1 Quantum Mechanics - Wiley-VCH · Quantum mechanics (QM), also called wave mechanics, is the modern theory of matter, of atoms, molecules, solids, and of the interaction of electromagnetic

1

1

Quantum Mechanics

Joachim Burgdorfer and Stefan Rotter

1.1 Introduction 31.2 Particle-Wave Duality and the Uncertainty Principle 41.3 Schrodinger Equation 61.4 Boundary Conditions and Quantization 81.5 Angular Momentum in Quantum Mechanics 91.6 Formalism of Quantum Mechanics 121.7 Solution of the Schrodinger Equation 161.7.1 Methods for Solving the Time-Dependent Schrodinger Equation 161.7.1.1 Time-Independent Hamiltonian 161.7.1.2 Time-Dependent Hamiltonian 171.7.2 Methods for Solving the Time-Independent Schrodinger Equation 191.7.2.1 Separation of Variables 191.7.2.2 Variational Methods 231.7.3 Perturbation Theory 241.7.3.1 Stationary Perturbation Theory 251.7.3.2 Time-Dependent Perturbation Theory 251.8 Quantum Scattering Theory 271.8.1 Born Approximation 281.8.2 Partial-Wave Method 291.8.3 Resonances 301.9 Semiclassical Mechanics 311.9.1 The WKB Approximation 311.9.2 The EBK Quantization 331.9.3 Gutzwiller Trace Formula 341.10 Conceptual Aspects of Quantum Mechanics 351.10.1 Quantum Mechanics and Physical Reality 361.10.2 Quantum Information 381.10.3 Decoherence and Measurement Process 391.11 Relativistic Wave Equations 411.11.1 The Klein–Gordon Equation 411.11.2 The Dirac Equation 42

Encyclopedia of Applied High Energy and Particle Physics. Edited by Reinhard StockCopyright 2009 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-40691-3

2 1 Quantum Mechanics

Glossary 43References 44Further Reading 45

3

1.1Introduction

Quantum mechanics (QM), also calledwave mechanics, is the modern theory ofmatter, of atoms, molecules, solids, and ofthe interaction of electromagnetic fieldswith matter. It supersedes the classicalmechanics embodied in Newton’s lawsand contains them as a limiting case (theso-called classical limit of QM). Its develop-ment was stimulated in the early part of thetwentieth century by the failure of classicalmechanics and classical electrodynamics toexplain important discoveries, such as thephotoelectric effect, the spectral density ofblackbody radiation, and the electromag-netic absorption and emission spectra ofatoms. QM has proven to be an extremelysuccessful theory, tested and confirmedto an outstanding degree of accuracy forphysical systems ranging in size from sub-atomic particles to macroscopic samplesof matter, such as superconductors andsuperfluids. Despite its undisputed successin the description of physical phenomena,QM has continued to raise many concep-tual and philosophical questions, in partdue to its counterintuitive character, defy-ing common-sense interpretation.

One distinguishes between nonrelativis-tic quantum mechanics and relativisticquantum mechanics depending on the

characteristic speed v of the constituentsof the system under study. The domainof nonrelativistic QM is characterized byspeeds that are small compared with thespeed of light:

v � c (1)

where c = 3 × 108 m s−1 is the speed oflight. For v approaching c, relativistic exten-sions are necessary. They are referred toas relativistic wave equations. Because of thepossibility of production and destruction ofparticles by conversion of relativistic energyE into mass M (or vice versa) according tothe relation (see chapter 2)

E = Mc2 (2)

a satisfactory formulation of relativisticquantum mechanics can only be devel-oped within the framework of quantumfield theory. This chapter focuses primar-ily on nonrelativistic quantum mechanics,its formalism and techniques as well asapplications to atomic, molecular, optical,and condensed-matter physics. A brief dis-cussion of extensions to relativistic waveequations is given at the end of the chapter.

The new ‘‘quantum world’’ will becontrasted with the old world of classicalmechanics. The bridge between the latterand the former is provided by the

Encyclopedia of Applied High Energy and Particle Physics. Edited by Reinhard StockCopyright 2009 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-40691-3


semiclassical mechanics, whose rigorousformulation and application have onlyrecently been developed.

1.2Particle-Wave Duality and the UncertaintyPrinciple

The starting point of modern quantummechanics is the hypothesis of the particle-wave duality put forward by de Broglie(1924): the motion of each particle of massM is associated with a wave (‘‘matter wave’’)of wavelength

λ = h

p(3a)

where h = 6.63 × 10−34 J s is Planck’s con-stant, discovered by M. Planck in 1900 inhis investigation of blackbody radiation,and p = Mv is the classical momentum ofthe particle. Similarly, the frequency ν ofthe wave is associated with the mechan-ical energy of the particle through theexpression

ν = E

h(3b)

These relations, ascribing wave propertiesto particles, are the logical complementsto postulates put forward 20 years earlierby Einstein (1905), assigning the charac-teristics of particles (‘‘photons,’’ particlesof zero rest mass) to electromagnetic radi-ation. In fact, Einstein employed, in hisexplanation of the ‘‘photoelectric effect,’’the expression E = hν (Eq. 3b) to find theenergy E of each photon associated withelectromagnetic radiation of frequency ν.Photons can eject electrons from the atomsof a material provided that the energyexceeds the binding energy of the elec-tron. This particle-wave duality of QM

unifies the description of distinct entitiesof classical physics: mechanics and elec-tromagnetism. Particles, the fundamentalconstituents of matter, possess, in additionto mechanical properties such as momen-tum and angular momentum, wave charac-ter. Conversely, electromagnetic radiation,successfully described by Maxwell’s the-ory of electromagnetic waves, takes on,under appropriate conditions, the prop-erty of quasiparticles, the photons, carryingmechanical energy and momentum. Theparticle and wave character are comple-mentary to each other. An object behaveseither like a particle or like a wave.

The hypothesis of particle-wave dualityhas been experimentally verified in remark-able detail. Interference of matter wavesdue to the indistinguishability of pathswas demonstrated first for electrons (Davis-son and Germer, 1927), later for neutrons,and, most recently, for neutral atoms. Theparticle-wave duality of electromagneticradiation and its complementarity havebeen explored through modern versionsof Young’s double-slit experiment usingstate-of-the-art laser technology and fastswitches. Hellmuth and colleagues (1987)succeeded, for example, in switching fromparticle to wave character and vice versaeven after the photon has passed throughthe slit (in the experiment, a beam split-ter) by activating a switch (a Pockels cell)in one of the two optical paths of theinterferometer. By use of the switch, theinterference patterns are made to appearor disappear, as expected for a wavelike orparticlelike behavior of the photon, respec-tively. This ‘‘delayed choice’’ experimentnot only demonstrates the particle-waveduality for photons but also elucidatesimportant conceptual aspects of the nonlo-cality of quantum mechanics. The decisionas to whether the photon will behavelike a particle or a wave need not be

1.2 Particle-Wave Duality and the Uncertainty Principle 5

made at the double slit itself but can bespatially separated and temporarily post-poned.

The consequences of the wave-particleduality for the description of physical phe-nomena are far reaching and multifaceted.One immediate consequence that repre-sents the most radical departure fromclassical physics is the uncertainty princi-ple (Heisenberg, 1927). One deeply rootednotion of classical physics is that all dynam-ical observables can be measured, at leastin principle, with arbitrary accuracy. Uncer-tainties in the measurement are a matterof experimental imperfection that, in prin-ciple, can be overcome. The wave natureof particles in QM imposes, however, fun-damental limitations on the simultaneousaccuracy of measurements of dynamicalvariables that cannot be overcome, no mat-ter how much the measurement processcan be improved. This uncertainty princi-ple is an immediate consequence of thewave behavior of matter. Consider a waveof the form

ψ(x) = Aei(kx−ωt) (4)

traveling along the x coordinate(Figure 1.1). Equation (4) is called aplane wave because the wave fronts formplanes perpendicular to the direction of

propagation. The physical interpretation ofthe amplitude A was provided by Born(1926) in terms of a probability density

P(x) = |ψ(x)|2 = |A|2 (5)

for finding the particle at the coordinate x.The matter wave (Eq. 4) is further charac-terized by the wave number k = 2π/λ= p/�and the angular frequency ω = 2πv = E/�.The ‘‘rationalized’’ Planck’s constant � =h/2π = 1.05 × 10−34 J s is frequently usedin QM.

The uncertainty principle is inherent toall wave phenomena. Piano tuners haveexploited it for centuries. They sound avibrating tuning fork of standard frequencyin unison with a piano note of the samenominal frequency and listen to a beat tonebetween the struck tune and the tuningfork. For a fork frequency of ν = 440 Hzand a string frequency of ν′ = 441 Hz,one beat tone will be heard per second(δν = ν′ − ν = 1 Hz). The goal of the tuneris to reduce the number of beats as muchas possible. To achieve an accuracy of δν =0.010.01 Hz, the tuner has to wait for atleast about δt = 100 s (in fact, only a fractionof this time since the frequency mismatchcan be detected prior to completion ofone beat period) to be sure no beat had

00

1

2

3

4−10

010

Traveling matter waveReψ(x, t)

t x Fig. 1.1 Matter wave travelingalong the x direction; the figureshows the real part of ψ(x),Eq. (4) (after Brandt and Dah-men, 1989).


occurred. Piano tuning therefore relies onthe frequency–time uncertainty

�ν �t � 1 (6)

Measurement of the frequency with infi-nite accuracy (�ν → 0) requires an infinitetime period of measurement (�t → ∞), orequivalently, it is impossible within anyfinite period of time to determine the fre-quency of the string exactly since timeand frequency are complementary variables.Similarly, the variables characterizing thematter wave (Eq. 4) are complementary:

�k �x ≥ 1

�ω �t ≥ 1 (7)

These relations, well known from classicaltheory of waves and vibrations, haveunexpected and profound consequenceswhen combined with the de Brogliehypothesis (Eq. 3) for quantum objects:

�p �x ≥ �

�E �t ≥ � (8)

The momentum and the position ofa particle, its energy and the time,and more generally, any other pair ofcomplementary variables, such as angularmomentum and angle, are no longersimultaneously measurable with infiniteprecision. The challenge to the classicallaws of physics becomes obvious whenone contemplates the fact that Newton’sequation of motion uses the simultaneousknowledge of the position and of thechange of momentum.

Since � is extremely small, the uncer-tainty principle went unnoticed duringthe era of classical physics for hun-dreds of years and is of little conse-quence for everyday life. Its far-reaching

implications become obvious only in themicroscopic world of atoms, molecules,electrons, nuclei, and elementary parti-cles. More generally, whenever the sizeof an object or mean distance between con-stituents becomes comparable to the deBroglie wavelength λ, the wave nature ofmatter comes into play. The size of λ alsoprovides the key for quantum effects inmacroscopic systems such as superfluids,superconductors, and solids.

1.3Schrodinger Equation

The Schrodinger equation for matter wavesψ describes the dynamics of quantumparticles. In the realm of quantum physics,it plays a role of similar importance to thatwhich Newton’s equation of motion doesfor classical particles. As with Newton’slaws, the Schrodinger equation cannot berigorously derived from some underlying,more fundamental principles. Its form canbe made plausible, however, by combiningthe Hamiltonian function of classicalmechanics,

H = T + V = E (9)

which equals the mechanical energy, withthe de Broglie hypothesis of matter waves(Eq. 4). The potential energy is denoted byV and the kinetic energy by T = p2/2M.Formally multiplying the wave functionψ(x,t) by Eq. (9) yields

Hψ(x, t) = p2

2Mψ(x, t) + Vψ(x, t) (10)

In order to connect the de Broglie relationsfor energy and momentum appearing inthe arguments of the plane wave (Eq. 4)with the energy and momentum in the

1.3 Schrodinger Equation 7

Hamiltonian function, H and p in Eq. (10)must be taken as differential operators,

H ←→ i�∂

∂t

p ←→ �

i

∂

∂x(11)

This substitution into Eq. (10) leads to thetime-dependent Schrodinger equation

i�∂

∂tψ(x, t) = Hψ(x, t)

=(

− �2

2M

∂2

∂x2+ V(x)

)× ψ(x, t) (12)

which describes the time evolution of thematter wave. For physical systems thatare not explicitly time dependent, i.e., thathave time-independent potentials V(x), theenergy is conserved and i� ∂/∂t can bereplaced by E, giving the time-independentSchrodinger equation

Eψ(x) =(

− �2

2M

∂2

∂x2+ V(x)

)ψ(x) (13)

If no interaction potential is present(V(x) = 0), Eq. (12) is referred to as thefree-particle Schrodinger equation. The planewaves of Eq. (4) are solutions of thefree-particle Schrodinger equation witheigenenergies E = (�k)2/2M.

The correspondence principle, first for-mulated by Bohr, provides a guide toquantizing a mechanical system. It statesthat in the limit of large quantum num-bers or small de Broglie wavelength, thequantum-mechanical result should be thesame, or nearly the same, as the classi-cal result. One consequence of this pos-tulate is that the quantum-mechanicalHamiltonian operator in the Schrodingerequation (Eq. 12) should be derived fromits corresponding classical analog, the

classical Hamiltonian function (Eq. 9),through the replacement of the classi-cal observable by the operator equivalents(Eq. 11).

The replacement of dynamical variablesby differential operators leads to thenoncommutativity of pairs of conjugatevariables in QM, for example,

[x, px ] ≡ x�

i

∂

∂x− �

i

∂

∂xx = i� (14)

with similar relations for other comple-mentary pairs, such as Cartesian coordi-nates of position and momentum (y, py),(z, pz), or energy and time. It is the noncom-mutativity that provides the basis for a for-mal proof of the uncertainty principle, Eq.(8). The Schrodinger equation (Eqs. 12 and13) is a linear, homogeneous, and (in gen-eral) partial differential equation. One con-sequence of this fact is that matter wavesψ satisfy the superposition principle. If ψ1

and ψ2 are two acceptable solutions of theSchrodinger equation, called states, a linearcombination

ψ = c1ψ1 + c2ψ2 (15)

is an acceptable solution, as well. The phys-ical implication of Eq. (15) is that the quan-tum particle can be in a ‘‘coherent superpo-sition of states’’ or ‘‘coherent state,’’ a factthat leads to many strange features of QM,such as destructive and constructive inter-ference, the collapse of the wave function,and the revival of wave packets.

