

e ik-TdX = 27ro(k).

In the last two formulas the integrals are understood in the principalvalue sense. The formula (9) is an analogue of the equality

(Wi,'Pk) = sik

for eigenvectors of the discrete spectrum.

We now construct the spectral function of the momentum opera-tor. Only here we use the letter II for the spectral function, so therewill be no confusion with the momentum operator.

In the momentum representation the operator IIp (A) is the oper-ator of multiplication by a function:

Hp(A) P(p) = O(A - p) SP(p)

Let us pass to the coordinate representation Using the formula (11.9)for the one-dimensional case, we get that

li a x - 1 OA - eTp(T-y) dp ( ) ( , y)27Th

( p) pR

-- e h dp = V. (x) WP (y) dP.27r h

_ y)

h _ -The derivative of the spectral function with respect to the pa

rameter A is called the spectral density. The kernel of this operatorhas the form

d rip y) = V,\ (x) V.\ (y) -27Th

eh a(X-y)da ( )(x,

We see that this is an analogue of the projection on a one-dimensionaleigenspace.

13. The energy, the angular momentum, andother examples of observables

We now proceed to a study of more complicated observables. Ourproblem is to associate with an arbitrary classical observable f (p, q)a quantum analogue Af of it. We would like to set Al = f (P, Q),


but there is no general definition of a function on noncommutingoperators. For example, it is no longer clear which of the operatorsQ2P, QPQ, or PQ2 one should associate with the classical observableq2p. However, for the most important observables such difficulties donot arise in general, since these observables are sums of functions ofthe mutually commuting components Q1, Q2, Q3 and P1, P2, P3.

We present some examples.

The kinetic energy of a particle in classical mechanics is

pi + p2 + p3T=2m

The corresponding kinetic energy operator has the form

Pl + P2 + P32rn

and in the coordinate representation

h(1) T,(x)

2m

where I. = 262 + 62+ a is the Laplace operator. In the momentum

I 2 3

representation the operator T is the operator of multiplication by thecorresponding function:

pi+p2+p327n

SA(P)

In exactly the same way it is easy to introduce the potentialenergy operator V (Q 1, Q2, Q3) . In the coordinate representation, Vis the operator of multiplication by the function

(2) V;c(x) = V (X) SP (x),

and in the momentum representation it is the integral operator withkernel

13

e (9- P)V(P q) =27rh R3

V(x) e h dx.

The total energy operator (the Schrodinger operator) is defined a.5

11 = T + V.

We write the Schrodinger equation in detail

ihdV(t) = I1 t .(3)dt ( }


In the coordinate representation the Schr6dinger equation for a par-ticle is the. partial differential equation

a(x, t} h2ih

at - - Av)(x, t) + V ( X )

and in the momentum representation the integro-differential equation

a0 (P, t) P2 0(pihat - 2m t) + V (p, q) b(q, t) dq.f 3

Along with (4) we consider the equation for the complex conju-gate wave function:

azl) (x, t) _ h(5) -ih

at-

2 _Oib(x, t) + V(x) P(x, 1).

Multiplying the equation (4) by 70 and the equation (5) by 0 andsubtracting one from the other, we get that

2-00 hih *-+V V))at 01) 2m

h2 div( grad 7P - V) grad

or

(6)

where

2

aatl + aivi = n,

- ih(,0 grad grad- 2m V)).

The equation (6) is the equation of continuity and expresses thelaw of conservation of probability. The vector j is called the densityvector of the probability flow. It is clear from (6) that j has thefollowing meaning: fs j,, dS is the probability that a particle crossesthe surface S per unit of tirrie.

The angular velocity plays an important role in both classical andquantum mechanics. In classical mechanics this observable is a vectorwhose projections in a Cartesian coordinate system have the form

11 g2p3 -43P2

12 = g3pi - g1p3,

13=gIP2--g2pI.

66 L. D. Faddeev and 0. A. Yakubovski"

In quantum mechanics the operators

(7) Lk = QlPm - QmP1

are the projections of the angular momentum. Here k, 1, in are cyclicpermutations of the numbers 1, 2, 3. The right-hand side of (7) con-tains only products of coordinates and momentum projections withdifferent indices, that is, products of commuting operators. It is in-teresting to note that the operators Lk have the same form in themomentum and coordinate representations:

(A)

h( a ali `C7xm - xm

C7x( (X)'

(i)z- m -Pm iP(P)

The properties of the operators Lk will be studied at length below.

Quantum mechanics can also describe systems more complicatedthan just a material point, of course. For example, for a system ofN material points the state space in the coordinate representationis the space17 L2 (R3N) of functions ;p(Xl, ... , XN) of N vector vari-ables. The Schrodinger operator for such a system (the analogue ofthe Hamiltonian functions (1.5)) has the form

N h2 N N

2mii+>1'j(Xi-Xj)+>Vj(X).(9) H=-

i=1 i<j i=1

The physical meaning of the terms here is the same as in classicalmechanics.

We consider two more operators for a material point which turnout to he useful for discussing the interconnection of quantum andclassical mechanics:

U(u) = e-i(U1 P1+U2 '2+u3P3)

V(V) ` e-2(v1Q1+v2Q2+V3Q3)

17 Below we shall see that for identical particles, the state space 'H coincides withthe subspace L2 C L2 of functions with a definite symmetry


where u (u x , u2 , u3) and v (vi , V2, v3) are real parameters. The opera-tor V (v) in the coordinate representation is the operator of niultipli-cation by a function-

V(v) V(X) = eL°XV(X)

We now explain the meaning of the operator U(u). For simplicitywe consider the one-dimensional case

U(u) = e- iuP.

Let (u, x) = e- iu'Sp(x) Differentiating Sp(u, x) with respect to theparameter u, we have

a;p(u, x)On

- X),

or

&(u, x) _ h 80(u' x)On ax

To find the function ;p(u, x), we must solve this equation with theinitial condition

co(u, x) I,L=o = Se(x)

The unique solution obviously has the form

p(u. x) = p(x - hu) -

We see that in the. coordinate representation the operator U (u)is the operator translating the argument of the function ,,p(x) by thequantity -hu. In the three-dimensional case.

(12) U(u) p(x) _ p(x - uh).

The operators U (u) and U (v) are unitary, because the operators1'j, P2, P3 and Qi , Q2, Q3 are self-adjoint. Let us find the coxninuta-tion relations for the operators U (u) and U (v) . In the coordinaterepresentation

V(V) U(U) O(X) = uh),

U(u) V(v) cp(x) = e-`°("-Dh) o(x - uh).

From these equalities it follows at once that

L°°(13) U(u) V (v) = V (v) U(u)e`


Of course, the relations (13) do not depend on the representation. Wenote further the formulas

U(ui) U(u2) = U(ui + U2),

V(vi) V(v2) = V(vi + v2);

that is. the sets of operators U(u) and 11(v) form groups: demotethese groups by U and V

We can now give a precise formulation of von Neumann's theorem.For simplicity we confine ourselves to a system with a single degreeof freedom

Theorem. Let (I and V be one-parameter groups of unitary operatorsU(u) and V(v) acting in a Ililberl, space H and satisfyinq the. condition

(14) U(u) V (V) = v(V) U(11) P.ivuh

Then 7-l can be represented as a direct sum

(15) f=7-l, mW2O ..,where each 7-ij is carried into itself by all the operators U(u) and V (v),and each li can be mapped unitarily onto 112 (R) in such a way thatthe operators V (v) pass into the operators i(x) e - ivXj(x) and theoperators U(u) pass into the operators O(x) -+(x - uh).

One can say that a representations of the relations (14) by unit aryoperators acts in the space W. If this representation is irreducible,they i the sum (15) contains only one term.

In the conclusion of this section we prove the irreducibility of thecoordinate represent,atioii for P and Q.

Let K be an operator commuting with Q and P:

[K, Q] = 0, [K. P] = 0.

It follows from the second equality that

KU(u) = U(u) K.

Applying the operators on both sides of the equality to an arbitraryfunction cp(x), we get that

ih) dy =JR

K(x - uh. y) (y) dy.JR

§ 14. Interconnection between quantum and classical 69

The substitution y --- uh y on the left-hand side lets us rewrite thisequality as

JRK (x, y + uh),r(y) dy = J K(x - uh,J) ,?(?/) dy.

RSince p is arbitrary,

K(.x,y+uh) = K(x-uh,y)from which it is obvious that the kernel K(x. y) depends only on thedifference x -- y, that is,

K(x. y) = k(x - y).We now use the fact t hat, K commutes with the coordinate operator:

x 00

xk(x - y) (y) dy = - y) y(y) dy,-x t:k(xfrom which it, follows that

(x-y)k(x-y) =0The solution of this equation has the. form

k(x - y) = c.S(x-y),

and the function on the right-hand side is the kernel of the operatorC,I.

Accordingly, we have shown 1 hat any operator commuting withQ and P is a multiple of the identity.

§ 14. The interconnection between quantum andclassical mechanics. Passage to the limitfrom quantum mechanics to classicalmechanics

We know that the behavior of a macroscopic system is described wellby classical xrrechanics, so in the passage to a macro-object, quantummechanics must reduce to the same results. A criterion for the "quan-tumness" of the behavior of a system can be the relative magnitude ofPlanck's constant h = 1.05.10-27 erg-sec and the system's character-istic numerical values for observables having the dimension of actionIf Planck's constant is negligibly small in comparison with the values


of such observables, then the system should have a classical behavior.The passage from quantum mechanics to classical mechanics can bedescribed formally as the passage to the limit as h - 0.18 Of course,under such a passage to the limit, the methods for describing themechanical systems remain different, (operators acting in a Hilbertspace cannot be transformed into functions on phase space), but thephysical results of quantum mechanics as h - 0 must coincide withthe classical results.

We again consider a system with one degree of freedom. Ourpresentation will be conducted according to the following scheme.

We first use a certain rule to associate self-adjoint operators A fwith real functions f (p, q) on the phase space (f - A1). Then wefind an inversion formula enabling us to recover the function f (p, q)from the operator A f (Af -+ f) Thereby, we shall have establisheda one-to-one relation between real functions on the phase space andself-adjoint operators acting in fl, (f H A f). They correct formulaturns out to be

1 TrA = dpdq() f JM f (p, q) 27rh

Finally, we determine what functions on the phase space corre-spond to the product A f o A. and to the quantum Poisson bracket(Aj, Aq } . We shall see that these functions do not coincide with theproduct fg and to the classical Poisson bracket {f, g }, but do tendto them in the limit as h -* 0. Thus, we shall see that the algebra ofobservables in quantum mechanics is not isomorphic to the algebra ofobservables in classical mechanics, but the one-to-one correspondencef Af does become an isomorphism as h - 0.

Suppose that a quantum system with Schrodiuger operator H isin a state M, and let A f be some observable for this system. Theone-to-one correspondence described enables us to associate with theoperators H, M, and A f, a Hamiltonian function H(p, q). a functionpi (p, q), and an observable f (p, q). Let p(p, q) = pi (p. q)/2irh. It

18Here there is an analogy with the connection between relativistic and classicalmechanics Relativistic effects can be ignored if the velocities characteristic for the sys-tem are much less than the velocity c of light The passage from relativistic mechanicsto classical mechanics can be regarded formally as the limit as r - no


follows from (1) that

(2) ly M = f P(P, 9) dp d9' = 1,

(3) TrMA fh ---*O

ff(mq)P(Pq)dPdq..tit

The formula (2) shows that the function p(p, q) has the correctnormalization, and the formula (3) asserts that in the limit as h - 0the mean value of an observable in quantum mechanics coincides withthe mean value of the corresponding classical observable.I9 Further,suppose that the evolution of the quantum system is described by theHeisenberg picture and the correspondence A f (t) f (t) has beenestablished for an arbitrary moment of time t. We show that ash - 0 the classical observable f (t) depends correctly on the time.The operator A f (t) satisfies the equation

(4)dA f (t)

= {H,Af(t)}h.Ih-dt

It follows from the linearity of the correspondence f H A f that !YLt Hd , and moreover, {H, A f } H {H, f I ash --+ 0, so that as h 0

the classical equation

is a consequence of the quantum equation (4).

We proceed to the details. Let us consider a function f (p, q) anddenote by f (u, v) its Fourier transform:2°

(5) f (q, p) = f (u, v)e-àgVe-iPu

du dv,27r W

1f(u,v) = f (p, q) dqdp.27r R,2

19We underscore once more that the left-hand side of (3) does not coincide withthe right-hand side when h 0, since a function different from f (p, q)p1(p, q) corre-sponds to the product MA f Moreover, we note that for h 0 0 the function p(p, q) canfail to be positive, that is, it does not correspond to a classical state, but in the limitas h - 0 it follows from (3) that p(p, q) satisfies all the requirements of a classicaldistribution function

20 We omit the stipulations which should be made in order for all the subsequenttransformations to be completely rigorous


The Fourier transform of a real funct ion f (p, q) has the property that

(7)WN ^

f(u,v) = f(-u.-v).

To construct the operator Af corresponding to the function f (p, q),we would like to replace the variables q and p iii (5) by the operatorsQ and P. However, it is riot ininiediately clear in what order weshould write the rioncommuting factors V(v) and U(u). which satisfythe commutation relations

(8)ihuvU(U) V(V) = V(V) U(U) e

A natural recipe (but by no rrieans the only one) ensuring self-adjointriess was proposed by Weyl and has the form

(z'

du dv.9) = 1

27rf(u, v)V (v)U(u)e f

h2

.RThe appearance of t he extra factor e`h",'/2 is connected with the non-c:onimutat ivity of V (v) and U(u) and ensures that, the operator Af isself-adjoirit. Indeed, using (7) and (8). we have21

*f27r

f= I ^ u v U* u V * v e- 'hte

du dv

1

-

1h III=

27rf (-u, _v) U(-u) V (-v) e--2

du dv

1 0111L

f (u. v) U(u) V (v) e- du dv27r

1 fJ(uv)= V(?)) U(u) e 2 dudv = Af.

At the end of this section we verify that the operators f (Q) and f (P)correspond to the functions f (q) and f (p) by virtue of the formula(9), in complete agreement with the assumptions made earlier.

We now find an inversion formula. In computing the trace of anoperator k we shall use the formula

(10) TrK = fK(xx)dx.

21 F3clow we omit the bymbol R" in the expression for an integral over the n-dinicrisional apace


where K(x, y) is the kernel of K. We find the kernel of the operatorV(v)U(u). It follows from the formula

V (v) U(u),p(x) = e-"'x p(x - uh)

that the kernel of the integral operator V(v)U(u) is the function

V (v) U(u)(x, x') = e-ivX8(x - uh - x').

By (lo),00

11 Tr V M U u e-ivx J -uh dx =27r6v u.

J-xWe verify that the inversion formula is

(12) J(u,v) = hTrAfV(--v)U(--u)etj2+ .

To this end we compute the right-hand side of the equality, using (8)and (11):

Tr A f V (-v) U(-u) ei`'2"

=27r Jf(u,.v,)v(v,)u(u')e',,v(-v)u(_u)eu,du l

Tr f(u , v) V (VI) V (_V) U(u) U(-U} e 2(v) du dv'2ir

Jfu',r I I'41'-22t'V) I I2zrv)V(v --v)U(u -u)e du, dv

1 r I Iff(u', v )b(I) --v}S(u ---n)c du dv

1--- h f (u, v)

We set u=v_=0in(12):

f (07 0) = h Tr Af

UU the other hand,

/(0,0) = d d ,f(p q) q27r

and we obtain the formula (1).

It now remains for us to find the functions on phase space corre-sponding to A f o A,, and {Aj, Ay } h using (12), and to see that these


functions tend to f g and {f, g} as h -* 0. We first find the func-tion F (p, q) corresponding to the nonsymxnetric product A f A9 . ItsFourier transform is

P(u,v) = h Tr A fA9 V (-v) U(-u) e i`'2"

12

f(ui,ihut z1

= hTr--

dul dvl du2 dv2 v1),q(u2,v2) V(vi) U(ui) e 2

x V (V2) U(u2) eth 2 2 V(-v) U(-u) e th2

2h Tr (±-)

27rfdul dv, du2 dv2 J(u1 , vi) (u2, vz)

x V (V1 + v2 - v) U(ul + uz - u)

x exp [!(u1v1 + u2v2 + uv + 2ulv2- 2uly - 2u2v)

Finally,

(13)

AfAg H 2 dul dvl due dvi f (ul, vl) (u2,v2)6(vi + vz - v)

x 5(u1 + U2 - u) e2tulv2-u2v1)

The exponent in the exponential was transformed with the o-functionsunder the integral sign taken into account. We recall that the Fouriertransform 4 (u, v) of the product (p, q) = f (p, q)g(p, q) of the twofunctions is the convolution of the Fourier transforms of the factors:

(u, v) =27f f dul dvl dug dv2

x f (u1, v1) .9(u2, V2) a(v1 + V2 - v) 6(ul + U2 - u).

The function F(u, v) differs from the function 4 (u, v) by the factore 2 (ul v2 -u2 vi) under the integral sign. This factor depends on theorder of the operators A f and Ag, and therefore the operators A fA9and A9 A f correspond to different functions on the phase space. Inthe limit as h - 0 we have e 2 (uiv2-u2v1) 1 and F(p, q) - 4(p, q).Of course, this assertion is valid also for the function Fs (p, q) corre-sponding to the symmetrized product Af o A9.


We denote by G(p, q) the function corresponding to the quantumPoisson bracket

Af, A., 1, (Af Aq - AgAf)-

From (13) we get, that

(u,v) 27rh

f dug dvl dul dv2 f(ui , vi) Xu2, v2) b(vi + V2 - v)

X b(u1 + u2 - u) a 2? (ul V2-u2v1) - eh («2v1-u1v2)

ahf dut A, du2 dvz .f (ul. V1) ,(U1, v2) a(v, + v2 - v)

x Oui + u2 - u) Sill' (U2V1 - ul v2),

andaash-f0

., A

G(u, v) - - dul dv1 dul dv2 (U2V1 -- ui v2) f (U1, v1) g(u2, V2)7r f

X Ovl + v2 - v) a(u, + u2 - u).

The integral on the right-hand side is the Fourier transform of theclassical Poisson bracket

{f,g} =afa9 - Of 'Nap aq q,

A

since the functions -iv f and -iu f are the Fourier transforms of therespective derivatives and "faq OP

Thus, we have verified all the assertions made at the beginningof the section.

In conclusion we present some examples of computations usingthe formulas in this section.

Let us find a formula for the kernel of the operator A f in thecoordinate representation. Using the formula

V (v) U (u) (x, xf) = eivx6(x - uh - x'),


we get that

A f (x, x') = f(u.v)e_itxo(x - uh - x') e 2 du dv.2-r

(14) A f(x. x') = f x - x , v e- 2 dv.27rh h

We show that f (q) H f (Q). For such a function on the phase space,

1f(u,v)=27r

f (q) e ivq eiUP dq dp = f (V) 6(u).

Here f (v) denotes the Fourier transform of the function f (q) of asingle variable. Further, by (14),

A f (x, x') =27rh h

f(v)e'dv;+x'= f 6(x - X') _ f(x) b(x - X'),

2

and the operator with this kernel is the operator of multiplication bythe function f (x). In exactly the same way it is easy to verify thatf (p) f (P) in the momentum representation.

We shall get an explicit formriula, for finding the classical statecorresponding to the limit of a pure quantum state as h - 0.

Using the inversion formula, we find pl (q, p) corresponding to theoperator P1k, and we construct p(q, p) = p1 (q, p)/(27rh). The vectori' is assumed to he given in the coordinate representation. From the

formula

V (-V)U(-u) e?VX f (x') v)(x') d r' ,4'(x + uh).

it follows that

V (-v) U(-u) 131, (x, x') = eit,r2,7(x + uh) V., (x')

and

1 ill «:1(it,r) _ hTrV(-r;) U(-u) 1 f' z

27Th

1 zz.c thuse l'(x + uh) v(x) e 2 dx

27r

§ 15. One-dimensional problems of quantum mechanics 77

If we introduce the function22

(15) F(x, u) =h

m V'(x + uh) G(x),

then

p(u, v) = f ezvx F(x, u) dx

is the Fourier transform of the classical distribution function corre-sponding to the limit of the state P.0 as h -> 0. The distributionfunction p(q, p) itself is given by

(16) p(p, q) = J e-iPUF(q, u) du.

Suppose that 0(x) is a continuous function and does not dependon h as a parameter. Then

F(x, u) = 2 I(x)I2

and

p(p, q) = 1,0(q) I26(p)

The state of a particle at rest with density J 1 2 of the coor-dinate distribution function corresponds to such a quantum state inthe limit

ipoxLet O (x) = SP (x) e , where p (x) is independent of h and iscontinuous. In this case in the limit as h --p 0, we arrive at theclassical state with distribution function

p(p, q) = J12a(p - po).

We see from these examples that in the limit as h - 0 a mixedclassical state can correspond to a pure state in quantum mechanics.

15. One-dimensional problems of quantummechanics. A free one-dimensional particle

In §§ 15-20 we consider one-dimensional problems of quantum me-chanics. The Hamiltonian function for a particle with one degree of

22This limit is not always trivial, since in physically interesting cases i/ (x) usuallydepends on h as a parameter (see the example below)


freedom in a potential field has the formz

H(P, 4) = 2m + V (q).

Corresponding to this Hamiltonian function is the Schr" inger oper-ator

H= P+ V(Q).

For a free particle V = 0. We begin with a study of this simplestproblem in quantum mechanics. Let us find the spectrum of theSchrodinger operator

(1)_ P2

The equation for the eigenvectors has the form

P2

(2)

or, in the coordinate representation,

h2d2,0

' E .(3)2m dx2 -

It is convenient to use a system of units in which h = 1 and m = 1/2.Then for E > 0 the equation (2) takes the form

(4) 011 + k2,0 =01 k2 =Ej k>O.The last equation has two linearly independent solutions

-- 12 fikx(5) bf k (x)

e=21r

We see that to each value E > 0 there correspond two eigenfunctionsof the operator H. For E < 0 the equation (3) does not have so-lutions bounded on the whole real axis. Thus, the spectrum of theSchrodinger operator (1) is continuous and positive and has multi-plicity two.

The functions (5) are at the same time the eigenfunctions of themomentum operator P = -i that correspond to the eigenvalues±k. We note that the functions (5) do not belong to L2 (R) and,

§ 15. One-dimensional problems of quantum mechanics 79

therefore, are not eigenfunctions in the ordinary sense. For subse-quent formulas it is convenient to assume that -- oo < k <k = ± VEK Then both the solutions (5) have the form

(s) ,W,/ 1 fkxk (x - (i;) 8The normalizing factor in (6) was chosen so that

(7)JR

Ok(x) 'Ok-(X) dX =a(k - k').

oo and

The solutions of (2) in the momentum representation are obtainedjust as simply. For m = 1, (2) has the form

(s) p2b=k2i',

and its solutions are the functions

(9) Ok(P) = 8(p - k), k =fem.The normalization of the functions (9) is chosen the same as for thefunctions (6):

JR= Jo(-. k) b(p - k') ap = a(k - k').

To explain the physical meaning of the eigenfunctions Ok, weconstruct a solution of the nonstationary Schrodinger equation for afree particle

dip (t)= HO(t)

dtunder the initial conditions

,0(0) holl = 1.

This problem is simplest to solve in the momentum representation

ft (p, 0 = p2o(p, t),

10 (A 0) _ 0 (P)Ji(p)l2dp = 1.

Obviously,

0(p, t) = SP(p) a-$Pat.

80 L. D. Faddeev and 0. A. Yakubovskri

In the coordinate representation the same state is described bythe function(10)

1

2 -pZt) -2t(x, t) = (27r) JR W(P) e" dP = JR (k) ?Gk(x) e A.

It is easy to verify that the normalization of the function *(x, t) doesnot depend on the time:

JRjjp(X7 t) 12dX = I I P (P) 12dp = 1.

The formula (10) can be regarded also as a decomposition of thesolution O(x, t) of the Schrodinger equation with respect to stationarystates, that is, as a generalization of the formula (9.13) to the case ofa continuous spectrum. The role of the coefficients c is played hereby the function p(k).

The state described by the functions O(x, t) or O(p, t) has thesimplest physical meaning in the case when cp(p) is nonzero in somesmall neighborhood of the point po. States of this type are generallycalled wave packets. The graph of the function Icp(p)12 is pictured inFigure 4.

We recall that I lp (p, t) 12 = I cp(p) 12 is the density of the momentumdistribution function. The fact that this density does not depend onthe time is a consequence of the law of conservation of momentum fora free particle. If the distribution IW(p)12 is concentrated in a smallneighborhood of the point po, then the state fi(t) can be interpretedas a state with almost precisely given momentum.

All10-

PO p

Figure 4

§ 15. One.dimensional problems of quantum mechanics 81

The function kt'(x, t)12 is the density of the coordinate distribu-tion function. Let us trace its evolution in time. First of all it followsfrom the Riemann- Lebesgue theorem 23 that (x, t) ---* 0 as Itl --> 00for smooth ap(p), and therefore the integral fn (x, t) 12 dx over anyfinite region f of the real axis tends to zero. This means that asItt --* oo, the particle leaves any finite region; that is, the motion isinfinite.

In order to follow more closely the motion of the particle for largeJt, we use the stationary phase method. This method can be used tocompute integrals of the form

(11)b

I (N) _ F(x) esNf (x) dxa

asymptotically as N -+ oo. For the case of smooth functions f (x)and F (x) and when there is a single point i in (a, b), where ,f (x) isstationary (f(X) = 0, f"() 0), we have the formula(12)

2IN= F xex iN +sn" 0 (h).(NI;)} 4Werewrite the expression for 0(x, t) in the form

Iz

x t = (i-) d0(') rJRV (p) p

The upper sign in the exponential corresponds to t > 0 and the lowerto t < 0. We find the asymptotics of b(x, t) as t --* ±oo from theformula (12). Then the stationarity is found from the condition

xPP) = T 2p + Ft = 0,

from which p = 2t . Further, f " (p) =::F22 and as t -* ±o,c

x t = x 1 2eix(x,t) 0(13) t(, )`;P +it- (21tj

where x is a real function whose form is not important for whatfollows.

23This theorem asserts that f°o f(x)e' dx - 0 as N --r oo if the function f (x)is piecewise continuous and absolutely integrable on the whole axis -oo < x < oo


By assumption, ap(p) is nonzero only in a small neighborhood ofthe point po, therefore, 1,0 (x, t) 12 = I p(it)12

2t +O( -It 1-3 , is nonzeroonly near the point x = 2pot. From this relation it is clear that theregion with a nonzero probability of finding a particle moves along thex-axis with constant velocity v = 2po; that is, the classical connectionp = my between the momentum and the velocity remains valid (recallthat we set m = 1/2). Further, from the asymptotic expression forI lli(x,t)12 it is clear that the square root D,,,x of the variance of thecoordinate for large values of ItI satisfies the relation

Owx = 20wPltl

or

A"X A"VItj.

This means that the region in which there is a laxge probability offinding the particle will spread out with velocity A,, v while movingalong the x-axis.

Even without computations it is not hard to see that the be-havior of a classical free particle in a state with nonzero varianceof coordinates and momenta will be exactly the same. Thus, quan-tum mechanics leads to almost the same results for a free particle asdoes classical mechanics. The only difference is that, according tothe uncertainty relations, there are no states with zero variance ofcoordinates and momenta in quantum mechanics.

We mention some more formal properties of a solution of theSchrodinger equation for a free particle. It follows from (13) that

xt 0 1(j ) Itj1/2

that is, for any x

kt'(x,t)i < jji/2'where C is a constant. It is thus natural to expect that the vectorsli(t) tends weakly to zero as ItI --> oo. It is simplest to prove thisassertion in the momentum representation. For an arbitrary vectorf E L2 (R),

(b(t),f) =JR

§ 16. The harmonic oscillator 83

In view of the Riemann-Lebesgue theorem, the last integral tends tozero as ItI -> oo.

In the coordinate representation the weak convergence to zero ofip(t) has a very simple meaning. The constant vector f is given bya function which is markedly different from zero only in some finiteregion SZ, and the region in which O(x, t) is nonzero goes to infinitywhile spreading out. Therefore, (i/.'(t), f) --+ 0 as It I --+ oo.

We verify that the asymptotic behavior of the expression (13) forthe function 0 (x, t) has the correct normalization. As ItI -+ oo, wehave

JRI'P(x, t) 12 dX

N ft 21tj I

0 (X)2tI2 - JR Ico(k)i2dk=l.

§ 16. The harmonic oscillator

In classical mechanics a harmonic oscillator is defined to be a systemwith the Hamiltonian function

H(Pj 9') = 2m + 2 g2

The parameter w > 0 has the sense of the

freq2

uency of oscillations.

The Schrodinger operator of the corresponding quantum mechan-ical system is

PZ w2Q2(1) H = 2 + 2

We use a system of units in which m = 1 and h = 1. Our problem isto find the eigenvectors and eigenvalues of H. We solve this problemby using only the Heisenberg uncertainty relations for the operatorsP and Q and not by passing to a concrete representation. To thisend we introduce the operators

(2)a =

1(WQ + ip),

a* = 1(WQ - ip).

Using the Heisenberg relation

(3) [Qj P] = ij


we get that

waa*= wl Q2 + P2+aw( 2

(-QP+I>Q)=H+-

2 2

wa*a = -GO 2Q2 + P2) + 2w(QP - PQ) = H --

ÌO

2 2 2or

(4)

(5)

H = waa* 2WH=wa*a+2

.

From (4) and (5) we find at once the commutation relation forthe operators a and a*:

(6) [a,a*] = 1.

From this it is easy to verify by induction that

(7) [n, (a*)n]=n (m )^-1.

Finally, we need the commutation relations of the operators a and a*with H. To compute the co mutator [H, a], it suffices to multiplythe formula (4) by a from the right, multiply (5) by a from the left,and find the difference between the expressions obtained:

[I-I, a] = -wa.

(9) [H, a*] = wa*.

Let, us now pass to a study of the spectrum of the operator H.Suppose that there is at least one eigcnvalue E of H. Let 'E denotea corresponding eigenvector. We show that the eigenvalues E arebounded from below. By assumption, the vector 'OE satisfies theequation H F = Ei/ , which by (5) can be rewritten in the form

wa*a +') OF = EV)E.

Multiplying this equality from the left by 'OE, we get thatwllalFll2+w

IIh/,E 112 =EII 112,2

which implies at once that E > w/2, with equality possible only ifaibE = 0 From the expression (5) for H it is clear that if some

§ 16. The harmonic oscillator 85

vector satisfies the condition a = 0, then it is an eigenvector of Hcorresponding to the eigenvalue w/2.

We show how to construct new eigenvectors from an arbitraryeigenvector 1E. Let us compute the expression HaE using (8):

Haz/iE= aH?/iE - wa'OE =(E - w)aViE.

It is clear from the last relation that either aE is an eigenvectorcorresponding to the eigenvalue E -- w, or aIPE = 0. If a'E 0, theneither a2 ,0,F is an eigenvector with eigenvalue E -- 2w, or a2 E = 0.Thus, from an arbitrary eigenvector bE we can construct a sequenceof eigenvectors ?PE, aE, ... , aNuE corresponding to the eigenvaluesE, E -- w, ... , E -- Nw. However, this sequence cannot be infinite,because the eigenvalues of H are bounded from below by the numberw/2. Therefore, there exists an N > 0 such that aN E 54 0 andaN+ 1 V)E = 0. Let the vector a N uE be denoted by 00. This vectorsatisfies the equations

w(10) ado = 0, HV)o = 2 V)o

We see that the assumption that there is at least one eigenvectorbE is equivalent to the assumption that there is a vector satisfying(10). The vector iU describes the ground state of the oscillator, thatis, the state with the smallest energy w/2.