A solution of a linear, homogeneousdifferential equation is determined onlyup to an overall amplitude factor, whichis arbitrary. The probability interpreta-tion of the matter wave (Eq. 5), however,removes this arbitrariness. The probabil-ity for finding the particle somewhere


is unity,

∫dxP(x) =

∫dx|ψ(x)|2 = 1 (16)

Equation (16) and its generalization tothree dimensions are referred to as thenormalization condition on the wave func-tion and eliminate the arbitrariness of themodulus of the wave function. A wavefunction that can be normalized accordingto Eq. (16) is called square integrable. Theonly arbitrariness remaining is the overall‘‘phase factor’’ of the wave function, eiφ ,i.e., a complex number of modulus 1.

1.4Boundary Conditions and Quantization

In order to constitute a physically accept-able solution of the Schrodinger equation,i.e., to represent a state, ψ must satisfyappropriate boundary conditions. Whiletheir detailed forms depend on the coor-dinates and symmetry of the problemat hand, their choice is always dictatedby the requirement that the wave func-tion be normalizable and unique and thatphysical observables take on only realvalues (more precisely, real expectationvalues) despite the fact that the wave func-tion itself can be complex. Take, as anexample, the Schrodinger equation for arotator with a moment of inertia I, con-strained to rotate about the z axis. Theclassical Hamiltonian function H = L2

z/2I,with Lz the z component of the angularmomentum,

Lz = xpy − ypx = �

i

∂

∂φ(17)

gives rise, according to the corre-spondence principle, to the following

time-independent Schrodinger equation:

−�2

2I

∂

∂φ2ψ(φ) = Eψ(φ) (18)

where φ is the angle of rotation about the zaxis. Among all solutions of the form

ψ(φ) = Aei(mφ+φ0) (19)

only those satisfying the periodic boundaryconditions

ψ(φ + 2π ) = ψ(φ) (20)

are admissible in order to yield a uniquelydefined single-valued function ψ(φ). (Theimplicit assumption is that the rotatoris spinless, as discussed below.) Themagnetic quantum number m thereforetakes only integer values m =−∞, . . . , −1,0, 1, . . . ∞. Accordingly, the z componentof the angular momentum

Lzψ(φ) = m�ψ(φ) (21)

takes on only discrete values of multiplesof the rationalized Planck’s constant �.The real number on the right-hand sidethat results from the operation of adifferential operator on the wave functionis called the eigenvalue of the operator.The wave functions satisfying equationsthat are of the form of Eqs (18) or (21)are called eigenfunctions. The eigenvaluesand eigenfunctions contain all relevantinformation about the physical system; inparticular, the eigenvalues represent themeasurable quantities of the systems.

The amplitude of the eigenfunctionfollows from the normalization condition

∫ 2π

0dφ|ψ(φ)|2 = 1 (22)

1.5 Angular Momentum in Quantum Mechanics 9

as A = (2π )−1/2. The eigenvalues of E inEq. (18) are Em = �

2m2/2I. Just like fora vibrating string, it is the impositionof boundary conditions (in this case,periodic boundary conditions) that leads to‘‘quantization,’’ that is, the selection of aninfinite but discrete set of modes of matterwaves, the eigenstates of matter, each ofwhich is characterized by certain quantumnumbers.

Different eigenstates may have the sameenergy eigenvalues. One speaks then ofdegeneracy. In the present case, eigenstateswith positive and negative quantum num-bers ±m of Lz are degenerate since theHamiltonian depends only on the squareof Lz. Degeneracies reflect the underlyingsymmetries of the Hamiltonian physicalsystem. Reflection of coordinates at thex–z plane, (y →−y, py → −py and there-fore Lz → −Lz) is a symmetry operation,i.e., it leaves the Hamiltonian invariant. Inmore general cases, the underlying symme-try may be dynamical rather than geometricin origin.

1.5Angular Momentum in Quantum Mechanics

The quantization of the projection of theangular momentum, Lz, is called directionalquantization. The Hamiltonian functionfor the interaction of an atom having amagnetic momentµ = γ L with an externalmagnetic field B reads

H = −µ · B = −γ L · B (23)

The proportionality constant γ betweenthe magnetic moment and the angularmomentum is called the gyromagnetic ratio.According to the correspondence principle,the Schrodinger equation for the magneticmoment in a field with field lines oriented

along the z axis becomes

−γ BLzψ(φ) = Emψ(φ) (24)

The eigenenergies

Em = −γ B�m (25)

are quantized and depend linearly on themagnetic quantum number. Dependingon the sign of γ , states with positive mfor γ > 0, or negative m for γ < 0, corre-spond to states of lower energy and arepreferentially occupied when the atomicmagnetic moment is in thermal equilib-rium. This leads to an alignment of themagnetic moments of the atoms along oneparticular direction and to paramagnetismof matter when its constituents carry a netmagnetic moment.

The force an atom experiences asit passes through an inhomogeneousmagnetic field depends on the gradientof the magnetic field, ∇B. Because ofthe directional quantization of the angularmomentum, a beam of particles passingthrough an inhomogeneous magneticfield will be deflected into a set ofdiscrete directions determined by thequantum number m. A deflection patternconsisting of a few spots rather thana continuous distribution of a beamof silver atoms was observed by Sternand Gerlach (1921, 1922) and providedthe first direct evidence of directionalquantization.

In addition to the projection of the angu-lar momentum along one particular axis,for example, the z axis, the total angu-lar momentum L plays an important rolein QM. Here, one encounters a concep-tual difficulty, the noncommutativity ofdifferent angular momentum components.


Application of Eq. (14) leads to

[Lx , Ly

] = i�Lz,[Ly, Lz

] = i�Lx[Lz, Lx

] = i�Ly (26)

As noncommutativity is the formal expres-sion of the uncertainty principle, theimplication of Eq. (26) is that differentcomponents of the angular momentumcannot be simultaneously determined witharbitrary accuracy. The maximum num-ber of independent quantities formed bycomponents of the angular momentumthat have simultaneously ‘‘sharply’’ definedeigenvalues is two. Specifically,

[L2, Lz] = 0 (27)

with L2 = L2x + L2

y + L2z being the square of

the total angular momentum. Therefore,one projection of the angular momentumvector (conventionally, one chooses the zprojection) and the L2 operator can havesimultaneously well-defined eigenvalues,unaffected by the uncertainty principle.The differential operator of L2 in sphericalcoordinates gives rise to the followingeigenvalue equation:

L2Y(θ , φ) = −�2[

1

sin θ

∂

∂θ

(sin θ

∂

∂θ

)

+ 1

sin2 θ

∂2

∂φ2

]Y(θ , φ)

= �2l(l + 1)Y(θ , φ) (28)

The solutions of Eq. (28), Yml (θ , ψ), are

called spherical harmonics. The superscriptdenotes the magnetic quantum numberof the z component, as above, whilethe quantum number l = 0, 1, . . . ∞ iscalled the angular momentum quantumnumber. Since |Lz | ≤ | L|, we have therestriction |m | ≤ l, that is, for given angularmomentum quantum number l, the zcomponent varies between −l and l : m =

− l, . . . , −1, 0, 1, . . . l. The total number ofquantized projections of Lz is 2l + 1. Theeigenvalue of the L2 operator is, however,�

2l(l + 1) and not �2l2. The latter can be

understood in terms of the uncertaintyprinciple and can be illustrated with thehelp of a vector model (Figure 1.2). Becauseof the uncertainty principle, the projectionsLy and Lx do not take a well-defined valueand cannot be made to be exactly zero,when Lz possesses a well-defined quantumnumber. Therefore, the z projection m�

will always be smaller than the magnitudeof L = �

√l(l + 1) in order to accommodate

the nonzero fluctuations in the remainingcomponents.

The spherical harmonics are given interms of associated Legendre functionsPm

l (cos θ ) by

Yml (θ , φ) = (−1)m

[(2l + 1)

4π

(l − m)!

(l + m)!

]1/2

×Pml (cos θ )eimφ (29)

The eigenfunctions of Lz (Eq. 19) can berecognized as one factor in the expressionfor Ym

l . The probability densities |Yml |2 of

the first few eigenfunctions are representedby the polar plots in Figure 1.3.

In addition to the orbital angularmomentum operator L, which is thequantum-mechanical counterpart to theclassical angular momentum, another typeof angular momentum, the spin angu-lar momentum, plays an important rolefor which, however, no classical analogexists. The spin is referred to as an inter-nal degree of freedom. The Stern–Gerlachexperiment discussed above provided themajor clue for the existence of the electronspin (Goudsmit and Uhlenbeck, 1925). Theorbital angular momentum quantizationallows only for an odd number of dis-crete values of Lz (2l + 1 = 1, 3, 5, . . .)and therefore an odd number of spots in

1.5 Angular Momentum in Quantum Mechanics 11

0

Lz

−h

−2h

2h

h

h6

h6

h6

h6

h6

L

Fig. 1.2 Vector model of angular momentum Lfor quantum number l = 2 and |L| = √

6 �.

the deflection pattern. Stern and Gerlachobserved, however, two discrete directionsof deflection (see schematic illustration inFigure 1.14). The spin hypothesis solvedthis mystery. The outermost electron ofa silver atom has a total orbital angularmomentum of zero but has a spin angularmomentum of s = 1

2 . The number of dif-ferent projections of the spin is therefore2s + 1 = 2 in accordance with the num-ber of components into which the beamwas magnetically split. Within nonrela-tivistic quantum mechanics, the spin mustbe introduced as an additional degree offreedom on empirical grounds. A concep-tually satisfactory description is provided

by relativistic quantum mechanics (seeSection 1.11). The identification of the spinas an angular momentum rests on thevalidity of the commutation rules (Eq. 26),

[Sx, Sy

] = i�Sz,[Sy, Sz

] = i�Sx ,[Sz, Sx

] = i�Sy (30)

identical to those of the orbital angularmomentum. Any set of the componentsof a vector operator satisfying Eq. (26)has a spectrum of eigenvalues (quantumnumbers) of an angular momentum withquantum numbers j = 0, 1

2 , 1, . . . and m =−j, −j + 1 . . . , + j. The quantum numberj is an integer in systems with an even


θ

= 0

= 1

= 2

m = 0

m = 0 m = ±1

m = 0 m = ±1 m = ±2

Fig. 1.3 Polar plots of the probabilitydistributions |Ym

l (θ , φ)|2 of angular momentumeigenfunctions.

number of spin 12 particles (including

zero) and half-integer in systems withan odd number of spin 1

2 particles. TheHamiltonian for the interaction of anelectron spin with an external magneticfield is given in direct analogy to Eq. (23)by

H = −γ gS · B (31)

The only difference is the ‘‘anomalous’’gyromagnetic ratio represented by thefactor g. An approximate value of g can bededuced from the relativistic wave equation(Dirac’s equation) for the electron (seeSection 1.11).

1.6Formalism of Quantum Mechanics

The quantum-mechanical description out-lined in the previous section with the helpof specific examples can be expanded toa general formalism of quantum mechan-ics. The starting point is the identifica-tion of wave functions with vectors in

an infinite-dimensional vector space, theHilbert space,

ψ −→ |ψ〉 (32)

with the vectors represented by a ‘‘ket.’’The ket notation was first introduced byDirac (1930). The coherent superpositionillustrated by Eq. (15) can be recognized asvector addition in Hilbert space. Physicalobservables such as the Hamiltonian or theangular momentum can be represented bylinear and Hermitian operators. Linearityof an operator B means

B(c1|ψ1〉 + c2|ψ2〉) = c1B|ψ1〉 + c2B|ψ2〉(33)

for all complex numbers c1, c2 and allvectors |ψ1〉, |ψ2〉. Hermitian operators aredefined as those for which

B† = B (34)

that is, the operator is equal to its Hermi-tian adjoint. The Hermiticity assures thatthe eigenvalue equation

B|ψ〉 = λ|ψ〉 (35)

1.6 Formalism of Quantum Mechanics 13

possesses real eigenvalues λ, which can beidentified with measurable quantities. Thetime-independent Schrodinger equation,Eq. (13), is of the form in Eq. (35) withB being the Hamiltonian operator H andλ being the eigenenergy E. The result ofa measurement of a physical observableyields an eigenvalue λ of the correspondingoperator, B.

The noncommutativity of operators (Eq.14) corresponds to the well-known non-commutativity of matrices in linear alge-bra. The overlap of wave functions can beidentified with the scalar product in Hilbertspace:

〈φ|ψ〉 ≡∫

d3r φ∗(r)ψ(r) (36)

The adjoint of a vector in Hilbert space〈ψ | is called a bra and the scalarproduct a bracket (Dirac, 1930). Thenormalization requirement imposed onthe wave function can be expressed interms of the scalar product as 〈ψ | ψ 〉 = 1.The requirement of normalizability (Eq.22) implies that the Hilbert space isspanned by square-integrable functions.Since eigenstates of Hermitian operatorsbelonging to different eigenvalues areorthogonal to each other, each physicaldynamical variable possesses a completeorthonormal set of eigenstates:

〈ψi|ψj〉 = δij ={

1 i = j

0 i �= j(37)

The operator B is a diagonal matrix ina basis consisting of its eigenstates, the‘‘eigenbasis’’ |ψ i 〉 with i = 1, . . . , ∞. In itseigenbasis, the operator B possesses thefollowing ‘‘spectral representation’’:

B =∑

i

λi|ψi〉〈ψi| (38)

containing only its eigenstates {ψ i} and itseigenvalues λi. Two operators possess thesame eigenbasis when they commute witheach other. Physical observables whoseoperators commute with each other arecalled compatible observables. A state of aphysical system is therefore specified tothe maximum extent possible, consistentwith the uncertainty principle, by onecommon eigenstate of a maximum setof compatible observables. For example,the set {L2,Lz} is the maximum set ofcompatible observables for the angularmomentum with the common eigenstate|lm〉.