Let us examine how the operator a* acts on eigenvectors of H.Using (9), we get that

(11) Ha*tE = a*HbE + wa* bE = (E + w)a* bE

We point out the fact that a*bE cannot be the zero vector. Indeed,it is obvious from the expression (4) for H that a vector satisfyingthe equation a* = 0 is an eigenvector of H with eigenvalue --w/2,which is impossible, since E > w12. Therefore, it follows from (15.11)that a*uE is an eigenvector of H with eigenvalue (E + w). Similarly,(a*)2pE is an eigenvector with eigenvalue (E + 2w). Beginning sucha construction from the vector 00, we get an infinite sequence ofeigenvectors Oo, a *o , ... , (a*)?z1/.'o,... corresponding to the eigenval-Ues w/2, w12 +w, ... , (n+ 112)w I .... Let the vector 00 be normalized:


III = 1. We compute the norm of the vector (a*)flt,o:

11(a*)nO0112 = ((a* )noo, (a* )nOO)

(bo,a1_l(a*)nabo) + n(V)o, (a*

Here we used (7) and (10).

Thus, the sequence of normalized eigenvectors of the operator Hcan be given by the formula

(12) On(a*)noo, n = 07 1127

The orthogonality of the vectors on corresponding to different eigen-values follows from general considerations, but it can also be verifieddirectly:

(Ok, fin) (0o, ak(a')"'+Po) = 0 for k # n.

The last equality is obtained from (7) and (10).

We now discuss the question of uniqueness of the vector Rio. Let?{ be the Hilbert space spanned by the orthonormal system of vectorsOn. The elements cp E 7-l have the form

Inn! E jCnj2 < 00.(13) O=i:C (an 7= jpO I

In this space we get a realization of the Heisenberg commutationrelations if, in correspondence with (2), we let

(14) Q =a+a*

PVrw(a - a*)

ofThe relation (3) is then a consequence of the formula (6). The space?[ is invariant under the action of the operators P and Q (more pre-cisely, under the action of bounded functions f (P) and f (Q), forexample, U(u) and V(v)), and does not contain subspaces with thesame property. Therefore, the representation of Q and P acting in Wby (14) is irreducible.

If in some representation there exist two vectors 00 and 00' sat-isfying the equation (10), then together with W we can construct aspace ?{' similarly from the vector 0o. This space will be invariant

§ 17. The oscillator in the coordinate representation 87

with respect to P and Q; that is, a representation in which there ismore than one solution of (10) will be reducible.

§ 17. The problem of the oscillator in thecoordinate representation

In the last section we found the spectrum of the Schr"finger op-erator for the oscillator in a purely algebraic way, under the singleassumption that the operator H has at least one eigenvector. This isequivalent to the existence of a solution of the equation ado = 0. Weshow that in the coordinate representation there does indeed exist aunique solution of this equation. Let us write the operators a and a*in the coordinate representation:

72= w- ((2) a WX

w= ( )The equation ado = 0 takes the form

wx'00(x) + Rio (x) = 0.

Separating the variables, we get that

dV'o - --wx dx00

and

by (x) = C'e

The constant C is found from the normalization condition

f00

00

1= IC12f--

e "s' dx = IC1200

This condition is satisfied by C = (w/ir)'/4, and the normalized eigenfunction for the ground state has the form

or:afo(x)

4= () e s


The eigenfunctions din (x) for the excited states are found by theformula (16.12) using (2):

n' (2w) - 1 d WX(W) 2

On (X) = - UjX -dx

e 2

7r Vn-1

Obviously, these functions have the form

2

,On (x) = Pn (x) e-"4

-1

where Pn (x) is an nth-degree polynomial. It can be shown thatPn(x) = w--x), where is the nth Chebyshev-Hermite poly-

nomial. It is known that the system of functions H e' -t2 is Com-

plete in L2(R). This assertion follows in general from the irreducibil-ity of the coordinate representation and results in the preceding sec-tion.

The function is the density of the coordinate distributionfunction in the nth state of the oscillator. It is interesting to comparethis distribution with the corresponding classical distribution. Thesolution of the classical problem of an oscillator has the form

(3) x(t) = A sin(wt + W) 7

where A is the amplitude of oscillation and Sp is the initial phase. Theenergy of the oscillations can be computed from the formula

_ x2 w2 x

2

2E=+ 2

and is equal to w2A2/2. The corresponding density of the coordinatedistribution function has the form

(4) F(x) = 6(x - x(t)).

It is clear that a stationary state of a quantum oscillator can underno conditions pass into a classical pure state given by the formula(3). It is natural to assume that the limit of a quantum state will bea classical mixed state that is a convex combination of solutions (3)with random phases cp. For such a state the density of the coordinate

§ 17. The oscillator in the coordinate representation 89

Figure 5

distribution function is obtained by averaging (4):27r

F(x) =2-7r

S(Asin(wt + gyp) - x) dcp0

0, x (-A, A),A}o, x ,

or, briefly,

F(x) =O(A 2 - X2)

r A2 - x2The graphs of the functions F(x) and k1'(x)I2 are shown in

Figure 5 for sufficiently large n under the condition that24 En =h(n + 112)w = w2A2/2. As n -+ oo, the quantum distribution willtend to the classical distribution. The condition n --+ oo can be sat-isfied for a given energy E if h --+ 0.

We point out an important difference between the functions F(x)and F(x) = 0 for IxI > A; that is, the classical particle is

24Here for convenience we write out Planck's constant h explicitly, not assumingthat h = 1

90 L. D. Faddeev and 0. A. YakubovskiY

always between -A and A (in the classically allowed region), whilethe function 17kn(x)12 does not vanish for Jxj > A. This means thatthe quantum particle can be observed with finite probability in theclassically forbidden region. For an arbitrary potential V (x) this re-gion is determined by the condition that the total energy E is < V (x),which corresponds to negative values of the classical kinetic energy.

§ 18. Representation of the states of aone-dimensional particle in the sequencespace 12

According to the results in § 16, any vector W E W can be expandedin a series of eigenvectors of the energy operator of the harmonicoscillator:

_ (a*)

0011: 'Cl2

Cnn n< 00.

Each vector p E 7I is uniquely determined by the coefficient se-quence {C}, that is,

O H

If ik +-+ {B}, then

(a) = 1: BnCnn

in view of the orthonormality of the system of vectors

(a*)n

On = 00 -ri

Let us consider how the operators act in such a representation.For this it suffices to construct the operators a and a*. Suppose that5O {C} and acp {C}. We must express Cm in terms of C, .

§ 18. Representation for a one-dimensional particle 91

This can be done as follows:

(1)

acp = al: C 1Po

nnnlasln-1

l J0

CnY

76!

* n--1n

E n G-rnlu' 1

n V(" n-- - ---1) i I P 0

vrn- -+1 Cn+1(a*)n

00 -

The calculation used the relations [a, (a*)f] = n (a *) n and at'c = = 0,and the last equality is obtained by the change n --+ n + 1 of thesummation index. It is clear from (1) that

(2)

Similarly,

n

Therefore, if a`V -> {C;;}, then

(3) Cn = Vn-Cn-l-

The formulas (2) and (3) become especially intuitive in matrixnotation. Let us write the sequence {C} as a column:

Co

Clcp H C2

Cn= Vn-+1 Cn.

a*

n

n

E (aIncn !00

n- Cn /a*)ntl

00nt

* n+ln

1: Vn-+l Cnu

00n

Y(a*)n'LCn-1

Tn !


Then the operators a and a* can be written as the matrices(4)

aH

o vfi- 0 0

0 0 o0 0 0

0 0 00 0 0

................. . . . . . . . . . . . . . .

:::

...

. . . . . .

a*

0 0 0

f 0 0 ..

0 ,./2 0 ..

................................We verify that these expressions are equivalent to the relations (2)and (3). Indeed,

0 f 0 0 ... Co f/Tci0 0 f2- 0 Ci f Ca

a`p 0 0 0 ... C2 vf3-Cs

which coincides with (2); the connection between the formulas (3) and(4) can be verified similarly. The commutation relation [a, a*] = 1 canbe verified immediately for a and a* in the representation (4). Theeigenvectors of the operator H in this representation have the form

1 0 0

0 1 00- of - o ,3= 1 , ....

The vector b0 obviously satisfies the equation abo = 0.

The operators a* and a are often called the operators of creationand annihilation of excitation, due to the fact that a* carries a statewith energy E to a state with greater energy E + w, while a carries astate with energy E to a state with energy E -- w (the ground state

o is annihilated by a). Sometimes the so-called excitation numberoperator N = a*a is introduced. In our representation it has the form

N+-+

0 0 0 0 ...0 1 0 0 ...0 0 2 0 ...0 0 0 3 ...

The eigenvalues of this operator coincide with the ordinal number ofthe excited state, on.

§ 18. Representation for a one-dimensional particle 93

Finally, we write the operators H, P, and Q in this representa-tion:

0 0 ...

(5) H=wa*a+wH O 2 0 ...2 0 0 Zw ...

............. ...

(6) P =VW--(a - a*)

if

M C) -

0 V-1 0 0 ...\/ _1 o z o ...0 \52 0 \F3

...0 0 v/3-- 0..............................................

It is clear from these formulas that II is represented by a diagonalmatrix; that is, this representation is an eigenrepresentation for theSchr"finger operator of the harmonic. oscillator. Further, it is at onceobvious that P and Q are self-adjoint operators, and it can easily beverified that they satisfy the commutation relation [Q, P] = = i.

In connection with the representation of states in the space 12,we would like to say something about the original Heisenberg matrixformulation of quantum mechanics. Finding the admissible values ofthe energy of a system was a basic problem in the initial stage of thedevelopment of quantum mechanics. For a system with one degree offreedom, Heisenberg proposed the following recipe. lie considered aclassical system with Hamiltonian function H (q, p) = pl /2m + V (q) .Self-adjoint matrices Q and P satisfying the relation [Q, P] = i wereconstructed (such matrices are not uniquely determined) . The matrixH = 1i2/2ni + V (Q) was constructed next. The last step consisted indiagonalization of this matrix, and the eigenvalues of II were identi-fied with the admissible values of the energy.

The formulas (6) and (7) give an example of matrices P andQ satisfying the Heisenberg commutation relations. These matrices

H

fO VfY n o ...

\A_O_

-vfl- 0 %/2- 0

V2_0 \/2'- 0 v/3--

0 0 \/3- 0

a+a* 1


were chosen so that the matrix of the operator H for the oscillator isat once diagonal. However, for an arbitrary H it is riot possible toavoid this diagonalization step.

We see that the Heisenberg formulation essentially coincides withthe contemporary formulation of quantum mechanics if 12 is taken asthe state space.

19. Representation of the states for aone-dimensional particle in the space V ofentire analytic functions

We consider the set of functions of a complex variable of the form

(1)00

f (Z) = 1: Cnzn

This set V of functions becomes a Hilbert space if the scalar productis defined by

(2) (11,12) ffl(z)ieI2d,L(z).7r

The integral is taken over the complex plane, and dp (z) = dx d y.

Let, us verify that the functions zn / n! form an orthonorrnalbasis in D. For this we compute the integral

2ao 21r

'nm = T'n-lz1 d(z) = fpdpfdpn+me(n_m)e_P2.

For n m we have I,nn = 0 due to the integration with respect to Sp.For n = m,

oc

Ix

Inn = 27r p2n+leP2 dp =7r the-t dt = 7rn!.U U

An arbitrary state p E fl with = En Cn n, Oo can be rep-

resented by the function f (z) = En Cn LL : cp f (z) . The eigenvectors On of the oscillator are represented by the basis functionszn / rt!.

We consider how the operators a and a* act in this representation.Using the computations which led us to the formulas (18 2) and (18.3),

§ 20. The general case of one-dimensional motion 95

we can write the vectors aco and a* P in the formn(a*)n-1

- E , * p

(a*)n+la;p Cn o a = 1: Cn *0

n n ni

These vectors are represented by the functions

nzn-1 dacp >Cn = f (z),

Vn-! dzn

zn+ 1

E Cn = Z f (z);n n.

that is, for a and a* we have obtained the representation

aH d-, a* +z.

Here are the corresponding formulas for the operators Q, P, H:

1 d +zQ H2wdz '

r- dP+-+

i v/2- \ dx - z/

HHw zd1dz

+2

All the basic relations ([a, a*] = 1, [Q, P] = i, H1,zw (n + 1/2) in) can easily be verified in this representation. The repre-sentation constructed can turn out to be convenient if the observablesunder study are polynomials in Q and P.

§ 20. The general case of one-dimensionalmotion

In the preceding sections we considered two one-dimensional problemsof quantum mechanics: the problem of a free particle and the problemof a harmonic oscillator. The free particle gives us an example ofa system with a continuous spectrum for the Schrodinger operator,and the harmonic oscillator gives an example of a system with apure point spectrum. In most real physical problems the spectrum is


more complicated. Let us consider the problem of the spectrum of aSchrodinger operator

ll=-d-d +V(x)

with very general assumptions about the potential.

The forces acting on the particle usually are appreciably differentfrom zero in some finite region on the x-axis and tend to zero asIxI --+ oo; therefore, it is most common to encounter potentials V(x)that tend to constant values as x --+ ±oo. For simplicity we confineourselves to the case when the potential is exactly equal to constantsfor x < -a and x > a. By the arbitrariness in the definition of apotential, we can always take one of these constants equal to zero.

We consider Schr6dinger equations(1) 0it+FO-= V(x) ik

under the condition that V (x) is a continuous function on the realaxis with V (x) = 0 for x < -a and V (x) = Vo for x > a. Fordefiniteness we assume that Vo > 0. The graph of the potential ispictured in Figure 6.

Figure 6


(2)

(3)

For x < -a and x > a the Schr"finger equation (1) simplifies:

b"+EV) =0, x<-a,+(E--Vo)ip =0, x>a.

For any values of E there are two linearly independent solutions of(1), which we denote by 01 and 02. The general solution of thisequation is

(4) V) = C101 + C202-

In our study of the spectrum of the operator H we are interested ineither square-integrable solutions of (1) that are eigenfunctions of Hor solutions that are bounded on the whole real axis. With the helpof the latter we can describe the continuous spectrum of 11.

We now consider three cases.

1)E<0.We rewrite the equations (2) and (3) in the form

0" -- x2ip = 0, x < -a, x2 = --E > 0, x > 0;

0"-xiV)=0, x>a. x? =--(E-Vo)>O, xl>0.

The functions for x < -a and ell, x for x > a are linearlyindependent solutions of these equations. Therefore, an arbitrarysolution (4) of the equation (1) in the region x < -a has the formC' e - x + C2 e" , and in the region x > a the form C1' e - xl X + C2 e xl X .

Here Ci, C2, Ci', and C2 are constants that depend linearly on Cland C, 2 in the formula (4). The solution i will be square integrableunder the conditions C' = 0 and C2 = = 0. Just one of these conditionsis enough for 0 to be determined up to a numerical factor. From thecondition Cl = 0, one can find the ratio of the coefficients CI and C2,which will depend on the parameter E:

Cl = F, (E).C2

Similarly, from the condition C2 = 0, we get that

ZT- = F2(E).2


A square-integrable solution 0 will exist only for values of E suchthat

F1(E) = F2(E).

The roots of this algebraic equation, if they exist, are the eigenvaluesof the operator H. From the foregoing it is natural to expect thepresence of a simple point spectrum for E < 0.

2)0<E< VO.We write the equations (2) and (3) in the form

"+k21p =0, x< --a, k 2=E>0, k>0;7P"-xj,0=0, x>a, 4=-(E-V0)>O, x1>0.

The functions e±i for x < -a and efx1x for x > a are linearly inde-pendent solutions. It is immediately obvious that there are no square-integrable solutions, but a bounded solution can be constructed if wechoose C1 /C2 so that 0 has the form C1 e-xx for x > a. Therefore,in the interval 0 < E < VO the spectrum is simple and continuous.

3) E > Vo.

In this case both equations (2) and (3) have oscillatory solutions±ikx for x < -a and e+ 1x for x > a, k1 = E - VO), hence any so-

lution of (1) is bounded, and there are no square-integrable solutions.For E > VO the spectrum of H is continuous and has multiplicity two.

In Figure 6 the eigenvalues of H are represented by horizontallines, the ordinary shading shows the region of the simple continuousspectrum, and the double shading shows the region of the spectrumof multiplicity two.

We discuss the physical meaning of the solutions of (1). Thesquare-integrable solutions describe the stationary states with en-ergy equal to an eigenvalue. These functions decay exponentiallyas lxi -* oo, and thus the probability of observing the particle out-side some finite region is close to zero. It is clear that such statescorrespond to a finite motion of the particle. The eigenfunctions ofthe continuous spectrum do not have an immediate physical mean-ing, since they do not belong to the state space. However, with theirhelp we can construct states of wave packet type which we consideredfor a free particle. These states can be interpreted as states with


almost given energy. A study of the evolution of such states showsthat they describe a particle that goes to infinity (infinite motion) asIt) --+ oo We shall return to this question when we study the theoryof scattering.

In classical mechanics, as in quantum mechanics, the motion isfinite for E < 0 and infiinite for E > 0. For 0 < E < V0 the particlecan go to infinity in one direction, and for E, > VO in two directions.We direct attention to the fact that the multiplicity of the continuousspectrum coincides with the number of directions in which the particlecan go to infinity.

In the example of a particle in a one-dimensional potential wellwe consider the question of the classical limit of quantum stationarystates For the computation of the limit (14.15) it is convenient touse the asymptotic form of the solution of the Schrodinger equationas h -* 0. Methods for constructing asymptotic solutions of theSchrodinger equation as h --+ 0 are called quasi-classical methods.We use one such method: the Wentzel Kraxners--Brillouin (WKB)method.

We write the Schrodinger equation in the form

(s)E --- V(x) - 0.

Here, as earlier, we use a system of units in which m = 1/2. Setting(x} = exp h fem. g(x) dx], we get an equation for g(x):

(6) ihgf -9

2 + E - V = 0.

The solution of this equation will be sought in the form of a powerseries in h/i:

(7)k=lo

Substituting (7) in (5), we have

00 h n oo n n

- E -7 9n -I E E -- 9k9n - 90n=1 (t)

n=1 k=O 2

9(X) = E (2 ) A (X) -


Equating the coefficients of (h/i )', we get a system of recursiveequations for the functions 9k (X):

(8) g02= E - V.

(y) 9n-1 = >29k9n-k.k=0

From (8) we find go(x):

go W V -(X) = ±P(X) -

1 sere p(x) with E > V (x) is the classical expression for the abso-lutes value of the momentum of a particle with energy E in the fieldV (x). For E < V (x) the function p(x) becomes purely imaginary.Setting n = 1, we get from (9) that

' = --2 = -1 g- _ -1 d to x .9'a .90.91, g32 9() 2 dx g 1p(

) I

Confining ourselves to these terms of the expansion (7), we get theasymptotic form as h --40 for the. two linearly independent solutionsof the Schrodinger equation:

r x

(10) *i.z =lp(x) I

exp I f Zf

p(x) + O(h)h o

with p(x) 0. The functions (10) are sometimes called WKB-solutions of the Schrodinger equation.

The precise theory of the WKB method is fairly complicated. Itis known that in the general case the series (10.7) diverges and isan asymptotic series. A finite number of terms of this series enablesus to construct a good approximation for the function V) if Planck'sconstant h can be regarded as sufficiently small under the conditionsof the specific problem.

In what follows we assume that V (x) = 0 for I x i > a and thatE < 0 We suppose that for min,, V (x) < E < 0 there are two pointsx 1 and x2 (-a < x i < X2,< a) satisfying the condition E --- V (x) = 0.These are so-called turning points, at which according to classicalmechanics the particle reverses direction. It is not hard to see thatin the classically forbidden region (x < x 1 or x > x2) one of theWKB-solutions increases exponentially while the other decays upon


going farther from the turning point into the depth of the forbiddenregion. For j > a, the WKB-solutions coincide with the exact so-lutions and have the form efxx, where E = --x`2. We recall thatthe eigenfunctioris of the discrete spectrum of H decrease exponen-tially as x --- ±oo. As h - 0, an eigcnfunction must coincide inthe allowed region with some linear combination Cl ip1 + C22 of theWKB-solutions. The construction of such a linear combination is afairly complicated problem since the WKB-solutions become mean-ingless at the turning points. It can be shown that the conditions fordecrease of the function O(x) hold as x --+ --oo if

(11) O(X) _ C sin hfX

, p(x) dx + + O(h).p(x)

Similarly, it follows from the conditions of decrease as x --> +oo that

+ O(h).(12) fi(x) = C sin (hf4)These two expressions for O(x) coincide if

X2

(13) xdx=-rh n+1 n=012...X,

This condition determines the energy eigenvalues in the quasi-classicalapproximation and corresponds to the Bohr Sommerfeld quantizationrule in the old quantum theory.

We proceed to the computation of the classical limit of a quantumstate. The limit as h --+ 0 can be found under various conditions.For example, we can consider a state corresponding to the energyeigenvalue En for a fixed value of n in the condition (13). It is easyto see that then En --+ Va = minx V (x) as h -- 0, and the state of aparticle at rest on the bottom of the potential well is obtained in thelimit. Let us analyze a more interesting case: h -. 0, n -+ oo, andthe energy E remains constant. (We remark that in the given casethe integral on the left-hand side of (13) does not depend on h, andh -+ 0, running through some sequence of values.) Substituting theasymptotic expression (11) for O (x) in the formula (14.15), we get

102

that

L. D. Faddeev and 0. A. YakubovskiI

F(x' u)27r

n m G(x + ufa) (X)

Cv

(1'x+hm (x + uh) p(er)sin ji

1

p(x) dx + 4

+uh1 fx/1 x C1cosx sin I p(x) dx + ) p(x)

him Lh J p(x) dx

- cos1

J ztuh p(x) + h fxp(x) + 2 J

1

We denote all normalization factors by C. The limit of the secondterm in the generalized function sense is zero, therefore,

F(x, u)C

coti(p(x)u).(X)

Using (14.16), we find the distribution function for the limiting clas-sical state:

C 00

p(q,p) =p(q) f__oo e-sue

Cos(P(9)u) du

Finally, we get that

C °° e-ipu(eip(q)u + e-ip(4')u) du.

p(q)

(14) P(R 4) - PC ) [a(p - P(4)) + b(P + p(4'))]

The state described by the function p(q, p) has a very simplemeaning. In this state the density of the coordinate distribution func-tion is inversely proportional to the classical velocity of the particle,and the momentum of the particle at the point q can take the twovalues ±p(q) with equal probability. The formula (14) was obtainedfor the allowed region. In the same way it is not hard to verify thatp(q, p) = 0 in the forbidden region.

§ 21. Three-dimensional quantum problems 103

§ 21. Three-dimensional problems in quantummechanics. A three-dimensional freeparticle

The Schrodinger operator for a free particle in the coordinate repre-sentation has the form

h2

H2m

A.

The equation for the eigenfunctions (for h = 1 and m = 1/2) is

(2) -AO = k 201 E=k2 > 01

and it has the solutions

(3) fi(x) = eikx

The normalization constant is chosen so that

fi(x) Oki (x) dx = a(k -- k').j3We see that the spectrum of H is positive and continuous, and

has infinite multiplicity. To each direction of the vector k there corre-sponds the eigenfiunction (3) with the eigenvalue k2. Therefore, thereare as many eigenfunctions as there are points on the unit sphere.

As in the one-dimensional case, the solution of the Cauchy prob-Iem

i

dt=HAG, (0)=

for the nonstationary Schrodinger equation is most easily obtained inthe momentum representation

80(P, t)8t = P2O(P, t), '0 (P, 0) = O(P)

It is obvious that

i(P,t) = w(P)e_ 2t.

Passing to the coordinate representation, we get that32 2 Z

x t) eitpX_P t) d (k) k x eik t A.R fR


Just as in the one-dimensional case, the functions p(x, t) or b(p, t)describe an infinite motion of the particle with a momentum distri-bution function independent of the time. Using the stationary phasemethod, we can show that a region in which there is a large probabil-ity of observing the particle moves with the classical velocity v = 2p{(we assume that the support of the function p(p) is concentrated ina neighborhood of the point p}) . We have the estimate

kL'(x,t)I <C

It13/2

where C is a constant. Finally, as in the one-dimensional case,

lim 0t b, 00

for any E 1I.

22. A three-dimensional particle in a potentialfield

In the coordinate representation, the Schrodinger operator for a par-ticle in a potential field has the form

hA+V(x).

The importance of the problem of the motion of a particle in a po-tential field is explained by the fact that the problem of the motionof two bodies reduces to it (as in classical mechanics). We show howthis is done in quantum mechanics. Let, us consider a system of twoparticles with masses m1 and m2 with mutual interaction describedby the potential V (x 1 - X2)- We write the Schrodinger operator ofthis system in the coordinate representation:

h2 h2H = - Al - 2m 02 + V (XI - X2)-2m, 2

Here 01 and A2 are the Laplace operators with respect to the coor-dinates of the first and second particles, respectively.

We introduce the new variablesmlxl + m2x2X= , x=x1 -x2;m1+m2

§ 22. A three-dimensional particle in a potential field 105

X is the coordinate of the center of inertia of the system, and x givesthe relative coordinates. Simple computations lead to the expressionfor H in the new variables:

h2 h2H -- --

YMx --

2A, + V (x) .

Here M = rnl + m2 is the total mass of the system, and jirn 1 m2 / (m 1 + m2) is the so-called reduced mass. The first term inH can be interpreted as the kinetic energy operator of the center ofinertia of the system, and the operator

H1h2

2A+V(x)

is the Schrodinger operator for the relative motion. In the equation

H4' =ET,the variables are separated, and the solutions of such an equation canbe found in the form

P (Xj X) = 7P (X)?Pl (X) -

The function.5 O(X) and 01(x) satisfy the equations (h = 1)

- ZM OlV (X) _ 160 (X)

i Alp, (x + V(X)'Oj x1 = 61 01 x12/u

with E = e + 61. The first of these equations has the solutions

2

X 1 eixx K _ .27r 2M

The problem reduces to the solution of the second equation, whichcoincides in form with the Schrodinger equation for a particle withmass It in the potential field V (x). We note that the spectrum of theoperator H is always continuous, since the spectrum of the operator-- 2 'A is continuous.

The most important case of the problem of the motion of a parti-cle in a potential field is the problem of motion in a central force field.In this case the potential V (x) = V (jx1) depends only on lx} = r. Theproblem of two particles reduces to the problem of a central field ifthe interaction potential depends only on the distance between the


particles. Before proceeding to this problem, we study properties ofthe angular momentum and some questions in the theory of represen-tations of the rotation group. This will enable us to explicitly takeinto account the spherical symmetry of the problem.

§ 23. Angular momentum

In quantum mechanics the operators of the projections of the angularmomentum are defined by the formulas

L 1 = Q2P3 - Q3 P2,

(1) L2 = Q3P1 - Q1P3,

L3 = Q1 P2 - Q2P1

We introduce one more observable, called the square of the an-gular momentum:

(Z) L2 = L, + L2 + L3.

Let us find the commutation relations for the operators L1, L2 ,

L3, and L2. Using the Heisenberg relations [Qj, Pk] = iSjk and theproperties of the commutators, we get that

[L1, L2] = [Q2P3 - Q3P2, Q3P1 Q1P3]

= Q2P1 [P37 Q3] + Q1P2[Q3, P3]

= i(Q1P2 -- Q2P1)= iL3.

Thus, the operators L1, L2, L3 satisfy the following commutationrelations:

(3)

[L1,L2] = iL3,

[L2, L3] = iL1

[L3, L1] = iL2.

It is not hard to verify that all the operators L1, L2, L3 commutewith L2:

(4) [Lj I L2] = 01 1,2,3.

§ 23. Angular momentum 107

Indeed,

[Li, L 21 = [L1, Li + Lz + L3]

= [Lil L2] + [L1, Ls)

[L1, L2] L2 + L2[L1, L2] + [L1, L3]L3 + L3[L1, L3]

iL3L2 + iL2L3 -- iL2L3 -- iL3L2 = 0.

The properties of the commutators and the formulas (3) were used inthe computation.

It follows from the commutation relations (3) and (4) that theprojections of the angular momentum are not simultaneously mea-surable quantities. The square of the angular momentum and one ofits projections can be simultaneously measured.

We write out the operators of the projections of the angular mo-mentum in the coordinate representation:

C lLl

=i a aX2 8xX3 a - xl2 3l

C lL2 -Z xl

aax -x3

a8xlI

C a lL3 - i x2

ax,-XI a

ax

It is not hard to see that the operators L1, L2, L3 act only onthe angle variables of the function O(x). Indeed, if the functiondepends only on r, then

L3'0(r) = i x2 axl - x,a2 (xi + x2 + x3)

= io'(xi + xi + x3)(x2 2x, - xl 2x2) = 0.

For what follows, it turns out to be useful to have the formula

(5)


which we also verify in Cartesian coordinates. To this end we computethe operator - J12, using the notation x, y, z for the projections of x:

2 a a 2 a a z a a 2-L = x19 -yax + yaz - za+ zax -x1922 2 2 192

192492 2

822 192 2Z a19z2= (x +y +z) + ay2 + a2 -x 2 _y` 2 -z 2a ay

192 _ 192 192 d d a- 2xy a2yz a az - 2zxaz ax - 2x

19x -2y

a_ 2z

az .JTaking into account the relations

a a 19 arar -xax+yay +zaz,

19 2 2192 2192 19`2 a 19 aVrar

-xaxe

+y a 2+Z2

1922+xax +yay +zaz

,92192

192

+xy + yz + 2zx

19X (9y ay az192

axJ

we get that - T.2 = r2 A - (r)2i9r 07-r9 =r 2A - dT(r2

c0 ).

§ 24. The rotation group

Denote by G the collection of all rotations of the space R3 aboutthe origin of coordinates. Each rotation g E G generates a lineartransformation

x'=gxof three-dimensional space, where g now denotes the correspondingmatrix IIgig 1 1 , called a rotation matrix. It is well known that g is anorthogonal matrix and det g = 1. The converse is also true: to eachsuch matrix there corresponds a definite rotation. Therefore, in whatfollows we identify rotations with their matrices. The group of realorthogonal matrices of order three with determinant 1 is denoted bySO(3) and called the rotation group

There are several ways to parametrize rotations. For example,in theoretical mechanics one often uses the ruler angles. For ourpurposes it is most convenient to specify a rotation by a vector a = na(j n J = 1) directed along the axis n of rotation with length a equal to

§ 24. The rotation group 109

the angle of rotation. Here it is assumed that the direction of rotationand the direction of the vector a form a right-hand screw and thati he angle a of rotation does not exceed ir (0 < a < ir). If we start the,vectors a from a single point, then their endpoints fill a ball of radiusTr. Different rotations correspond to different interior points of thisball, and diametrically opposite points of the boundary correspondto the same rotations.

Thus, the rotation group is topologically equivalent to a hall withopposite points identified.

We shall establish a connection between the rotation group andthe Lie algebra of skew-symmetric matrices of order three.

A matrix A is said to be skew-symmetric if the transposed matrixA' is = -A. An arbitrary skew-symmetric matrix is given by threeparameters and has the form

0 a12 a13A= --a12 0 (123

(-a13 -a23 0

The collection of skew-symmetric matrices becomes a Lie algebraif the commutator [A, 13] is taken as the Lie operation. This assertionfollows from the properties of commu t ators and the equality [A, f3]' =-[A, B]. The latter can be verified directly:

[A, B]'= (AB-BA)'= 13'A'-A'B' =BA- AB = -[A, B].

It is convenient to choose the matrices

0 0 0 U 0 1 0--1 0

A,= 0 0 -1 ill = 0 0 U A3 = 1 0 0

0 1 0 \-i 0 0 U 0 0

as the generators of the Lie algebra. Any skew-symmetric matrix canbe represented in the form

A(a) = a, Al + a2A2 + a3A:;.


A3:

We find the comirlutation relations for the matrices A1, A2, and

[A, ,As]= AIA2 - A2A1

0 0 0 0 1 0 0 -1 U

= 1 0 0- U 0 0= 1 0 0= A3,U 0 0 0 0 0 U 0 0

and the desired relations are

(1)

A direct verification shows that

[A(a), A(b)] = A(a x b).

Let us consider the matrix eI A3 . To get an explicit expressionfor this matrix, we compute the integer powers of the matrix A3:

-1 0 0

A2= 0 -1 0 ---13, A3 =-A3, A4=I3,3 3 3

0 0 0

Expanding e'7 A3 in a series, we get that

A - x arcA; 0O (1)k2k (-1}k+1c2k-1e 3 - E ` = I + I>

(2k) 43 (2k-i)!n=O k=1 k=-1

= I+I3(cosa - 1)+A3sina,

or

cos a - since 0

e, A3 - = Sin cx Cos a 0

0 0 1

We see that e'A, is a rotation about the third axis through theangle a; that is, g(0, 0, a) = It can be verified similarly thatg(a, 0, 0) = e°A' and g(0, a, 0) = C,A2. F o r rotations about the coor-dinate axes we shall use the abbreviated notation g (a), i = 1, 2, 3.

§ 25. Representations of the rotation group 111

The matrices 8q(a) are called infinitesimal generators of the8a; LOrotation group. Using the formulas for rotations about the coordinateaxes, we get that

ag(al , 0, 0)

a=n salae°`1 A,

a, =0 0a1 a1=0= Al .

Similarly,

ag(a)

8a2A2

a=0

Og(a)= A3-

(9a3 a=o

We see that the skew-symmetric matrices A1, A2, As are infinitesimalgenerators of the rotation group.

Now we can easily get a formula for an arbitrary rotation g(na).It is obvious that the product of the rotations about a single axis nthrough angles a and f3 is the rotation through the angle a +,3 aboutthe same axis:

g(nj3) g(na) = g(n(a +,3)).

Differentiating this identity with respect to /3 and setting /3 = 0, weget, that,

+ n A+ n A na) = dg(na).(niA 1 22 3 3} g( di

We have found a differential equation for g(na) which must be solvedunder the initial condition g(nO) = I. The unique solution of thisproblem is

(2) .9(na) = eXP[(niAi + n2Aa + nsAs)a].

§ 25. Representations of the rotation group

A representation of the rotation group G on a Hilbert space E isdefined to be a mapping W that carries elements g of G into boundedlinear operators W (g) on E depending continuously on g and suchthat

1) W(I) = I,

2) W(9i92) =W(9i)W(9a)


In the first condition the first I denotes the identity rotation, and thesecond denotes the identity operator on C. It follows at once fromthe conditions 1) and 2) that

W(g-1)=W-1(g).

A representation is said to be unitary if the operators W (g) areunitary; that is, W *(g) = W -'(g).

In group theory it is proved that any representation of the rota-tion group is equivalent to some unitary representation, and thereforewe can confine ourselves to the study of unitary representations.

We recall the two equivalent definitions of an irreducible repre-sentation:

1) A representation is said to be irreducible if there are nooperators other than those of the form C1 that commutewith all the operators W (g) :

2) A representation is said to be irreducible if 6 does not havea proper nontrivial subspace 6 invariant under the actionof the operators W (g).

It is easy to construct a representation of the rotation group onthe state space ' = L2 (W). To do this we define the operators bythe formula

(1) W(g) OX = 0(g-'X).

The condition 1) of the definition is obvious, and the condition 2) isverified as follows:

W (9192) (92 '911 x) = W(91) /' (.9 ' x) = W (91) W (92) zb (x).

The operators W (g) are one--to-one mappings of 1-1 onto itself, andtherefore to verify that they are unitary it suffices to see that theypreserve the norm of a vector ':

IIW(g)II2 = 1,0(g-IX)12dX = I= IIV9112,R3 JRJ

where we have used the equality det g = 1. Let us show that

(2) W (a) = exp[-i(I,i al + Lla2 + L3a3)]-

§ 25. Representations of the rotation group 113

If ere we have denoted W(g(a)) by W (a), and the. projections of theangular momentum by L1, L2, L3- To prove (2) we first consider arotation about the third axis and show that

W3((-t) = e -iL3CY'

or, equivalently,

(3) b(93 1(a) X) = e_iL3a'0 (X)

for any fi(x) E R. We verify the last equality by showing that thefunctions on the left- and right-hand sides satisfy one and the sameequation with the same initial conditions.

The function

'0(X, a) -- e- iL30t v)(X)

obviously satisfies the equation

80(x,a)as - -iL30(x,a),

or, in more detail,

(4)a)

2 - X1 1P (x, a) -aa (X ax, aX2 )The function on the left-hand side of (3), which we denote by

ip (x, a), has the form

01 (x, a) ='(x1 cos a + X2 sin a, -x1 sin a + x2 cos a, x3).

We verify that this function satisfies the equation (4):

O,1 a 1 a(--xi sin a + x2 cos a) + (--x1 cos a- x2 sin a),

as Ox1 Ox2

x2 =8x1

x28x1

cos a - x2axe

sin a,

x 1901 = x 8- sin a+ x 0 cos a1Ox2 iax1 iax2

from which (4) follows for the function 01(x, a). Finally, both func-tions satisfy the same initial condition

0(XI 0) = *I (XI 0) = *(x) -


Thus, the operators of rotation about the coordinate axes havethe form

(s) Wj(a) = e-iLjcx, j=1, 213-

The infinitesimal operators of the representation are found at oncefrom (5):

8W (a)Oaf

aWj(a)

a=O da (=0= --iL3 .

Repeating the arguments that led us to the formula (24.2), wenow get (2). We note that the operators - i Li have the same com-mutation relations as the matrices A3 , j = 1,2,3. Indeed, it followsfrom

[L1 , L2] = iL3

that[= --iL3.

In group theory it is proved that the commutation relations be-tween the infinitesimal operators of a representation do not depend onthe choice of the representation. Moreover, if some operators actingin a space E satisfy the same commutation relations as the infinites-imal generators of a group, then they are the infinitesimal operatorsof some representation acting in E.

In connection with the rotation group this means that if wefind operators M1, M2, M3 satisfying the relations [M1, M2] = M3,[M2, M3] = M1, [M3, M1] = M2, then we can construct a representartion W by the formula

W(a) = exp(a1M1 +a2M2 +a3M3)-

§ 26. Spherically symmetric operators

An operator A is said to be spherically symmetric if it commutes withall the operators W (g) in a representation of the rotation group.

Obviously, an operator A is spherically symmetric if [A, La] = 0fori=1,2,3.

Here are some examples of spherically symmetric operators.

§ 26. Spherically symmetric operators 115

1) The operator of multiplication by a function f (r). Indeed, wehave seen that the angular momentum operators act only on the anglevariables. Therefore, L i f (r) * (x) = f (r) L i iI (x) for any '0 E R.

2) The operator L2. The spherical symmetry of this operatorfollows from the relations [L2, Li] = 0 obtained earlier for i = 1, 2, 3

3) The kinetic energy operator T = -(1/2m)A. The sphericalsymmetry of this operator is clear at once in the momentum repre-sentation, where it is the operator of multiplication by the functionp'2 / (2m). In the momentum representation the angular momentumoperators Li have exactly the same form as in the coordinate repre-sentation.

4) The Schrocliuger operator

H_ -- 10+ V(r)2m

for a particle in a central field, as a sum of two spherically symmetricoperators.

We direct attention to the fact that the very existence of spher-ically symmetric operators different from CI implies the reducibilityof the representation of the rotation group constructed on the statespace H.

For the spectrum of the Schrodinger operator in a central fieldwe now clarify some features connected with its spherical symmetry.Let 0 be an eigenvector corresponding to the eigenvalue E:

H = E.Then

HW(g) ?P = W(g) H = EW(g)1,

from which it is obvious that W (g) is also an eigenvector of Hcorresponding to the same eigenvalue.

We see that the eigenspace HF of H corresponding to the eigen-value E is invariant under rotations (that is, under the action of theoperators W (g)) . The representation W of the rotation group on thespace h induces a representation WE, g --+ WF (g), on the subspace7Y E, where WE (g) is the restriction of W (g) to HF (below we use theflotation W (g) also for the operators W,; (g)). There are two cases:


either the representation induced on HE is irreducible, or R F, con-tairis subspaces of smaller diniensiori that are invariant under tiW(g),and then this representation will be equivalent to a direct suns ofirreducible represenitations.2'

We see also that the multiplicity of our eigenvalue of a sphericallysymmetric Schrodinger operator is always no less than the dimen-sion of some irreducible representation of the rotation group, and itcoincides with this dimension in the first case mentioned.

In physics the appearance of multiple eigenvalues of energy iscalled degeneracy, and such energy levels are said to be degenerate.If on each of the eigenspaces the induced representation is irreducible,then one says that the operator II does not have accidental degen-eracies. In this case the multiplicity of the spectrum is completelyexplained by the chosen symmetry of the problem. In the presenceof accidental degeneracies there may exist, a richer symmetry groupof the Schrodinger equation. This is the case for the Schrodinger op-erator for the hydrogen atom, which, as we shall see, has accidentaldegeneracies with respect to the rotation group.

We remark that for a spherically symmetric operator H with purepoint spectrum there are eigenvalues with arbitrarily large rrlultiplic-ity. Indeed, in this case 7-1 can be represented in the form

`l = "'E1 ®nE2 a...

On the other hand,

fl=Rl.®7-12e ,

where the N,, are subspaces on which irreducible representations ofthe rotation group act. In studying such representations in § 30 weshall see that among the 7-1, there are subspaces of arbitrarily largedimension. But for any ln, at least one of the intersections 7-1,, n HE,,

25 Here we use a well-known theorem in group theory if g - W (g) is a unitaryrepresentation of the rotation group on a Hilbert space E, then there exist finite-dimensional subspaces El, e2, that are invariant with respect to W (g) and suchthat the representation W is irreducible on each of them. they are pairwise orthogonal,and they have sum E, that is, E = El ®E2 9

§ 27. Representation by 2 x 2 unitary matrices 117

is nonernpty, and then it contains l-( entirely, so that the eigenvalueEk has multiplicity not less than the dimension of 7n .26

If the system does not have accidental degeneracies, then thecigenvalues of H can be classified with the help of irreducible rep-resentations of G in the sense that each eigenspace H E is also aneigenspace of the corresponding representation. Therefore, the prob-lent of finding all the irreducible representations of the rotation groupis important. We dwell on this question in the following sections.

In concluding this section we remark that such a relatively simplequantum mechanics problem as the problem of motion in a centralfield could be solved in general without invoking group theory. Ourgoal with this example is to show how group theory can be usedin solving quantum mechanics problems. The xnicroworld (atoms,molecules, crystal lattices) is very rich in diverse forms of symmetry.The theory of group representations makes it possible to take thesesymmetry properties into account, explicitly from the very beginning,and it is often the case that one can obtain important results for verycomplex systems only by using an approach based on group theory.

§ 27. Representation of rotations by 2 x 2unitary matrices

We shall construct a representation of the rotation group G acting inthe space C2. To this end we introduce three self-adjoint matrices

26A very Simple example of a spherically symmetric operator with pure pointspec trump is the Sehrodiriger operator

f f =Pj 4 P2 4- P2 + ,02

1 1 ± -2 ±3

)

2 2

or, in the coordinate representation,

'Fl=_ l A+

2r2

The problem of separation of variables reduces to the one-dimensional case, and forthe eigenval ues one obtains the formula

En1n2n3 = n1 + n2 + TL3 +3),

ni, n2. n3 = 0. 1, 2,2

To each triple of numbers n1, n2, n3 there corresponds an eigenvector 'Pn 1 n2 n3 It isclear that the multiplicity of the eigenvalues E grows unboundedly as E ---> oo

118 L. D. Faddeev and 0. A. Yakubovski i

with trace zero:0 1 _ 0 -i 1 0of _ 1 0' 2 i 0' 3--' 0 -1

These are called the Pauli matrices. Let us compute the commutationrelations for them:

_U - 0' O = 2 0 - _2 0 =2i 1 0[a1o2] 1

0 -i 0 i 0 -1

that is, [a', o-2] = 2io3; similarly,

[o, (73] = 2icr1, [Q3, (71] = 2io2.

It is not hard to see that the matrices --iaa /2, j = 1. 2, 3, havethe same commutation relations as the infinitesimal generators Aj ofthe rotation group

2Q1 192 2Qg

2' 2 2

Therefore, we can construct a representation g --+ U(g):

(t) U(g) = exp [_(o-iai2 + 2a2 + Q3as)J .

It should be noted that this is not a representation in the usualsense of the word, since instead of

(2)

we have

(3)

U(91) U(92) = i1(9192)

U(91) U(92) = aU(9192),

where a = ± 1. It is not hard to see this in a simple example, com-puting the product U (gl) U (g2) when gi = g2 is the rotation throughthe angle 7r about the x3-axis:

U(.9,) U(.92) e e; e0 i

At the same time, U(0, 0, 0) = I corresponds to the identity ele-ment of the group according to the formula (1). A mapping g - U(g)satisfying (3) with J = 1 is called a projective representation witha multiplier. If we nevertheless want to preserve (2), then we shallhave to consider that to each rotation there correspond two matrices

§ 27. Representation by 2 x 2 unitary matrices 119

U differing in sign. In physics such representations are said to betwo-to-one. These representations play as important a role in quan-tum mechanics as the usual representations. In what follows we shallnot stress this, distinction. We remark further that the appearance ofsuch representations is explained by the fact that the rotation groupis not simply connected.

We consider the properties of the matrices U (g) . It is obviousthat they are unitary, since the of are self-adjoint matrices. It is nothard to see that they have determinant equal to 1 Indeed, U(g)has the form eis, where S is a self-adjoint matrix with trace zero.This matrix can always be reduced to diagonal form by a similarity

transformation, and it takes the form 0. Correspondingly,0 _

eza 0the diagonal form of the matrix will be 0 e)_ i.X . The trace

and determinant are invariant under a similarity transformation, sodet U (g) = 1.

Let us determine the general form of a unitary matrix with de-terminant equal to 1. The condition for unitarity is

a b) (a c + bb a+ bd _(c d) (b d) -

(aca + db cc + ddJ

From the equality cu + db = 0, we have that d = -cd/b, and from theconditions det U = I and ad + bb = 1, we get that

ad - be = - ab - be = - b (aa + bb) _ - b = 1;

that is,

c= --b, d =a.

Thus, unitary matrices with determinant 1 have the form

U= (-6a bJ, ja12-FJb12=1.a

The group of such matrices is denoted by SU(2).


§ 28. Representation of the rotation group on aspace of entire analytic functions of twocomplex variables

In t his section we construct all irreducible represent ations of the rotest ion group. As the space of the representations we choose t he Hilbertspace D2 of functions f E C,'q E C) of the form

ni n2f = C n C 2 < oc .f q) 1 M2 nin2

r

nl,n2=() ni! ni. nl,n2

with the scalar product

(f,.9) = 2 f r1} r)) e212 dlL(r!)

Just as in 19, the functions f n 1 n2 = ,t i ??Tb2 //nj ! n2! can beseen to form an orthonormal basis in this space: (f12, fn' n2) _6nin; Sn2n2. Taking into account the connection between the groupsSO (3) and SU (2), we can construct a representation of the groupSU(2). In what follows, it is convenient to represent f rz) as f ('),

where = E C2 The representation U -* W(U) is defined by

the formula

(1) W (U) f ((') = f (U-10-

We. shall denote these operators also by W (a) or W (g), and the op-erators of rotation about the axes by Wj (a), j = 1, 213.

To get an expression for W (g), we find the infinitesimal operatorsof the representation, which we denote by -i Mj, j = 1, 2, 3:

a Wai =dWl(a)(),a=ofif (() fda8a1

daf (Ui-I(a) 010=0

n=o

d(ei)afda o=

a f 4 (a) a f drl(a)- a da 4-0 + ada n=o

Here we have used the definition (1) and have denoted by (a) andii(a) the components of the vector e 2' a (. The last derivatives are

§ 28. Representation of the rotation group

computed as follows:

d i l0 1

doe e a-0

and therefore

2a, e2°10c2 ct=0

i77, dI(ada JQ=0 2 da

2 77

2 2(()'

As a result we get that

i of of-iMi f (0 = 2 (T n+ a

The operators M2 and M3 are found in exactly the same way.write out the expressions for these operators:

(2)

2 (77 d+ a }M,

77

M=_i2 2 na + a

a aM32 77077)

121

We

It is easy to verify that the operators Mj have the same com-mutation relations as the angular momentum operators, and -iMM,j = 1,2,3, have the same commutation relations as the matricesA2 , A2, A3. For the operators W (a) we get that

(3) W(a) = exp[-i(Mial + M2a2 + M3a3)]

The basic convenience of the representation space D2 is that itis very easily decomposed into a direct sum of invariant subspacesin which the irreducible representations act. Indeed, the invarianceof some subspace with respect to the operators W (a) is equivalent tothe invariance with respect to the action of the operators MI, M2. Ltd3.From the formulas (2) it is clear that the subspaces of homogeneouspolynomials of degree n = nl +n2 are such invariant subspaces. Thesesubspaces have dimension n + 1, n = 0, 1, 2, .... It remains for itsto show that such subspaces do not contain invariant subspaces of


smaller dimension. To do this we introduce the operators

/AI

=- aM =M iM2

M- = M -iM = --

and consider how they act on the basis vectors fn1 n2 :

a Cnj It2M+fnin2 = -nl o 1f .n21n lf .n2I . .

ns-lnn2tl\/n- -(n

/(ni - 1)!(n2 -F 1)!

that is,

(5)

M+fnin2 = - nl(n2 + 1) fni-1,n2+1,

M_ fnin2 = - (n1 + 1)n2 fnl+l,n2-1

It is obvious that

M+fort = 01 M fnu = 0.

From (5) it is clear that the subspaces of homogeneous polynomialsdo not contain invariant subspaces of smaller dimension.

Let us show that the basis vectors are eigenvectors of the operatorM3 and the operator MI = Mi + M2 + M32. We have

nt r1712M3 -- -1 a fin' n2

=-1(1

n -n2)fnln2 a a%/-n--j!. n2, .

2Vn n2.2

that is,

(6) -{n1 - n .1n22

2)fn1n2

For the operator M2 we have the formula

M2=M+M_+M3 -M3.

Indeed,

M+ M- = (M1+iML)(Ml-iM2) = M12 +M.2 -i(M1M2-M2M1)

=M2-M2 +M3.

§ 29. Uniqueness of the representations D3 123

Next, we have

M 2 f,,1 n2 = (M4-M- + M3 - 3) frt, rt 2

-- (v"ni + 1)n2 n2 (n, + 1)V

+ 1 (n, -- n2)2 + 2(nI - n2)fnin21

so that

(7) M2 fn1 712 = ((ni + n2)2 -F' 2 (nl + n2) J fni nz.

It is convenient to rewrite all the relations obtained, replacingni, n2 by j, m according to the formulas

ni+n2 m_- (ni - n=0113...2 2 2 2

m=-3,-j+1,...,j,or

n j =j - m, n2 = j + m.

Then the formulas (5) -(7) take the form

(8) M+fjm = - (j - rn) Cj + m + 1) fj,rn+i

(9) M _ f m = -- (j - m + 1 } (, + m} f j,m-i

(10) M3,fj9n = mf jrri,

(11 M2fjm = . (j + 1)fjrri

where f j,, denotes fra, n2 = f j -rra,j+m.

The new indices j and m are convenient in that to each in-dex j there corresponds a representation of dimension 2j + 1, j =0. 1/2,113/21 .. . This representation is usually denoted by D,, andj is called the representation index. The formulas (8) (10) make itpossible to easily construct the explicit form of the matrices MI, M2,M3 for each Dj. Thus, we have constructed the finite-dirnerisionalrepresentations 1). of the rotation group for all dimensions.

§ 29. Uniqueness of the representations Dj

We prove that the irreducible representations Dj constructed areunique (up to equivalence). In the course of the proof we shall see


to what extent the spectrum of the angular momentum operatorsis determined by their commutation relations, and how an arbitraryrepresentation of the rotation group is decomposed into irreduciblerepresentations.

Suppose that some irreducible representation is defined on ann-dimensional space E, and denote by - i Jr . --- i J2 , - i J3 the infinites-imal operators of this representation. They satisfy the commutationrelations

[Ji, J2] = ieJ31

iJ2, Jai] = iJ1

[J3,Ji] = 2J<2.

The equivalence of this representation to some representation Djwill be proved if we can prove that for a suitable choice of basis inE the matrices Jk, k = 1. 2, 3, coincide with the matrices Mk . Wenote first of all that a representation of the rotation group is at thesame time a representation of its subgroup of rotations about the x3-axis through angles a. This subgroup is Abel i an, and therefore allits irreducible repiesentations are one dimensional and have the forme-'rn a (we do not assume anything about the numbers mk, allowingthe possibility of "multivalued" representations). This means thatfor a suitable choice of basis in E the matrix of a rotation about thex3-axis has the form

e-iha =

e irnla

l

0 ... 0

0 e-im2 0

......... ..........

0 ... 0 e- im"Q

Thus, E has a basis of eigenvectors of the operator J3:

J3erra = Trtem .

Further. it follows from the irreducibility of the representation thatthe operator J2 = '12 + J22+ J,21, which commutes with Jk for allk = 1.2, 3, must be a multiple of the identity on the space E This ispossible if all the basis vectors e, are eigexivectors of J2 correspondingto one and the same eigenvalue, which we denote by j (j + 1) (for thepresent this is simply a notation). The basis vectors will be denoted

§ 29. Uniqueness of the representations D. 125

by elm

(2) J2ejm =X(.7 + 1) e)m

We introduce the operators JJ = J1 ± i J2 . It is easy to verifythe formulas

(3)

(4)

(5)

[J2, jam] -- o,

[J3, J±] = ±J±,J2=J±J::F +J3 j3.

We find a bound for the possible values of 17,nl for a given eigenvaluej (j + 1). To this end we take the scalar product with e j m of theequality

(J,2 + J2) elm = (J(1 + 1) - m2)

and get that

((J? + Jl) ejrr1, ejrn) = j(3 + 1) - m2.

The left-hand side is nonnegative, therefore

(6) lmi < /j(j + 1).

From the relations (3) and (4) it follows that

+1)Jfe J3Jte =(rn±1)JCjj(j m t mTherefore, the vectors J± e j,, (if they are nonzero) are eigenvectors ofthe operator J2 with eigenvalue j (j + 1) and of the operator .13 witheigenvalues m ± 1. Thus, for an arbitrary basis element e7 r, we canconstruct a chain of eigenvectors with the same eigenvalue of J2 andwith eigenvalues ml, m] + 1, ... , 7,n2 of J3. Here m 1 and m2 denotethe smallest and largest eigenvalues, respectively. The existence ofnil and rn 2 follows from the inequality (6): the chain of eigenvectorsinnst break off in both directions.

Let us compute the norm of J± e j,n, using the facts that ikjm 11 = 1and J-L = JT.

iiJ±C,m 112 = (JTJ±ejm. (('12 - 3 T J3) ejm, ejm)_ j(j + 1) -rn(rat+1)= (.j rn)(j±m+1).

126 L. D. Faddeev and 0. A. Yakubovskij

Therefore. we can write

(7) J=e.in = - (j T m) (j ± m + 1) ej.r1at

This formula enables us for any unit basis vector to construct newunit basis vectors e j,7 ± 1 satisfying all the requirements The minussign in front of the square root is writ ten for convenience.

We have not yet explained what values the numbers j and m cantake. We start from the equalities

(8) J_ ejmi = 0, J+ejr2 = 0

Multiplying these equalities by J +. and by J_ and using (5), we getthat,

(J2J+J3)jmi=0. (J2-JJ.3)jm2=0,3

or

J(j + 1) _ M2 + mi = 0,.

j(j +1) -rn -m2 =0From (9) we get at, once that (mi + M2) (MI - m2 - 1) = 0. We takethe solution ml = -m2 of this equation, since m2 > rnl. Further,rn2 - rn z = 2rn2 is an integer or zero. Therefore, rrm2 can take thevalues 0, 1/2. 1, 3/2, .... Finally, we see from (0) that j can be takento be m2. This means that the eigenvalues of 'I 2 have the formj (j + 1), where j = 0, 1/2, 1, 3/2, . . , and for a given j the eigenvaluesm of J3 ruri through the (2j + 1) values -j, + 1, ... , j - 1. j. Theriuinbers j and in are simultaneously either integers or half-integers.We emphasize once more that these properties of the spectrum of theoperators J2 and J have been obtained using only the commutationrelations.

To finish the proof it remains for us to see that the eigerivectorsconstructed form a basis in £. This follows from the irreducibility ofthe representation. Indeed, the subspace E` spanned by the vectorse37 n with m = -j, -j + 1,. .. , j will be invariant with respect to theoperators Jk, k = 1, 2, 3, and therefore must, coincide with e, and thedimension of the representation is n = 2j + 1. The formulas (1) and(7) show that for this choice of basis the matrices Jk coincide withthe matrices Mk.

30. Representations of the rotation group on LZ(S2) 127

We note that at the same time we have constructed a way ofdecomposing an arbitrary representation into irreducibility represen-tations. Suppose that a representation of the group of rotations -iJkand its infinitesimal operators acts in some space e. To distinguishthe invariant subspaces, we must find the general solutions of theequations

(10) J2e.rn + 1) ejm. J3e3m = mejm.

F o r given j and rn = -j, --3 + 1, ... , j the vectors ejm form a basisof an irreducible representation of dimension 23 + 1. The problem offinding the general solutions of (10) is solved most simply as follows.First a vector e. ,j satisfying the equations

J+ejj = 0, J3ej.j = jejj

is found, and then the vectors a j m are constructed using the formula

J_ejn = -- (j + m}(j -- rn + 1) ej,m-1,

which makes it possible to find all the vectors e j,,, frorri e j j .

§ 30. Representations of the rotation group onthe space L2(S2). Spherical functions

In § 25 we constructed a representation of the rotation group on thestate space f = L2 (R3) by the operators

W (a) = exp[--i(L1a1 + L2a2 + ,3a3)]

where L1, L2.13 are the angular momentum operators. We recall thatthese operators act only on the angle variables of a function (x) EL2 (R3) , and therefore it is convenient to regard the space L2 (R3)a s 1,2 (R+) ® L2 (S2) . Here L2 (R4) is the space of square-integrablefunctions f (r) with weight r2 on R+, and L2(S2) is the space ofsquare-integrable functions i(n) = 0(0, co) on the unit sphere. Thescalar products in these spaces are given by the formulas

xfr2fm(r)Jdr, f V), (n) GZ (n) dn,

2

where do = sin 0 dO dcp is the surface element of the unit sphere.


Somewhat cumbersome computations lead to the following formfor the angular momentum operators in spherical coordinates:

Lt got e Cos 0a I ,

L2 = -i cos O To cot ©sixld00

}

aL3

Ow

In these variables the operators L± = L1 ± iL2 have the form

8L = ei-P i cot B L_ = e-- i Cot o

The common eigenfunctions of the operators L2 and L3 will he de-noted by lm (n) (the number j for L is usually denoted by 1). Inspherical coordinates the equations L, Y11 = 0 and 13 Y11= l Y11 havethe form

ni

OY/I-i- =lYtt,SP

aYl+ iCoto y =0.

80

From the first equation in (1) we see that

Yt (0, P) = Fat (0),

where 1 can take only the integer values l = 0, 1, 2, .... The secondequation in (1) enables us to get an equation for F411(0):

(2)OF,, (e)

- i cot e F11 (e) - v.ae

Solving this equation, we get that

F11(0) = C since 0.

We note that for each 1 there exists a single solution of the equation(2). Thus, we have found that

Y1(0, co) = C sin1 0 ei10 ,

§ 30. Representations of the rotation group on L' (S2) 129

where the constant C can be found from the normalization condition.The remaining functions Ym (n) can be computed by the formula

Y -- - rl-i42(-570-+

19 icote aV(1 + rn)(l-m+1) a, tm

We shall riot go through the corresponding comput ations. The funs--t ions Yrra (n) are called the normalized spherical functions of order 1.They can be expressed as

1 irnSpYrrm (e1 ) _ - e 1u1rn(cos O)

2ir

where the functions

P1, (A) - (l_m)!{2] 1 z 1)1

11! 1- )dot -"d

are called the normalized associated Legendre polynomials.

Accordingly, a basis of an irreducible representation in the spaceL2(S2) consists of the spherical functions Ym (n) for a fixed 1 andm = -1, -1 + 1, ... -, 1The space L2 (S2) contains a subspace for anirreducible representation for each odd dimension 21 + 1 (1 an integer).The theorem on decomposition of a representation of the rotationgroup into irreducible representations is equivalent in this case to theassertion that the spherical functions are complete in L2 (S2) . Anyfunction 0(n) E L2(S2) can be expanded in a convergent series

00 1

(3) (n)= E 1: CimYm (n).1=0 m=-l

We recall that irreducible representations of both even and odddimensions acted in D2. The space D2 can be represented as a directsum D2 eV2 , where D2 and D2 are orthogonal subspaces of even andodd functions, respectively. In E)2+, as in L2 (S2), only representationsof odd dimensions act. Any element f (z) E D2 can be represented inthe form

00 1 l-mil+m(4) f (Z) = E E C1M-

1=0 M=-1 V(1 - M)! (I + M)!

130 L. D. Faddeev and 0. A. Yakubovskjr

It is clear that the one-to-one correspondence

f(z) V) (n)

establishes an isomorphism between the spaces Di and L2 (S2) forwhich

l l rrt n)(l -- rrt)!(l + m)}.

MkHLk, k=1,2,3.In the conclusion of this section we consider a representation of

the rotation group on the state space L2 (R3) = L2 (S2) 0 L2 (R+).Lot {f,L(r)} be an arbitrary basis in L2 (R+). Then {fit(r)Yirn(n)} isa basis in the space L2 (R,"), and any function b (x) e L2 (R3) can beexpanded in a series:

00 1

V(x) = 1: E 1: Cnlmfrt(r) Yim(0, o)n l=O m=-l

From this formula it is clear that 12 (R3) can also be decomposed(in many ways) into subspaces in which the irreducible representationsof the rotation group of order (21+1) act, and each representation DIis encountered infinitely many times. Any of the invariant subspacesin which the irreducible representation Dl acts is a set of functions ofthe form f (r) Elrt=-1 C Yl., (0, o), where f (r) E L2(R+).

§ 31. The radial Schrodinger equation

Let us return to the problem of the motion of a particle in a centralfield We shall look for solutions of the equation

--I D

+ V(r) 0 = E01

or, using the formula (23.5), the equation

1 a 2 8 L2(1) 2 r2 r ar 0+ 2 r2 + V (r) 0= F.

u arWe have seen that in the absence of accidental degeneracies the eigen-spaces of the Schrodinger operator H must coincide with the sub-spaces of the irreducible representations D1, while in the presence ofaccidental degeneracies they are direct sums of such subspaces. It is

§ 31. The radial Schrodinger equation 131

clear that all the independent eigenfunctions of H can be constructedif we look for them in the form

(2) /'(r, n) = Ri (r) Ym (n).