The probability amplitude for finding thesystem in a particular state |ψ1〉 is given bythe projection of the state vector onto thisstate, i.e., by the scalar product 〈ψ1 | ψ〉.The corresponding probability is

P(ψ1) = |〈ψ1|ψ〉|2 (39)

Formally, Eq. (39) can be viewed as theexpectation value of the projection operatorP =|ψ1 〉〈ψ1| in the state |ψ〉. The processof a measurement is therefore associatedwith a projection in Hilbert space. Thisprojection is sometimes referred to asthe collapse of the wave function. Theprocess of measurement thus influencesthe quantum system to be measured.

The state of a physical system is oftennot completely specified by a ‘‘pure’’state described by a single Dirac ket|ψ〉 or a coherent superposition of kets∼c1 |ψ1 〉 +c2 | ψ2〉. The limited informa-tion available may only allow us to specify astatistical mixture of states. Such a mixtureis described by the statistical operator ordensity operator

ρ =∑

i

Pi|ψi〉〈ψi| (40)


with Pi denoting the probability for thesystem to be found in a particular state|ψ i〉. The result of the measurement of anobservable B is then given by the statisticalexpectation value

〈B〉 = Tr(ρB) =∑

i

〈ψi|B|ψi〉Pi (41)

i.e., by the diagonal elements of the opera-tor B, each weighted with the probability tofind the system in the corresponding state.The preferred set of states, |ψ i〉, for whichthe density matrix is ‘‘diagonal,’’ that is, ofthe form in Eq. (40), is sometimes referredto as the set of pointer states.

In light of the expression for theprobability as a projection, Eq. (39), wecan rewrite Eq. (5) as

P(r) = |ψ(r)|2 = |〈r|ψ〉|2 (42)

where |r〉 is formally treated as a vector(‘‘ket’’) in Hilbert space. This definitionleads to the difficulty that |r〉 cannot bedefined as a square-integrable function.Nevertheless, it can be consistently incor-porated into the formalism by employingthe concept of generalized functions (‘‘dis-tributions’’; see Lighthill, 1958; Schwartz,1965). One defines a scalar productthrough

〈r|r′〉 = δ(r − r′)

= δ(x − x′)δ(y − y′)δ(z − z′) (43)

where the right-hand side is called Dirac’sδ function. The δ function is zero almosteverywhere:

δ(x) ={

0, x �= 0

∞, x = 0(44)

with the strength of the singularity suchthat the area underneath the peak is∫

dxδ(x) = 1 (45)

While the δ function is not an ordinary(Riemann integrable) function, it can beapproximated as a limit of Riemannintegrable functions, for example,

limε−→0

1

π

ε

x2 + ε2= δ(x) (46)

Equation (46) satisfies Eq. (45) providedone evaluates the integral first and takesthe limit ε → 0 afterward. As the order ofthe operations of integration and taking thelimit is not interchangeable, Eq. (46) is saidto converge ‘‘weakly’’ toward a δ functionin the limit ε → 0. δ functions play animportant role in the quantum theory ofscattering.

The time evolution of the Hilbert-spacevectors is given by Schrodinger’s equationwritten in Dirac’s ket notation:

i�∂

∂t|ψ〉 = H|ψ〉 (47)

For systems with a classical analog, theHamiltonian operator is given, accordingto the correspondence principle, by theclassical Hamiltonian function H(r, p)upon treating the canonical variables (r, p)as noncommuting operators (see Eq. 14).There are, however, additional dynamicalvariables in quantum mechanics, such asspin, for which no classical counterpartsexist.

A vector in Hilbert space can be expandedin terms of any basis (a ‘‘representation’’),

|ψ〉 =∑

i

ci|φi〉 =∑

i

〈φi|ψ〉|φi〉 (48)

where ci = 〈φi | ψ〉 are the expansion coef-ficients of the ket |ψ〉 in the basis

1.6 Formalism of Quantum Mechanics 15

{|φi 〉}. Switching from one representationto another |ψ 〉→ |ψ ′ 〉 = U | ψ〉 amountsto a unitary transformation, U, sincethe norm of a state vector must beconserved,

〈ψ ′|ψ ′〉 = 〈ψ|U†U|ψ〉= 〈ψ|ψ〉 (49)

or U†U = 1. Transformations among rep-resentations play an important role in QM.While the maximum number of compat-ible observables is determined by com-mutation rules, the choice of observablesthat are taken to be diagonal or to have‘‘good’’ quantum numbers depends onthe specific problem under consideration.Unitary transformations provide the toolto switch from a basis within which oneset of observables is diagonal to another.The transformation law for state vectors|ψ ′ 〉 = U | ψ〉 implies the transformationlaw for operators

A′ = UAU† (50)

The latter follows from the invarianceof the expectation value 〈ψ ′ | A′ |ψ ′ 〉 =〈ψ | U†A′U |ψ〉 under unitary transfor-mations.

Classical Hamiltonian functions aswell as quantum-mechanical Hamiltonianoperators possess discrete symmetries. Forexample, the Hamiltonian function of theharmonic oscillator,

H(x, p) = p2

2M+ 1

2 Mω2x2 (51)

is invariant under reflection of all spatialcoordinates (x →−x, p →−p) since Hdepends only on the square of eachcanonical variable. In classical mechanics,this symmetry is of little consequencesince the solution of Newton’s equation of

motion depends on the initial conditionsx(t = t0), p(t = t0), which are not subjectto any symmetry constraint. They may‘‘break’’ the symmetry and usually do so. Inquantum mechanics, however, the parityof a state vector becomes a dynamicalobservable.

We can define a parity operator p through

pH(x, p)p† = H(−x, −p) (52)

Since p2 is the unit operator (two successivereflections restore the original function),we have p = p† = p−1. Eigenfunctions ofp, ψ(x) = 〈x|ψ〉, can now be characterizedby their parity:

〈x|p|ψ〉 = 〈−x|ψ〉 = ±〈x|ψ〉 (53)

The ± sign in Eq. (53) is referred toas indicating positive or negative parity. Awave function cannot always be assigned awell-defined parity quantum number ±1.However, when p commutes with H,

pH(x, p)p = H(x, p) (54)

there exists a common eigenbasis inwhich both the Hamiltonian operatorand the parity operator are diagonal and,consequently, ψ is an eigenstate of p with awell-defined parity. A similar argument canbe developed for symmetry with respectto the time-reversal operator T : t → −t,p →−p, r → r.

An additional discrete symmetry, whichis of fundamental importance for thequantum mechanics of identical particles,is the permutation symmetry. Considera wave function describing the stateof two identical particles ψ(r1, r2). Thepermutation operator p12 exchanges thetwo particles in the wave function:

p12ψ(r1, r2) = ψ(r2, r1) (55)


The spin-statistics postulate of quantummechanics makes a general statementconcerning the permutation symmetry. Amany-body system consisting of identicalparticles of integer spin (‘‘bosons’’) pos-sesses wave functions that are even underall interchanges of two particles, whereasthe wave function is odd for half-integerspin particles (‘‘fermions’’). The eigenvalueof pij, the operator of permutation for anytwo particles of an N-body system, is +1 forbosons and −1 for fermions. Wave func-tions are totally symmetric under particleexchange for bosons and totally antisym-metric for fermions. An immediate con-sequence is that two fermions cannot bein identically the same state for whichall quantum numbers agree (the so-calledPauli exclusion principle) since the anti-symmetric wave function would vanishidentically.

1.7Solution of the Schrodinger Equation

1.7.1Methods for Solving the Time-DependentSchrodinger Equation

1.7.1.1 Time-Independent HamiltonianIf the Hamilton operator is not explic-itly time dependent, i.e., the totalenergy is conserved, the time-dependentSchrodinger equation can be reduced to thetime-independent Schrodinger equation.Equation (47) can be formally solved by

|ψ(t)〉 = U(t, t0)|ψ(t0)〉 (56)

where U(t, t0) is called the time-evolutionoperator, a unitary transformation describ-ing the translation of the system in time.Substitution of Eq. (56) into Eq. (47) shows

that

U(t, t0) = exp[−iH(t − t0)

�

](57)

where the exponential function of the oper-ator H is defined through its power-seriesexpansion

U(t, t0) = 1 − it − t0

�H

−1

2

(t − t0

�

)2

H · H + · · · (58)

If one is able to solve the time-independentSchrodinger equation

H|ψn〉 = En|ψn〉 (59)

where n labels the eigenstate, one can cal-culate the time evolution of the eigenstate:

U(t, t0)|ψn〉 = e−iH(t−t0)/�|ψn〉= e−iEn(t−t0)/�|ψn〉 (60)

using the definition in Eq. (58). The phasefactor in Eq. (60) is called the dynamicalphase. The explicit solution for an arbitrarystate can then be found by expanding|ψ(t0)〉 in terms of eigenstates of theHamilton operator:

|ψ(t0)〉 =∑

n

|ψn〉〈ψn|ψ(t0)〉

|ψ(t)〉 = U(t, t0)|ψ(t0)〉=

∑n

e−iEn(t−t0)/�|ψn〉

×〈ψn|ψ(t0)〉 (61)

Hence, for time-independent systems, thesolution of the energy eigenvalue problem(Eq. 59) is sufficient to determine the timeevolution for any initial state.

1.7 Solution of the Schrodinger Equation 17

1.7.1.2 Time-Dependent HamiltonianFor a time-dependent Hamiltonian H(t),the solution of the time-evolution operatorU in terms of the exponential functionEq. (57) must be redefined in order totake into account that the Hamiltonianoperator at different instances of time maynot commute, [H(t), H(t′)] �= 0. We have

U(t, t0) = 1 − i

�

∫ t

t0

dt′H(t′)

− 1�2

∫ t

t0

dt′H(t′)

×∫ t′

t0

dt′′H(t′′) · · ·

= T exp(− i

�

∫ t

t0

H(t′)dt′)

(62)

Equation (62) is called the time-orderedexponential function and T is calledthe time-ordering operator. In two lim-iting cases, the evolution operator fortime-dependent Hamiltonian systems canbe easily calculated by reducing the prob-lem to the time-independent Hamiltonian.One is the adiabatic limit. If the temporalchanges in H(t) are very slow comparedwith the changes in the dynamical phasesof Eq. (60) for all t, the U matrix is diagonalin the slowly changing eigenbasis of H(t)at each instant t,

|ψn(t)〉= eiγn(t)exp[− i

�

∫ t

t0

dt′En(t′)]

|φn,t〉(63)

The dynamical phase factor is determinedby En(t′), the energy eigenvalue of thestationary Schrodinger equation for theHamiltonian operator H(t′) at time t′. In Eq.(63), the state |ψn(t)〉 denotes the evolvedstate and the state |φn,t〉 represents thestationary eigenstate for the HamiltonianH = H(t) at the time t, where the time playsthe role of a parameter.

The additional phase γ n(t) is calledBerry’s phase (Berry, 1984) or the geometricphase, since it records the information onthe excursion of the Hamiltonian H(t′) inparameter space in the case of adiabaticevolution. The role of γ n(t) is to relatethe phases of the adiabatic wave function|ψn(t)〉 at different points in time. Insertionof Eq. (63) into Schrodinger’s equationyields the equation that defines the realphase function γ n(t),

γn(t) − i

⟨φn,t

∣∣∣∣ d

dtφn,t

⟩= 0 (64)

The analogy to the parallel transport of vec-tors on a sphere (Figure 1.4) can give aglimpse of the concept underlying Berry’sphase. Consider a quantum-mechanicalstate to be represented by a vector at pointA. By infinitely slowly (adiabatically) vary-ing an external parameter, for example, thedirection of the magnetic field, one can‘‘transport’’ state vectors successively frompoint A to B, then to C, and finally backto A. After completion of the closed loop,the vector does not coincide with its orig-inal orientation but is rotated by an angleγ that is proportional to the solid anglesubtended by the loop. This angle givesrise to a geometric phase, exp(iγ ), whichdepends on the geometry of a loop, butunlike the dynamical phase, Eq. (60), it isindependent of the time it takes to com-plete the transport of the state along theclosed loop. If one is now able to form in aphysical system a coherent superposition oftwo states, one accumulating the geomet-ric phase along the adiabatic change of theexternal field while the other is subject to aconstant field, an interference pattern willarise. The Berry phase has been observedin interference experiments involving neu-tron spin rotation, in molecules, in lighttransmission through twisted optical-fibercables, and in the quantum Hall effect.


C

A

γ B

Fig. 1.4 Parallel transport of a vector on a spherealong a closed loop leads to a rotation of the vector.The vector’s orientation relative to its local environmentremains unchanged during parallel transport. The angleof rotation is related to Berry’s phase.

In the opposite limit, where a suddenchange of the Hamiltonian from H(t) =H0 for all t < 0 to another HamiltonianH(t) = H0 for all t > 0 occurs, no dynamicalor geometric phase can be accumulatedduring the instantaneous change of theHamiltonian at t = 0. The initial eigenstateof H0, ψn, in which the system is preparedat t = 0, evolves for t > 0 as

|ψn(t)〉 =∑

n′exp

(− i εn′ t

�

)|ψn′ 〉〈ψn′ |ψn〉

(65)In the sudden limit, the matrix U is non-diagonal. The eigenstate belonging to the‘‘old’’ Hamiltonian forms a coherent super-position of eigenstates |ψn〉 of the newHamiltonian. The ‘‘sudden’’ formation ofcoherent states causes ‘‘quantum beats,’’observable oscillations in amplitudes andprobabilities.