These functions are already eigenfunctions of the operators L2and L3:

L2TP =l(l+l)b, L3ip="IV),

and therefore they describe states of the particle with definite valuesof the square of the angular momentum and of its third projection.

Substitution of (2) in (1) gives us an equation for Ri (r):

1 d (r2)dR1(r) 1(1 + 1 ) R r V r R r= ER r.2 r2 dr dr + 2 r2 , ( ) + () t () I ( }

We introduce a new unknown function by

R1 r =fa(r)rand the equation for fa (r) takes the form

_ 1 d2.fi 1(1+1)(3)

Z dr2 + 2 r2 fl + V (r) fl = Et fl.

This equation is called the radial Schrodinger equation. We point outsome of its features. First of all, the parameter m does not appear inthe equation, which physically means that the energy of the particledoes not depend on the projection of the angular momentum on thex3-axis. For each 1 its own radial equation is obtained. The spectrumof the radial equation is always simple (this can be proved), andtherefore accidental degeneracies are possible if the equations (3) withdifferent 1 have the same eigenvalues.

The radial equation coincides in form with the Schrodinger equa-tion

1 d`2 /

,i dx2+VVY = F,

2l

for a one-dimensional particle if we introduce the so-called effectivepotential

Veff Y V M + a


Figure 7

However, there is one essential difference. The function p(x) is definedon R while f (r) is defined on R+, and hence the radial equation isequivalent to the one-dimensional Schrodinger equation for the prob-lem with the potential V (x) under the condition that V (x) = oo forx < 0.

Figure 7 shows the graphs of the functions V (r) , 1(1 + 1) / (2/ tr2 ),

and V0 ff (r), with V (r) taken to be the Coulomb potential -a/r ofattraction (a > 0).

The expression 1(1 + 1) / (2i,ir2) can be interpreted as the poten-tial of repulsion arising due to the centrifugal force. Therefore, thisexpression is usually called the centrifugal potential.

In quantum mechanics one has to solve problems with very diversepotentials V (x). The most important of them seem to be the Coulombpotential V (r) = a/r describing the interaction of charged particlesand the Yukawa potential V (r) = g e r

roften used in nuclear physics.

§ 31. The radial Schrodinger equation 133

One usually considers potentials that are less singuar than 1/r 2_eas r - 0 (e > 0). In dependence on their behavior as r --* oc, decay-ing potentials (V(r) --+ 0) are divided into the short-range potentials,which satisfy V (r) = o( 1l r24c ) for some e > 0, arid the long rangepotentials, which do not satisfy t his condition The Yukawa potentialis a short,-range potential, while the Coulomb potential is a long-rangepotential. The spectrum of the radial Schrodinger operator

(2 t(l + 1) 1

dr2 + r2 + V ( r ) (p=)2

is well known for a very broad class of potentials. In the case ofan increasing potential with V (r) - oo as r --+ oc, the spectrumis a pure point, spectrum arid is simple In the case of a decayingpotential the interval 0 < E < oc is filled by the continuous spectrum,and the negative spectrum is discrete. For a short-range potentialthe positive spectrum is simple and continuous, while the negativespectrum consists of a finite number of eigenvalues.

We shall give some simple arguments enabling us to understandthe basic features of the spectrum of H1. To this end we consider howthe solutions of the radial equation behave as r - 0 and r --p oo. If asr -+ oo we ignore the term Veff in the radial equation, then it reducesto

1+F;fi=0.

2p

For E > 0 this equation has the two linearly independent solutionse -ikr said eik'", where k2 = 2 jiE > 0 and k > 0. For E < 0 thelinearly independent solutions have the form e-"' and e". wherex2=--2pE>0and x>0.

In the case r --- 0, we can hope to get the correct behavior of thesolutions of the radial equation if in this equation we leave the mostsingular r {2i ,' f j of the terms linear in fl, so that

fill - aft + i)fi = o.r2

This equation has the two linearly independent solutions r`l andrl--i


Let us now consider what conditions should reasonably he im-posed on the solutions of the radial equation. We are interested inthe solutions V) (x) = f` } Yrn (n) of the Schrtidiiiger equation. Thefunctions b(x) must be continuous in R3 and either square-integrableor bounded on the whole space. In the first case they are eigenfunc-tions in the usual sense of the word, and in the second case they canbe used to describe the continuous spectrum.

The continuity of bb(x) implies the condition fl (0) = 0. Therefore,only solutions of the radial equation that behave like Crl+1 as r - 0are of interest. This condition determines fl(r) up to a numericalfactor. Next, for E < 0 we must, find a solution f (r) that behaveslike Ce x" as r - oo (otherwise, the solution will be unbounded).For arbitrary negative E these conditions turn out to be incompatible.Those values of E for which we can construct a solution having thecorrect behavior at zero and at infinity are precisely the eigenvalues.

For any E > 0 the solution fl (r) is bounded, and thus it sufficesthat it have the correct behavior at zero. The spectrum is continuousfor E > 0.

The eigenfunctions of the discrete spectrum describe a particlelocalized in a neighborhood of the center of force; that is, they cor-respond to a finite motion. The eigenfunctions of the continuousspectrum can be used to describe the states in which the motion ofthe particle is infinite.

The eigenfunctions of the discrete spectrum of the radial equationwill be denoted by fkl (r), where k indexes the eigenvalues Ekl of theequation for a given 1

Hlfkl = Eklfkl

The eigenfunctions of the continuous spectrum corresponding tothe energy E will he denoted by f hl :

Hl f F,l = Ef EI.

For a broad class of potentials it has been proved that the system{ f kl , f r, t } of functions is complete for each 1 = 0, 1, 2,. .. . This means

§ 31. The radial Schrodinger equation 1 35

that an arbitrary function27 f (r) E L2 (0, c) has a representationoc

f (r) = Ck f kl (r) + C(E) IE1(r) dE,k c

where

Ck = fx f (r) .fki (r) dr, C(E) = f .f (r) .fFt (r) dr.v o

Let us return to the three-dimensional problem. The functions ofthe discrete spectrum have the form

x ` f kt (r)nktrra ( } r lira }

and the multiplicity of the eigenvalue Eki is equal to 21 + I (in theabsence of accidental degeneracies). For the eigenfunctions of thecontinuous spectrum we have

fEt(r)()Et7n

The multiplicity of the continuous spectrum is infinite, since for anyE > 0 there are solutions of the radial equations for all l and, more-over, m = -l,--1+ 1,...,1.

The parameters k, 1, and m that determine the eigenfunctionsof the point spectrum are called the radial, orbital, and magneticquantum numbers, respectively. These names go back to the oldBohr-Soxnnmerfeld quantum theory, in which a definite classical orbit(or several such orbits) corresponded to each admissible value of theenergy. The numbers k and 1 determined the size and shape of theorbit, and the number m determined the orientation of the plane ofthe orbit in space. The number m plays an essential role in magneticphenomena, which explains its name.

The completeness of the system {fkz , f El } of functions in L2 (0, oc)implies the completeness of the system {'kjm(X), LE1m (x) } in L2 (R3) .

For brevity of notation we consider the case when the point spec-trum of the operator H is absent. In this case an arbitrary function

27L2(0, oc) denotes the space of square-integrable functions (without the weight7 2) on R+ If f E L2(0,00), then H = f /r E L2 (R-"')


Z;' (x) E L2 (R3) can be represented in the form

x I xVI(x) _ 1: Cim(fE) ,O Fim(x) dE.

1-0 m--l U

I t is clear that p is determined by the sequence {C'im(E)} offunctions, and therefore we get the representation

H {Clrn(E)}

in the space of sequences of functions. In this space the scalar productis given by

cc t x(1,i/'2)= E Ctrn C(" dE.

1=0 m=-l 1(0)17n

From the fact that JBam (x) is an eigenfunctiori of the operators H,L2, and L3, it follows easily that these operators act as follows in therepresentation constructed:

IICim(E) = ECjm(E),

(5) L2CIm(E) =1(l + 1) Cim(E),

L3Cim(E) = mCim(E).

Therefore, this representation is an eigenrepresentation for the threecommuting operators H, L2, and L3. (The equation (5) should notbe confused with the equations for the eigenvectors.)

§ 32. The hydrogen atom. The alkali metalatoms

The hydrogen atom is a bound state of a positively charged nucleuswith charge e and an electron with charge -e (e > 0 is the absolutevalue of the charge of an electron). Therefore, the potential V (,r) hasthe form

e2V(r) = --.

We consider the problem of motion in the field

V(r) Zee.r

§ 32. The hydrogen atom. The alkali metal atoms 137

Such a potential corresponds to a hydrogen atom with Z = 1 andto the hydrogen-like ions He+, Li+-, ... with Z = 2,3,. .. In thecoordinate representation the Schrodinger operator has the form

H1 it Zee=- - ,

2/4 rwhere It = m M/ (m + M) is the reduced mass, and m and M arethe masses of the electron and nucleus, respectively. We shall solvethe problem in the so-called atomic system of units in which h = 1,A = 1, and e2 = 1. Then the radial Schrodinger equation takes theform

_1 "r 1(1+1)ft () + 2 f1--f1=Ef1.

We are interested in the discrete spectrum, so we consider thecase E < 0. It is convenient to use the notation -2E = x2. Then

1 "2Z -ly+1) -2 =0.() fit + r fl r2 ft f t

The arguments in the previous section about the behavior of asolution as r -+ 0 and r -* oo suggest that it is convenient to look fora solution in the form

(2) fft (r) = rt+1 e-x' At (r).

If we can find Al (r) representable by a convergent power series,

(3)

00

A,(r) = >ajr'i=u

with a0 0 and such that f j (r) satisfies (1), then the correct behaviorof ft (r) as r 0 will also be ensured Of course, the behavior of ft (r)as r -- oo depends on the asyxnptotics of the function At (r) as r --+ oo.

It is convenient to make the substitution of (2) into the equation(1) in two steps. Introducing the function g by

f=e-xrg,f , = e-X' (x2q - 2xg' + y"),

we have2Z - 1(1 +1) =0.9 r 9 r2 ,9


Further. setting

g = r1+1 A,

tr1 A(l+1) +A'=rg

1+1

rl(1 + 1) A 2(1 + 1) A' '=g r + + A

r2 r

we get that

n + 2(1+1)_ 2x I A' + r2Z _ 2x(l + 1)1 Al = 0.r / `r r

Let us look for a solution of this equation in the form of a series(3).

aa[i(i- l )ri-2+2(l+l)ira-2 -2ixr'-1 0.i=O

We make the change of summation index i --> i + I in the first twoterms in the square brackets, and then00

E ri {ai+1 [(i + 1)i + 2(i + 1) (1 + 1)] - ai [2x(i + l + 1) -- 2Z] } = 0.z=O

Equating the coefficients of the powers of r, we get that

(4)1 ) ( i 2 1 +

= 21)

From d'Alembert's criterion it is clear that the series convergesfor all r. We estimate the behavior of the series with the coefficientsdefined by (4) for large r. Of course, the asymptotic behavior asr -p oc is determined by the coefficients of the higher powers, butthen

2xai+1

+1 ai;

that is,

and

(2x)zai ti C

it

2Al Ce .=


Thus. for the solution f we get that

flCri+lexr

as to _+ 30 (This argurzient could be made more precise, of course.)

We see that a solution of the radial equation with the correctbehavior as r - 0 grows exponentially as r -+ oo. However, it is clearfrom. (4) that there are values of x such that the series breaks off at,some term. In this case the function Al turns out to be a polynomial,and the solution fl (r) is square integrable. Let k denote the indexof the highest nonzero coefficient. that is, ak 34 0 and ak+1 = 0 fork = 1, 2..... It is obvious from (I) that t his is possible if

zx= xk t =

k+7_471- .

From the formula -2E = x2 . we get that

Z(5) Ekl _

-2(k + l +1 ) 2

The parameter k is the radial quantumn number introduced ear-liei We see that the eigenvalues Eki depend only on n = k + 1 + 1.This number is called the principal quantum number. Recalling thatk = 0, 1, 2.... and l = 0, 1, 2, ... we get that, n -- 1, 2, 3, .... Fur-rherruore, for a given n the quantumn number l can take the values0, 1.2. ..,n - 1

Accordingly, we have obtained the, following results The eigen-values E are given by the formula

Z2(6) Eit

2n2

and the eigenfunctions have the form

(7) Ortlrrt = rte-x"''A.t(r) Ym(O. ).

where Ar1 1 is a polynomial of degree n. -- I - 1 whose coefficients arefound from the formula (4), with ae found from the normalization con-dition. We see that the number of eigenvalues is infinite and has accu-mulation point E = 0. It is not hard to determine the multiplicity ofthe eigenvalire E,, To each E,, there correspond eigenfunctions


differing by the quantum numbers 1 and m, with 1 = 0. 11 2..... n - 1and m = -1, -t -- 1,. .. ,1. For the multiplicity q we have

n-12q=j:(21+1)=n

t=v

The multiplicity of the eigenvalues for the Coulomb field turns outto be greater than in the general case of a central field: there is anadditional degeneracy with respect to 1. We have already mentionedthat this "accidental" degeneracy is explained by the presence of asymmetry group richer than SO(3) for the Schrodinger operator forthe hydrogen atom.

Let us now consider what physical information is given to us bythe solution of the Schrodinger equation for the physical atom. Firstof all, we have found the admissible values of the energy, which it isreasonable to give in the usual units. For this it suffices to multiplythe expression (6) for F,t by the atomic unit of energy equal to

= 4.36. 10erg = 27.21 eV.h2

We assume that Z = 1, that is, we consider the hydrogen atom, andthen

(s)le4

Ert = - 2n2h2 .

For the energy of the ground state of the hydrogen atom (n = 1), wehave

El = 2h2_ -13.6 eV

The absolute value of I his energy is called the ionization potential orthe binding energy of the electron in the atom and is equal to thework that must he done to strip the electron from the atom.

The formula (8) enables us to compute the frequencies of t he spec-tral lines for the hydrogen atom. Quantum electrodynamics confirmsBohr's hypothesis that, the frequency of a spectral line is determinedby the formula

hwrrt n = Err - Ern 7 Fn > Err, 1


and moreover, there is absorption of a quantum of light if the atompasses from a state with less energy to a state with greater energy,and emission of a quantum for the reverse transition. 28

For the frequencies of the spectral lines we have the formulajie4

1 1(9) Wnm = 2h3 rte m2

n < m.

This is called Balmer's formula and was discovered by him empiricallylong before the creation of quantum mechanics.

We direct attention to the dependence of the frequencieson the reduced mass p. In nature there are two forms of hydrogen:the ordinary hydrogen H whose nucleus if a proton with mass M =1836 m (m is the mass of the electron), along with a small quantityof heavy hydrogen or deuterium D whose nucleus is twice as heavyas the proton. Using the formula it = rnM/(m + M), we easilycompute that /2D11LH = 1.000272, that is, the reduced masses arevery close. Nevertheless, the precision of spectroscopic measurements(wavelengths can be measured with an accuracy of 7-8 significantfigures) make it possible to accurately measure the ratio W D /W H forthe corresponding lines. This ratio is also equal to 1.000272 (forsome lines there can be a deviation in the last significant figure). Ingeneral the values computed theoretically according to the formula(9) coincide with the experimental values of the frequencies to fivesignificant figures. The existing deviations, however, can be removedby taking into account relativistic corrections.

Along with the transitions between stationary states of the dis-crete spectrum there can be transitions from the discrete spectrumto the continuous spectrum and the reverse transitions; physically,they correspond to ionization and recombination processes (captureof an electron by the nucleus). In these cases a continuous spectrumof absorption or emission is observed.29

28An absorption spectrum (dark lines on a bright background) arises if a lightbeam with continuous spectrum passes through a medium containing atomic hydrogenAbsorption lines are observed in the spectra of stars A line spectrum of emissionwill be observed, for example, if an cbs trical discharge takes place in a medium withatomic Hydrogen Under the action of impacts with electrons, the hydrogen atoms willthen pass into excited states The transitions to levels with less energy lead to theappearance of bright lines

29Tlie word `spectrum" is used here in two senses the spec trump of an operatorand the admissible values of the frequency of electromagnetic radiation

142 L. D. Faddeev and 0. A. Yakubovsk j

The spectral lines of hydrogen on spectrograms are grouped intoseries corresponding to a definite value of n in the formula (9) andm = n + 1, n + 2,. -.The first few series have been given names: theLymian series (n = 1), the Balmer series (n = 2), and the Paschenseries (n = 3). The lines of the Lyman series lie in the ultravioletpart of the spect ruin, the first four lines of the Balmier series lie in thevisible part of the spectrum, and the lines of the Paschen series andsubsequent series lie in the infrared part of the spectruni. Towardthe end of each series the lines converge to the so-called limit of theseries, after which the continuous spectrum begins.

In Figure 8 the horizontal lines represent the energy levels ofa hydrogen atom, and the vertical segments represent the possibletransitions between there. The region of the continuous spectrum isshaded.

9E

18

1_2 `-V--Lyman series

Paschen s

Balmer series

eriesn--3

n=2

n=1

Figure 8


Figure 9

In Figure 9 we represent schematically the form of a spectralseries, with the limit of the series represented by the dashed line.

The probabilities of transitions between states are important char-acteristics of atoms. The intensities of the spectral lines depend on thetransition probabilities. The transitions that take place are sponta-neous from an upper level to a lower level with emission of a quantum,forced (under the action of a light beam), or, finally, due to collisionswith charged particles. Formulas for computing the probabilities ofspontaneous and forced transitions are given by quantum electrody-namics, and transitions due to collisions are studied in the quantumtheory of scattering. To compute all these characteristics, one mustknow the wave functions. Moreover, knowledge of the wave functionsmakes it possible to judge the size of an atom, the distribution ofcharge in the atom, and even the shape of the atom. We recall thatI is the density of the coordinate distribution function. Bythe size of the atom we understand the size of the region in whichk'(x)I2 is not negligibly small. It is clear that the size of an atom isa conditional concept.

As an example let us consider the ground state of a hydrogenatom (n = 1,1 = 0, m = 0). Since Yoo(n) = const and x, = 1, we getby the formula (7) that

P]OUX = CC-r.

We find the constant C from the normalization condition30

,012 dx = IC12 47r e-2rr2 dr = IC12 7r =1,JR3 (l

from which C = 1 / f and

1V)lUU(X) =

OF8-r.


Figure 10

It is easy to see that

p(r) = 4irIb100(r)12r2 = 4e' 2rr2

is the density of the distribution function of the coordinate r. Thegraph if this function is pictured in Figure 10. The maximum ofp(r) is attained for ro = 1; that is, r0 = 1 is the most probabledistance of the electron from the nucleus. In the usual units, ro =h2lpe2 = 0.529. 10-8 cm. It is interesting to note that this numbercoincides with the radius of the first Bohr orbit. We see that the sizeof a hydrogen atom has order 10-8 cm.

By the density of the charge in an atom we understand the quan-tity -- e b (x) 2, that is. we consider that because of its rapid motionabout the nucleus, the electron is as if smeared over the volume ofthe atone, forming an electron cloud.

Finally, the form of the function (7) shows that for 1 34 0 thedensity of the coordinate distribution is not spherically symmetric.The dependence of this function on the angles lets us say somethingabout the shape of the atom in various states.

In this same section we consider a simple model of alkali metalatoms, based on the assumption t hat their optical properties are ex-plained by the motion of the valence electron in a certain central fieldV (r) . This potential V (r) can be written as a suns of two terms,

V(r) =-Z +V r ,r


where the first term describes the interaction of the electron with thenucleus, and Vl (r} can be interpreted as the potential of the inter-action of the electron with the negative charge of the other electronsdistributed over the volume of the atom. The reasonableness of sucha model for atoms of the alkali metals becomes clear only after webecome acquainted with the properties of complex atoms and the_l lend eleev periodic table.

We know very little about the potential V (r), but neverthelesswe can assert that

V(r) - 11 forr --+ocr

andL

V (r) - -- for r -* 0.rThe first condition follows from the obvious fact that when the valenceelect ron is removed to infinity, it finds itself in the field of an ionwith a single positive charge. The second condition follows from thecontinuity of the potential Vl (r) of the volume distribution of charges.

As a model potential we choose

(10) V(r)21

a>0.

Despite the fact that this potential has the correct behavior at infin-ity, its behavior at zero is different from that of the "true" potential.At the same time, the model potential correctly reflects the fact thatupon approach to the nucleus the field becomes stronger than theCoulomb field -1 /r. We assume that the parameter a is small (inwhat sense is indicated below). The numerical values of this parame-ter for the different alkali metal atoms is most reasonably chosen from.a comparison of the results of computations of the energy levels withthose found experimentally.

The radial equation for such a potential is very simple to solve.Indeed, it has the form

112 2a _l(1+1} x2

+ r fr + r2 f, r2 .fi .f i - 0.

We introduce the number 1' satisfying the equation

l'(l'+1)+2a-l(1+1) =0


and the condition lima--.o 1'= 1, so that 1' = --1/2 + /fl1/2)2 --- 2a.The equation (11) can be rewritten in the form

, 2 _ 11(11 +1) _x2 =0fit + r ,ft r2 .ft .ft

that is, it coincides formally with the equation for the Coulomb field.All this can make sense only under the condition that 1(1 + 1) + 1 /4 --2a > 0 Otherwise we get complex values for 11.30

If a < 1/8, then the condition 1(1 + 1) + 1 /4 -- 2a > 0 holds forall 1. One usually writes 1' up to terms of order a2; that is,

a1+1/2

Using the formula (5) with Z = 1, we then get that1Ekl _

2(k + 1 - a.+1)2 1

or, introducing the principal quantum number n = k + 1 + 1,

F_. - - Int 22(n--aj)It

is clear from (12) that for the potential (9) the Coulomb degen-eracy with respect to 1 is removed. The energy levels Ent lie deeperthan the levels E7 of the hydrogen atom, and the levels Ent and Enbecome close as n increases. The formula (12) describes the energylevels of the alkali metal atoms fairly well for an appropriate valueof a. This formula was first obtained by Rydberg by analyzing ex-perimental data. We remark that for the alkali metal atoms, as forhydrogen, the principal quantum number takes integer values, butthe minimal value of n is not 1 but 2 f o r Li, 3 f o r Na, ... , since thestates with smaller principal quantum number are occupied by theelectrons of the inner shells of the atom (this assertion will becomeclear after we become familiar with the structure of complex atoms).

In conclusion we note that this model illustrates a semi-empiricalapproach to the solution of complex quantum mechanics problems.

'ieit can be shown that for 2c, - 1(1 + 1) > 1/4 the radial Sc hrodinger operator

H1 d2 1(t+1)-2Q - 1

2 dr2 + 2r2 rbecomes unbounded below

§ 33. Perturbation theory 147

Such an approach consists in the following: instead of solving theproblem in the exact formulation, one uses physical considerations toconstruct a simplified model of the system. The Schri dinger operatorfor the model problem usually depends on parameters which are justas difficult to find theoretically as it is to solve the problem in fullscope. Therefore, the parameters are found by comparing the resultsof computations of the model problem with experimental data.

§ 33. Perturbation theory

In quantum mechanics there are relatively few interesting problemsthat admit the construction of exact solutions. Therefore, approxima-tion methods play an important role. Approximation theories oftenturn out to be more valuable for understanding physical phenomenathan exact numerical solutions of the corresponding equations. Themain approximation methods in quantum mechanics are based onperturbation theory and the variational principle.

We describe the formulation of a problem in perturbation theory.Suppose that A is a given self-adjoint operator whose spectrum isknown. It is required to find the spectrum of the operator B = A+ Cunder the condition that the self-adjoint operator C is small in somesense. We do not specify what is meant by smallness of C, since weare considering only a formal scheme in perturbation theory. Giving arigorous basis for such a scheme requires the solution of some complexmathematical problems.

We shall analyze the case when A has a pure point spectrum, andwe begin with the problem of perturbation of a simple eigenvalue. Letus consider the one-parameter family of operators

(1) Ae = A+eC.

It is clear that A o = A and A 1 = B. We know the cigenvectors V)nand eigenvalues A,, of A, which satisfy the equation

(2) Ann =ArY)n

I t is assumed that the spectrum of A is simple; that is, to each Anthere corresponds one eigenvector V)n.

14 8 L. D. Faddeev and 0. A. YakubovskiT

The equation for the eigenvectors of the operator A, is

(3) A,Ei'e = ae )e .

Our main assumption is that ip, and a, depend analytically oii F; thatis, they can be represented in the form

(4) a, = A(O) fi ea(1) +62A(2) + .. .

(5) V)(0) + F (i) + c2V)(2) + .

Substituting (4) and (5) in (3),

(A+6C)(,0(0) +eV) + - - -) = (A(O) +EA(l) + .)(0(()) +e*(') +

and equating the coefficients of like powers of e, we get the system ofequations

A,o (0) = A(0)V) (0)

A,0(1) + Cv(0) =A(0)v(1) +A(1) I(0)

...............................

AV (k) + c ('-1) = A(O) O(k) + A( 1),,(k-1) +... + A(k),O(c}).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

which are more conveniently rewritten as

A'(°) =A(0)V)(0)

(s)

. . . . . . . . . . . . . .

(A A(())) V)(1) = (A(1) C)00).

(A \(O)) *(2) = (,\(l) C) V(1) + \(2)V)(0),

........................... .......................

( _ \(O)) V)(k) = (,\(l) _ C) (k-1) +... + A(k)V)(0)

................................................

From the first equation in (6) it fol lows31 that V)(0) is an cigenvec-tor of A, and from the assumption about simplicity of the spectrumwe have

,1(0) = \n, (0) _ 'Vn

3' Below we should equip the eigenvectors w,, the eigenvalues A, and A"), andV M with the subscript n, but for brevity of notation we do not do that


Before turning to the subsequent equations in (6) we choose anormalization condition for the vector V),. It turns out, that the con-dition

(7) (V)C,,(O)) = 1

is most, convenient. We assume that ip °°) is nornialized in the usualway: lbo) II = 1, and thus the condition (7) is equivalent to theconditions

(8) ({1}, {{}}) = 0, .. (,(k}. ) = o, ... .

Thus, we can look for the corrections 00), .. , (k) .... in the sub-spawe orthogonal to the vector 0(0) = , .

Let us now consider the second equation in (6). This is an equa-tion of the second kind with a self-adjoint operator A, and a(°) isan eigenvalue of A. This equation has a solution if and only if theright-hand side is orthogonal to the vector 0(0) It follows at, oncefrom the condition

(0(0). (A(') - C) *(())) = 0

thatV) = (C'O(())j,0(0))7

or, in more detail,

(9) ant) = (Cb, On).

The formula (9) has a very simple physical interpretation. The first-order correction to the eigenvalue An coincides with the mean valueof the perturbation C in the unperturbed state V), n.

We consider what the second equation in (6) gives for the vector0{1). It might seem that we should write

(10) ,(1) = (A - A(°) J)-1(a(1) - C),O(°).

However, this formula needs to be made more precise. To understandwhy this is so, we consider more closely the operator (A - AI) -1,which is called the resolvent of A. The operator A can be written inthe form

A=> AmPm,M

150 L. D. Faddeev and 0. A. Yakubovskiii

where P,,, is the projection on the eigenvector 0,,; that is, P,n'p _(Sp. ,,n } 1bm . Then for the operator (A - Al)-' we have

(11) (A - A1)-1 =In mE Pin

.I - a

From (11) it is clear that the resolvent, loses meaning for A = an , thatis, precisely for the values of A of interest to us. However, we recallthat the right-hand side of the second equation in (6) is orthogonal toV)(0) = Vn and ((1), p(O)) = 0. So in fact we need not the operator(A - A])-' but the operator (A - Al) -1 P acting in the subspaceorthogonal to ion, where P is the projection I -- Pn onto that subspace.

The operator (A - A1)-1P can he represented in the form

(12) (A-AI)1P= E Prna amfa M

which preserves its meaning for A = An. Instead of (10) we mustwrite

(13) V) (1) = (A - A (0) 1) - 1 P (A (1) - C) V) (0).

This expression can be transformed as follows-

(A - A7, I) 'P[(C n, 1Jn} bn ...' C n]

= (A - AnI)-'P(Pn - I)C = -(A - AnI)-' PCB'n,

and therefore

(14) (11) _ -(A - and) -' PCV),L

Using (12), we get that

(15) (1) = (Cn,'m)

An - Arn71

rrz5n

Let us consider corrections of subsequent orders. Froxri the or-thogonality to (0} of the right-hand side of the third equation in (6),we get at once that

(16) A(2) = (Clp(l) 1 0(0)).


Using the form of V)(1), we find an explicit formula for the secondcorrection to the eigenvalue An:

(2) _ I(CV)nI

m) I2(17) ra -min An m

We shall not present the detailed computations for ,(2) and thesubsequent corrections ?,,(k) and A(k), but note only that theycan be found by the formulas A(c) = (Ci5(lt), (O)) and 0(k)(A -- A(°) I) -' P x [the right-hand side of the corresponding equationin (6)].

We now discuss the theory of perturbation of a multiple eigen-value, confining ourselves to construction of the first-approximationcorrection AM. Let An = A (we omit the index n) be an eigenvalueof A of multiplicity q:

A i = AV)i i=1,2,...,q.

Denote by N. the eigenspace of A corresponding to the eigenvalue A,and by Q the projection on this subspace.

We turn again to the system of equations (5). As earlier, it fol-lows from the first equation that a(°) = A. As for the vectors 00),we can only assert that 0(0) E NA. We now show that additionalrestrictions are imposed on the vectors (°), and therefore they donot coincide with the eigenvectors Oi in the general case. Indeed, thesecond equation in (6) has solutions if its right-hand side is orthogonalto the subspace NA,; that is,

Q(A(1) - C) 0(0) = 0.

Taking into account that Q*(O) = V)(°), we can rewrite the last equa-tion in the form

(18) QC'Q,0(°) = A(1),00)

We see that the 0(0) are eigenvectors of the q-dimensional op-erator QCQ, and the AM are eigenvalues of it. In practice theproblem reduces to the diagonalization of a matrix of order q. In-deed, substituting 0(0) = E?j ai i in (18) and using the fact that


/ we get that

ai (Ct 1, iPj )'jA(1)

aj V j .j i J

so thatCjiaz = a(l)aj,

where Cji = (C, V)i , V)j) . The matrix Jis self-adjoins. and thuscan always be reduced to diagonal form. Denote the eigenvalues ofthis matrix by a j(1 ), j = 1,2.....q. To the multiple eigenvalue Aof the unperturbed operator A there correspond q eigenvalues of theoperator B = A+ C. which in the first approximation of perturbationtheory have the form A + A(') , j = 1, 2, ... , q. One usually says thatthe pert urhation removes the degeneracy. Of course. the removal ofthe degeneracy can turn out to be incomplete if there are duplicatesamong the. numbers that is, if the operator QCQ has multipleeigenvalues.

Example. We consider a system with the Schrodinger operator

H= ----aBL.(19) 2

r3

Such a Schrodinger operator describes a hydrogen atom located in aconstant homogeneous magnetic field with induction vector directedalong the third axis.32

As the unperturbed operator it is reasonable to take the operator

11_ -10 - 1

° 2 r

32In electrodynamics the magnetic moment of a particle with charge c is definedto he the vector

P e E'M= -XXV= 2cµxxp= 2ù1

Here is N the mass of the particle. v is its velocity, 1 is its angular momentum, and cis the speed of light The Hamiltonian function of the particle in a constant homoge-neous magnetic field B contains the additional term -MB For a hydrogen atom in amagnetic field directed along the third axis, the Hamiltonian function has the form

Jf (q, p) = P2 - e2 - e Bl32;z r 2cM

The corresponding Schrodinger operator in atomic units coincides with (19) for n =1/2r


that is, the Schrodinger operator for the hydrogen atom, and to regard

Ali = -aBL3

as the perturbation. From the physical point of view off is small,since the magnetic force acting on an electron of an atom in attain-able magnetic fields is less by several orders of magnitude than theCoulomb force of attraction to the nucleus. We recall that the eigenfunctions nlrn (x) of the operator Ho are also eigenfunctions of theoperator L3:

' 3 n t rn = MOnl rn

The matrix of the perturbation MMI is at once diagonal, and itsdiagonal elements are equal to --amB. Therefore, for the energy ofthe hydrogen atom in a magnetic field we have the formula33

(20) Ercm -- -` 2n2- aBm.

We see that the magnetic field removes the degeneracy with respectto the magnetic quantum number in, but leaves the degeneracy withrespect to l that is characteristic for the Coulomb field.

The phenomenon consisting in a splitting of the energy levels ofatoms in a magnetic field and in a corresponding splitting of theirspectral lines is called the Zeernan effect.

It is interesting to look at this phenomenon from the point of viewof group theory The degeneracy with respect to m is explained bythe spherical symmetry of the Schrodinger operator. A magnetic fielddirected along the x3-axis disturbs this symmetry. The symmetrygroup of the Schri;dinger operator of the atom in the magnetic fieldis the group of rotations about the third axis. This group is Abelian,aind all its irreducible representations are one dimensional. Therefore,the presence of such a symmetry group does not cause degeneracy;any degeneracy will be accidental.

3';This is not the most satisfactory example. because the fig rictions are exactt'igenfunctions of H with eigenvalues (20)


§ 34. The variational principle

We consider the functional

1E (HO7 0) E l.() ,(01)

This functional has a simple physical meaning: E is the mean valueof the energy of the system in the state given by the vector V5111,011.If 0 = , where V)n is the eigenvector of H corresponding to theeigenvalue E, then E = E, Let us compute the variation of thefunctional (1):

6E= I(H6*jV))+(HV)j6V))

2Re((H E) V).

0

(V) I 1P)(V) I V)) 2

It is easy to see that the condition

(2) SE = 0

for the functional E to he stationary is equivalent to the Schrodingerequation

(3) = Ei.

Indeed, (3) implies (2). To get the converse it suffices to considerb*i = ibi along with the variation &P. Then it follows from thecondition (2) that

((HE),60)-o,(&,p)

and we have (3), since 60 is arbitrary.

We point out one more important property of the functional E.For any vector 0 E' we have E > E0, where EO is the smallest eigen-value, and equality holds only for 0 = Ci0. This is almost obvious,since the mean value of the energy is not less than the minimal possi-ble value. We verify this formally for an operator H with simple purepoint spectrum. Suppose that the eigenvalues are numbered in in-

CnOncreasing order: E0 < En < F2 < . Substituting v = En=OO

§ 34. The variational principle 155

in (1). we get that

1 - E = 1:71

E IC - E _ En (En- Eo) iCr1

12

>(4) v C 2 o E rt I C 2 0.

because E,, -- Eo > 0. Equality is attained in (4) if C,, = 0 forn = 1, 2, .... In this case ip = CoPo. It can be verified similarly that

E > E1. if ,c) 0,

(5) E> E 2 if (V5,io) =0,(,L,01) = 0,

..................... ...... ......

The property E > Eo makes the variational principle especially ef-fective for computing the ground state of the system. Substitutingan arbitrary vector / E 1-1 in (1), we get, an upper estimate for E0;of the two values E' and E" of the functional (1) the smaller one isclosest to E0. The use of the properties (5) to estimate En encountersdifficulties, since we do not know the eigenvectors oo, .. , iPz _ x .

There is a second formulation of the variational principle assertingthat the Schrddinger equation (3) is equivalent to the condition thatthe functional (Hit', ) be stationary for (,0, 0) = 1. Using Lagrange'smethod of undetermined multipliers, we can write the last, conditionin the form

(6) a[(H, ?) ---1;(x,')1= 0,where E is a Lagr ange multiplier. The equivalence of (6) and (3) isverified like the equivalence of (2) and (3).

Variational principles can be used in two ways to obtain approxinmate solutions of the equation (3). The first way is to look forat, approximate wave function in the class of functions of a certainanalytic form that depend on several parameters a1, ... , ak ThenE = E(a1, .. , ak). and the parameters are found from the condi-tions

aE(a,....,ak)c9cxi

The second way is to construct, for a complex system (for exam-ple. a complex atom), an approximate eigenfiinction i'(xt,. . , xN)of H depending on several variables by using unknown functions offewer variables (most often the approximation is represented as aproduct VG 1 (x1) '2 (x2) N (XN) or as a linear combination of such


products). Equations for the functions Zi'1, .... ON are found from thevariational principle. We shall become familiar with this way whenwe study complex atoms.

Example. Let us use the variational principle for an approximatecomputation of the ground state of a helium atom. The Schrodingeroperator for helium in atomic units has the form

H--1Q 1Q 2 2 1

2 2 r1 r2 r12

As a test function we take34are crr2

V)(X x2, a = e e

Computations which we omit give a simple expression for the func-tional

.

E(a) = a2 --27

a.8

The minimum of this expression is attained for cc = 27/16, and theapproximate theoretical value of the energy of the ground state is

E0 = E(27/16) = -(27/16)2 -2.85.

The experimental value is Ec eXp = -2.90. We see that such a sim-ple computation leads to very good agreement with experiment. Aswould be expected, the theoretical value. is greater than the experi-mental value.

We remark that e-°r is an eigenfunction of the ground stateof a particle in a Coulomb field -a/r. Therefore, the approximateeigenfunct ion a -270 t tr2) / 1 H is an exact eigenfunction for the operator

11,1Q -1Q 27 y 27

2 1 22 l6r1 16r2

The interaction between the electrons is taken into account in theapproximate Schrc dinger operator H' by replacing the charge Z = 2of the nucleus by Z' = 27/16, and by the same token the screening ofthe nucleus by the electron charge is taken into account.

'This choice of a test function can be ccxpl.tined by the fact that the functione 27,j.-2r2 is an exact eigerufunction of the operator H - 1 /r12 Indeed, if the term1/r12 is removed from H, then by separation of variables the problem can be reducedto the problem of a hydrogen-like ion, and, as has been shown, the eigenfunc tion ofthe ground state of such an ion is e f' , where Z is the charge of the nucleus

§ 35. Scattering theory. Physical formulation 157

In conclusion we remark that the computations of the heliumatom used test functions with a huge number of parameters, and theaccuracy attained was such that the existing deviations from exper-inient can be explained by relativistic corrections. Such an accuratesolution of the problem of the ground state of the helium atone hashad fundamental significance for quantum mechanics and confirmsthe validity of its equations for the three-body problem.

§ 35. Scattering theory. Physical formulation ofthe problem

We begin with a physical formulation of the scattering problem. Sup-pose that a beam of particles of type a obtained from an acceleratorhits a target consisting of particles of type b. The scheme of such anexperiment is represented in Figure 11

The particles a and b can be either elementary (for example, elec-trons, protons, neutrons) or compound (for example, atoms, mole-cules, atomic nuclei). The experimenter studies the physical charac-tei istics of the particles going out from the target. If they differ fromthe corresponding characteristics of the impinging particles, then onecan say that, a particle a has been scattered.

One usually tries to choose the thickness of the target so thata sufficiently large number of particles a are scattered by the parti-cles b, and at the same time, so that the portion of the particles aexperiencing multiple collisions is negligibly small. In this case toexplain the experimental results it suffices to study the problem ofscattering of a particle a by a particle b. This is a two-body problem

Counteram-

a

Figure 11


if a and b are elementary particles, and a many-body problem if aand b are compound particles. We recall that the two-body problemcan be reduced to the problem of the niotion of a particle in the fieldof a fixed force center by separating out the motion of the center ofinertia. This is the simplest problem of scattering theory.

In scattering by a force center, a particle can change only its di-rection of motion due to the law of conservation of energy. In ibis caseone speaks of elastic scattering. More complex processes are possiblewhen compound particles collide. For example. when an electron col-lides with a hydrogen atom, elastic scattering is possible (the stateof the atone does not change), scattering with excitation is possible(the electron t rannsfers part, of its energy to the atorri, which thenpasses into a new state), and finally, ionization of the atom by theelectron is possible Each such process is called a scattering channel.Scatt Bring of a particle by a force center is single-channel scattering,while scat tering of compound particles is usually nrnult ichannel scat-tering. If, however, the colliding particles a and b are in the groundstate and the energy of their relative motion is less than the energyof excitation, then the scattering is single channel.

The basic characteristic of the diverse scattering processes mea-sured in experiments is their cross-section, which we define below.Some set of possible results of scattering is called a scattering pro-cess. The following are processes in this sense:

1) Elastic scattering into a solid angle element do constructedabout the direction n;

2) Elastic scattering at an arbitrary angle;

3) Scattering into a solid angle do with excit ation of the targetfrom the ith level to the kth level:

4) Scattering with ionization of the target;

5) A process consisting in scattering in general taking place. andso on.

The probability N of some scattering process of a by b dependson a certain quantity characterizing the accuracy of using particlesa to "shoot" a particle b. To introduce such a characterization ofthe state of an impinging particle a, we construct the plane passing

§ 36. Scattering of a one-dimensional particle 159

through the point where the scatterer b is located and perpendicularto the momentum of a. The probability dW of a crossing an areaelement dS of this plane is proportional to dS: that is, dW = I dS.It is clear that the probability N will be proportional to the quantityI computed at the point where the scatterer is located.``'

It now seems natural to define the cross-section a byN_l'

For the five processes listed above, the cross-sections have thefollowing names: 1) the differential cross-section of elastic scattering;2) the total cross-section of elastic scattering. 3) the differential cross-section of excitation; 4) the ionization cross-section: 5) the total cross-section The concept of cross-section becomes especially intuitiveif we assume that there is complete deterniinisrri of the scat teringresults. In the case of such determinism. the scattering result wouldbe determined by the point of the transverse cross-section of the beamthrough which the particle would pass in the absence of the scatterer.To the scattering process would then correspond sonic region in theplane of the transverse cross-section, and the cross-section would beequal to the area of this region.

There are two problems in scattering theory: from known inter-action potentials between the particles determine the cross-sections ofthe various processes (the direct problem), and from the known cross-section obtain information about the interaction of the particles (theinverse problem).

In these lectures we confine ourselves to a study of the directproblerri of scattering of a particle by a potential center, beginningwith the simplest one-dimensional case.

§ 36. Scattering of a one-dimensional particle bya potential barrier

We structure the exposition of this problem according to the followingplan First we formulate the so-called stationary scattering problem.

35Here the scatterer b is assumed to be distant, since I characterizes the state ofa freely moving particle a

160 L. D. Faddeev and 0. A. Yakubovskj

For this we study the solutions of an equation Hb = Eik of a certainspecial form. The physical meaning of such solutions will be explainedlater, after we have used them to construct solutions of the nonsta-tionary (or time-dependent) Schrodinger equation i dt = H'41 Forsimplicity we employ a system of units in which h = 1 and m = 1/2.

In the coordinate representation, the Schrodinger operator hasthe form

(1) Hd 2

+ V W.dX2

We take the function V (x) to be compactly supported (V(x) = 0 for1x1 > a) and piecewise continuous.

Let us denote the regions x < -a, x > a, and -a < x < a of thereal axis by 1, 11, and 111, respectively. The scattering problem is aproblem of infinite motion of a particle. Such snot ion is possible forE > 0, and we know that for E > 0 the spectrum of the Schrodingeroperator is continuous. From a mathematical point, of view, scat-tering problems are problems about the continuous spectrum of theSchrodinger operator.

The stationary Schrodinger equation with the form

(2) --W"+V(,r)yP=k2v, E=k2. k>0,

on the whole real axis simplifies in the regions I and II:

(3) W' + k2b = 0.

The equation (3) has the two linearly independent solutions eikzand a-ikx. The solutions of (2) on the whole axis can be constructedby sewing together the solutions in the regions I III. In doing this,we must use the conditions of continuity of the solutions and theirfirst derivatives at the points -a and a. This imposes four conditionson the six arbitrary constants appearing in the expressions for thegeneral solutions in the regions I III. The fifth condition for theseconstants is the normalization condition Therefore, we can construct


linearly independent solutions of (2) with the form 36

(4)

I II

V), (k, x), eikx + A(k) e-ikx. B(k) eikx

iP2 (k. x), D(k) e-ikx, a-ik2' + C(k) eika'.

in regions I and If. For example, in constructing lpi we use theai bit rariness of one of the constants to get that the coefficient of(I- ikx is equal to zero in the region 11. We then choose the coefficientof eikx to be equal to 1 in the region I, thereby determining thenormalization of the function 'l . The coefficients A and B are foundfrom the conditions for sewing together, along with the constants mand n. where mcpl + n'P2 is the general solution of (2) in the regionIII. It is not hard to see that the conditions for sewing togetherlead to a nonhoiriogeneous linear system of equations for A, B, m,and n with determinant that is rionzero if c1 and cp2 are linearlyindependent. The solution *2 is constructed similarly. The linearindependence of 'I and /2 follows from the fact that the Wronskianof these solutions is not zero.

Let us determine the properties of the coefficients A, B, C, , andD. For this we note that the Wronskian W (W1 ,'p2) = ;pi p2 -'p1 Cpl ofthe solutions Pi and 'p2 of (2) does not depend on x. Indeed, supposet hat V, and V2 satisfy (2). Then

dpi -(V-12) i -=0, Sp2-(V-k2)Sp2=0.

Multiplying the first equation by 'P2 and the second by 'p1 and sub-tracting one from the other. we get that

d -'' =0.dx

,F cp2}

Using this property, we can equate the Wronskian for any pairof solutions in the regions I and H. Choosing (i1,'2), (fir, 1).(ii'2. i'2), and ('i,1P2) as such pairs successively, and using the equal-ities W (e-ikx, eikx) = 2ik and Me±ikx,efikx) = 0, we arrive at the

361f the potential is not compactly supported but decreases sufficiently rapidly asx1 - 0o, then these expressions for 01 and 02 in I and II should he regarded as the

asymptotics of 'i and W2 as x - tx We shall not study this case

162 L. D. Faddeev and 0. A. Yakubovsk j

following relations between the coefficients A, B. C, and D:

(5) 13=D,

(6)JA12 + 1B12

= 1,

(7) 1B2 + ICI2 = 1,(8) At --BC-=0.

(}nor example, W (/11,1) = -2ik(1 - AA) in I and W(1 , 01)-2ikBB in H. Equating these expressions, we get (6).)

The relations (5) (8) show that the matrix S consisting of thecoefficients A, B = D, and C, that is,

A BS B C '

is symmetric and unitary. This matrix is called the scattering matrixor simply the S-matrix. We shall see below that all the physicallyinteresting results can be obtained if the S-matrix is known, andtherefore the computation of its elements is the main problem in one-dimensional scattering theory.

Let us consider the question of normalization of the functions01 (k. x) and '02(k, x). We have the formulas

(9) V), (ki - x)V)2( k2, X)dx = 01 ki > 01 k2 > 01

00

(10) i1,2 (ki , x) b1.2 (k2, x) dx = 27r6(ki - k2),

that is, the same relations hold as for the functions ei kr and e - ikX

and the normalization does not depend on the formal of the potentialV (x) The integrals in the formulas (9) and (10) are understood inthe principal value sense. Let us verify (10) for the function,01. Theremaining two relations are verified in the same way. Substitution ofi11 (k1, x) and -01 (k2, x) in (2) leads to the equalities

(11) p'(k1, x) + k20(k1, x) = V(x) '(k1, x),

(12) 011(k2, x) +k22 '(k2, x) =V (x) ib(k2, x).

(For brevity we do not write the index 1 in the notation of the so-lution ?)1 .) Multiplying (11) by '(k2, x) and (12) by O(kl, x) and


subtracting the first from the second, we get that

d W (?p (k Zx), V) (k2, X))= (k1 2- k2 ? ) 1w(/ki> ,r) V( 2 X)-C x

this equality, we haveintegratingv 1

(13) f Vi(ki, x) (ka, x) = 2 2 W x), G(kanr k1 - k2

N

-N

We are interested in the limit as N -+ oo of the integral on theleft-hand side of (13) in the sense of generalized functions. It is clearalready from (13) that this limit depends only on the form of thesolutions in the regions I and 11 (for the case of infinite potentials itdepends only on the asymptotics of the solutions as x -* ±oo).

After simple computations we get that

N

V (k1, x) (k2,x) dx-N

2A A + 13 k1 k2 e`(k, -k2)N - e-z(k' - k2)1Vk _ (ki) (k2) ( ) 13 ( ))

1 2

2 [A(k2) e-i(ki +k2)N +A(k1) et(k1 +k2)Nj

k1 + k2

By the Rienlaun-Lebesgue theorem, the second terns tends to zero (inthe sense of generalized functions). The similar kissertion is not truefor the first term, since it is singular for k1 = k2. The singular partof this term does not change if we replace A(k2) and B(k2) by A(k1)and B(kl ), respectively.37 Using (6), we have

2*1(kt, x) i11(k2, x) dx =

k;siri(k1 - k2) N + F(k1, k2. N),

-N 1 -k2where

lim F(ki , k2, N) = 0.

Finally, using the well-known formula

limsin Nx = rrb(x)

,N--oc x

'?For this it suffices that A (k) and B(k) be differentiable functions of k, whichcan he proved


we get thatv

lim (ki, x) 01(k2. x) dx = 27rb(k1 - k2),N-'3c -N

which coincides with (10).

§ 37. Physical meaning of the solutions V), and IP2

To clarify the physical meaning of the solutions i, and 02, we usethem to construct solutions of the nonstat ionary Schrodingcr equation

dtFor this equation we consider the solution

x(xj t) -

21f J C(k) V), (k. x) e-£k2l A

constructed from the function im (k. x). The function C(k) is assumedto be rionzero only in a small neighborhood of the point ko. In thiscase. of (x, t) has the simplest physical meaning. Moreover, we assumethat,

00

IJand then it, follows from (36.10) that,

f_:l(x,t12dx=1,that is, the solution Pi (x, t) has the correct normalization. Using theconcentration of the function C(k) in a neighborhood of ko and thecontinuity of time functions A(k) and I3(k), we can write approximateexpressions3s for the function 5P1 (x, t) in the regions I and 11:

T : cp1(x, t) p (x, t) + A(ko) SP- (x, t)II.,1(x, t) ,.' B(ko)+(x, t),

where

(2) 0--- (X, t) C k eikzt dkJ°°

38These expressions can be regarded as having any desired degree of accuracy ifthe interval Lk inside which C, (k) is nonzero is sufficiently small However, we cannotpass to the limit, replacing C(k) by a 6-function, because we do not get a square-integrahle solution of the Schrodinger equation

§ 37. Physical meaning of the solutions W1 and '02 165

The functions (pf (x, t) are normalized solutions of the Schrodingerequation for a free particle, and were studied39 in § 15. In the samesection we constructed asymptotic expressions for these solutions ast - ±x:

^T t = 1C;' ± e'x + 0(') 2t t/2 -It I

where x is a real function whose form is not important for us. Fromthis expression it is clear that for Itl --+ oc the functions cp j (x, t) arenonzero only in a neighborhood of the points x = ±2k0t. Therefore,p4 describes the state of a free particle moving from left to right withvelocity40 v = 2k0}, and :p_ describes a particle having the oppositelydirected velocity (recall that m = 1/2)

It is now easy to see what properties the solution Ppn (x, t) has ast ±oc. Suppose that, t -oc. Then in the regions I and 11 wehave

I: y,1 (x, t) = cps (x, t),

II: P1 (x. t) = Q,since as t -*-ocwehavecp_(x,!)=0forx<-a andpp-(x.t)=0for x > a. Similarly, as t -- +oc

1: p (x, t) = A(kc) _ (x t),If: ,(x,t) = B(ko) +(x,t).

We see that long before the scattering (t --+ -oc) the probabilityis 1 that the particle is to the left of the barrier and is moving towardthe barrier with velocity 2k0.

Let us compute the probabilities W1 and W11 of observing theparticle as t -, +oc in the regions I and II, respectively. We have

Wi = f I = II00= I1 = iA(ko)i2.

39There is an inessential difference in the notation Here it is more convenientfor us to take k > 0 and to write out the sign of the momentum in the exponentialetikx explicitly Moreover, the integration in (2) can be extended to the whole realaxis, since C(k) 0 for k < 0

40More precisely, the region in which there is a nonzero of finding the particlemoves with velocity 2k0


t -"X

t - +3c

v= -2k0

(pl (x.t) I2

I

-a a x

(p I(.r.t)I2

v = 2k0

1-a

Figure 12

It is possible to replace the region I of integration by the whole realaxis, because for t -* +o0 we have 9_ (x, t) 0 only in I. In exactlythe same way we can verify that

W11 = IThe

graphs of the function (x, t) (2 as a function of x for t ±ooare shown in Figure 12.

Thus, the solution yp, (x, t) of the Schrodinger equation describes aparticle that approaches the potential barrier with velocity 2ko beforethe scattering and then with probability I it is reflected fromthe barrier or with probability 1112 it passes through the potentialbarrier. 41

We point out that the result does not depend on the form of thefunction C(k). it is important only that the interval ok in which

41 to the one-dimensional problem the concept of cross-section loses its meaningAll information about the scattering is contained in the probabilities IA12 and Bill

§ 38. Scattering by a rectangular barrier 1 67

C(k) is nonzero be small. Physically, this requirement is understand-able if we want to experimentally determine the dependence of, say,the reflection coefficient J on k: we must use particles in a statewith variance of k as small as possible (states with zero variance donot exist). In concrete computations of the reflection and transmis-sion coefficients IA(k)12 and IB(k)12, it is not necessary to solve thenonstationary Schrodinger equation; it suffices to find the solution'1(k, x) . We remark that the solution ;02 (x, 1), which can be con-structed from the function 2(k, x), has the same meaning, exceptthat the particle is approaching the barrier from the right.

Let us recall the properties of the scattering matrix S. The equal-ity B = D leads to equality of the probabilities of passing through theharrier in opposite directions and, as can be shown, is a consequenceof the invariance of the Schrodinger equation with respect to timereversal. The equalities

IAl2 + 1B12 = 1, IB12 +IC12 = 1

express the law of conservation of probability Indeed, the normal-ization of the solutions o 1(x, t) and c02 (x, t) does not depend on thetune, and as t ----> oo we have

xflira

J l'j(x,t)l2dx = I+ I= 1.coo 00

§ 38. Scattering by a rectangular barrier

We consider the concrete problem of scattering of a one-dimensionalparticle by a rectangular potential barrier. Let

Vol lxla,Vo>O,V(x) =0, lxl>a.

In this case the equation Hb = k2?k has the following form in theregions I -III:

(1)

I and II: 1V' + k2 = 0,

III: " + a2 = 0,

where a = k -- Vo (for definiteness we assume that a > 0 fork2-Vo>Oand--is>Ofork2-Vo <0).


We construct a solution 11(k x). In the regions I -III this solu-tion has the form

(2) 1. eikx + Ae-ika-

(3) If: Beik' .

III: meickx + ne z`Yx .

The coefficients A, B. m. n are found from the conditions that Sb1and its derivative be continuous at the points a and -a

e-ika + APika = In(,-ina + rtei(tia

(e-ika - Aeika) _ a(rne-iara - neictia).k

rne? + ne,,-icva = Beika

ce(mezCYa - rte - 2`xa) = kBeika .

We wi ite the expression for the coefficient B:

/ -1f(a + k)2 2i(k-(x)a - (a - k)2

4ake

4oke

It is not hard to verify that B -* 0 under any of the conditions

1) k-0, 2) Vooc, 3) a - oc, k2 <Vo.

On the other hand. B -* I under one of the conditions

4) k -+oc, 5) V0--+ 0. 6) a-40.

We see that in the limit cases 1) 5) the results obtained on the basisof quant um mechanics coincide with the classical results.42 Figure 13shows the graph of the function I B(k)12. From this graph it is clearthat for certain finite values of k the probability of transmission islB2l = 1. It is interesting to note that the equations (1) togetherwith the conditions (2) and (3) describe the passage of light wavesthrough transparent plates. Here I B 12 and J 2 are proportional tothe intensities of the transmitted and reflected waves. In the caseI B 1 2 = 1, lAl2 = 0 the reflected wave is absent. This phenomenon isused for coated optics.

42 According to classical mechanics a particle passes through the harrier withprobability I when k2 > V0 and is reflected with probability 1 when k2 < V0

§ 39. Scattering by a potential center

1

IB(k)12

------------ --- -

k

Figure 13

§ 39. Scattering by a potential center

169

For the problem of scattering by a potential center, the Schrodingeroperator has the form

(1) H_-0+V(x).(We again assume that in = 1/2 and It = 1.) In what follows weusually assume that the potential V (x) is a compactly supportedfunction (V(x) = 0 for lxi > a). The exposition is according to thesame scheme as in the one-dimensional case, that is, we first considersome solutions of the stationary Schrodinger equation, and then weexplain their physical meaning with the help of the nonstationarySchrcidinger equation

Our first problem is to formulate for the solut ion of the equation

-AO(x) + V (x) i' (x) = k2/'(x)

an asymptotic (as r - oo) condition that corresponds to the physi-cal picture of scattering and which is an analogue of the conditions(36 4) for the one-dimensional problem. It is natural to expect thatone terni will correspond asymptotically to the particle impinging onthe scattering center in a definite direction, while the second terrawill correspond to the scattered particle, which can have differentdirections of motion after the scattering and which goes away fromthe center. The analogues of the functions eik.c and a-ik, in the


three-dimensional case are the functions eik' /r and e-ikr/r. There-fore, it is reasonable to assume that the terms (e - i kr /r) b (n + w) and(ei kr /r) S(k, n, w) correspond to the particle before and after scat-t eying. We use t he following notation: k is the momentum of theimpinging particle. w = k/k. n = x/r, S(n - no) is the s-function onthe unit sphere, defined by the equality

f(n)h(n--no)dn = f(no), dn=sin0dOdV;.S2

and S(k, n, w) is a certain function which will be shown to contain allinformation about t he scattering process and which must be found insolving the problem We shall see t hat S(k, n, w) is the kernel of acertain unitary operator S, which is called the scattering operator

We come to the following statement of the problem of scatteringby a force center: it is required to find for the equation

(2) -A(x, k) + V(x) i(x. k) = k2V)(x, k)

a solution which has the asymptotic behaviore- ikr ,ikr i

(3) V1 (x, k) =r

S(n + w)r

S(k, n, w) + o -r

asr -moo.This formulation of the problem caxi be justified only with the

help of the nonstationary formalism of scattering theory, and thiswill be done in the next section. The question of the existence ofa solution of (2) with the condition (3) will also be discussed later.However, this question has a simple answer for the case V (x) = 0.We show that the function

x k) = k eikx = k ei kr nwvc ( 2iri 2iri

which is obviously a solution of (2) for V (x) = 0. also satisfies thecondition (3). and we find the form of the function S(k. n, w) for thiscase.

We are looking for the asynnptotics of the function ip0 (x, k) in theclass of generalized functions, and therefore we must find an asymp-totic expression for the integral

I= f(n)(n),Oo(rn,J2

§ 39. Scattering by a potential center 171

as r -> oc, where f (n) is a smoot h function. Using the explicit formof (x, k) and introducing a spherical coordinate system wit Ii polaraxis directed along the vector w, we have

27H

j27rI = d f (Cob B. ) eikr cos a sin 9 dO

U

k2i

f1

dp dr1 f (rl, V) eikr,.2ir2

Integrating by parts, we get that

k 2-/r 1

d,^ -- f (rl, ) eik"I27ri ilr

k r I l

21ri

f27djp di f' (77, V)

eikr7j.ikr

I ntegradrig by parts again, we easily see that the second term hasorder 0(1/r 2), and hence

2iI d(p (eikl f(w) _ eikr f(-w)) + 0

f r2e-ikr eiki

r f(-w) - r f (w} + nr2

Thus, we have shown that

e iki eikr(4) V)o(x. k) = - o(n + w) - b(n - w) + 0 2r r r2

Comparing (4) with (3), we find the function S for a free particcle:

SO(k, n. w) = 6(n - w).

Using (4), we can rewrite the asymptotic condition (3) in theforma

k ikr

(i).(5) (x. k) = eik+ f(k, n. )+ o2 ri r r

where the function f (k, n, (A;) is connected with the function S(k. n. W)by the relation

(6) S(k, n, w =bn - w+knw.


The function ,f (k, n. w) is called the scattering amplitude. Obviously,f (k, n, w) = 0 for V (x) = 0. We shall see that the scattering cross-section is most simply expressed in terms of this function.

Instead of the asymptotic condition (5). one often uses

eikr 1

(7) (x, k) - eik7 nw + f(k. n.w)

-+0 - .r (r)

which leads to a more convenient normalization of the function zit(x, k).

Let us now consider the properties of the function S (k, n, W). Thefollowing auxiliary assertion turns out to be convenient for the studyof these properties.

Suppose that the functions '1 (x) and IP2 (x) satisfy the equation(2) and the asymptotic conditions

e- ikr e2kr. ,(x) = A, (n) r + B1(n) r +o

eA

V)2(x) = A2(n)1,r

+ 0 (r)as r --+ coo, which can be differential end once with respect to r. Then

(8) fAi(n)B2(n)dn= A2 (n) B1(n) doS2 fS22

This assertion is easily proved with the help of Green's formula

- ' dX t - d.(9) z i) 1j(11b2Z st

We should choose the domain of integration sl to be the ball of radiusR and take the limit as R - oo, obtaining

e- ikr ,ikr l

c'i (x) = /'(x, k) = -b(n + w) - -S(k, n. w) + or r

j'2(x)-= '(x,k')=e- ikr ,ikr l

b(n+w') - S(k.n,w')+o - ,I,, r-ikr eikr 1

1 (x,k} - --S(k.n.w/)+

bt(n+w')+o ,r r

where k' = kw'. Applying the formula (8) to the functions vI and'V2, we get that

S(k, --w, w') = S(k, - w', w),

§ 39. Scattering by a potential center 173

or. replacing w by -w,

(10) S(k, w, w') = S(k. -w', -w).

This formula is an analogue of the equality B = D for the one-dimensional scattering problem and expresses the fact that the valuesof S for the direct (w' w) and time-reversed (-w ---+ --w') collisionprocesses coincide. It can be shown that this property (just like thesymmetry of the S-matrix in the one-dimensional case) is a conse-quence of the invariance of the Schrodinger equation with respect totime reversal.

Further, applying the formula (8) to the functions ipx and 1/12, weget that

(11) f S(k, n, w') S(k, n, w) do = 8(w - w').2

If we consider the function S (k, n, w) as the kernel of an integraloperator

S;p(w) = S(k, w, w') ;p(w') dw'S2

acting in L2 (S2), then (11) can be rewritten in the form

SE`S = I

In view of (10) it follows from (11) that

SS* = I.

The last two relations imply that the operator S is unitary.