Consider, for example, an atom that isexcited at the time t = 0 to a state that is acoherent superposition of a state |ψ1〉, withenergy E1, and a state |ψ2〉, with energy E2,that is, |ψ 〉 = c1 | ψ1 〉 + c2 | ψ2〉. At a laterpoint in time, t, the state acquires a phasefactor as a function of time according toEq. (65):

|ψ(t)〉 = c1e−iE1t/�|ψ1〉 + c2e−iE2t/�|ψ2〉(66)

The projection onto a final state |f 〉representing, for example, the atom inthe ground state and the emission of aphoton,

Pf (t) = |〈 f |ψ(t)〉|2= |c1|2|〈 f |ψ1〉|2 + |c2|2|〈 f |ψ2〉|2

+ c1c∗2exp

[it

�(E2 − E1)t

]×〈 f |ψ1〉〈 f |ψ2〉∗+ complex conjugate (67)

will display oscillations (‘‘quantum beats’’)due to the time dependence of the relativephases of the two states. These quantumbeats due to dynamical phases have beenobserved, for example, in the Lyman-α (n =2 → n = 1) radiation of hydrogen excited inthe n = 2 level subsequent to fast collisionswith a foil (Figure 1.5).

The standard computational method ofsolving the time-dependent Schrodingerequation for time-dependent interactionsconsists of expanding a trial wave function|ψ(t)〉 in terms of a conveniently chosentruncated time-independent basis of finitesize:

|ψ(t)〉 =N∑

n=1

an(t)|ψn〉 (68)


0 10 20 30 40 50 mm

Inte

nsity

Fig. 1.5 Quantum beats in the Lyman-α photonintensity of hydrogen after collision with a carbonfoil: •, experimental data from Andra (1974); – ,theory from Burgdorfer (1981). The theory curve isshifted relative to the data for clarity.

with unknown, time-dependent coeffi-cients an(t). Insertion of Eq. (68) into Eq.(47) leads to a system of coupled first-orderdifferential equations, which can be writtenin matrix notation as

i�a = H(t) a(t) (69)

using the vector notation a = (a1, . . . an)with H the matrix of the Hamiltonianin the basis {|ψn 〉}, Hn, n′ = 〈ψn | H | ψn′ 〉.The solution of Eq. (69) is thereby reducedto a standard problem of numerical mathe-matics, the integration of a set of N coupledfirst-order differential equations with vari-able (time-dependent) coefficients.

1.7.2Methods for Solving the Time-IndependentSchrodinger Equation

The solution of the stationary Schrodingerequation, Eq. (59), proceeds mostly inthe {|r 〉} representation, i.e., using wavefunctions ψ(r). Standard methods includethe method of separation of variables andvariational methods.

1.7.2.1 Separation of VariablesThe energy eigenvalue problem, Eq. (59),requires the solution of a second-order par-tial differential equation. It can be reducedto an ordinary second-order differentialequation if the method of separation ofvariables can be applied, i.e., if the wavefunction can be written in a suitable coor-dinate system (q1, q2, q3) in factorized form

ψ(r) = f (q1)g(q2)h(q3) (70)

with obvious generalizations for an arbi-trary number of degrees of freedom.Equation (70) applies only to a very spe-cial but important class of problems forwhich the Hamiltonian possesses as manyconstants of motion or compatible observ-ables as there are degrees of freedom (inour example, three). In this case, each fac-tor function in Eq. (70) is associated withone ‘‘good’’ quantum number belongingto one compatible observable. This classof Hamiltonians permits an exact solutionof the eigenvalue problem and is calledseparable (or integrable).


Consider the example of the isotropicthree-dimensional harmonic oscillator.The Hamiltonian is

H = p2x

2M+ p2

y

2M+ p2

z

2M+ 1

2 Mω2(x2 + y2 + z2) (71)

In this case, one suitable set of coordinatesfor the separation are the Cartesiancoordinates (q1 = x, q2 = y, q3 = z). Eachof the three factor functions of Eq. (70)satisfies a one-dimensional Schrodingerequation of the form

(− �

2

2M

∂2

∂x2+ 1

2 Mω2x2

)f (x) = Exf (x)

(72)and E = Ex + Ey + Ez. The simplest solu-tion of Eq. (72) satisfying the boundaryconditions f (x →±∞) → 0 for a normal-izable wave function is a Gaussian wavefunction of the form

f (x) =(

Mω

π�

)1/4

e−(Mω/2�)x2(73)

It corresponds to the lowest possible energyeigenstate, with Ex = 1

2 �ω. Ground-statewave functions are, in general, nodefree, i.e., without a zero at any finite x.The Schrodinger equation possesses aninfinite number of admissible solutionswith an increasing number n = 0, 1, . . . ofnodes of the wave function and increasingeigenenergies

Ex = �ω(n + 1

2

)(74)

The fact that wave functions belonging tohigher states of excitation have an increas-ing number of zeros can be understood as aconsequence of the orthogonality require-ment, Eq. (37). Only through the additionalchange of sign of the wave function can

all contributions in the orthogonality inte-gral (Eq. 36) be made to cancel. The wavefunctions for excited states of a harmonicoscillator are given in terms of Hermitepolynomials Hn by

fn(x) = (2nn!)−1/2(

Mω

�π

)1/4

× e−(Mω/2π )x2Hn

(√Mω

�x

)(75)

Examples of the first few eigenstates areshown in Figure 1.6 With increasing n,the probability density |f n(x) | 2 begins toresemble the classical probability densityfor finding the oscillating particle near thecoordinate x, Pcl(x) (Figure 1.7). The latteris proportional to the time the particlespends near x or inversely proportionalto its speed:

Pc1(x) =(ω

π

) (2Ex/M − ω2x2)−1/2

(76)

The increasing similarity between the clas-sical and quantal probability densities asn → ∞ is at the core of the correspon-dence principle. This example highlightsthe nonuniformity of the convergence tothe classical limit: the quantum probabil-ity density oscillates increasingly rapidlyaround the classical value with decreasingde Broglie wavelengths λ (compared withthe size of the classically allowed region).Furthermore, the wave function penetratesthe classically forbidden region (‘‘tunnel-ing’’). While the amplitude decreases expo-nentially, ψ(x) possesses significant non-vanishing values outside the domain ofclassically allowed trajectories of the sameenergy over a distance of the order of thede Broglie wavelength λ. Only upon aver-aging over regions of the size of λ canthe classical–quantum correspondence be


0

2

1

3

4

5

E

−4 −2 0 2 4x

ψn(X)Fig. 1.6 Wave functions f n(x) for the first fiveeigenstates of the harmonic oscillator.

Turning point a Turning point b

Classical probability density

Quantum probability density

n = 39

0.4

0.3

0.2

0.1

0.0

−0.1−10 0 10

x

Fig. 1.7 Probability density|f n(x)|2 of a harmonic oscillatorin n = 39 compared with theclassical probability densityPcl(x) = (ω/π)(2E/M − ω2x2)−1/2

indicated by the dashed line.Classical turning points aredenoted by a and b.

recovered. As λ tends to zero in the classi-cal limit, classical dynamics emerges. Itis the smallness of λ that renders themacroscopic world as being classical forall practical purposes.

The complete wave function for thethree-dimensional isotropic oscillator (Eq.71) consists of a product of three wavefunctions each of which is of the form ofEq. (75),

ψnxnynz (x, y, z) = fnx (x) fny (y) fnz (z) (77)

with eigenenergies

Enxnynz = �ω(nx + ny + nz + 3

2

)(78)

The energy depends only on the sum of thethree quantum numbers,

n = nx + ny + nz (79)

but not on the individual quantum num-bers nx , ny, and nz, separately. Differentcombinations of quantum numbers yieldthe same energy; the spectrum is thereforedegenerate. The degeneracy factor gn, thenumber of different states having the sameenergy En, is

gn = (n + 1)(n + 2)

2(80)


Only the ground state (n = 0) is nonde-generate, a feature generic to quantumsystems and the origin of the third law(Nernst’s theorem) of statistical mechan-ics. The degeneracy signals that alternativecomplete sets of commuting observables(instead of the energies in each of the threeCartesian coordinates Ex, Ey, and Ez) andcorresponding alternate sets of coordinatesshould exist in which the problem is sep-arable. One such set of coordinates is thespherical coordinates.

Consider the example of the hydrogenatom. One of the most profound earlysuccesses of quantum mechanics was theinterpretation of the line spectrum emittedfrom the simplest atom, the hydrogenatom, consisting of an electron and aproton. The Schrodinger equation (in thecenter-of-mass frame) for the electroninteracting with the proton through aCoulomb potential V(r) = −e2/4πε0r readsas

[−�2∇2

2me− e2

4πε0r

]ψ(r) = Eψ(r) (81)

where me = 9.1 × 10−31 kg is the reducedmass of the electron, r is the vector thatpoints from the nucleus to the electron,e = 1.6 × 10−19 C is the electric charge ofthe nucleus, and −e is the charge of theelectron. If we express the Laplace operator∇2 (defined as ∇2 = ∂2/∂x2 + ∂2/∂y2 +∂2/∂z2) in polar spherical coordinates (r,θ , φ), the Schrodinger equation becomes

[−�2

2me

(∂2

∂r2+ 2

r

∂

∂r

)+ L2

2mer2

− e2

4πε0r

]ψ(r, θ , φ) = Eψ(r, θ , φ) (82)

Since the angular coordinates enter onlythrough the square of the angular momen-tum operator (see Eq. 28), the Schrodinger

equation is separable in spherical coordi-nates and the wave function can be writ-ten as a product ψ(r, θ , φ) = Rnl(r)Ym

e (θ , φ)with one factor being the spherical har-monics Ym

l and the Rnl(r) being the radialwave function satisfying the second-orderdifferential equation

[−�2

2me

(∂2

∂r2+ 2

r

∂

∂r

)+ �

2l(l + 1)

2mer2

− e2

4πε0r

]Rnl(r) = ERnl(r) (83)

The negative-energy eigenvalues of boundstates can be found as

En = − mee4

2�2n2(4πε0)2= − EH

2n2(84)

where EH = mee4/�2(4πε0)2 is called aHartree or an atomic unit of energy. Theenergy depends only on the ‘‘principal’’quantum number n, being independent ofthe angular momentum

l = 0, 1, . . . , n − 1 (85)

and of the magnetic quantum number m.The m independence can be found for anyisotropic potential V(|r|) that depends onlyon the distance of the interacting particles.The degeneracy in l is a characteristicfeature of the Coulomb potential (and of theisotropic oscillator), and it reflects the factthat the Schrodinger equation is separablenot only in polar spherical coordinates butalso in parabolic coordinates. The Coulombproblem possesses a dynamical symmetrythat allows the choice of alternativecomplete sets of compatible observables.The separability in spherical coordinatesis related to the set {H, L2, Lz} whilethe separability in parabolic coordinatesis related to the set {H, Lz, Az} whereA is the Runge–Lenz (or perihelion)vector, which, as is well known from


Nucleus

L×Av

v

L

Electron

A

Fig. 1.8 The Runge–Lenz vector A and the angularmomentum L for the classical orbit in an attractive−1/r potential.

astronomy, is a constant of motion for a1/r potential (Figure 1.8). The emissionspectrum follows from Eq. (84) as

ν = (En − En0 )

h

= EH

2h

(1

n20

− 1

n2

)(86)

with n0 = 1 and n = 2, 3, . . . giving rise tothe Lyman series of spectral lines, n0 = 2and n = 3, 4, . . . giving rise to the Balmerseries, and so on. The radial probabilitydensities r2Rnl(r)2 are shown in Figure 1.9

1.7.2.2 Variational MethodsFor all but the few separable Hamilto-nians, the time-independent Schrodingerequation is only approximately and numer-ically solvable. The accuracy of approxi-mation can, however, be high. It mayonly be limited by the machine preci-sion of modern computers. The mostsuccessful and widely used method ingenerating accurate numerical solutionsis the variational method. The solutionof the Schrodinger equation can be rec-ognized as a solution to the variationalproblem of finding an extremum (in prac-tice, a minimum) of the energy functional〈E 〉 = 〈ψ | H |ψ 〉 /〈ψ |ψ〉 by varying |ψ〉until 〈E〉 reaches a stationary point at anapproximate eigenvalue. The wave functionat the minimum is an approximate solutionto the Schrodinger equation. In practice,one chooses functional forms called trialfunctions ψ(r) that depend linearly and/or

nonlinearly on parameters (c1, . . . ,ck). Thetrial function is optimized by minimizingthe energy in the k-dimensional parameterspace according to

d

dci〈E〉 = 0 (1≤i≤k) (87)

Provided the trial function is sufficientlyflexible to approximate a complete set ofstates in Hilbert space, ψ becomes exact inthe limit as k →∞. The ground state of thehelium atom consisting of an α particle asa nucleus (nuclear charge number Z = 2)and two electrons provides one of themost striking illustrations of the success ofthe variational method. The Hamiltonian(neglecting the motion of the nucleus)

H = − �2

2me(∇2

1 + ∇22 ) − Ze2

4πε0r1

− Ze2

4πε0r2+ e2

4πε0|r1 − r2| (88)

is nonseparable, an analytic solution isunknown, and no perturbation solutionis possible since all interactions areof comparable strength. The variationalmethod pioneered by Hylleraas (1929)and further developed by Pekeris (1958,1959) achieves, however, a virtually exactresult for the ground-state energy, E =2.903 724 377 03 . . . EH , for which the accu-racy is no longer limited by the methodof solution but rather by the approxi-mate nature of the Hamiltonian itself(for example, the neglect of the finite


0

−0.1

−0.2

−0.3

−0.4

−0.5

0

−0.05

−0.1

0

−0.05

−0.1

0 5 10 15 20 25r

r2R2n,l(r), l = 1

r2R2n,l(r), l = 0

r2R2n,l(r), l = 2E

E

E Fig. 1.9 Radial densities r2Rnl(r) for n = 1, 2,3 and l = 0, 1, and 2 of the hydrogen atom.

size of the nucleus and of relativisticeffects).

1.7.3Perturbation Theory

A less accurate method, which is, however,easier to implement, is perturbation theory.The starting point is the assumption thatthe Hamiltonian H can be decomposedinto a ‘‘simple,’’ separable (or, at least,numerically solvable) part H0 plus a

perturbation gV ,

H = H0 + gV (89)

which is nonseparable, but ‘‘small’’ suchthat it causes only a small modificationof the spectrum and of the eigenstates ofH0. The control parameter g (0 ≤ g ≤ 1)is used as a convenient formal device forcontrolling the strength of the perturbationand keeping track of the order of theperturbation. At the end of the calculation,g can be set equal to unity if V itself is small.


Two cases can be distinguished: stationaryperturbation theory and time-dependentperturbation theory.

1.7.3.1 Stationary Perturbation TheoryThe Rayleigh–Schrodinger perturbationtheory involves the expansion of both theperturbed energy eigenvalue E0(g) and theperturbed eigenstate |ψ0(g)〉 in powers ofthe order parameter g (or, equivalently, inpowers of V),

E0(g) = E0 + gE(1) + g2E(2) + · · ·|ψ0(g)〉 = |ψ0〉 + g|ψ (1)〉 + g2|ψ (2)〉 + · · ·

(90)Inserting Eq. (90) into the Schrodingerequation

H(g)|ψ0(g)〉 = E0(g)|ψ0(g)〉 (91)

yields recursion relations ordered inascending powers of g for the successivecorrections to E0(g) and |ψ0(g)〉. In prac-tice, the first few terms are most important:Thus the first-order corrections for a non-degenerate eigenvalue E0 are

E(1) = 〈ψ0|V |ψ0〉 (92)

and

|ψ (1)〉 =∑n �=0

|ψn〉〈ψn|V |ψ0〉E0 − En

(93)

where the sum over {|ψn 〉 n�=0} extendsover the complete set of eigenstates ofthe unperturbed Hamiltonian excludingthe state |ψ0〉 for which the perturbationexpansion is performed. The second-ordercorrection to the eigenenergy is

E(2) = 〈ψ0|V |ψ (1)〉 =∑n �=0

|〈ψ0|V |ψn〉|2E0 − En

(94)

Alternative perturbation series can bedevised. For example, in the Brillouin–Wigner expansion, the perturbed E0(g),rather than the unperturbed energy E0,appears in the energy denominator of Eq.(94). The energy is therefore only implicitlygiven by this equation.

1.7.3.2 Time-Dependent PerturbationTheoryTime-dependent perturbation theory dealswith the iterative solution of the system ofcoupled equations, Eq. (69), representingthe time-dependent Schrodinger equation.One assumes again that the Hamiltonianconsists of an unperturbed part H0

and a time-dependent perturbation V(t).Expansion in the basis of eigenstates of theunperturbed Hamiltonian (Eq. 68) yields amatrix H consisting of a diagonal matrix〈i | H0 | j 〉 = Ejδij and a time-dependentperturbation matrix Vij = 〈i | V(t) | j〉.

The dynamical phases associated withthe unperturbed evolution can be easilyincluded in the perturbative solution of Eq.(69) by making the phase transformation

aj(t) = cj(t)e−iEjt/� (95)

This transformation, often referred toas the transformation to the interactionrepresentation, removes the diagonal matrix〈i | H0 | j〉 from Eq. (69), resulting in thenew system of equations

i�d

dtC(t) = VI(t) C(t) (96)

with matrix elements

〈i|VI(t)|j〉 = exp(

i

�t(Ei − Ej)

)〈i|V(t)|j〉

(97)and C = (c1,c2, . . . ,cN). The perturbativesolution of Eq. (96) proceeds now byassuming that initially, say at t = 0, the


f (ω, t )

t2

−6πt

−2πt

2πt

4πt

6πt

−4πt

0 ω

Fig. 1.10 The function f (ω, t) as afunction of ω.

system is prepared in the unperturbed state|j〉 with unit amplitude, cj(0) = 1. Sincethe perturbation is assumed to be weak,the system at a later time t will still beapproximately in this state. However, smalladmixtures of other states will develop withamplitudes

ci(t) = 1

i�

∫ t

0dt′〈i|VI(t′)|j〉cj(t′)

≈ 1

i�

∫ t

0dt′〈i|VI(t′)|j〉 (98)

Therefore, to first order, the time-dependent perturbation induces during thetime interval [0, t] transitions |j 〉→ | i〉 withprobabilities

Pi(t) = |ci(t)|2 = 1

�2

∣∣∣∣∫ t

0dt′〈i|VI(t′)|j〉

∣∣∣∣2

(99)In the special case that V is timeindependent, Eq. (99) gives

Pi(t) = 1

�2|〈i|V |j〉|2 |eiωijt − 1|2

ω2ij

= 1

�2|〈i|V |j〉|2 4 sin2(ωijt/2)

ω2ij

(100)

with ωij = (εi − εj)/�. Plotted as a functionof ω, the function f (ω,t) = 4 sin2(ωt/2)/ω2

(Figure 1.10) is strongly peaked aroundω = 0. The peak height is t2 and thewidth of the central peak is given by�ω � 2π/t. In other words, f describesthe energy–time uncertainty governingthe transition: The longer the availabletime, the more narrowly peaked is thetransition probability around ωi − ωj = 0,that is, for transition into states that arealmost degenerate with the initial state.In the limit t →∞, the time derivative ofEq. (100), that is, the transition probabilityper unit time, or transition rate w, has thewell-defined limit

w = d

dtPi(t)t−→∞ = 2π

�|〈i|V |j〉|2δ(εi − εj)

(101)where the representation of the δ function,πδ(ω) = limt→∞ (sinωt/ω), has been used.Equation (101) is often called Fermi’s goldenrule and provides a simple, versatile toolto calculate transition rates to first order.The physical meaning of the δ function issuch that in the limit t → ∞ only thosetransitions take place for which energy isconserved, as we would have expected since

1.8 Quantum Scattering Theory 27

the interaction in Eq. (100) was assumed tobe time-independent (after being switchedon at time t = 0). For example, in scatteringprocesses, many final states with energiesvery close to that of the initial states areaccessible. The δ function is then evaluatedwithin the sum (or integral) over these finalstates.

1.8Quantum Scattering Theory

Scattering theory plays, within the frame-work of quantum mechanics, a specialand important role. Its special role isderived from the fact that scattering dealswith unbound physical systems: two (ormore) particles approach each other frominfinity, scatter each other, and finallyseparate to an infinite distance. Obvi-ously, one cannot impose upon the wavefunction the boundary condition ψ(r =|r2 − r2 | →∞) → 0 since the probabilityof finding the scattered particles at largeinterparticle distances r is, in fact, nonva-nishing. This has profound consequences:energies are not quantized since the corre-sponding boundary condition is missing.The energy spectrum consists of a con-tinuum instead of a set of discrete states.The states describing the unbound motion,the ‘‘scattering states,’’ are not squareintegrable and do not, strictly speaking,belong to the Hilbert space. They canonly be orthonormalized in terms of a δ

function,

〈ψE |ψE′ 〉 = δ(E − E′) (102)

Mathematically, they belong to the spaceof distributions and can be associated withstates in Hilbert space only upon form-ing ‘‘wave packets,’’ that is, by convolutingthe scattering state with a well localized

function. Nevertheless, most of the timeone can manipulate scattering states usingthe same rules valid for Hilbert-space vec-tors (see Section 1.6) except for replacingthe ordinary orthonormalization condition,Eq. (37), by the δ normalization condition,Eq. (102).

The strategy for solving the Schrodingerequation, which governs both boundedand unbounded motion, is different forscattering states. Rather than determiningthe energy eigenvalues or the entirewave function (which is, in many cases,of no interest), one aims primarily atdetermining the asymptotic behavior ofthe wave function emerging from theinteraction region, which determines theflux of scattered particles. The centralquantity of both classical and quantumscattering is the cross section σ , defined as

σ =number of scattered particles

per unit timenumber of incident particles

per unit area and per unit time(103)

A cross section σ has the dimension ofan area and can be visualized as an effec-tive area that the target presents to theincoming projectile. Using the probabilis-tic description of quantum mechanics, onecan replace in the definition Eq. (103) allparticle currents by corresponding prob-ability currents. Accordingly, σ can beexpressed in terms of the transition ratew as

σ = w

j(104)

where j = |ψ | 2v represents the incomingprobability current density. For an incom-ing plane particle wave (Eq. 4), j = �k/M(with M the reduced mass and the ampli-tude A set equal to unity). Calculations of


total cross sections σ proceed typically byfirst calculating differential cross sectionsfor scattering into a particular group of finalstates and then summing over all possiblefinal states. For elastic scattering of parti-cles with incident wave vectors kin pointingin the z direction, a group of final statesis represented by a wave vector kout of theoutgoing particles pointing in a solid anglecentered around the spherical angles (θ , φ),

dσ

d�= M

�kinwkin−→kout (105)

Equation (105) gives the differential crosssection per unit solid angle.

1.8.1Born Approximation

The evaluation of dσ /d� is most straight-forward in the Born approximation. Thefirst Born approximation is nothing butthe application of Fermi’s ‘‘golden rule,’’Eq. (101), to scattering:

dσ

d�= 2πM

�2kin

∫ ∞

0

k2out

(2π )3dkout (106)

|〈kout|V |kin〉|2 × δ(Ein − Eout)

where the integral is taken over themagnitude of kout but not over thesolid angle and where k2

outdkout d�/(2π )3

represents the number of final statesavailable for the scattered particles inthe interval dkoutd�. Since k2

outdkout =(M/�

2)koutdEout, we have

dσ

d�= M2

�4(2π )2|〈kout|V |kin〉|2 (107)

with |kout | = | kin|. The first Born approx-imation to elastic scattering, Eq. (107), isonly valid if the potential V is sufficientlyweak to allow for a perturbative treatment

and vanishes sufficiently rapidly at largedistances for the matter wave to become aplane wave. In such cases, the matrix ele-ment in Eq. (107) can be expressed as theFourier component of the potential,

〈kout|V |kin〉 =∫

d3r eir · (kout−kin)V(r)

(108)

For a Coulomb potential V(r) = Z1Z2e2/4πε0r representing two charged particleswith charges Z1e or Z2e scattering eachother, Eq. (107) yields – coincidentally –the Rutherford cross section(

dσ

d�

)R

= Z21Z2

2e4

16E2(4πε0)2 sin4(θ/2)(109)

This expression provided essential cluesas to an internal structure and chargedistribution of atoms. Two remarkablefeatures should be noted: Prior to theadvent of quantum theory, Rutherford(1911) used classical mechanics instead ofquantum mechanics to derive Eq. (109). Hecould do so since, for Coulomb scattering,classical and quantum dynamics happen toagree. Keeping in mind that the classicallimit entails the limit � → 0, the QMand classical cross sections should agree,since (dσ /d�)R does not depend on �.The equivalence of the classical and theQM Rutherford cross sections is due tothe long range (in fact, infinite range)of the Coulomb potential. In this case,the de Broglie wavelength of the particlewave is negligibly small compared to the‘‘size’’ of the target at all energies. Asecond coincidence lies in the fact thatthe first Born approximation yields theexact quantum result even though theBorn approximation is, strictly speaking,not even applicable in the case of aCoulomb potential. The infinite range ofthe Coulomb potential distorts the matter

1.8 Quantum Scattering Theory 29

wave at arbitrarily large distances so thatit never becomes the plane wave that isassumed in Eq. (108). It turns out thatall high-order corrections to the Bornapproximation can be summed up to givea phase factor that drops out from theexpression for the cross section.

1.8.2Partial-Wave Method

For elastic scattering by a spherically sym-metric potential, that is, a potential thatdepends only on the distance between theparticles, V(r) = V(|r|), phase-shift analy-sis provides a versatile nonperturbativemethod for calculating differential andintegral cross sections. For each angularmomentum l (which is a good quantumnumber for a spherically symmetric poten-tial), the phase shift for a radial wavefunction u(r) = rR(r) can be determinedfrom the radial Schrodinger equation

(− �

2

2M

d2

dr2+ �

2

2M

l(l + 1)

r2+ V(r)

)u(r)

= Eu(r) (110)

The idea underlying the phase-shift anal-ysis can be best illustrated for the phaseshift δl for l = 0. In absence of the scat-tering potential V(r), Eq. (110) representsa one-dimensional Schrodinger equationof a free particle in the radial coordinate,

− �2

2M

d2

dr2u(r) = Eu(r) (111)

whose solution is u(r) = sin(kr) with k2 =2ME/�2. Since the matter wave cannotreach the unphysical region of negativer, we have to impose the boundary condi-tion u(0) = 0, which excludes the alternativesolution ∼cos kr. In the presence of the

potential V(r), the wave function will bemodified within the region of the nonvan-ishing potential. At large distances r → ∞,however, when V(r) → 0, the wave func-tion will again become u(r) = sin(kr+δ0)since in this region it must satisfy Eq.(111). The only effect that the potentialcan have on the scattered wave functionat large distances is to introduce a phaseshift δ0 relative to the unperturbedwave.

Simple arguments give a clue as to theorigin and the sign of the phase shift. Foran attractive potential (negative V), thewave number k(r) = √

2M(E − V(r))/�2

increases locally and the de Brogliewavelength λ(r) = 2π/k(r) becomes shortercompared with that of a free particle ofthe same energy. The phase of the matterwave is therefore advanced compared withthe wave of a free particle and δ0 ispositive (Figure 1.11). Similarly, if Vis a repulsive potential, λ increases inthe interaction region causing a phasedelay and, hence, a negative phase shiftδ0. Including all angular momenta, orall ‘‘partial waves,’’ the differential crosssection can be expressed in terms of thepartial-wave phase shifts δl as

dσ

d�= 1

k2

∣∣∣∣∣∞∑

l=0

(2l + 1)eiδl sin δlPl(cos θ )

∣∣∣∣∣2

(112)A particularly simple expression can befound for the total cross section:

σ = 4π

k2

∞∑l=0

(l + 1) sin2 δl (113)

For short-ranged potentials, δl decreaseswith increasing l such that only a smallnumber of terms must be included inthis sum. The scattering phase shifts δl(E)depend sensitively on the energy E of theincident particle.


0

1

−1

0 10 20r

Potential

Free–particle wave

Scattered wave

δ0

Fig. 1.11 Comparison betweenthe scattered wave in anattractive potential and afree-particle wave (the scatteringphase δ0 is shown modulo π).