We see that for the problem of scattering by a potential cen-ter the operator S has the same properties as the S-matrix for theone-dimensional scattering problem. Since S is unitary, we have theimportant relation

(k,n,w)I2dn= 4k Imf(k7 w,),(12) 152 If


which is called the optical theorem. Indeed, using (6) and (11), weget that

S z

LS(n - w') -21r f (k' n, [(n - w) + 2f (k, n, w)J do

= 8(w - w') + 2 [f (k, w , w) - f (k, w, w')]2 r

+ k2J f(k,n,w)f(k,n,w')dn=S(w-w').4'rs2

Setting w = w', we immediately arrive at the formula (12).

In § 42 we shall see that the integral on the left-hand side of (12)coincides with the total cross-section a for scattering of a particle bya potential center. Therefore, (12) can be rewritten as

4rrcr= Tinf(k,w,w).T

This formula connects the total scattering cross-section with the imag-inary part of the zero-angle scattering amplitude.

By using the unitarity of the operator S, it is easy to show thatthe functions (x, k) satisfying the asymptotic condition (7) have thenormalization

(13)k(x,k) (x, k') dx = (27r)sb(k - k'):IR3

that is, they are normalized just like the free-particle solutions eakx.

To verify (13) we multiply the equalities

o1(x, k) + k2L(x, k) = V(x) O(x, k).

o(x, k') + k'2 7P (x, k') = V(x) iP(x, k')

by 0 (x, k') and i (x, k), respectively, subtract the first from the sec-ond, and integrate over the ball of radius R. Then

J y)(x, k) (x. k') dxS2R

k2 k12R

1 [(x, k xk' "'' xk' O xk dx.

§ 40. Motion of wave packets in a central force field 175

With the help of Green's formula, this equality can be rewritten inthe form

O(x, kb(x,k')dxfb(x,1

k2 - k'a

j(ii(x,k)k')igr 49r

where SR is the sphere of radius R. The formula (13) is obtained fromthe last relation by passing to the limit as R -+ oo, and the integralon the right-hand side is computed with the help of the asymptoticexpression for O(x, k). We omit the corresponding computations,since they repeat literally those which led us to the formulas (36.9)and (36.10).

§ 40. Motion of wave packets in a central forcefield

With the help of the function O(x, k), we construct for the nonstationary Schrodinger equation

iab(x't) = H x t

a solution of the form

1) (x,t) = 1 2 C(k) V)(x, k)e-tk2t A.

C21r) .R,3

If the function C(k) satisfies the condition

(2)1R3

then O(x, t) has the correct normalization

J3 Iib(x,t)I2dx = 1

in view of (39.]3).

176L. D. Faddeev and D. A. YakubovskiI

As in the one-dimensional case, it is reasonable to consider asolution with the function C (k) concentrated in a small neighbor-hood of the point ko = kowo. By the Riemann Lebesgue theorem as

fj (x, t) 2 dx 0 for any finite- oo, we have V)(x, t) --+ 0, and fregion Q. Therefore, the function P (x, t) describes an infinite motionof a particle. We are interested in the behavior of this solution asIt I --+ oo and r --- oo, and we can replace ' (x, k) in the formula (1)by its asymptotic expression

tai /e-ikT eikr \ /1\I r f .(x, k) _

kw) - r S(k, n, w)) +o

Asr ---+oo,wehave

3

1 2 00

27rax t kdk dwC k w

o s2

f -ikr ikrx e(-5(n+') _ e S(k, n, w) a-ik2t

r r= V11 (X, t) + 02 (X, 0,

where '1(x, t) and2 (x, t) correspond to the converging wave e-ikr/rand the diverging wave eikr /r in the asymptotic..

The function 01(x, t) can be rewritten in the form

(3) x t= i xkC(k. -n) e-i1CTe-ik2 t dk

r 2i -o o

(we consider integration over the whole real axis, setting C(k, n) =0 for k < 0). Computing the integral (3) by the stationary phaseYnethod, we get that

(4)1

(X, t) - C\-2t,-n) etx +0

\ItJ2/

here x, as always, denotes a real function that is not of interest to us.


To compute 12 (x, Owe use the formula (39.6) for the function S.

02(x,t) = - Z J11)0

kdkJr 2 x SZ

(5)

x [S(n- w+ ikf k n, w eikre- ik2t

0"k A

r V"2--,-7 r _o c

Here we have introduced the notation C, (k) = fC(k, w) dw, and,taking into account that C(k, w) is S-shaped, we have replaced thefunction f (k, n, w) by its value at the point w = wo. The integral in(5) can be computed by the stationary phase method:

1p2(x, t) _

x [Gry\2t'nl+47rtC, 2t/ f(2t'n'w0)Je`x,+0(

ItI2/

x I Qk, n) + :Cl (k) .f (k, n, wo) Ct(kr-k2t)L

Finally, since C1 (k) is S-shaped (it is concentrated in a neighborhoodof ko), we get that

(6)

V;2 (Xi 0 =

32

x [+4 Cl \ 2t / f(ko, n, WO) e'xj .+-p I\P /It is clear from (4) and (6) that 01 contributes to 0 (x, t) only as

t - -oc, and 02 only as t --> +oo. For the density Ii/'(x, t)2 of thecoordinate distribution function, we have

r l 2(7) 1b(x,t)I2 ='

81t13 IC \ 2t' -n/ 1 t --> -oo,1

178 L. D. Faddeev and 0. A. Yakubovskj

(g) I/'(x,t)I2 -81t13

1IC(2tr,n)IZ

- ko ImCl ()c(j-,n)f(ko,n,wo)

+ 41r2 I Cl \2tl I

If(ko,n,)o)I2], t --> +oo.Z

It follows from (7) and (8) that the asymptotic expressions obtainedfor (x, t) have the correct normalization as t --+ ±oo. (This followstrivially for the case t - -oo and is a consequence of (39.12) for thecase t -' +oo.)

Recalling that C(k, w) is nonzero only in a small neighborhoodof the point k0w0 and Cl (k) is nonzero only in a small neighborhoodof k0, we see that as t -* -oo the density IV) (x, t)12 of the coordinatedistribution function is nonzero in a neighborhood of the point r =- 2 k0 t, n = -WO. As t --+ +oo the density 1,0 (x, t) 12 is nonzero interiorto a thin spherical shell of radius r = 2k0 t. The angular probabilitydistribution can be obtained43 by integrating (8) with respect to thevariable r with the weight r2 . It is clear that the first two terms in(8) contribute to this distribution only in directions close to WO. Theangular distribution with respect to all the remaining directions isproportional to If (ko, n, wo)12.

We can now easily see how the motion takes place for a particle inthe state described by the function(x, t). Long before the scattering(t --+ -oo) the particle approaches the scattering center with velocity2k0, moving in the direction w0. After the scattering (t - +00) itmoves away from the scattering center with the same velocity, andit can be observed at any point of the spherical shell of radius r2kot with angular probability distribution depending on C(k) andf (k0, n, wo). In Figure 14 we have shaded the regions in which theprobability of observing the particle is large as t --p ±oo. The regions

43 We do not write out the exact formulas for the angular probability distribution,since it is essentially dependent on the form of the function C(k) and is therefore nota convenient characteristic of the scattering process (the function C(k) correspondingto a concrete scattering experiment is never known) The cross-section is a suitablecharacteristic As we shall see, the cross-section turns out to be insensitive to the formof C(k) it is only important that this function be concentrated in a small neighborhoodof ko Physically, this requirement means that the momentum of the impinging particlemust be almost specified


t -+ --oo

r = -- 2k0C

v = 2kow0

Figure 14

w{)

in which this probability is nonzero even in the absence of a scatteringcenter is crosshatched.

The three terms in the formula (8) admit the following interpre-tation. The integral Wl over the whole space of the sum of the firsttwo terms is the probability that the particle goes past the force cen-ter without scattering. This probability is less than 1 because of thesecond term. The integral W2 of the third term is the probability ofscattering. We have already noted that the asymptotic expression (6)for 4 (x, t) has the correct normalization, and hence

W1+W2=1.

We see that the solution i&(x, t) of the Schrodinger equation con-structed with the help of the function O(x, k) correctly describes thephysical picture of the scattering. This justifies the choice of theasymptotic condition for ip(x, k) .

We mention some more features of the solution iP(x, t). It is nothard to see that as t -p -oo this function has the same asymptoticsas the solution

3

(x, t)2

=1

C(k) ei(kx-k2t) dkJR

I /

I

scatteringcenter

/ / / I I \\\ \v=2k()

t -4 +00


of the Schrodinger equation for a free particle. (Here C(k) is the samefunction as in the integral (1).) Indeed, the diverging waves eikr /rdo not contribute to the asymptotics as t --+ -oo. and the coefficientsof e -ikr/r in the asymptotic expression for the functions ekx and7'(x, k) coincide.

We show that, also for t -* +oc the solution V )(x, t) tends asymp-tot ically to some solution of the Schrodinger equation for a free par-ticle. Using the fact that for t +oc the converging waves a-ikl/rdo not contribute to the asymptotics, we get that

3

V/ (Xj t=(1)2

C k x k e-ik2t kf3

1 2x

k 2 dk d W C w)27ri

S(k, n, w)eikr

_ik2t-- - eJO s2 r

32

Joy k2 A Js dw

x27ri

S k.kS2

2kT

w', W) C(k,w) S(w' - n) e e_ik2tr

_ - I1

1

2 00

fok2 dk I

2'riC(k, w') S(w' - n)\2r/ sz k

e-ik2tr

3lO(k) ei(kx-k2t) dk = AX, t).

2ir J 3

Here

(9) C(k) = O(k, ') = S(k,w,w') C(k,w') dw'.S2

We see that the solution i'(x, t) of the Schrodinger equation tendsasymptotically to the solution yp(x. t) for a free particle as t - oo. Thefunction C(k), which determines the final state of the free motion, isobtained from the function C(k) giving the initial state as a result ofthe action of the operator S. The unitarity of S ensures the correctnormalization of (x, t), since

fR3IO(k)I2dk= 1

in view of the unitarily.

41. The integral equation of scattering theory 181

41. The integral equation of scattering theory

Most approaches to the construction of the solutions L(x, k) and thescattering amplitude f (k. n, w) are based on the integral equation

ikjx-yj(1) b(x, k) = euI c

47rV(y) V) (y, k) dy,

R 3 Ix-ylwhich is often called the Lippmann-Schwinger equation.

We verify that a solution of this equation satisfies the equation(39.2) and the asymptotic condition (39.7).41 Indeed, using the for-mulas

(O + kz) e:t« = 01 (Q + k 2 ) eikr/r = -4a8(x)-

we get that

3a(X - y) V(y) V)(y, k) dy = V(x) Vi(x, k).(o + k 2) O(X, k) = o + f

R

We verify the asymptotic condition for the case of a compactlysupported potential (V(x) = 0 for r > a). We have

1 1 1+0 , r = lxl, 13'! < a,lx - yl

=r (r2

s /Ix -yI=r 1-2xY+Y =r-ny+Ol 1r)

,r2 r2

eikIX-YI = ei(kr-kny) +0 r7'1 ,

Therefore, for the function O(x, k) we get that

,O(X, k) = etc - cr r f 3 0(y, k) dY + 0 (-k)R

Comparing the last formula with (39.7), we see that the solution ofthe integral equation has the correct asymptotics, and moreover, weget that

V) (x, kw) dx.(2) .f (k, n, w) = -4 3

440f course, it is also possible to verify the converse assertion


The formula (2), in which the scattering amplitude is expressed interms of the solution of (1), often turns out to be useful for theapproximate determination of f (k, n, w).

One of the approximation methods of scattering theory is basedon the use of a series of iterations of the equation (42.1).

00

(3) (x, x) = E V)(n) (x, k),n=0

where

(°) (x, k) = eikx,

eik x-yl(x, k) = _ V(Y) V) (n) (yj k) dy.

47r frR,,J 1X-Y1

The series (3) is called the Born series, and substitution of (3) in(2) gives the Born series for the scattering amplitude. The Bornseries for the problem of scattering by a potential center has beenthoroughly studied. It is known, for example, that it converges underthe condition that

dy < 47r.InXXR4 IV(Y) Ix-yl

It is also known that under this condition on the potential theoperator H does not have a discrete spectrum. If a discrete spectrumis present, then there are values of k for which the series (3) does notconverge. At the same time, for sufficiently large k it does convergefor a very broad class of potentials.

The simplest approximation for the scattering amplitude is ob-tained if in place of O(x, k) we substitute 0(°) (x, k) = eikx:

k n, w = - 1e-iknxv x eikwx dx.f 3

This approximation is called the Born approximation. The preciseassertion is that for large k

f(k,n,w) - f (k,n,w) = o(1)

uniformly with respect to n and w.

§ 42. Derivation of a formula for the cross-section 183

The integral equation is used for mathematical investigation ofthe scattering problem. For a suitable choice of function space, thisequation can be reduced to an equation of the second kind

0 +AV)

with a compact operator. Therefore, the Fredholm theorems hold forthe equation (1). For a broad class of potentials it has been shownthat the corresponding homogeneous equation can have solutions onlyfor imaginary values of the parameter k (k9, = ix7z, x, > 0). By re-peating the calculations carried out in the beginning of this section,it is easy to see that the solution V,, (x) of the homogeneous equationsatisfies the Schrodinger equation [-+V(x)]i/'(x) = -x/'(x) andhas the asymptotic expression 0,, (x) '=-' f (n) e - xn T /r, where f (n) isa function defined on the unit sphere. This means that the solutionsof the homogeneous equation are eigenfunctions of the discrete spec-trum of H. The completeness of the collection {fin (x), (x, k) I ofeigenfunctions of H was proved with the help of the integral equation(1)

§ 42. Derivation of a formula for thecross-section

The main characteristic of the process of scattering of a particle bya potential center is the differential cross-section. In accordance withthe general definition of the cross-section, the differential cross-sectionis defined by the formula

1dor = dN

where dN is the probability of observing the particle after the scat-tering (t -+ oo) in the solid angle element do about some direction n,and I is the probability of a free particle crossing a unit area elementoriented perpendicular to the motion of the particle. This proba-bility is defined at the point where the scattering center is located.We stress that dN is a characteristic of a particle in the field of thescattering center, while I is a characteristic of a free particle.

Little is known about the state w of a free particle in the beam.Indeed, it is known that the particle has momentum approximately

1 84 L. D. Faddeev and 0. A. Yakubovskii

equal to ko = kowo, and the variance (Ak1)avg+(Ok2)avg+(k3)vg of the momentum is known; moreover, for the scatteringexperiments to admit a simple interpretation, one tries to use beamsof particles with a small variance of the. momentum. Very little isusually known about the coordinates of a particle in the beam. Butnevertheless it is known that at some moment of time (which can betaken to be t = 0), the particle is in a macroscopic region between theaccelerator and the t arget. The transverse size of this region can beidentified with the diameter of the beam. We remark that accordingto the Heisenberg uncertainty relation, the existence of such a regionimposes lower bounds on the variance of t he momentum.

In deriving the formula for the differential cross-section, we maketwo assumptions about the smallness of the square root ok of thevariance of the momentum:

1) Ok<<ko,

2) I - f (k'. n, w') I < 1f(k.n')1 if Ikw -- k'w'l < Ak.In principle these requirements can always be satisfied if f (k, n, w)

is a continuous function of k = kw (the continuity of f can be provedfor a broad class of potentials). These conditions usually hold in prac-tice. However, there are cases when a small change in the variable kleads to a sharp change in the function f (k, n, w) (the case of narrowresonance). In the final analysis the quantity ok is determined by theconstruction of the accelerator and in practice cannot be made arbi-trarily small. Therefore, in some experiments the condition 2) maynot hold. In this case one cannot use the formula obtained below forthe cross-section.

We show that under the conditions 1) and 2) the differential cross-section depends on the state of the impinging particle only in termsof k0, and that

do- = I

The most general state for a free particle is a mixed state, given bya density matrix Mo (t) which we represent as a convex combinationof pure states:

Mo(t) = EasPPSt ), Ics = 1.8

§ 42. Derivation of a formula for the cross-section 185

The pure state Pp, (t) is determined by a wave function Vs (x, t) withthe form

3

X, 0 =1

(-_)2fCs k ei(x-k2t) dk.Ps ( '

27t 3

It can be asserted that an appreciable contribution is given only bythe pure states in which the momentum variance does not exceed themomentum variance in the state M0 (t) . Therefore, we assume thatall the functions C., (k) are nonzero only in a small neighborhood ofthe point k0 = k0w0, and that the diameter of this neighborhood doesnot exceed Ak. We do not have any information about the concreteform of the function45 Cs (k) nor about the weights as, and we do notmake any assumptions about them.

We first compute the probability I. Let us choose a coordinatesystem with origin at the force center and with x3-axis directed alongthe vector w0. For a pure state Spa (t) it is easiest to compute theprobability I(s) by using the density vector

j (.s) (X, t) = i (;p. O S03 - S0.14 V SPs }

of the probability flow, and then00

{s} = 33s)

-(x,t)

oo

Computing

(s) assJ3 (x,t)IX = i

3 3 x.=0

dt.

1 dk dk/ + k, i(k 2 -k2)t

3 3(k3 3) Cs (k} C's (k) e

R1!2 s

R

45 Corresponding to different functions C, (k) are both different forms and differentarrangements of wave packets in the configuration space at some moment of time Forexample. suppose that some function CY (k) gives a wave packet whose c enter of gravity(x(t) moves along the x3-axis directed along the vector w0 and passes through thescattering center Then corresponding to the function C1 (k) - eik"1 C4 (k) is the packetshifted by the vector u1 in the configuration space (ezk" is the shift operator, writtenin the momentum repres('ntation) As a classical analogue (though not completelyaccurate) of the state p. (x, t) we can take the state of a particle moving on a rectilineartrajectory that passes through the scattering center Then corresponding to the stateSP1(x, t) is a trajectory passing by the scattering center at a distance p equal to theprojection of the vector ul on the plane of the (rocs-section of the beam In classicalscattering theory the distance p by which the particle would go by the center if therewere no interaction is called the aiming parameter


we get that

ei(2 - 2)t

I (v) = 1 A dk'°°

dt C C (k')(2ir)3

(k3 3) s (k) sR3 R3 1-00

(2ir)2 f 3 A JIF3 ' (k3 + IC3) Cs(k) CS(k') k2)

1

(27r)2 fR A R3 the 3 Cs(k) k)

1(27r)2 f A I A' Cs(k) CY(k') d(k' - k).

The computations use the equality

(2) 6(k2 - k'2) = 2k(6(k - k') + b(k + k'))

and the condition Ok K ko, which enables us to replace (k3 + k3)/2kby 1. The second term in (2) dog not contribute to the integral,because k > 0 and k' > 0 in the region of integration.

It is obvious that for a mixed state Mg(t)

I -1

as (21r)2 J, A f ,s

dk' C.,(k) Gs(k') 8(k - k').

Let us now compute dN. In the computation we must use thedensity matrix M (t) describing a state which long before the scat-tering (t -+ -oo) tends asymptotically to Mo(t). The operator M(t)can be written in the form

M(t) _ as (t),

where the pure states Pw ,(t) are determined by the wave functions3

- 12 -ik2t

(3) 03(t)21r

3(k),O(x k) eR3

dk C'

The density of the coordinate distribution function in a pure statePP,

K(t) is f'3 (x, t )12 , and for the probability d N(-) of observing the

particle as t --+ oo in a solid angle element do in the state P(t) weget

(4) dN(3) = lien do r2 dr IV)5(x, t)12.t-'+00

A

§ 42. Derivation of a formula for the cross-section

We know that as t --+ oc the. particle goes to infinity, and thereforewhen we substitute (3) into (4) we can replace b(x. k) by its asymp-totic expression, obtaining

xdN(s) = urn do r2 drt-+oc e

3

1 2 A2-7r 3

CK (k)R

ikrx eikx + f (k, n, w)

a e-ik2tr

187

2

The function (2,7r) -312 fR3 dk C(k) eikx-- ik2 t is a wave packet fora free particle and is nonzero only in that, part of the configurationspace where there is a finite probability of observing an unscatteredparticle of the beam, that is, inside a narrow cone constructed aboutthe direction wo. For all the remaining directions, only the divergingwave gives a contribution, and for such directions5)

00

dN(q) = line-dn

drt- (27r) 3 Jof Cs (k) f (k, n, w) eikr-ik2t dk

R3

2

Using the condition 2), we can replace the function f (k. n, w) un-der the integral sign by its value at the point k = ko, w = woand take it out from under the integral sign. Further, the integralJR3 Cs (k)eu1'2t dk becomes a one-dimensional wave packet afterintegration with respect to the angle variable of the vector k, andas a function of r it is nonzero only for large positive r as t --+ oc.Therefore, the integration with respect to r in (5) can be extendedover the whole real axis. Then

d N (`s) = limndo Ift-y+oc (2ir)3

00

X- 00

dr A Al 8(k) C,(k') ei(k-k')re -i(k2-k'2)t

;

lR

1= do If (ko, n, wo)I2 ()2JR C. (k) Cs(k') 6(k - k').


Finally, for the mixed state M(t)

dN = do I f (1o, n, wo)I2 as (27r)2

1

x dk dk'C8(k)C8(k')o(k-k'),R3 JRs

or

dN = dnIf(ko,n,wo)12I,

from which we get at once that

(6) do = If(ko,n,wo)f2dn.

In conclusion we note that we have obtained (6) for all directionsn except for n = wo. The forward scattering cross-section cannot bedefined by the formula (1), because the probability of observing theparticle as t -+ oo in a solid angle do constructed about the directionn = wo is not proportional to I (moreover, we cannot in generaldistinguish a particle that is scattered forward from an unscatteredparticle). When people speak of the forward scattering cross-section,they always mean the extrapolation to the zero angle of the cross-sections of scattering through small angles. With this understanding,the formula (6) is correct for the forward scattering cross-section.

For many problems a convenient characteristic of scattering turnsout to be the total cross-section, defined by the formnula46

or = fS I.f(1o,n,O)Ildn.2

§ 43. Abstract scattering theory

In this section we shall use the following notation. The Schrodingeroperator for a free particle is denoted by Ho, and the Schr6dingeroperator for a particle in a field is denoted by 11,

11 = Ho + V,

where V is the potential energy operator. Any solution of the nonsta-tionary Schrcidinger equation for the particle in a field is denoted by

'6Tn classical mechanics the total (ross-section a is = x if the potential is notcompactly supported A peculiarity of quantum mechanics is the finiteness of thecross-section a for sufficiently rapidly decaying potentials

§ 43. Abstract scattering theory 189

p(t). This vector is uniquely determined by its value at t = 0, whichwe denote by 0: that is,

iP (t) = e-'11ti

(we recall that a-'HI is the evolution operator).

Similarly, any solution of the Schrodinger equation for a free par-ticle is denoted by Sp(t), and its value at t = 0 is denoted by cp:

sp(t} = e-iHo ti0-

Different solutions (p (t) can he supplied with additional indices. Thewave functions corresponding to the vectors (t), , p (t) , So are writ-ten as 0(x, t), b(x), ;p (x, t), cp(x) in the coordinate representation,and as (p, t), /(p), cp(p, t), W(p) in the momentum representation.Finally, the eigenvectors of the operator 11 (if there are any) will bedenoted by X,,: HXn = FnXn.

For the Schrodinger equation we constructed a solution 0 (x, t) =e-iHti'(x) in § 40 tending asymptotically to some solutions Spy (x, t) _e- iHo t cps (x) of the Schrodinger equation for a free particle as tmoo, and therefore we can expect that

(1} lim IIe-i?tcp - e-iftotcp II = 0

for such a solution 0(t).

The physical picture of scattering can be represented as follows.Long before the scattering, the particle moves freely far from thescattering center, then it falls into the zone of action of the potential(scattering takes place), and finally the motion of the particle againbecomes free over a sufficiently long period of time. Therefore, thefollowing formulation of the nonstationary scattering problem seemsnatural.

1. For an arbitrary vector ;p- in the state space f, construct avector 0 such that (1) is valid as t --' -oc.

2. For the vector 0 constructed, find a vector + e 1 such that(1) is validast -+oo.

The vector V )(t) = e '1110 describes a state of the particle thatcoincides with cp _(t) = e - i Ho t p _ in the distant past and becomesp+ (t) = e -z Ho t o+ as t -* +oo. Physics is interested in the connection


between the vectors cp_ and Sp+. Therefore, the following can beadded to the items I arid 2 of the formulation.

3. Show that there exists a unitary operator S such that

+ = S`p-.

Let us begin with item I. We pose the problem in a somewhatbroader form and see whether it is possible, for arbitrary vectorsW_, W+ E fl, to construct vectors 0 such that (1) is valid as t --+ --00

arid t - +00, respectively (of course, the vectors constructed from(p_ and W+ do not have to coincide).

Rewriting (1) using the unitarity of the operator e-"" in theform

urn I_ete_iH0tpII = 0-t- ++0C

we see that the problem posed reduces to the existence of the stronglimits

(2) lim Ud.t>±oc

If they exist, the operators UU are called the wave operators. If U_has been constructed, then the vector = U_ p_ satisfies item 1 inthe formulation of the problem.

We shall find a simple sufficient condition for the existence of thewave operators. Let us consider the operator

U(t) = c,iHt e-Wo t

arid compute its derivative'7dU(t) iHt -illnt Hit ill(Itdt

= ae kn - no) e = ze v e ° .

Obviously, U (O) = I, and hencet

U(t) = I + i dt,

=oc(3) U ± = J + i eilltVe-itb0t dt.

u

The question of the existence of the operators UU has been reducedto the question of the convergence of the integrals (3) at the upper

17 Note that the expression for the derivative takes into account that II and II()do not (commute


limit A sufficient condition for the convergence of (3) is the existenceof the integrals

(4)In

toeiIVe_10t coil dt

for any Sp E N (we have taken into account that ei Ht is a unitaryoperator). Finally, the integrals (4) converge at the upper limits iffor any yo EN

(5) I= ° (1t1i+F) 1e > 0, iti-oo.

Let us see which potentials satisfy (5). In the coordinate repre-sentation we have an estimate48

Ck< ItI32

uniform with respect to x, for the wave function cp(x, t) = e-iH{cOx).

Then

iiVe_i0twii2 = JR I v(X) (X, t) RIV(X)I2dx.

3

We see that the condition (5) holds if the potential is squareintegrable. Of course, this condition is not necessary. The class ofpotentials for which UU exist is broader, but there axe potentialsfor which the wave operators cannot be constructed. An importantexample of such a potential is the Coulomb potential V (r) = a/r.The reason for the absence of wave operators for this potential isthat it decays too slowly at infinity. The solutions of the Schriidingerequation for the Coulomb potential do not tend to solutions for afree particle as t --+ ±oo (the particle "feels" the potential even atinfinity). In connection with this, both the nonstationary and thestationary formulations of the scattering problem in a Coulomb fieldrequire serious modifications. The asymptotic form of the Coulombfunctions '(x, k) is different from (39.7).

48'This estimate is obtained by the stationary phase method for the asymptoticcomputation of the integral p(x, t) = (27r) -3/2 fR3 cp(k)et(kx-k2t) dk The stationaryphase method can be used if W(k) satisfies certain smoothness conditions But the setV of smooth functions is dense in 7-l and the operators U(t) and Ut are hounded, soit suffices to prove the existence of the strong limits of U(t) on the set V


Let us return to our study of potentials for which the wave op-erators UU exist, and discuss item 2. We can ask whether for anyvector '0 E 7 there are vectors p+ and ;p- such that (1) is valid ast-*±00.

It turns out that if H has eigenvectors Xn, llXnH = 1, then forthem the equality (1) is false for all o+ and V- . Indeed, in this caseone can easily compute that for any ;p E 7 with 1

lim lle_tXn _ e-ilfot.pll = lim lie -iEntXn --t--.±oo

lim VIIXn 112 + 11;0112 - 2 Re a-iEnt V2.t->±oc

We have taken into account that

(s) lim (Xn, e'Hot0) = 0,t-+fx

since the vector a-iHot.p tends weakly to zero as t - ±oo.

From physical considerations it is also easy to see why the statee - i H t Xn does not tend asymptotically to some state of free motionfor e-"illot;o. The vector e-iHtXn describes the state of a particle thatis localized near the scattering center, while for any ;p E 7-1 the vectore - i Ho, t describes the state of a particle going to infinity as t - ±oo.

It is clear that (1) fails also for any vector X E B, where B isthe subspace spanned by the eigenvectors of H. We shall call B thesubspace of bound states.

Accordingly, we see that for an arbitrary vector 0, there does notin general exist a vector ;p+ satisfying (1) as t ---i +oc To determinewhether for a constructed vector V) = U_ p_ it is possible to find acorresponding vector ;p+, we need to study the properties of the waveoperators.

Let R± denote the ranges of the operators U±. We show thatRf18.

It suffices to verify that the vectors U± ,p are orthogonal to theeigenvectors X,, for any p E R. From (6) we get that

(U± V, Xn) = lim (euhute_uhbot'p, Xn) = lixxi eiL' t (e` Wot701 X71) =0-t- +±O t -+f oc 1 1


We state the following property without proof. It turns out that

(7) 1Z+=R_=R, R©S=N.

The proof of this assertion in abstract scattering theory is very com-plex. The subspace R which coincides with the ranges of the operatorsU± is often called the subspace of scattering states.

We show further that the operators Uf are. isoxrmetries. Indeed.from the strong convergence of the operator U (t) as t --* ±oo andfrom its unit arity it follows that

(Ucp, U±V) =t

limx(U(t) W, U(t) c0) = (V, SP):

that is, the operators UU preserve the norm of a vector cp, and hence

(8) UU UU = I.

The operators UU are unitary only in the absence of eigenvectorsof the discrete spectrum of H. In this case the operators UU map 1-1bijectively onto itself, and then along with (8) we have the equality

(9) U± U; = I.

If H has a discrete spectrum and E R, then there are vectorsp+ and ;p_ such that

0 = U±±.Acting on this equality by UU and using (8), we obtain

(10) U10 = cp±,

(11) UUUj =b, I' E R.

On the other hand, for x E 8 and any cp E H,

(Uy,cp) = (x,U±ca) = 0,

and therefore

UUx=0, UUUUx=O, XEB.

Any vector V E H can be represented as

Sp=X+,0, XEB, ER,and

V) = (I - P) 01


where P is the projection on the subspace B of bound states. Thus,in the general case we have

(12) UUUU - I - P

instead of (9), and therefore the operators UU are not unitary in thepresence of a discrete spectrum.

Finally, we show that

(13) f(H)UU = U+f(HO)

for any hounded function f (s), s E R. Passing to the limit as t --+ ±ooin the equality

eiHreiHte-iH0tSp = eiH(t+r)e_ Ho(t+r)eiHorSp, 'p E 1, T E R,

we get thate iHrU± = U±eiHUr

which at once gives us (13).

Let us return to the nonstationary scattering problem formulatedat the beginning of this section. If for some potential V the waveoperators UU exist and (7) holds, then the nonstationary scatteringproblem has a unique solution.

The vector 0 is found for any Sp- E 1-1 from the formula

'0=U_cp-.

This vector is in R; therefore by (10),

P+=U+10

or

0+ = U+ U- O - ,which can be rewritten in the form

(14)

where

0+ = S'p-

S=U+U-.The unitarity of the scattering operator S follows from (8), (12), andthe obvious equalities PUf = 0. Indeed,

S*S= U-**U+U+U- = U* (I - P) U- = I,


and similarly

SS* = I.

Further, it follows from (13) that

(15) S f (11o) = f (Ho) S.