1.8.3Resonances

Often, one phase shift δl(E) increases byπ within a narrow energy range. Thiscorresponds to a phase shift of one-halfof a wave length, or to the addition ofone node to the wave function. A suddenincrease of the scattering phase by π isa signature of a ‘‘resonance’’ in the crosssection. A resonance is characterized by asharp maximum in the partial-wave crosssection at the energy Eres = �

2k2res/2M:

(σl)max =(

4π

k2res

)(2l + 1) (114)

where the phase shift passes through π/2(or an odd multiple thereof). Resonancesappear when the scattering potential allowsfor the existence of ‘‘virtual levels’’ orquasi-bound states inside the potentialwell, whose probability density can leakout by tunneling through the potentialwell. The additional node that the scatteringwave function acquires as the energy passesthrough Eres reflects the presence of aquasi-bound state. An isolated resonancecan be parametrized by a formula dueto Breit and Wigner. One assumes that

the rapid change of the scattering phasenear the resonance can be modeled by

δl(E) � tan−1 �

2(Eres − E)(115)

and that the energy dependence of the non-resonant (smooth part) of the phase shiftcan be neglected. The term � denotes thewidth over which the phase jump occurs.Inserting Eq. (115) into the lth term of Eq.(113) yields, for the shape of the resonantcross section, the Breit–Wigner formula

σl(E) = (σl)res�2/4

(Eres − E)2 + �2/4(116)

The shapes of resonant cross sectionsbecome more complicated when inter-ferences between the resonant and thenonresonant parts of the scattering ampli-tude come into play. The cross sectioncan then exhibit a variety of shapes(Beutler–Fano profile; Fano, 1961), includ-ing asymmetric peaks or dips (‘‘win-dow resonance’’) depending on the rel-ative phases of the amplitudes. In thelimit that a large number of reso-nances with energies Eres,i (i = 1, . . . ,N)

1.9 Semiclassical Mechanics 31

are situated within the width � (‘‘over-lapping resonances’’), the cross sectiondisplays irregular fluctuations (‘‘Ericsonfluctuations’’) (see Chapter 10). Theseplay an important role in scatteringin complex systems, for example, innuclear compound resonances (Ericsonand Meyer-Kuckuk, 1966) as well as inscattering in chaotic systems Burgdorfer etal., 1995).

1.9Semiclassical Mechanics

While the correspondence principle pos-tulates that in the limit of large quan-tum numbers (or, equivalently, � → 0),quantum mechanics converges to classi-cal mechanics, the approach to this limitis highly nonuniform and complex. Thedescription of dynamical systems in thelimit of small but nonzero � is calledsemiclassical mechanics. Its importance isderived not only from conceptual but alsofrom practical aspects since a calculationof highly excited states with large quan-tum numbers involved becomes difficultbecause of the rapid oscillations of thewave function (see Figure 1.7). Semi-classical mechanics has been developedonly recently. The old Bohr–Sommerfeldquantization rules that preceded the devel-opment of quantum mechanics are nowunderstood as an approximate form ofsemiclassical mechanics applicable to sep-arable (or integrable) systems.

The wave function of the time-indepen-dent Schrodinger equation for a singleparticle,

Eψ(r) = −(

�2

2M

)∇2ψ(r) + V(r)ψ(r)

(117)

can be written as

ψ(r) = A exp[

iS(r)

�

](118)

where the unknown function S(r) replacesthe exponent k · r of a plane wave (Eq.4) for a free particle in the presence ofthe potential. The idea underlying thesemiclassical approximation is that in thelimit � → 0, or likewise, in the limit of smallde Broglie wavelength λ→ 0 (comparedwith the distance over which the potentialchanges), ψ(r) should behave like a planewave with, however, a position-dependentde Broglie wavelength. Inserting Eq. (118)into Eq. (117) leads to

E = 1

2M[∇S(r)]2 + V(r) − i�

2M∇2S(r)

(119)with p =∇S. Equation (119) is wellsuited for taking the classical limit � → 0and yields the classical Hamilton–Jacobiequation. In this limit, S is the action orHamilton’s principal function. Quantumcorrections enter through the term linearin �. Approximate solutions of Eq. (119)can be found by expanding S in ascendingpowers of �. The apparent simplicity of theclassical limit is deceiving: the principalfunction S appears inside the exponent Eq.(118) with the prefactor �

−1 and this causesthe wave function to oscillate infinitelyrapidly as � → 0.

1.9.1The WKB Approximation

For problems that depend only on onecoordinate, retention of the first two termsof the expansion of Eq. (119),

S(x) = S0(x) + �S1(x)


with

S0(x) = ±∫ x

p(x′)dx′

S1(x) = 12 i ln|p(x)| + c (120)

leads to the Wentzel–Kramers–Brillouin(WKB) approximation where c is anintegration constant to be determined bynormalization of the wave function,

ψWKB(x) = A√|p(x)|exp[±i

�

∫ x

p(x′)dx′]

(121)Several features of Eq. (121) are worth not-ing: the WKB wave function is an asymp-totic solution of Schrodinger’s equation inthe limit of small λ and satisfies all rules ofQM, including the superposition principle.All quantities in Eq. (121) can be calculated,however, purely from classical mechan-ics. In particular, the quantum probabilitydensity |ψWKB(x) | 2 can be directly associ-ated with the inverse classical momentum∼ | p(x) | −1 or, equivalently, the time theparticle spends near x. This result wasalready anticipated for the highly excitedstate of the harmonic oscillator (Figure 1.7).Furthermore, S0(x) is a double-valued func-tion of x corresponding to the particlepassing through a given point movingeither to the right or to the left with momen-tum ±p(x). It is the multiple-valuedness ofp(x) or S(x) that is generic to semiclas-sical mechanics and that poses a funda-mental difficulty in generalizing the WKBapproximation to more than one degree offreedom.

For describing bound states, a linearcombination of ψWKB with both signs ofthe momentum in Eq. (121) is requiredin order to form a standing rather than a

traveling wave:

ψWKB(x) = A√|p(x)| sin

×(

1

�

∫ x

ap(x′)dx′ + π

4

)(122)

Here a is the left turning point, theboundary of the classically allowed region(Figure 1.7). The additional phase of π/4is required to join ψ(x) smoothly withthe exponentially decaying solution in theclassically forbidden region where p(x)becomes purely imaginary (‘‘tunneling’’).With equal justification, we could havestarted from the classically forbiddenregion to the right of the turning pointb and would find

ψWKB(x) = A√|p(x)| sin

×(

1

�

∫ b

xp(x′)dx′ + π

4

)(123)

The uniqueness of ψWKB(x) mandates theequality of Eqs (122) and (123). The two areequal if the integral over the whole periodof the trajectory (from a to b and back to a)equals

(∫ b

a+

∫ a

b

)p(x)dx =

∮p(x)dx

= (n + 12 )2π� (124)

Equation (124) can be recognized as theBohr–Sommerfeld quantization rule of the‘‘old’’ quantum theory put forward priorto the development of quantum mechan-ics. Quantization rules were successful inexploring the discrete spectrum of hydro-gen, one of the great puzzles that classicaldynamics could not resolve. The additionalterm 1

2 × 2π� in Eq. (124) was initiallyintroduced as an empirical correction. Onlywithin the semiclassical limit of QM does

1.9 Semiclassical Mechanics 33

Path B

Path A

Fig. 1.12 Confinement of trajectories of anintegrable Hamiltonian for constant energy Eto the surface of a torus, shown here for asystem with two degrees of freedom (N = 2)(after Noid and Marcus, 1977).

its meaning become transparent: it is thephase loss in units of h = 2π� of a matterwave upon reflection at the turning points(in multidimensional systems, at caustics).These phase corrections are of the formα(π/4) with α = 2 in the present case. Thequantity α is called the Maslov index and itis used to count the number of encounterswith the caustics (in general, the numberof conjugate points) during a full period ofthe classical orbit.

1.9.2The EBK Quantization

The generalization of the WKB approxi-mation to multidimensional systems facesfundamental difficulties: the momentumvector p(r) is, in general, not smooth but it isa highly irregular function of the coordinatevector r and takes on an infinite numberof different values. Take, for example, abilliard ball moving in two dimensions ona stadium-shaped billiard table. A ball ofa fixed kinetic energy T = p2/2M can passthrough any given interior point r witha momentum vector p pointing in everypossible direction. Consequently, the gen-eralization of Eq. (121),

ψ(r) = √|D(r)| × exp[

i

�

∫ r

r0

p(r′) · dr′]

(125)

is well defined only under special circum-stances, that is, when p(r) is smooth andthe action S(r0, r) = ∫ r

r0p · dr′ is either sin-

gle valued or (finite-order) multiple valued(called a Lagrangian manifold; Gutzwiller,1990). Einstein pointed out, as early as1917, that the smoothness applies only tointegrable systems, i.e., systems for whichthe number of constants of motion equalsthe number of degrees of freedom, N. Forsuch a system, classical trajectories are con-fined to tori in phase space (as shown inFigure 1.12 for N = 2) and give rise tosmooth vector fields p(r) (Figure 1.13). Thecorresponding quantization conditions are

∮Ci

p(r) · dr = 2π�

(n + α

4

)(i = 1, . . ., N)

(126)where Ci (i = 1, . . . , N) denote N topolog-ical distinct circuits on the torus, that is,circuits that cannot be smoothly deformedinto each other without leaving the torus(for example, the paths A and B inFigure 1.12). The conditions of Eq. (126)are called the Einstein–Brillouin–Keller(EBK) quantization rules and are the gen-eralization of the WKB approximation(Eq. 124). The amplitude of Eq. (125) isgiven by the projection of the point den-sity of the torus in phase space onto thecoordinate space. Since the point density is


−2

−2

0

0

2

2 4

Y

X

Fig. 1.13 Regular trajectory in coordinate space foran integrable system. The vector field p(r) is multiplevalued but smooth (after Noid and Marcus, 1977).

uniform in the classical angle variables θ i

conjugate to the action (Goldstein, 1959),the amplitude is given by

|D(r)| ∝ det

∣∣∣∣∂θ1∂θ2∂θ3

∂x∂y∂z

∣∣∣∣ (127)

For a system with one degree of freedom(N = 1), Eq. (125) reduces to Eq. (121) since

∣∣∣∣∂θ

∂x

∣∣∣∣1/2

=∣∣∣∣ ∂t

∂x

∂θ

∂t

∣∣∣∣1/2

=∣∣∣∣M

pω

∣∣∣∣1/2

∝ 1|p|1/2

(128)

1.9.3Gutzwiller Trace Formula

As already anticipated by Einstein (1917),tori do not exist in many cases ofpractical interest – for example, for themotion of an electron in a stadium-shapedsemiconductor heterostructure (Marcus etal., 1992). WBK or EBK approximations arenot applicable since the classical motionof such an electron is chaotic, destroyingthe smooth vector field p(r) as in thebilliard discussed above. The semiclassicalmechanics for this class of problems wasdeveloped only recently (Gutzwiller, 1967,1970, 1971). The starting point is the

semiclassical Van Vleck propagator (VanVleck, 1928),

KSC(r1, r2, t2 − t1)

= √|D(r1, r2, t2 − t1)|

×exp[

i

�

∫ t2

t1

L(r(t′))dt′ − iαπ

2

](129)

where L is the classical Lagrange functionfor the particle traveling along the classi-cally allowed path from r1 at time t1 tor2 at time t2 and α counts the number ofsingular points of the amplitude factor D(similar to the Maslov index). If more thanone path connects r1(t1) with r2(t2), thecoherent sum of terms of the form in Eq.(129) must be taken. The Van Vleck prop-agator was recognized only 20 years later(Feynman, 1948) as the semiclassical limitof the full Feynman quantum propagator

K(r1, r2, t2 − t1)

=∫

{Dr} exp[

i

�

∫ t2

t1

L(r, t′)dt′]

(130)

where {Dr} stands for the path integral,that is the continuous sum over allpaths connecting r1 and r2 including allthose that are classically forbidden. Inthe semiclassical limit � → 0, the rapidoscillations in the phase integral cancel allcontributions except for those stemmingfrom the classically allowed paths for which

1.10 Conceptual Aspects of Quantum Mechanics 35

the actions∫

Ldt possess an extremum or,equivalently, for which the phase in Eq.(130) is stationary.

Equation (129) provides a method forinvestigating the semiclassical time evo-lution of wave packets in time-dependentproblems. The Fourier–Laplace transformof Eq. (129) with respect to t yields thesemiclassical Green’s function G(r1, r2, E).The spectral density of states of a quantumsystem can be shown to be

n(E) =∑

i

δ(E − Ei)

= − 1

πTr Im G(r1 = r2, E) (131)

where the trace implies integration over thevariables r1, r2 with constraint r1 = r2. Notethat each of these eigenenergies causesa sharp peak (resembling the singularityof the δ function) in n(E). Calculation ofn(E) therefore allows the determination ofeigenenergies of quantum systems. It ispossible to evaluate Eq. (131) in the semi-classical limit by using the semiclassicalGreen’s function and evaluating the tracein the limit � → 0 by the stationary-phasemethod. The result is the trace formula

n(E) = nav(E) + Im1

i�π

×∑

periodicorbits

Ti

2 sinh(Tiλi/2)

× exp[

i

(Si

�− αi

π

2

)](132)

The sum extends over all unstable classicalperiodic orbits, each characterized by theperiod Ti, the instability (or ‘‘Lyapunov’’)exponent λi, and the action Si, all of whichare functions of the energy E. An orbitis unstable if its Lyapunov exponent ispositive. The term λi measures the rate ofexponential growth of the distance from the

orbit when the initial condition is ever soslightly displaced (Gutzwiller, 1990). Thesmooth average density of states,

nav(E) = d

dE

(V(E)

(2π�)N

)(133)

is given by the rate of the change of theclassical phase-space volume with energyin units of the volume of Planck’s unitcell (2π�)N associated with each quantumstate. The derivation of Eq. (132) assumestwo degrees of freedom and makes explicituse of the fact that the physical system iscompletely chaotic and integrable regionsare absent. Only (hyperbolic) unstableperiodic orbits are assumed to exist,each with a positive Lyapunov exponentλi > 0 that measures the instability ofeach periodic orbit. Equation (132) istherefore complementary to the EBKquantization formula (Eq. 126). Thisformula of ‘‘periodic orbit quantization’’ isan asymptotic series, that is, the sum overall periodic orbits is, in general, divergent.For several systems, the sum over the firstfew orbits has been shown to reproducethe position of energy levels quite well.The mathematical and physical propertiesof the Gutzwiller trace formula are stilla topic of active research in quantummechanics.