Let us write (15) in the momentum representation for f (s) = s:

S(k,k') k'2 = k2S(k,k').

We see that the kernel of S can be written in the form

(16) Skk' =2S(k,w,w')Sk2-k'2 k=kw

and the relation (14) takes the form

k'2dk'2S(k,w,w') 6(k - k') kk w = dw'P( ) SP ( )

.s2 jSS(k, w, w') cp_ (k, w') dw'.

1S2

omparing this formula with (40.9), we get that the functionS(k, w, w') in (16) coincides with the function S(k, w, w') in § 39. Thisconnection between the S-operator of nonstationary scattering theoryand the asymptotics of the wave functions of the stationary scatteringproblem can also be established in the framework of a rigorous theory,of course.

There is a simple connection between the wave operators UU andthe solutions (x, k) introduced in § 39 of the Schrodinger equation.We recall that for the nonstationary Schrodinger equation the solution

(x, t) constructed from the function (x, k) has the form3

(x,t)2

-- 1 _ (k) (x, k e-ik2t dk.JR3

Here we have denoted the function C(k) by ;p_ (k). We know that ast -oo the function (x, t) tends asymptotically to

3

1 _ k eikx a-ik2t dk.- x t =2

Sp ( , )(27r R3

Sp ( )


Setting t = 0 in these equalities, we get that3

x = 1

2 k _dk( ) 27r(x, ) (k)JR3''

2

p_ (x) _ A.27r JR3

Comparing these formulas with p = U_p_, written in the mo-mentum representation

V)(k) - U- (k, k') p- (k') dk' IJR3

we get that

U_ (k, k') = 1 e-ikx (x, k') dx.(27r)3

JRIt can also be shown that

The connection between the operators Uf and the eigenfunctionsof the continuous spectrum of the operator H becomes especiallyclear if we write (13) with f (s) = s together with (8) and (12) in themomentum representation

(17) HUt (p, k) = k2Uf (p, k)

(H is the Schrodinger operator in the momentum representation),

(18) f -k2),

(19) fR3 U(P,k)U(P',k) A +Xn(P)Xn(P')=b(P-P')n

The formula (17) shows that the kernels of the operators UU, asfunctions of p, are eigenfunctions of the continuous spectrum corre-sponding to the eigenvalue k2. Then (18) is the condition of "or-thonormality" of the eigenfunctions of UU (p, k), and (19) is the con-dition of completeness of the systems {Xn(p), UU (p. k)}.

§ 44. Properties of commuting operators 197

§ 44. Properties of commuting operators

In this section we consider the properties of commuting operators,discuss once again the question of simultaneous measurement of ob-servables, and introduce a concept important for quantum mechanics.the notion of a complete set of commuting observables.

To begin, let us consider two self--ad joint commuting operators Aand 13 with pure point spectrum. We show that such operators havea common complete set of eigenvectors. Let

(1)

ASp{m) _ Am;p{,n}, i = 1, 2, .-. ,

B j = 1, 2, ... ,

where the collections {'P} and {t4f} of vectors are complete in thestate space H.

Let Nm denote the eigenspace of A corresponding to the eigen-value Am, and let P m be the projection onto this subspace. Let Vand P,, be the analogous objects for B. From the condition AB = BAit follows that PmPn' = PnPm. Let Pmn = PmPP. Obviously, Pmn isthe projection onto the subspace flmn = Nm f1 Nn. If SO E Nmn, thenit satisfies both the equations in (1). Further, Nmn _.L Nm' n, if theindex mn is different from the index m'n'. Let {p}, k = 1,2,...,be a basis in the subspace fmn. To prove the completeness of thesystem f W n 1,9 k, m, n = 1, 2, ... , of vectors in H it suffices to verifythat no nonzero vector SP in N is orthogonal to all the subspaces Nmn.It follows from the completeness of the set of eigenvectors of A thatfor any nonzero 'P there is an index m such that PmSp 34 0. Similarly,there is an index n such that R' PmW 0; that is, Pmn;p 34 0 for anynonzero Sp for some m and n.

Thus, we have shown that {4} is a basis in h consisting ofcommon eigenvectors of the operators A and B.

The converse also holds. If two operators have a common com-plete set of eigenvectors, then they commute. Indeed, let {'P} be acommon complete set of eigenvectors of A and B:

Acp{mn = AmS'(mn,(2)

k) =._ tn'P k) k = 1, 2, ... .


Then Am iLn cprnn and H AcpM,}, = Am/1n'. From thecompleteness of the collection it follows that AB;p = BA'p for anyypEH.

The assertions proved are valid for an arbitrary number of pair-wise commuting self-adjoint operators A1, . . , An, ... with pure pointspectrum and with certain stipulations for operators with a continu-ous spectrum.

We also have the following assertion. If the self-adjoint operatorsA1,. .. , An commute, then there is a self-adjoint operator R such thatall the operators Ai, i = 1. 2, ... , n, are functions Ai = FZ (R) of it.

It is quite easy to construct such an operator R for operatorswith pure point spectrum. Let us consider the case of two commutingoperators A and B. They have a common complete system I ofeigenvectors which satisfy the equations (2). We define the operatorR by the equalities

(In)n - ryn i mnwhere the r( km are distinct real numbers. Obviously, R is a self-adjointoperator with a simple pure point spectrum. We now introduce realfunctions F W and G (x) satisfying the conditions Am = F (r ,,7) and/-In = G (rmn ). For x different from a all the numbers the valuesof F (x) and G (x) are not important. It follows at once from thedefinition of a function of an operator that A = F (R) and B =G(R). The general theorem for commuting operators A, ..... An witharbitrary spectrum was proved by von Neumann.

We now discuss the physical consequences of the assertions for-mulated for commuting operators. We recall that the Heisenberguncertainty relation does not impose any restrictions on the varianceof the commuting observables, and in this sense we have called themsimultaneously measurable observables. The concept of simultane-ous measurability can now be made more precise. The last assertionshows that to measure the numerical values of commuting observablesit suffices to measure the single observable R; that is, it is possiblein principle to find out the numerical values of all the observablesA1, ... , An with a single measurement. The existence of a system ofcommon eigenvectors implies the existence of an infinite set of statesin which all these observables have definite numerical values. Finally,

§ 44. Properties of commuting operators 199

it will follow from results in the next section that for any state it ispossible to construct a common distribution function for the numer-ical values of simultaneously measurable observables.

We shall introduce the concept of a function of commuting oper-ators A and B. For a real function f (x, y) it is possible to constructa self-adjoint operator f (A, B) by setting

f(A'B) Vmn -where the vectors prn n satisfy (2). This definition agrees with thedefinition given earlier for a function of simultaneously measurableobservables.

We have the following assertion. If the operator D commuteswith any operator C commuting with all the operators A1, .. , Art

and [Ai, Aj] = 0, then D is a function of these operators. In theproof we again confine ourselves to the case of two operators withpure point spectrum.

Suppose that to each pair Am, An of eigenvalues there correspondsa single eigenvector:

ASpmn = Am Pmn, Bcomn = /1 n(pmrzn

In this case it suffices that D commute with just the operators A andB. Indeed, if [D, A] = 0 and SO E fm, then also Dcp E Wm, sinceADcp = DAcp = Am Dep. Similarly, from the condition [D, B] = 0,we get that if Sp E 1l;t, then Dip E H. Therefore, if cp E f mn, thenDcp E xrrara. By assumption, the spaces flmn are one dimensional,and DScrnn is only a numerical multiple, which can be different fromOmn; that is, D;p,nn = XrrznjOmn. Choosing the function f (x, y) suchthat Xmn = f (Am, Pn), we see that D = f (A, B).

Let us proceed to the more complicated case when the subspacesxrrtn are not one dimensional

..A'rrrnn = Am'prnn, B"pmmn = inSO(k) 7k = 1121.

We consider the collection of eigenvectors corresponding to thepair Am, Ian . For brevity we omit the index mn. The operators CU)


and C(ji) are introduced by specifying that(k) _ Sjk(P(k).

CUO(P

(k)

0

cu) P = Cw) O = 0,

where f n is the orthogonal complement of the. subspace H,nn. It isIrri

easy to verify that all the operators CO) and COO commute with Aand B. From the condition [D, CU)] = 0 we get that

DC(j) p(j) = C{j) AP(j )

from which it follows that D' p(j) is proportional to Sp(j) :

We show that all the numbers x(j) coincide:

X(j)(PU) = DW(j) = C(jt)Dcp(d)

= CUO X(0) 0(l) = X(l) cP(3 }

Thus, the vectors areare eigenvectors of D, and the eigenvaluesdo not depend on the index k: D'pmn = Therefore, D -f (A. B) as in the first case.

We point out that commutativity of D with just A and B wouldnot be sufficient here. What is more, if the commutativity of D withA and B implies that D = f (A, B), then we can assert that to eachpair of eigenvalues Am and jn there corresponds a single eigenvector'P,nn . Indeed, if there are several such vectors, then we can alwaysconstruct an operator that commutes with A and B but is not afunction of them: for example, the operator C(pl).

We can now define the important concept of a complete system ofcommuting operators. A system of self-adjoint operators A1, ... , A,,,is called a complete system of commuting operators if:

1) the operators Ai commute pairwise, that is, [As. A3] = 0 fori, = 192,...,n;

2) none of the operators Ai is a function of the others;

§ 45. Representation of the state space 201

3) any operator commuting with all the Ai is a function of theseoperators.

From the assertions proved above and the conditions 1) and 3) ofthe definition of a complete collection A1, ... , An, it follows that thereis a common complete system of eigenvectors of all these operators,

A1'Pai , .Q n = a1 'Pa i , ass i=1,2,...,n,

and to each collection of eigenvalues a1,. .. , an there corresponds asingle eigenvector Vai, a,,. The condition 2) implies that the lastproperty is not enjoyed by a common complete system of eigenvectorsfor only a part of the operators A1,.. . , An. In fact, 2) means thatthere are no "superfluous" operators among A1, ... , An-

If as a result of measurements it is known that the numericalvalues of a complete set of observables A1,.. . , Ay, in some state arecertainly equal to a,, ... , an, then we can assert that this state isdescribed by the vector 'pai, an .

In conclusion we note that if a set of pairwise commuting inde-pendent operators is not complete, then it can be supplemented inmany ways -to form a complete system.

§ 45. Representation of the state space withrespect to a complete set of observables

Suppose that A1,. .. , A,7 is a complete set of operators with purepoint spectrum. These operators have a common complete set ofeigenvectors 'Pal, an, and corresponding to each collection of eigen-values is a single eigenvector Va i , an . An arbitrary vector /' E ?lcan be represented as a series

11 = E VI(al, ... , a,t} (pat, are , 2/(al, ... , an) = (Z/J, SPai, an ).a,, an

This formula determines a one-to-one correspondence between thevectors 0 and the functions '(al, .. , an) defined on the spectrum ofthe operators A1, ... , An:

(ai,...,an).


Obviously,

(PI,'2) _ ii (a, . .. , an) 12(a] , . .. I an)Yal aAiy) H aii/J(al, ... , an);

that is, the representation constructed is an eigenrepresentation forall the operators A1, ... , An (the action of these operators reduces tomultiplication by the corresponding variables).

The function ,&(a1,.. . , an) is called a wave function. To clarifyits physical meaning we construct, as in the preceding section, anoperator R such that Ai = F i (R) f o r i = 1, 2, ... , n,

i , an. = ra, , .a n SPu, , ,a,:

where ra,, are distinct real numbers and ai = Fi (ra, , Weknow that J ,a,,) I2 = l'I'(ai.. . . , an) j2 is the probability of ob-taining the numerical value ra, , a,z for the observable R by a measurentent. Therefore, liI'(ai,.. . , an) I2 is the probability of obtainingthe values al , . . . , an as a result of simultaneous measurement of theobservables A 1, . . . , Are .

All these results generalize to the case of a complete set of opera-tors A1, . . . , An with arbitrary spectrum. We formulate the theoremwit hout proof.

Theorem. Let A1,. .. , An be a complete set of commuting operators.Then there exists a representation of the state space such that a vectorb E fl is represented by a function *(a,, ..,an) defined on some set

'A (a = (al, ..,an) E 2). A measure duc(a) is given on the set %.,and a scalar product is defined by the formula

(11, )2) = jIPI(a)1k2(a)d/2(a).

In this representation the operators A, , ... , An are the operators ofmultiplication by the corresponding variables

AiVY(al,...,an) = ai (ai,...,an), i = 1,2,...,n.

The function I?J(al, ...,an) 12.. , an )12 is the density of the common distribu-tion function for the observables Al, .... An with respect to the mea-sure dp(a).

§ 46. Spin 203

Above we already had examples of complete sets of commutingoperators and corresponding representations of the state space R.

For a structureless particle the coordinate operators Q1, Q2. Q3form a complete set. The coordinate representation corresponds tothis set The momentum representation is constructed similarly fromthe complete set P1, P2, P3. A complete set is also formed by the oper-ators H, L2, I,3, where H is the Schrodinger operator for a particle ina central field. The representation corresponding to this complete setwas described in § 31. For a one-dimensional particle the Schrodingeroperator II for the harmonic oscillator by itself represents a completeset. The corresponding representation was constructed in § 18.

§ 46. Spin

tip to this time we have assumed that the electron is a material pointwith mass ni and charge -e; that is, it is a structureless particle whosestate space H can be realized, for example, as the space L2 (R3) ofsquare-integrable functions O(x). On the basis of this representationof the electron, we calculated the energy levels of the hydrogen atomand obtained results that coincide with experimental results to a highdegree of accuracy. Nevertheless, there are experiments showing thatsuch a description of the electron is not complete.

We have already mentioned the experiments of Stern and Gerlach.These experiments showed that the projection on a certain directionof the magnetic moment of the hydrogen atone in the ground statecan take two values. In § 34 we constructed the quantum observable"project ion of the magnetic moment" of a structureless charged par-ticle, and we saw that the third projection of the magnetic momentis proportional to the projection L3 of the angular momentum. Fromthe calculation of the hydrogen atone, we know that the numericalvalue of L3 is zero for the ground state. Therefore, the magneticmoment of the atom in the ground state must also be zero. Thiscontradiction can he explained if we assume that the electron itselfhas a magnetic moment and a mechanical moment whose projectionson a certain direction can take two values

204 L. D. Faddeev and 0. A. Yakubovskiii

The electron's own angular momentum is called its spin, in con-tradistinction to the angular momentum associated with its motionin space, which is usually called its orbital angular momentum.

The existence of an observable, which can take two numericalvalues, leads to the necessity of considering that the electron canbe in two different internal states independently of the state of itsnotion in space. This in turn leads to a doubling of the total number

of states of the electron. For example, to each state of the electron ina hydrogen atom (without counting spin) there correspond two statesthat, differ by the projection of the spin on some direction.

If we assume that there is no additional interaction connectedwith the spin, then the multiplicity of all the energy eigenvalues turnsout to be twice that of a spinless particle. But if there are such inter-actions, then the degeneracies connected with the spin can disappear,and there can be a splitting of the energy levels. Experiments showthat such a splitting does indeed take; place.

In § 32 we described a model for an alkali metal atomn, basedon the assumption that the valence electron of the. atom moves in acentral field. This model is not had for describing the configurationof the energy levels of alkali metal atoms, but neither this model itselfnor any of its refinements can explain the observed splitting of thelevels into two nearby levels when 1 0. The hypothesis of spinenables us to easily explain this splitting. In atomic physics there aremany other phenomena which can be explained on the basis of thishypothesis. We shall see later that only the doubling of the numberof electron states that is connected with spin makes it possible toexplain the length of the. periods in the periodic table

We would like. to stress that the hypothesis of spin for the electronis a hypothesis about the nature of a concrete elementary particleand does not concern the general principles of quantum mechanics.The apparatus of quantum mechanics turns out to be suitable fordescribing particles with spin.

Let us begin with the construction of the state space for an elec-tron. Without spin taken into account, L2 (R3) is the state space fin the coordinate representation. The introduction of spin requires

§ 46. Spin 205

us to extend the state space, since the number of states of a particlewith spin is larger than for a spinless particle.

A doubling of the number of states without changing the phys-ical content of the theory is easily attained by replacing the statespace 'H = L2 (R3) by HS = L2 (R3) 0 C2 and associating with eachobservable A acting in R the observable A 0 I acting in 7-Is.

We explain this in more detail. The elements of the state spaceof a particle with spin are pairs of functions

101 (X)()02

The scalar product in the space 7_Is is given by

(2) (q, f) =R3

V)1 (x) W, (x) dx + ?F12 (X) 2 (X) dX.R

For the observables A 0 I acting in 7-Is we use the same notationand terminology as for the observables A acting in H. Thus, thecoordinate operators Q1, Q2, Q3, the momentum projection operatorsP1, P2. P3, and so on, all act in 7-Is. For example,

P1w=P1 (?P1 (X)

0z(X)

h oiy i (x)i axhi Ox,

To each pure state V)(x) in 7-I there now correspond the two orthogonalstat es

7P(X) T = 00 2 V)(x)

in ?-Is, or any linear combination of them. It is clear that the meanvalue of any observable A in a state b will be equal to the meanvalue of the observable A 0 I in the states IF1 and ' 2 . Therefore, byintroducing the space 7-Is and confining ourselves to observables ofthe form A R I, we have indeed attained a doubling of the number ofstates while preserving all the physical consequences of the theory.

However, along with the observables A 0 1 there are also otherobservables acting in the space 7Is, for example, of the form 1 0S, where S is a self-adjoins operator on C2. Of course, there areobservables acting in 7Is that cannot be represented in the form A ® Ior 10 S, for example, sums or products of such observables.


Let us consider observables of the type I ® S. First of all, it isclear that any such observable commutes with any observable A 0 1and is not a function of observables of the latter type. Therefore, onthe space RS the observables Q1, Q2. Q3 do not, form a corriplete betof commuting observables, arid we will have to supplement this set toobtain a complete set.

Any self-adjoint operator S on C2 is representable, as a self-adjoint2 x 2 matrix and can be expressed as a linear cornbinat ion of fourlinearly independent matrices. It, is convenient to take these matricesto be the identity matrix I and the Pauli matrices ar , (72, ar3.

1 01=n 1

,

U 1 0 -i 1 Ua1 = 1 0 , a2 = i 0 , a2 = 0 -1

The properties of the Pauli matrices were discussed in § 27. Thematrices S3 = o-3/27 j = 1, 2, 3, have the same commutation relationsas for the orbital angular momentum:

[S1, S2] = iS3, [S2. S3] = iSl, [,S3- S1 ] = iS2.

We have seen that the commutation relations are one of the mostimportant properties of the angular momentum operators Therefore,it is reasonable to identify the operators 1 ® Sj, j = 1, 2, 3, with theoperators of the spin pro jections.49 These operators will usually bedenoted by S3 in what follows. The operators S j have eigenvalues±1/2, which are the admissible values of the projections of the spinon some direction.`} The vectors %P ] _ (lp(x)) and W2 = (V (X) }are eigenvectors of the operator S3; with eigenvalues + 1 /2 and - 1/2.Therefore, these vectors describe states with a definite value of thethird projection of the spin.

49 De eper considerations are based on the fact that for a system with sphericallysymmetric Schrodinger operator the operators I., 4 S. are quantum integrals of motionWe shall discuss this question later

`"We write all fornnmlas in a system of units in which h = 1 In the usual systemuof units. the operators of the spin projections have the form SJ - (h/2}a;, and theadmissible numerical values of these projc ctions are equal to ±h/2 We remark that inpassing to classic al muecha,iics h -- 0, and correspondingly the projections of the spintend to zero Therefore, the spin is a specifically quantum observable

§ 46. Spin 207

For what follows, it is convenient for us to change the notationand write a vector 'P E 7s as a function W (x, s3 ), where x E R3 and83 takes the two values -+-1 /2 and --1 /2. Such notation is equivalentto (1) if we let

T X1 - 7P, (X). X. - - = '02 (X) -( 2) 2)

From the equality

S3

it follows that

(Pi(x))0l(X)-

1101W

\-2VYX))

S3' (X, s3) = S3 %P (X, s3);

that is, the operator S3, like the operators Q, , Q2, Q3, is an oper-ator of multiplication by a variable. We see that the representationconstructed for the state space of a particle with spin is an eigenrep-resentation for the operators Q1, Q2, Q3 and S3, and these operatorsform a complete set of commuting operators acting in 7is.

It is now easy to understand the physical meaning of the func-tions T (x, S3). According to the general interpretation, Is3) 12is the density of the coordinate distribution function under the con-dition that the third projection of the spin has the value s3, andR3 W (x, 83) 1 2 dx is the probability of obtaining the value 83 as aresult of measurement.

Along with the observables S1, S2, S3 we can introduce the op-erator S2 = Sj +S22 +S32 of the square of the spin. SubstitutingSi = o j in this expression and considering that oj = I, we get thatS2 = 4I. We see that any vector E 7s is an "eigenvector" for theoperator S2, with eigenvalue 3/4. This eigenvalue can be written inthe form51 s (s + 1). where s = 1 /2 Therefore, one says that the spinof the electron is equal to 1/2.

Let us construct a representation of the rotation group on thespace fs. We recall that the representation of the rotations g act inthe space H = L2(R3) as the operators W (g) = e- i(f,ial+L2a2+L3a3

511n §29 we showed that from the commutation relations for the angular mo-mentum it follows that the eigenvalues of the operator of the square of the angularmomentum have the form j (j + 1), where j is an integer or half-integer For the spinoperator this number is commonly denoted by s


and in the space C2 as the operators U(g) = e-i(s1a1 }.S2a2+.S3a3). The

mapping g ---+ WS (g) with W,5 (g) = W(g) U (g) is a representationon the space 1-Is, the product of the representations W and U.

An operator acting in 1-CS is said to he spherically symmetric ifit commutes with all the operators Ws(g). If the Schrodinger oper-ator H is spherically symmetric, then the operators WS (g) and theinfinitesimal operators

i)wvas j a

= -?VLj Z9 1 + I '11)j), = 11Z151

are integrals of motion. Therefore. for a system with spherically sym-metric Schrodinger operator we have the law of conservation of thetotal angular moment urn, whose projections are Jj = L j + SS.

We remark that in the general case of a spherically symmetric op-erator 11, there are no separate laws of conservation of orbital angularmomentum and spin angular momentum. However, if a sphericallysymmetric Schrodinger operator acting in Ns commutes also withall the operators W (g) 0 1, then it commutes with all the operatorsI 0 U (g) , and there are laws of conservation for the observables L jand Sj separately. An example of such a Schrodinger operator is theoperator 1101, where H is the Schrodinger operator for a particle ina central field.

§ 47. Spin of a system of two electrons

The space C2 introduced in the previous section is often called thespin space for the electron. For a system of two electrons the spinspace is the space C4 = C2 0 C2. In C2 we choose a basis consistingof the eigenvectors

rl; = (0I)and u_

of the operator S3 with eigenvalues 1/2 and -1 /2, respectively. Asbasis vectors in C4 we can take U(1)U(2), U(1)U(2), U+1)U(2), and

U (1) U(2) , where the indices (1) and (2) below indicate the spin sub-spaces of the electrons.

§ 47. Spin of a system of two electrons 209

However, it turns out to be more convenient to use another or-thonormal basis consisting of the vectors

Wi = U+

iV2 = 0)U (1) U (2)

W3 =72= (U

- +

W4 = IU(I)U(2) _ U(I)U(2))(' + +

The convenience of the new basis consists in the fact that the vectorsWi, i = 1, 2,3,4, are eigenvectors of the operators

S3 = S31) + 53(2)

S2=S +52+53.

Here the operator S3 is the third projection of the total spin of thetwo electrons, 52 and S1 and S2 have the analogous meanings. Theoperator S2 is the square of the total spin.

To verify our assertion about the vectors Wi, i = 1,2,3,4, wefind the results when the operators U1, o2, 03 are applied to the basisvectors U+ and U-:

Ulu+ = 0 1) (1) - 0 U-I(1 0) O (1)

cr1 U_ = U+,

a2 U+ = W-,

U2 U_ _ -- i U4-,

U3U+ = U,,

U3U_ _ -U--

52 More precisely, S3 = z Q3 0 I + 119 03 The operators a j 9 I and 19 oj willbe denoted below by Q 1 and Q 2 ) , respectively


Using these formulas, we obtain

S3W1 = 1 (T(1) + Q(2))U(')U(2) = U(1)U(2} = 1W2( 1 3 + + +

(2)S3W2 = -1W2.

S3W3 = OW3,

S3W4 = OW4.

Along with (2) we have the formulas

S2 W3 = 2W31 = 1,2;3,

S2W4 = OW4

We verify (3) for the vector W;:

S2W3=(52+1 S2-1 "32 + - S3 W3

=4Rai1) (0,0) + 022)2 + (o1) +v32))2]Ws

= 3I + 1(011 al + U2 172

+ .(1)x(2) U(1)U(2) + U(1)U(2)2 2(1 1 a 2 3 3) (+ +}

= 3W + 1 W3 + 1W -1W3= 2W3.2 3 2 2 ' 2

Here we have used (1) and the equalities rrj2= 11 j = 1, 213. Thus, thefirst three vectors W1, W2, W3 describe states in which the square ofthe total spin of the system of two electrons is equal to 2. The number2 can be written in the form 2 = S(S + 1), where S = 1, and thereforethe total spin in these states is equal to 1. In correspondence withthe general properties of the angular momentum, the projection ofthe total spin takes the values ±1 and 0 in these states. The vectorW4 describes a state with total spin equal to zero. One sometimessays that in the states W1, W2, W3 the spins of the electrons areparallel, while in the state W4 they are antiparallel.

We discuss the result obtained from the point of view of grouptheory. We know that the irreducible representation of the rotation group by the operators U(g) = e 2 (o,,a,+12a2+a3as) acts in thespace C2. It is clear that the mapping g -+ U(g) = U (g) 0 U (g) =e -i (S 1 a 1 + S2 a.2 S3 a3) is a representation of the rotation group on thespace C4. The representation U is the tensor product of the sametwo representations U, and it is reducible. According to results in

§ 47. Spin of a system of two electrons 211

§ 29, the space C4 is representable as a direct sum of two subspacesthat are invariant under the operators U (g), and irreducible repre-sentations act in these subspaces. The first subspace is spanned byW1, W2, W3 and the second by W4. Using the notation in § 29, wecan write this result in the form

D2 , x D2 , = Doc3Di.

We remark that we have proved a particular case of the theorem ondecomposition of a tensor product of irreducible representations ofthe rotation group. We state this theorem without proof:

If the irreducible representations Dj, and D j, of the rotationgroup act in the spaces E1 and £2. then the tensor product of therepresentations can be represented as the direct sum

Dj, ®Dj2 = Dlji-j21 0 DIj1 -j2I+1 e ... ®Dj1 +j2

of irreducible representations.

The last formula is called the Clebsch-Gordan decomposition.This decomposition is obtained by passing from the basis consistingof the vectors e j1 nt, ej2m2, mk = -jk, -- jk + I.... ti,, k= 1, 2, where

the e jk m k are eigenvectors of the operators (J(k))2 and J3k) , to thebasis of vectors ej1 j2 /A17 J = h- j21, hi - j21 + 1, ... J1 +j2, M-J, -J + 1, .... J. The vectors f'j1J2JM are eigenvectors for the fourcommuting operators (J(1)) 2, (J(2) )2,

J2 = (j(l)+ 1(2))2+( J21)+ J22)}Z+( J31}+ J22)}Z, J3 = Al)3+J32)

The vectors e j1 j 2 , f M are represent able in the form

(4) ejlj2JM = E Cjij2 1M,j1j2rra1rn2ejlrn, ej2rn27m1,m2

where the summation indices Mk run through the values -3k, -3k +1, ... , j,, k = 1, 2, and the coefficients C in the decomposition (4)are called the Clebsch-Gordan coefficients. We remark that in passingto the basis W1, W2, W3, W4 we found these coefficients for the casej1=j2=1/2.

From the stated theorem it follows also that if in some state theangular morrienturn J M 1) has the value j, and the angular momentumj(2) has the value j2, then the total angular momentum J can take


the values 131 - j21. hi -- j21 + 1, .... jl + j2. Both the orbital orspin angular rrlorrlenta for different particles and the orbital and spinangular momentum for a single particle can combine in this way.

In concluding this section we mention a property of the basiselements I'V1. i = 1, 2, 3, 4, that, is important for what follows. Intro-ducing the spin variables s31) and s3Z), each of which can take the twovalues ±1/2, we write the W as

W'V1(.531), S(2)) U+(3 }3 )).U'2(eS31},.532})

U_(.5(31)) 1_(8(32)).

(5) W3 s(2) = 1 U U+ U- 91U .S(2(S(1) )4

(41)) (.3(2))3 ) +( 3

VL

W, s(1) s(2) (2)1( 3 3 (S3 3 3 U+ (.s3

VG

where U+.(2) = 1, U+(-2) = O. U ()=O,U_(-)=1.It follows from (5) that the functions W1, ti'V2, W3 are symmetric

functions of the spin variables; that is,

Wi(83l),.531)} = tiili(s31), s3Z)}, i = 1, 2, 3.

while W4 is an antisymmetric function4(eS3 . 831}) - -W4(s31} 532}

§ 48. Systems of many particles. The identityprinciple

So far we have studied mainly the behavior of a single quantum par-ticle. In the coordinate representation, the state space of a spinlessparticle is the space L2 (R3), and for a particle with spin 1/2 it isL2 (R3) ® C2 . A natural generalization of such spaces to the case ofa system of n particles is the space

L2 (R3n) = L2 (R3) , ... ® L2 (R3)

for spinless particles, and the spaceL2(R3n) g C2n = L2(R3) o C2 ... L2(R3) g C2

for particles with spin 1/2.

§ 48. Systems of many particles. The identity principle 213

A comparison of theory with experiment shows, however, thatthis assumption about the stat e spaces of systems of rt particles turnsout to be valid only in the case when no two particles in the systemare alike. In the presence of identical particles there are peculiaritiesin the behavior of quantum systems that can be explained on thebasis of the so-called identity principle. We note at once that theidentity principle is a new principle of quantum mechanics It cannotbe derived frorri the other fundamental tenets of quantum mechanicsformulated earlier and must be postulated.

We use the single symbol f to denote the spaces L' (R3,) andL2 (RU) 0 C2*,, , and we write the elements of these spaces in the form

( , .. , r}, where z = x(i) for a spinless particle and j = ((), 8(`) )for a particle with spin, s being the spin variable of the ith particle.For a particle with spin 1/2, the variables s3(i) take the two values±1/2.

The scalar product in the spaces R can also be written in theuniform way

(TI D) = f T( I.. . I jj 0 * .. -rj) 41, * a

where it is assumed that, for a particle with spin the integration withrespect to the variable & is integration with respect to the spacevariable x(a) and summation with respect to the spin variable

Before passing to the statement of the identity principle, we con-sider the permutation group S, also called the symmetric group. Theelements of this group are the permutations

7r1, 2, ...=il, i2. ...

nint

and the identity element is the identity permutation

I- 1, 2% ..., n1. 2, ..., n

The product it = 7r2 7r1 of two permutations is the permutationobtained as a result of the successive application of the permutations7r1 and 7r2.


It is easy to construct a representation of the group S" on thespace W. We introduce the operators P, by

P,r'P(fit, .....n) =

Obviously, 7 is a unitary operator, and the mapping 7r --+ P, is arepresentation of the symmetric group on the space W.