1.10Conceptual Aspects of Quantum Mechanics

QM has proved to be an extraordinarilysuccessful theory and, so far, has neverbeen in contradiction to experimentalobservation. The Schrodinger equationhas cleared every test carried out so far,including a precision test of its linearity:the upper bound for corrections due tononlinear terms in the wave functions,


ψ(x)n (n > 1), in the Schrodinger equationstands currently at one part in 1027

(Majumder et al., 1990). Even so, theconceptual and philosophical aspects ofQM have remained a topic of livelydiscussion stimulated, to a large part,by the counterintuitive consequences ofQM that apparently defy common-senseinterpretation. The debate, which can betraced to the origins of QM and Einstein’sobjections summarized in his famousquote ‘‘Gott wurfelt nicht’’ (‘‘God does notplay dice’’), is centered around two aspects:the superposition principle, Eq. (15), andthe probability interpretation of the wavefunction, Eq. (39).

1.10.1Quantum Mechanics and Physical Reality

The counterintuitive aspects of the super-position principle are highlighted by thethought experiment of ‘‘Schrodinger’s cat.’’The coherent superposition of two states,each of which is presumably an eigen-state of an extremely complex many-bodySchrodinger equation, describes the stateof the cat, being either dead, |d〉, or alive,|a〉:

|ψcat〉 = cd|d〉 + ca|a〉 (134)

with |cd | 2+|ca | 2 = 1. Only by way ofmeasurement, that is, by projection ontothe states |a〉 or |d〉,

Pa = |〈a|ψcat〉|2, Pd = |〈d|ψcat〉|2 (135)

can we find out about the fate of the cat.While the meaning of two sharply differentoutcomes of Eq. (135), ‘‘dead’’ or ‘‘alive,’’ isdeeply engraved in our classical intuition,the quantum state of a coherent super-position, Eq. (134), defies any classicalinterpretation. Yet it is an accessible state

of the system. The point to be emphasizedis that it is not our incomplete knowledgeof an independently existing objective real-ity, but the fate of the cat itself, that canonly be resolved by the measurement. Sucha measurement is therefore, quite literally,a ‘‘demolition’’ measurement leading tothe ‘‘collapse’’ of the wave function to oneof the two possible states. In hypotheti-cal thought experiments for macroscopicsystems such as Schrodinger’s cat prob-lem, the average over a huge number(∼1023) of uncontrolled degrees of free-dom will, for all practical purposes, destroyany well-definded phase relation or coher-ence between the ‘‘states’’ dead and alive.In contrast, the axioms of QM can be put toa rigorous test for microscopic systems, forexample, by performing correlation mea-surements on two particles emitted fromthe same source. Examples along theselines were, for example, put forward byEinstein, Podolsky and Rosen (1935) tochallenge the purely probabilistic interpre-tation of QM.

Consider the emission of two elec-trons from, for example, a helium atom(Figure 1.14). The spin of each electroni = 1,2 can be in a coherent superpositionstate of spin up |+〉 and spin down |−〉,

|ψi〉 = αi|+〉 + βi|−〉 (136)

relative to one particular Cartesian axis,for example, the z axis. One possiblespin state of the electron pair (ignoringhere the antisymmetrization requirementfor fermions) would be an uncorrelatedproduct state,

|ψ〉 = |+〉1|−〉2 (137)

Already Schrodinger pointed out thatquantum theory allows to be alternatively


L2L1

z

He++

y

x

z

x

ee

Fig. 1.14 Two-electron emission from helium with theelectrons being spin-analyzed in two Stern–Gerlach-typefilters with different orientations (along the x and z axis,respectively). The difference in length, L1 < L2, makes thespin selection in the left detector happen first (inducing a‘‘wave function collapse’’) and in the right detector later.

in an entangled (or quantum-correlated)superposition state,

|ψ〉 = α|+〉1|−〉2 + β|−〉1|+〉2 (138)

In fact, the electron pair in the (singlet)ground state of helium with total spin S = 0forms such an entangled pair with α =−β = 1/

√2. The hallmark of entangled

states of the form Eq. (138) is that theycannot be represented as a factorizedproduct in the form of Eq. (137).

The consequences of such quantum cor-relations are profound: when measuringthe spin of particle 1 to be up (+), weknow with certainty that particle 2 hasits spin down (−). The ‘‘collapse’’ of theentangled wave function when measur-ing particle 1 forces particle 2 to switchits spin from a completely undeterminedstate to a well-defined value. We thereforeobtain information about the orientationof spin 2 without actually performing ameasurement on this particle. This corre-lation, often referred to as the nonlocalityof QM, is a consequence of the probabilis-tic description and superposition of wavefunctions in QM. The emergence of a defi-nite physical property or physical reality by‘‘postselection’’ through the measurementof an entangled state goes even further. Forexample, the entangled state Eq. (138) with

α = −β = 1/√

2,

|ψ〉 = 1√2

(|+〉1|−〉2 − |−〉1|+〉2) (139)

is rotationally symmetric and equallyapplies when we inquire about the spinorientation along the x rather than alongthe z axis. If we postselect the spin-updirection (+) along the x axis for the firstparticle, the second particle will have spindown (−) along the same axis. If we,instead, measured for the second parti-cle the polarization relative to the z axisas before (Figure 1.14), we would findboth spin up and down with 50% prob-ability, and thus a vanishing average spin.Different projection protocols for the firstspin imply entirely different results forthe second. The emergence of a specific‘‘physical reality’’ out of a specific mea-surement context is often referred to asquantum contextuality and is unknown inclassical physics. It has therefore beenquestioned whether the quantum mechan-ical description of a state such as in Eq.(138) can be considered a complete pic-ture of physical reality (Einstein, Podolskyand Rosen, 1935). In particular, it wasconjectured that particles may have addi-tional properties that are underlying thequantum-mechanical picture but which arehidden from our view. Such ‘‘hidden vari-ables’’ would ensure that physical systems


have only properties that are ‘‘real,’’ thatis, independent of a measurement andthat the result of any measurement isonly dependent on ‘‘local’’ system prop-erties rather than on a remote detectionof another observable. The knowledge of‘‘hidden variables’’ would allow one toremove the probabilistic aspects of QM.The role of probability in QM could thenbe the same as in statistical mechanics:a convenient way to quantify our incom-plete knowledge of the state of a system.The hidden-variable hypothesis, or alterna-tively, the quantum correlation built intoentangled states, Eq. (138), has been probedby testing the violation of Bell’s inequality(Bell, 1969; Clauser et al., 1969). Experi-ments with correlated pairs of particles, inparticular photon pairs – for example, byAspect and coworkers (1982) – have fal-sified the hidden-variable hypothesis byshowing that entangled states can violateBell’s inequality while being in full agree-ment with the predictions of QM. Morerecent experiments have succeeded in mea-suring entangled photons at a distance ofseveral hundred meters apart and withlight polarizers (used as photon analyz-ers), which switch their axes so rapidlythat during the photons’ time of flightthe polarizers have no chance to exchangeinformation on their respective orientation.(The ‘‘collapse’’ of the wave function bymeasuring one of the two photons occursinstantaneously for both photons, but can-not be used to transmit any informationand therefore does not violate the relativis-tic causality relations.) Even under suchstringent conditions, a violation of Bell’sinequality by more than 30 standard devia-tions was observed experimentally (Weihset al., 1998). Also so-called noncontex-tual hidden-variable theories and realis-tic models that assume a certain type

of ‘‘nonlocality’’ are untenable (Hasegawaet al., 2006; Groblacher et al., 2007).

1.10.2Quantum Information

The most fundamental unit of informa-tion in classical information theory is theso-called bit, which can take on only oneof two mutually exclusive values (typicallylabeled as ‘‘0’’ and ‘‘1’’). In conventionalcomputers, all numbers and operationswith them are encoded in this binary sys-tem. The corresponding quantum analogof the bit – named qubit – is given by atwo-state system with the two states |+〉and |−〉 from Eq. (136), now labeled as |0〉and |1〉. What sets the qubit apart from theconventional classical bit is precisely theability to be in a coherent superposition ofboth basis states,

|Q〉 = α|0〉 + β|1〉 (140)

with |α | 2+|β | 2 = 1. In quantum informa-tion theory, the intrinsic indeterminacy ofthe qubit’s value before a measurement isexploited as a resource rather than treatedas a deficiency.

One potential application of qubits issubsumed under the name ‘‘quantumcomputation’’. The basic building blockof a so-called quantum computer wouldbe a number of N entangled qubitsthat can be in any superposition of2N basis states. An example for N =2 is just the maximally entangled pairin Eq. (139). While a collection of Nclassical bits can only be in a singleone of the 2N possible states at agiven time, N qubits can be in anysuperposition state and thus can clearlyhave a much higher information content.Owing to the superposition principle, anyof the (unitary) logic operations of the


quantum computer performed on theinitial state could process all the involved2N basis states simultaneously. Severalalgorithms have been suggested (Shor,1994; Grover, 1996) for how this inherentlyquantum feature of ‘‘parallel processing’’could be used to solve fundamentalcomputational problems much faster, i.e.,with much fewer operations than with thecorresponding classical computations.

The manipulation of quantum stateswithin the framework of quantum com-putation and quantum information pro-cessing involves a set of rules as, forexample, the so-called no cloning theorem(Wootters and Zurek, 1982), which statesthe impossibility to create identical copiesof an arbitrary unknown quantum state.Although only imperfect ‘‘clones’’ of aquantum state are possible, a transfer ofa quantum state from one system to asecond, possibly distant, system is allowedaccording to QM. Such a ‘‘teleportation’’ ofa quantum state involves the destructionof the original state during the teleporta-tion procedure and is typically performedthrough an intermediate entangled stateof three particles (Bennett et al., 1993;Bouwmeester et al., 1997).

The laws of QM can also be used forsecure information transfer. In classicalcryptography, secret messages are enci-phered by way of a code that the recipient ofa message must know in order to deciphera message’s content. To prevent the break-ing of the code by a third unauthorizedparty, the key for the encoding algorithm ispreferably defined as a random sequenceof numbers, which is generated anew foreach message transmitted.

Challenges to classical cryptography arethe need of a truly random key and the riskof key corruption through the interceptionby a third party. Both of theses challengescan be met by intrinsic quantum features.

A truly random sequence of numbers canbe generated, for example, by repeatedlymeasuring the spin orientation (or photonpolarization) of an unpolarized spin, asituation that may be realized by firstprojecting the first spin of an entangledpair along a perpendicular direction (seethe example of Fig. 1.14 above).

Interception of a message shared withthe recipient through an entangled pairby a third party would require a mea-surement, which leads to a wave functioncollapse and to a detectable reduction inquantum correlations. Using the princi-ples of QM for the generation and trans-mission of a secure encryption key – atechnique that is referred to as quantumcryptography – has been successfully imple-mented for setups involving very largedistances (144 km) between the sender andrecipient (Ursin et al., 2007).

1.10.3Decoherence and Measurement Process

One fundamental obstacle in practicallyimplementing quantum computation andquantum information protocols is the chal-lenge to keep the coherent superpositionof a quantum state, in particular of entan-gled pairs, ‘‘alive’’ for extended periodsof time. Any unintentional or intentionalinteraction of the system inevitably andirreversibly converts the quantum super-position into a classical probability distri-bution and destroys the operating principleof a quantum information device. Themeasurement process and the destructionof coherence, frequently called decoher-ence, are closely intertwined. In the earlydays of quantum theory, the measure-ment device that caused the ‘‘collapse’’of the wave function was considered tobe a ‘‘classical’’ apparatus decoupled fromthe quantum dynamics to be observed


(Bohr, 1928). The notion of open quan-tum systems has superseded this viewand provides a widely accepted unifieddescription of measurement and decoher-ence within the realm of quantum theory.The starting point here is the fundamentalinsight that the quantum system (S) to beobserved is never completely isolated fromthe environment (E), which itself is quan-tum rather than classical. The Hamiltoniandescribing both system and environmentis given by H = HS+HE+HS−E, contain-ing the Hamiltonian of the system (HS)and its environment (HE), as well as theinteraction between the two (HS−E). Astate describing both system and envi-ronment, |ψ〉, even when at the initialtime t = 0 prepared in a factorizable state,|ψ(t = 0)〉 = |ψ i

S〉|ψ jE〉, will evolve in the

course of time under the influence of theinteraction HS−E into an entangled, non-factorizeable state

|ψ(t)〉 =∑

i,j

aij|ψ iS〉|ψ j

E〉(t) (141)

Here, |ψ iS〉 (and |ψ j

E〉) are basis states ofthe system (of the environment), whichoften, but not always, are eigenstates ofthe Hamilton operator of the system HS

(of the environment HE). ‘‘Tracing out’’the (unobserved) environment degrees offreedom of the entangled state by takingexpectation values leads to decoherencein the (observed) system and mimicsthe effect of a classical measurementdevice.