In N we can immediately distinguish two invariant subspaces withrespect to the operators P,r, namely, the subspace 7-(s of symmetricfinict ions, for which

Src) ='P(f 1, i n)i

and the subspace NA of antisynmmet ric functions, for which

P,r"P i - - - 9 W = (-1)[7r] qP (f1,...7Wi

where [7r] denotes the parity of the permutation 7r. It is clear thatR=NA _-?is.

In the case of two particles. H = 7-HA 0 N. Indeed, any functionT 1, 2) can be written in the form

2) +T( 2, 1) l 1 2} -`1- 2} +2 2

A where %PS E Ns and %PA E NA.

in the case of a large number of particles there are also invariantsubspaces more complicated than NS and HA, but these subspacesare not of interest. The identity principle asserts that the state spaceof a system of n identical particles is either the space WA or the space7-Is. The choice of one of these spaces as the state space depends onlyon the type of particles.

one says that particles whose states are described by symmet-ric functions are subject to Bose- Einstein statistics, while particlesdescribed by antisymmetric functions are subject to Fermi-Dirac sta-tistics. The first particles are called bosons, and the second particlesare called fermions

It turns out that the spin of a particle determines the statistics towhich it is subject. Particles with integer spin (including those with-out spin) are bosons, and those with half-integer spin are fermions.in nonrelativistic quantum mechanics there is no explanation for the

§ 49. Symmetry of the coordinate wave functions 215

connection between spin and statistics. This connection is explainedin part in relativistic quantum mechanics. Electrons, protons, andneutrons are fermions with spin 1/2, photons are bosons with spin1, and mesons are bosons with spin zero. The statistics of identicalcompound particles (for example, atomic nuclei) is determined by theparity of the fermions in their structure, since a permutation of thesame complex particles is equivalent to a permutation of several pairsof elementary particles. For example, deuterons, which are composedof a neutron and a proton, are bosons. We note that a deuteron hasinteger spin,53 since the spins of a proton and a neutron are equal to1 2.

We write the Schrodinger operator for a system of pairwise inter-acting particles in an external field:

n1

n n

1I= Ai+E V(x{i} ) + 1: U(x(i) - x{k)).2rni^x i=1 i<k

The first term is the kinetic energy operator of the system of particles,the second term describes the interaction of the particles with theexternal field, and the third term is the interaction of the particleswith each other. We point out that the masses of all the particles arethe same, and the interaction potentials V (x) and U(x) do not dependon the numbers of the particles. As a consequence, the Schrodingeroperator 11 commutes with all the operators P,.- [H, P,.] = 0.

49. Symmetry of the coordinate wavefunctions of a system of two electrons.The helium atom

Electrons are fermions, and therefore the wave function for a systemof two electrons must be antisymmetric:

T (6 1 W = -'P (6 1 W -

531t is known from experiments that the spin of a deuteron is I


We decompose the function with respect to the basisfunctions W1, W2, W,-3, 1V4 introduced in §47:

4

3 3), x(2)S(2)) = qji (X(l). X(2)) IV,. (.531). 832}(1) T(X(1 ).s(1

i=1

The first three terms in this sump correspond to the states with totalspin 1. and the fourth tei in describes a state with total spin zero. Thefunctions T i (x(1) , x(2) ), i = 1,2,3,4, introduced by the relation (1)are called the coordinate wave functions in coat rast to qP (fl , 2), whichis called the total wave function. In § 47 we saw that the W; (s31), 42))

with i = 1.2.3 are symmetric with respect to a permutation of thespin variables, and W4 (s31), s32)) is an ant isymurietric function. Thenit follows from the antisymmetry of the total wave function that

i(X21X(l )=-Tj(xl ,X2 ), i=1,2,3,4(x(2) , x(1)) = W4 (X(1), X(2) )j

t hat is, the coordinate wave functions for the states with spin I areant isyrmrietric. and those for states with spin zero are symmetric.

Let us apply this result to the helium atom. The Schr{ dingeroperator for a helium atom without considering spin interactions hasthe form 54

H _ -1Q - 1Q _ 2 _ 2 1

21

22

r1 r2 r12If is a solution of the equation

W,then the coordinate functions 'i (x(1) , x(2)) also satisfy the Schrodingerequation with the same eigenvalue E. Therefore, the problem reducesto finding solutions of the equation

(2) Hip (X(1), X(2)) = Eq'(x(1). X(2))

in subspaces of symmetric or antisymmetric functions. It is clear thatthere are fewer solutions of (2) in each such subspace than in the spaceL2 (R6) . The values of E for which (2) has a solution W (x(') , X(2))

54 In a precise formulation of the problem, the Schrodinger operator contains termsdepending on the spin, but it is possible to derive an expression for the spin interactiononly in relativistic quantum mechanics Moreover, for a helium atom these terms playthe role of small corrections and are always taken into account according to perturba-tion theory

§ 50. Multi-electron atoms. One-electron approximation 217

in the subspace of antisymrnetric functions correspond to states withspin 1, and the values of E for which there are symmetric solutionsof (2) correspond to states with spin zero.

We see that the energy levels of a helium atom depend on thetotal spin even when we ignore the spin interactions in the Schrodingeroperator. This dependence is a consequence of the identity principleand arises through the symmetry of the coordinate wave functions.

It can be shown that a symmetric coordinate wave function cor-responds to the ground state of a helium atom: that is, the spin ofthe helium atom in the ground state is equal to zero.

It is interesting to note that transitions with emission or absorp-tion of quanta between states with S = 0 and S = I turn out to havesmall probability. Therefore, the optical spectrum of helium is as ifthere are two sorts of helium, with S = 0 and S = 1. The first sortof helium is called parahelium, and the second sort is called orthohe-lium. Corresponding to each energy level of parahelium is the singlespin state W4, and corresponding to an energy level of orthoheliumare the three spin states W1, W2, W3. Therefore, the states of para-helium are called ringlet states, and those of orthohelium are calledtriplet states. Taking into account the spin interactions leads to asplitting of the triplet energy levels into three nearby levels. 5'5

§ 50. Multi-electron atoms.One-electron approximation

The Schrodinger operator of a multi-electron atom or ion with chargeZ of the nucleus has the form

1 rt n Z Il1H = - + > - .

2 ry,

i-1 i=1 i i<,7 L.-

This operator is written neglecting the motion of the nucleus and thespin interactions. Neglecting the motion of the nucleus of a complexatom is completely valid, since the corrections arising when it is takeninto account are less by several orders of magnitude than the errors

55Tt can be shown that such a splitting takes lace if the total orbital angularrrioinentiirn L of the two clue troiis is tionzero, and therefore it is not for all tripletenergy levels that splitting is observed


in comtemporary niethods of computation for complex atomns. Thespin interactions are almost always taken into account according toperturbation theory. and their role will be discussed below.

The operator H corresponds to a neutral atom when Z = n,to a positive ion when Z > n, and to a negative ion when Z < n.The state space of a multi-electron atom or ion is the space RA ofantisymmetric functions T f n) . As in the already analyzedcase of two particles, this space decomposes into a direct sum of spaceswith a definite total spin. The corresponding coordinate functionssatisfy certain symmetry conditions that are more complex than in thecase of two electrons, and we do not describe them here. The problemof finding the spectrum of H acting in HA reduces to the problems ofthe spectrum of H acting in the subspaces 1(k of coordinate functionswith certain symmetry conditions

The spectrum of H acting in the subspaces 7(k has been thor-oughly studied for Z > n. In this case the spectrum is continuouson an interval -p < E < oo with some generally positive ,,c andconsists of an infinite series of negative eigenvalues accumulating at-ju. The values of u for different subspaces can be different. Further,the discrete spectrum for one of these subspaces can be superimposedon the continuous spectrum for another. Thus, on part of the nega;tive semi-axis the spectrum of H is mixed. and eigenvalues lie on thecontinuous spectrum.

The case Z < n has been investigated less thoroughly. Examplesshow that the discrete spectrum can be finite or it can be absent. Theabsence of a discrete spectrum means that such a negative ion does nothave stable states. For example. it is known from experiments thatthere is a stable state of a negative ion of hydrogen (Z = 1, n = 2).Only the finiteness of the discrete spectrum for such a system hasbeen proved rigorously.

The problem of constructing eigenfunctions of H for a multi-electron atom is extremely complex, since the equation .11 = ETdoes not admit separation of variables. This complexity is connectedwith the last term in H, which takes into account the interactionbetween the electrons The attempt to take this term into account


according to perturbation theory leads to poor results, since the in-teract ion between the electrons has the same order as the interactionof the electrons with the nucleus.

One simple niethod for approximately taking into account theinteraction between the electrons is to replace the last terns in II bythe sum J:'-j V (ri) . where the potential V (r) can be interpreted asthe potential of the interaction of an electron with the charge of theremaining electrons. distributed over the volume of the atom. Thepotential V (r) is said to be self-consistent, since it depends on thestate of the atom, which in turn depends on V (r) .

Replacing the last term in 11 by J:', V (r;) gives the approxi-mate Sch r{id i nger operator

it n

Ai + 1: U(ri),i=1 i=1

where U(r) = -Z/r + V (r). A priori, we know about the potentialU(r) only that

U(r)z^=-. r, r--b0, U(r)^'- I

, r --goo.

In what follows we shall see that, there are methods enabling us tofind this potential. However, to understand the structure of complexatones it suffices to assume that H can be fairly well approximated bythe operator H', and the difference H' - 11 = W can be taken intoaccount by the perturbation method.

The approximate Schrodinger equation

(1) H' E; yt)

admits separation of variables. Let us look for a solution of thisequation in the form

(2) 41 fin) = i (ei) ... Vn (fin) .

Substitution of (2) in (1) shows that if V)1( are eigen-functions of the Schrodinger operator for a particle in a central fieldU(r),

(3)1

A Vyk (0 + U (r) Vk W =F1k Ok W


then the function (2) is an eigenfunction of the operator H' corre-sponding to the eigenvalue E = E 1 + + En . However, the function(2) does not satisfy the identity principle. The total wave functionmust be antisymnietric. W e remark that if ' satisfies (1), then I-' 14F

is also an eigenfunction of H', with the same eigenvalue E. Therefore,as an eigenfunction satisfying the identity principle, we should takethe antisymmetric combination of the functions PEW. that is,

7p1( ,) ... 01 (W(4) 'T!n(i .....r:) = C .......

V/17t IL

where C is a constant found from the normalization condition. Thefunction (4) is antisymYnetric, because a permutation of the coordi-nates is equivalent to a permutation of the columns of the determi-nant.

The functions (2) and (4) can be interpreted as follows. Thefunction (2) describes a State of a system of elect rons in which thefirst electron is in the one-electron state V)1, the second electron is inthe state V2, and so on. The function (4) corresponds to a state ofthe system in which the n electrons fill n one-electron states, and itdoes not make sense to speak of which electron is in which state Weremark that the construction of the function (4) is not necessarily pos-sible for an arbitrary solution (2). The determinant in (4) is nonzeroonly under the condition that no two of the functions,01 , ... , ,0,z arethe same. This result is called the Pauli principle, and it can be for-mulated as follows: no xxiore, than one electron can be in the sameone-electron state.

The representation of the state of an individual electron is con-nected with the one-electron approximation. For an arbitrary stateof the system the wave function cannot be represented in the form(2) or (4), and the concept of the state of a single electron becomesmeaningless. There is a formulation of the Pauli principle that is notconnected with the one-electron approximation, but in the generalformulation the Pauli principle loses its clearness, and we shall notpresent this formulation. We remark that the Pauuli principle is aconsequence of the antisymmetry of the wave function and is correctonly for fernmions.


Let us consider the question of classifying the energy levels of amulti-electron atom. The exact Schrodinger operator for the atomcan be written in the form

H =H'+WC+WS,

wheren

1n

WC = E -- E V (ri ),i<j rij i=1

and WS describes the spin interactions. We do not need the ex-plicit form of the operator W. Calculations for atoms show that thecorrections introduced by the operators WC and WS can be foundwith good accuracy by using perturbation theory, and for the atomsin the first half of the periodic table the main contribution is givenby the corrections from WC. Therefore, the possibility arises of arepeated application of perturbation theory; that is, as the unper-turbed operator we can first take H' and regard the operator WC asa perturbation, and then we can regard WS as a perturbation of theoperator H'+ WC.

The operator H' has a very rich collection of symmetries, andtherefore its eigenvalues usually have fairly large multiplicity. Let usconsider why the multiplicity arises. Without spin taken into account,the eigenfunctions of the equation (3) are classified by the three quan-tum numbers n, 1, m. The spin states can be taken into account byintroducing a spin quantum number n t, the two values ± 1/2according to the formula

Ums (s3)ins= i1,

Then the eigenfunctions of (3) can be written in the form

(5) ?Pnlmin (X, .S3) _ Onlm(X) U,n4 (S3).

The eigenvalues of (3) depend only on the quantum numbers n and 1.Therefore, the eigenvalue E of the operator H' depends on the set ofquantum numbers n and l for all the electrons. This set of quantumnumbers n and l is called the configuration of the atom. To expressthe configuration it is customary to assign the letters S, p, d, f,... tothe values I = 0, 1, 2, 3, .. . Then the one-electron state (5) with


n = 1, 1 = 0 is called the 1s-state, the state with n = 2, 1 = I iscalled the 2p-state, and so on. For the lithium atom, for example,possible configurations are (ls)22s, (ls)22p, l s2s2p, .... In the firstof these configurations two electrons are in the is-state and one is inthe 2s-state.

A collection of states with the same quantum numbers n and 1is called a shell. The states of the electrons in a single shell differ bythe quantum numbers m and m,,. For a shell with quantum numberl the number of such states is equal to 2(21 + 1), since in takes thevalues -1, -1 + L. ..,1 and m3 = ±1/2. A shell in which all 2(21 + 1)states are occupied is said to be filled. A completely determined setof functions (5) corresponds to a filled shell. For an unfilled shell adifferent choice of functions (5) is possible, differing by the numbersin and ms . Therefore, the multiple eigenvalues of H' turn out tobe those corresponding to configurations containing unfilled shells.For example, the multiplicity of an eigenvalue corresponding to theconfiguration (is)2 (2p) 2 is equal to (6) 2= 15, since there are six 2p-states, which are occupied by two electrons.

The corrections from the perturbation We can be found upon di-agonalization of the matrix of this perturbation, which is constructedwith the help of all possible functions (4) corresponding to the givenconfiguration. However, such a diagonalization is usually not neces-sary. In the theory of atomic spectra it is proved that if one replacesthe eigenfunctions (4) for a given configuration by linear combinations of them which are eigenfunctions of the square L2 of the totalorbital angular momentum operator, the square S2 of the total spinangular momentum operator, and the operators L3 and S3,

L2'P=L(L+1)W,S2 q, = S(S + 1) W,

1.3'I'=MT.W,

%F ,

then the perturbation matrix turns out to be diagonal with respect tothe quantum numbers L, S, ML, Ms, and its elements do not dependon the numbers ML and Ms. Thus, a degenerate energy level corre-sponding to a configuration K splits into several levels corresponding

§ 51. The self-consistent field equations 223

to the possible values of L and S. A collection of (2ML + 1) (2 Ms + 1)states with a given configuration and given numbers L and S is calleda term.56

Taking the perturbation WS into account can be done for eachterm separately. Here it turns out that the energy level correspond-ing to a term splits into several nearby levels The collection of theselevels is called a multiplet. It can be shown that the states cor-responding to different levels of a multiplet differ by the quantumnumber J. This number characterizes the eigenvalues of the squareof the total angular momentum, which is the sum of the total orbitalangular momentum and the total spin angular momentum.

Finally, corresponding to each level of a multiplet are severalstates differing by the projection Mj of the total angular momentum.This degeneracy can be removed by putting the atom in a magneticfield.

We see that the classification of the energy levels of a complexatom (the configuration, L, S, J) corresponds to the hierarchy ofsurnmands H', WAY, W,S of the Schrodinger operator.

§ 51. The self-consistent field equations

Though the approach in § 50 to the study of the spectrum of complexatoms makes it possible to understand the classification of energylevels, it is not convenient for practical calculations. Most efficientfor our purpose is the self-consistent field method (the Hartree- Fockmethod), which is based on the use of the variational principle. Inthis section we discuss the main ideas of this method.

The Hartree-Fock method is also based on the one-electron ap-proximation. The wave function of a complex atom is approximated

calf there are more than two electrons in unfilled shells, then several terms withthe same configuration and the same L and S can appear In this case the matrixof the perturbation We is quasi-diagonal, and to compute the corrections of the firstapproximation, the matrix has to be diagonalized


by a product T = 01 (6) 'On (fin) of one-electron functions (the iden-tity principle is not taken into account here), or by a determinant

V)i (6) ...

V)n(Sl On (W

or by a linear combination of such determinants. From the conditionof stationarity of the functional (HW, T) under the additional condi-tion ('I', T) = 1, we obtain a system of integro-differential equationsfor the one-particle functions 1(x),. . . , ,, (x).

Let us illustrate this approach with the example of the heliumatom, for which the Schrodinger operator has the form

1 H=--Al-1-A2- 2- 2 1 _H H 1() 2 2 r r -r J2

+ 2 + r'1 2 J2 12

where

Hz _.... -1Az- 2 i=1 2.2 ri

We look for an approximate wave function in the form

T(X(i), X(2)) _'01(x(1))'02(x(2)).

We remark that the condition

(%F,1R6

= I j (X1- -can be replaced by the two conditions

(2)JR3

1,41, (X) 12 dx = 1, I z(X)1l lx = 1,fR

which does not narrow down the class of functions to be varied. Thefunctional (H %P, %P) has the form

(3) (HW,W)= 'i(11H1b)1 dx 1}Ii4'2I2dx2)

R3 R3

+ 02H2 02 dX(2) kbI2dx'+ Iyl I I02I_dx1dX2.

. R3 .J 3 fRG r12

§ 51. The self-consistent field equations 225

Varying this functional with respect to the functions 01 and 02under the conditions (2) and using the Lagrange method of urndeter-mined multipliers, we have

(4)

10212

dX(2)

(IFR3 r12IV), I 2

1 =H202 + (11R3r72

x(1)V)2 F2'02

We have obtained as, ystem of nonlinear integro-differential equa-tions for the functions 1l (x) and /2 (x) . The equations (4) admit avery simple physical interpretation. For example, the first equationcan be regarded as the Schr6dinger equation for the first electronin the field of the nucleus and in the field created by the charge ofthe second electron. This charge is as if it were spread out over thevolume of the atom with density I02 (x(2) ),2 (recall that the chargeof the electron is e = 1), and the integral in the second term in (4)is the potential of this volume distribution of charge. We note thatt his potential is unknown and will be found in solving the system(4) Moreover, in contrast to the model in the previous section, eachelectron is in its own potential field, which depends on the state ofthe other electron.

The equations (4) were first proposed by Hartree, who wrote themstarting from the physical considerations given above. Fock estab-lished a connection between these equations and the variational prin-ciple, and he proposed a refinement of the Hartree method that takesinto account the identity principle. The Fock equations turn out tobe somewhat more complex, and some of the "potentials" (exchangepotentials) in them no longer admit such a simple physical interpre-tation. These potentials axise because of symmetry properties of thewave function.

In practical calculations the Fock method is usually employedfrom the very start to find the one-electron functions k (6 in theform of functions

qY l m (n) U , , , , 9 (s3) of the central

field.

The main steps of calculation consist in the following. An ex-pression is first found for the wave function of a definite term in the


form of a linear combination of determinants. Then an expression isformed for the functional (H'41, 'P). Finally, this functional is variedwith respect to the radial functions 1d (r) (the techniques of all theseoperations have been worked out in detail). The result is a system ofintegro-differential equations for functions of one variable. The num-ber of unknown functions in this system is equal to the number ofshells in the particular configuration of the atom.

Comparison of the computational results with experimental datashows that the accuracy of computation of the energy levels of lightatoms by the self-consistent field method is about 5%.

§ 52. Mendeleev's periodic system of theelements

The periodic law was discovered by Mendeleev in 1869 and is oneof the most important laws of nature. Mendeleev based this systemon the fact that if one arranges the elements in order of increasingatomic weight, then elements with closely related chemical and phys-ical properties occur periodically.

At the time of discovery of the periodic law only 63 elementswere known, the atomic weights of many elements had been deter-mined incorrectly, and Mendeleev had to change them.57 He left aseries of cells in the table empty, considering that they correspondedto yet undiscovered elements. He predicted the properties of threesuch elements with astonishing accuracy. Finally, in several casesMendeleev deviated from a strict arrangement of the elements in or-der of increasing atomic weight and introduced the concept of theatomic number Z.

The law discovered by Mendeleev was originally purely empiri-cal. In his time there was no explanation for the periodicity of theproperties of the elements; indeed, there could not be, since the elec-tron was discovered by Thomson in 1897, and the atomic nucleus byRutherford in 1910.

57For example, the atomic weight of Cerium was reckoned to be 92, but Mendeleevassigned the value 138 to it (the modern value is 140)

§ 52. Mendeleev's periodic system of the elements 227

An explanation of the periodic law in full scope is a complicatedproblem in quantum chemistry, but it is possible to understand the pe-riodicity of the properties of the elements already in the framework ofthe simplified model of the atom described in § 50. We recall that theeffective potential V (r) for an electron in an atom is not the Coulombpotential, and the eigenvalues of the one-electron Schrodinger oper-ator depend on the quantum numbers n and 1. Computations showthat for a typical effective potential of an atom, the eigenvalues E,,,increase in the order

Is, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, ... .

It is in this order that the electron shells of the atom fill up (in thissection we consider only the ground states of atoms). However, thisorder is not strict, since each element has its own effective potential,and for some elements slight deviations are possible in the order inwhich the shells are filled.

To understand the principle according to which the elements aredivided into periods, we consider the following feature of the d- and f -electrons which distinguishes them from the s- and p-electrons. Com-putations show that the density IO(X)12of the coordinate distributionfunction for d- and f -electrons is concentrated in regions of smallersize than for s- and p-electrons with close energies. This means thaton the average the d- and f -electrons are considerably closer to thenucleus than the s- and p-electrons. Therefore, the elements in whichthe d- and especially the f -shells are filling have similar chemicalproperties. (The chemical properties depend mainly on the states ofthe peripheral electrons of the atom. An explanation of this assertionis provided by the quantum theory of valence.)

The first element of each period of the Mendeleev table is anelement for which an s-shell begins to fill. All these elements, withthe exception of hydrogen, are alkali metals. The last element of eachperiod is an element for which a p-shell has been completely filled (anexception is the first period, whose elements, hydrogen and helium, donot have p-electrons). The last elements of the periods are the noblegases. The configurations of atoms of noble gases consist of filled


shells. It can he shown that the corresponding states transform underrot at ions according to the irreducible representat ion D0, that is, arespherically symmetric. The same property is enjoyed by the singlycharged positive ions of alkali metals. whose configurations coincidewith the configurations of the preceding noble gas. This is the reasonthat the optical properties of alkali metals are well described by themodel of a valence electron in a central field.

The sequence of electron shells is divided into periods as follows:

I Is. 2 elementsTT 2s. 2p. 8 elementsIII 3s. 3p. 8 elementsTV 4s. 3d, 4p, 18 elementsV 5s. 4d, 5p, 18 elementsVI 6s. 4 f , 5d, 6p, 32 elementsVII 7.s. 6d, 5f, ... ,

The number of a period is denoted by a roman numeral. The numberof elements in the period is indicated to the right. This number iseasy to determine if we recall that the number of one-electron stateswith quantum numbers n and l is equal to 2(21 + 1). For example, thenumber of elements in the sixth period is 2 + 14 + 10 + 6 = 32. Thenumber of a period coincides with the principal quantum number ofthe s- and p-shells which are being filled.

The elements containing filled d- and f -shells (or not containingsuch shells) are called elements of the principal groups. The elementsin which d- and f -shells are being filled are called elements of theintermediate groups. The elements of the intermediate groups haveclose chemical properties; this is especially true for the groups inwhich f -shells are filling (the shell 4f is filling in the group of rare-earth elements).

We remark that when d- and f -shells are filling, there is some-times a deviation from the order indicated above. For example, theconfigurations of the ground states of the atoms V, Cr, and Mn are(3d)3 (4s)2, (3d)54s, and (3d)5 (4s)2, respectively (we have not writtenout the configuration of the filled shells, which are the same for thesethree atoms: (ls)2(2s)2(2p)s(3s)2(3p)6). It turns out that the config-uration (3d) 54s for the atom Cr is energetically more advantageous

§ 52. Mendeleev's periodic system of the elements 229

than the configuration (3d)4(4S)2 which we would have assigned tochromium guided by our niodel. It is clear that this deviation can-not he explained in the framework of a model in which the effectivepotential is the same for all the electrons.

In conclusion we consider the variation in the binding energy`8 ofan electron in an atone. This parameter is closely connected with thechemical properties of the elements, and characterizes the capacityof an atom to "give up" an electron when entering into a cheniicalcombination.

We recall that for an atom the effective potential satisfies

U(r)^='-1/r asr ---roc

and

U(r)^-_'-Z/r asr --+ 0.Therefore, it is natural to expect (and this is confirmed by computa-tions) that with increasing Z the effective potential becomes stronger;that is, the corresponding one-electron energy levels for the next atomlie deeper than for the previous atom. Thus, the binding energy must:increase in a series of elements corresponding to the filling of someshell. However, upon passage to the filling of the next shell, the bind-ing energy can fall. An especially strong fall of the binding energy isobserved at the beginning of each period, when a new s-shell beginsto fill. Figure 15 shows experimental values of the binding energyfor various elements. The atomic number Z of an element is plottedalong the horizontal axis. From the figure it is clear that for eachperiod the alkali metals have the minimal value of the binding energyand the noble gases have the maximal value. The chemical inertnessof the noble gases is explained to a significant degree by the largevalue of the binding energy.

'"The binding energy is defined as the absolute value of the difference betweenthe energy of the ground state of an atom and the energy of the ground state of thecorresponding singly charged positive ion; in other words, the binding energy is equalto the minimal work required to remove an electron from the atom


Ne

Ar

Kr

Xc'

U

I I I I I I I I I I I J I I I I

1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Z

Figure 15

Appendix

Lagrangian Formulationof Classical Mechanics

For the convenience of the reader we present here, in a succinct form,the Lagrangian formulation of classical mechanics. For the detailedexposition of these standard facts we refer to the classic texts by R.Abraham and J. E. Marsden, Foundation of Mechanics (Addison-Wesley, Reading, MA, 1978; Amer. Math. Soc., Providence, RI,2008), and V. I. Arnold, Mathematical Methods of Classical Mechanics(Springer-Verlag, New York, 1989).

Classical mechanics describes systems of finitely many interact-ing particles. The position of a system in space is specified by thepositions of its particles and defines a point in a smooth, finite-dimensional manifold M, the configuration space of the system. Co-ordinates q = (q,. .. , qn) on M are called generalized coordinatesof the system, and the dimension n = dim M is called the numberof degrees of freedom. For the system of N interacting particles,considered in §1, the generalized coordinates are the Cartesian coor-dinates in R3N . The motion is described by the classical trajectoryq(t) = (qit),... , qn (t)) in the configuration space M, and the cor-responding time derivatives qz (t) are called generalized velocities andform a tangent vector q(t) = (41 (t),.. . , 4, (t)) to the manifold M atthe point q(t). The state of a classical system at any instant of time

231

232 Leon Takhtajan

is described by a point q in M and by a tangent vector v to M at,this point.

It is a basic fact, in classical mechanics that, the motion of a closedsystem is completely described by the Lagrangiarn function L - afunction L(q, q) of the generalized coordinates and velocities - a

smooth real-valued function on the tangent bundle T M to the man-ifold M. For mechanical systems describing the interacting particlesconsidered in §1, L = 7' - V - the difference between total kineticenergy and total potential energy of the system. The general principleof the least action was formulated by Hamilton (1834). It states thatthe classical trajectory which describes the motion between a pointq0 and at time. to and a point q1 at time t1 is the ext remurn of theaction functional

(1) S(q(t)) L(q, q)dt

on the space of all smooth paths q(t) such that q(t0) = q0 and q(t1) =q1

Finding extrenia of the action functional (1) is a classical problemof the calculus of variations.' Namely, let Sq(t) = (Sqi (t),. . . , hqn (1))

be the infinitesimal variation of the classical trajectory q (t) - - asmooth function on the interval [to, t1 ] satisfying Sq(to) = bq(t1) = 0.Since q(t) is the extremurn of the action functional S, its infinitesimalvariation at q (t) vanishes: that is, for all 6q(t)

(2) 6S = S(q(t) + 6q(t)) - S(q(t)) = 0

up to the terms quadratic in 6q(t). Keeping only linear terms in 6q(t)and 84(t) and using integration by parts, we can rewrite condition (2)

'For the introduction to this mathematical discipline, we refer the reader to theclassic textbook by I M Gelfand and S V Fomin. Calculus of Variations (Dover,New York, 2000)

Appendix 233

as

n ti(L(q(t) + 8q(t)(t) + bq(t)) - L(q(t)(t))dt0 = SS = 1: Ito

{_1

t1 aL aL

- i=1 J dtco 5qi

fti

n ti 8L d aL nOL

t'

E - aqi d t + ftto aqi dt aqi q=i t

i=1 i=1 0

The second sum in the last line vanishes due to the property 5q(to) =Sq (t i ) = 0, and we obtain that

nt1 OL d OL_ o idt = 0

i- f aqi dt O`1gi

for arbitrary smooth functions ft on the interval [to, tl ] which vanishat the endpoints. This implies that for each term in the sum, theintegrand is identically zero, and we obtain that the classical trajec-tories - extrema of the action functional (1) are described by theEuler-Lagrange equations

OL d OL(3)

Oq (T,dt= 0.

When L = T - V the difference between the kinetic energyand the potential energy -- the Euuler Lagrange equations reduce toNewton's equations. Thus for a particle in a force field with potential

V W, considered in §1. we havern, 2L = 2x _V(x).

Denoting byF=OL=_OVT

axthe force acting on the and using

aL = rn. we see that the.g particlea.+

Euler Lagrange equations (3) give Newton's equations (1) in §1:

rrty=F, x=v.

In general,OL

(4) P = 5-7 (q,q

234 Leon Takhtajan

are called generalized momenta, and introducing the generalized forcesF - aL

aqone can rewrite the Euler Lagrange equations (3) in Newton's form

. - Fp

The relation thus outlined between the Lagrangian formulationof classical mechanics and the Hamiltonian formulation, presented in§1, is given by the Legendre transform. Namely, suppose that insteadof the coordinates (q, 4) one can use the coordinates (q, p), where pare the generalized momenta given by (4). By the implicit functiontheorem, locally this amounts to the condition that the correspondingHessian matrix

a484-jil&M3 J _

is non degenerate. and such a Lagrangian function is called non-dege-nerate. The corresponding transformation (q, q) (q, p) is calledthe Legendre transformation.2 When the Legendre transformation isglobally defined and invertible, one can introduce the so-called Hainil-tonian function H (q, p) by

H(q, p) = p4 - L(q, ),

where 4 is expressed in terms of q and p by using (4) and the implicit function theorem. Then the Euler-Lagrange equations (3) areequivalent to Hamilton's equations

aH . aH= .

For the systems with the Lagrangian function L = T - V, derivationof these results is straightforward. Thus, for the Lagrangian function

1: 1: 1:i=1 i<j i=1

the corresponding Hamiltonian function is given by formula (5) in §1,and Hamilton's equations immediately follow from Newton's equations.

L a2L l

2Onc needs to impose further conditions to ensure that the Legendre transforma-tion is a di#feomorphism between tangent and cotangent bundles to M

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