Consider, as an illustrative example,a situation in which the system andthe environment are each representedby a two-dimensional Hilbert space anddescribed by the following entangled state:

|ψ(t)〉 = α|+〉S|−〉E + β|−〉S|+〉E (142)

The expectation value of any physicalobservable represented by an operator AS

acting only on the states in the Hilbertspace of the system S gives in the presenceof entangled environmental degrees offreedom,

〈AS〉 = 〈ψ|AS|ψ〉= |α|2S〈+|AS|+〉S

+|β|2S〈−|AS|−〉S

= Tr(ρAS) (143)

with ρ = |α|2| +〉S S〈+ | + |β|2|−〉S S〈−|.It is the coherent entanglement with theenvironment, Eq. (142), that leads to (par-tial) decoherence within the system (S) andto a statistical mixture described by thedensity operator introduced above (Eq. 40)with probabilities P+ = |α|2 and P− = |β|2.The ubiquitous coupling of a physicalsystem to the environment provides thenatural setting for the transition fromthe quantum indeterminism of superpo-sitions to the definiteness of the classicalmeasurement. The classical limit inducedby decoherence is conceptually differentfrom the classical limit of quantum sys-tems of large quantum numbers, which isthe focus of semiclassical mechanics (seeSection 1.9). In spite of recent progress, itis generally still considered an open ques-tion if decoherence concepts alone can fullyexplain the quantum-to-classical transitionand provide definite answers to the widelydiscussed questions regarding the ‘‘quan-tum measurement problem.’’

The above considerations demonstratethat it is very unlikely for a macroscopicobject such as Schrodinger’s cat to ever bein a superposition state ‘‘dead and alive’’ asin Eq. (134) because it is extremely difficultto isolate the cat from its environment.For all practical purposes, an exchange ofparticles and heat between the cat and

1.11 Relativistic Wave Equations 41

its environment will prevent the cat, foritself, to be in a pure quantum state.It will rather be described by a densitymatrix with strongly reduced quantumcorrelations. As both the cat and theenvironment are macroscopic objects withmany degrees of freedom, decoherencesets in at an extremely short timescale.Recent experiments have been able tosufficiently decouple ‘‘mesoscopic’’ objectsas superconducting rings (Friedman et al.,2000) from their environment to observethese systems in superposition states.

1.11Relativistic Wave Equations

When the characteristic speed v of particlesin a physical system becomes comparableto the speed of light, v ≤ c, the Schrodingerequation, which incorporates nonrelativis-tic quantum dynamics, fails. Generaliza-tions to relativistic wave equations face,however, fundamental difficulties due tothe possibility of particle creation anddestruction in relativistic dynamics (see

Chapter 2). For example, if photons haveenergies in excess of twice the relativis-tic rest energy of electrons, Eγ > 2mec2,a pair consisting of an electron and itsantiparticle, a positron, can be created.Therefore, the interpretation of |ψ(r)|2 asa single-particle probability density is nolonger possible. A consistent resolution ofthe difficulties can be accomplished withinthe framework of quantum field theory,which is outside the scope of this chapter.The relativistic analogs to the Schrodingerequation discussed in the following sectionare only applicable to relativistic systems inwhich particle production and destructionare not yet important.

1.11.1The Klein–Gordon Equation

We use the same recipe that led us inSection 1.3 to the Schrodinger equation,i.e., we replace the quantities H and p bydifferential operators (Eq. 11), motivatedby the de Broglie hypothesis. In thiscase, however, we use the relativisticrelation between energy and momentum(see Chapter 2) instead of the nonrelativisticone, Eq. (9):

E − V(r) =√

M2c4 + p2c2 (144)

Following this recipe, we are led to

i�d

dtψ(r, t) =

[√−�2∇2c2 + M2c4

+V(r)]ψ(r, t) (145)

Equation (145) has a major drawback:it treats the space and time derivativesasymmetrically. The Lorentz invarianceof relativistic dynamics requires, however,that space and time coordinates be treatedon equal footing. We thus start with thesquare of Eq. (144) and arrive at

[i�

∂

∂t− V(r)

]2

ψ(r, t)

= (−�2∇2c2 + M2c4)ψ(r, t) (146)

This is the time-dependent Klein–Gordon equation, applicable to particleswith integer spin. The corresponding sta-tionary Klein–Gordon equation follows byreplacing i�∂/∂t by E. Equation (146) givesan eigenvalue equation in E2 rather thanin E itself. Consequently, upon takingthe square root of the eigenvalue, solu-tions with both signs, ±√

p2c2 + M2c4,will appear. The physical meaning ofnegative-energy solutions is discussed


below. For a charged particle with charge qin the presence of an electromagnetic field,both the scalar potential φ(V = qφ) and theelectromagnetic vector potential A must beincluded by ‘‘minimum coupling’’:

p −→ p − qA =(

�

i

)∇ − qA (147)

With this coupling, the Klein–Gordonequation is invariant under gauge trans-formations of the electromagnetic field aswell as under Lorentz transformations.

1.11.2The Dirac Equation

Instead of taking the square of Eq. (144),Dirac pursued a different approach indeveloping a Lorentz-invariant relativisticwave equation. He ‘‘linearized’’ the squareroot

√c2p2 + M2c4 −→ (

cα · p + βMc2)(148)

with the subsidiary condition that thesquare of the linearized operator satisfies

(cα · p + βMc2)2 = c2p2 + M2c4 (149)

It turned out that solutions to this equationcan be found only if all αi (i = x, y, z)and β are taken as matrices of minimumdimension 4 × 4. Equation (149) is satisfiedif the matrices αi, β satisfy the relations

αiαk + αkαi = 2δik

αiβ + βαi = 0 (150)

Accordingly, ψ is now a four-componententity (called a spinor), ψ = {ψ i, i =1, . . . , 4}. The resulting Dirac equationdescribes spin 1

2 particles. When a charged

spin 12 particle interacts with an electro-

magnetic field as described by minimalcoupling, the stationary Dirac equationbecomes

[E − qφ(r)]ψ(r)

=[

cα

(�

i∇ − qA

)+ βMc2

]ψ(r) (151)

Dirac spinors permit the probability-density interpretation, that is, |ψ(x)|2 =∑4

i=1 |ψi(x)|2. The physical meaning of thefour components can be easily understoodby considering a free particle. The energyeigenvalues are E = ±√

p2c2 + M2c4. Inaddition to the positive energies we findnegative eigenenergies, which are a featureof relativistic quantum mechanics andwhich result from the fact that thesquare-root operator permits both thepositive and negative sign of E. Thefour-spinor can be decomposed into two2-spinors ψ = (φ1, φ2) one of whichrepresents the positive-energy solution andthe other the negative-energy solution.Each component of the two-spinor wavefunction can be understood as a productof a configuration-space wave functioneikr and a spin 1/2 eigenstate |ms =±1/2〉. A major success of the Diractheory was that the intrinsic degree ofan electron spin, which was a mereadd-on within the Schrodinger theory,is incorporated into the Dirac equationfrom the outset. Furthermore, the gfactor for the magnetic moment of theelectron as well as the spectrum of thehydrogen atom can be derived from theDirac equation to a very high degree ofapproximation. The pieces still missingare the quantum electrodynamic (QED)corrections that lead to small energyshifts of atomic levels (‘‘Lamb shift’’)and the deviations of the g factor forthe anomalous magnetic moment from

1.11 Relativistic Wave Equations 43

E = 2mec2

EFig. 1.15 Particle–antiparticle production visualized as the excitation of anegative-energy particle to positive energies leaving behind a hole.

2, g = 2 × 1.001 16. These corrections canbe determined only within quantum fieldtheory, the theory of the interactionsof charged particles with the quantizedelectromagnetic field.

The negative-energy states, accordingto Dirac, can be interpreted in terms ofantiparticle states. The vacuum state ofempty space is represented by a fullyoccupied ‘‘sea’’ of negative-energy states.While this ‘‘sea’’ itself is not observ-able, ‘‘holes’’ created in the sea by excit-ing negative-energy electrons to posi-tive energies (Figure 1.15) are observ-able and appear as antiparticles. Forexample, a missing electron with charge−e, energy −E, and momentum p repre-sents a positron with charge e, energy E,and momentum −p. The negative-energystates of electrons therefore offer a sim-ple explanation of particle–antiparticlepair production in terms of the excita-tion of a negative-energy particle to apositive-energy state (the particle) leavinga hole in the negative-energy ‘‘sea’’ (theantiparticle) behind.

Glossary

Coherence and Decoherence: In QM,the term coherence stands for the abil-ity of a particle’s state to interfere.Perfectly coherent states (‘‘pure states’’)are subject to the superposition principle of

QM, whereas imperfections in the coher-ence are described by a density matrix with(partially) classical correlations. Decoher-ence stands for the loss of coherence asinduced, for example, by coupling an iso-lated quantum system to its environment.

Correspondence Principle: The princi-ple that quantum-mechanical and classicaldescription of physical phenomena shouldbecome equivalent in the limit of vanishingde Broglie wavelength λ → 0, or equiva-lently, of large quantum numbers.

de Broglie Wave: According to de Broglie,the motion of a particle of mass M andvelocity v is associated with a matter wavewith wavelength λ = h/(Mv). h is Planck’squantum.

Delta (δ) Function: A particular case of adistribution, frequently used in quantummechanics to formulate the orthogonalityand normalization conditions of states witha continuous spectrum of eigenvalues.

Distributions: Also called generalized func-tions. Linear-continuous functionals oper-ating on a metric vector space. They formthe dual space of the metric vector space.In the context of quantum mechanics, thespace is a Hilbert space and the operation ofthe functional on an element of the Hilbertspace is called formation of a wave packet.

Eigenvalue Problem: The solution ofthe eigenvalue equation for the matrixA, Aψ =λψ where λ is a number called


an eigenvalue and ψ is the eigenvector.In quantum mechanics, ψ is a vector inHilbert space.

Entanglement: An intrinsic feature of QM,occurring when particles or subsystems (ineither case more than one), are correlatedbeyond a degree allowed by classicalphysics. Entangled systems, even when farapart, cannot be described independentlyof each other.

Hamilton–Jacobi Equation: A particu-lar canonical transformation in classicaldynamics that leads to the vanishing of theHamiltonian function in the new coordi-nates. The transformation equation for theoriginal Hamiltonian function is called theHamilton–Jacobi equation and is equivalentto equations of motion.

Hilbert Space: Infinite-dimensional com-plete vector space over the complex num-bers endowed with a metric induced bya scalar product that satisfies Schwartz’sinequality.

Laser: Light amplication by stimulatedemission of radiation. Modern source forhigh-intensity coherent electromagneticradiation in the infrared, visible, andultraviolet spectral region.

Schrodinger Equation: Fundamentalequation governing the dynamics of anonrelativistic quantum system, playing asimilar role in quantum mechanics as doNewton’s equations of motion in classicaldynamics.

State: A mechanical system is at any giventime completely characterized in quantummechanics by the state ψ . The projectiononto coordinate space, ψ(r), is called a wavefunction. States are vectors in Hilbert space.

Superposition Principle: Any superposi-tion in Hilbert space, ψ = aψ1 + bψ2, oftwo states ψ1 and ψ2 forms another physi-cally realizable state of a physical system.

Uncertainty Principle: Two canonicallyconjugate variables, such as position x andmomentum p, can be simultaneously mea-sured only with an intrinsic uncertainty ofat least �x�p � �/2.

Wave Function: The representation of thestate |ψ〉 in coordinate space, ψ(r) = 〈r |ψ〉.

References

Andra, H. (1974) Phys. Scr., 9, 257–80.Aspect, A., Dalibard, J., and Roger, G. (1982)

Phys. Rev. Lett., 49, 91–94; 1804–807.(a) Bell, J.S. (1965) Physics (New York), 1,

195–207; (b) see also Clauser, J.F., Horner,M.A., Shimony, A., and Holt, R. (1969) Phys.Rev. Lett., 23, 880–83.

Bennett, C.H., Brassard, G., Crepeau, C., Jozsa,R., Peres, A., and Wootters, W.K. (1993) Phys.Rev. Lett., 70, 1895–99.

Berry, M. (1984) Proc. R. Soc. London, Ser. A, 392,45–57.

Bohr, N. (1928) Nature, 121, 580–90.Bouwmeester, D., Pan, J.-W., Mattle, K., Eibl, M.,

Weinfurter, H., and Zeilinger, A. (1997)Nature, 390, 575–79.

Born, M. (1926) Z. Phys., 38, 803–27.Brandt, S. and Dahmen, H.D. (1989) Quantum

Mechanics on the Personal Computer,Springer-Verlag, Berlin.

de Broglie, L. (1924) Thesis; reprint (1926), J.Phys. (Paris), 7, 321–337.

Burgdorfer, J. (1981) Phys. Rev. A, 24, 1756–67.Davisson, C.J. and Germer, L.J. (1927) Phys. Rev.,

30, 705–40.Dirac, P.A.M. (1930) The Principles of Quantum

Mechanics, Oxford University Press, Oxford,Reprint (1986) Oxford Scientific Publications,Oxford.

Einstein, A. (1905) Ann. Phys. (Leipzig), 17,132–48.

Einstein, A. (1917) Verh. Dtsch. Phys. Ges., 19,82–92.

Einstein, A., Podolsky, B., and Rosen, N. (1935)Phys. Rev., 47, 777–80.

Ericson, T. and Mayer-Kuckuk, T. (1966) Annu.Rev. Nucl. Sci., 16, 183–206.

Fano, U. (1961) Phys. Rev., 124, 1866–78.Feynman, R. (1948) Rev. Mod. Phys., 20, 367–87.

Further Reading 45

Friedman, J.R., Patel, V., Chen, W., Tolpygo,W.K., and Lukens, J.E. (2000) Nature, 406,43–46.

Goldstein, H. (1959) Classical Mechanics,Addison-Wesley, Reading, PA.

Goudsmit, S. and Uhlenbeck, G. (1925)Naturwissenschaften, 13, 953–54.

Groblacher, S., Paterek, T., Kaltenbaek, R.,Brukner, C., Zukowski, M., Aspelmeyer, M.,and Zeilinger, A. (2007) Nature, 446, 871–75.

Grover, L.K. (1996) Proceedings, 28th AnnualACM Symposium on the Theory ofComputing (STOC) 212–19.

Gutzwiller, M. (1967) J. Math. Phys., 8,1979–2000.

Gutzwiller, M. (1970) J. Math. Phys., 11,1791–1806.

Gutzwiller, M. (1971) J. Math. Phys., 12, 343–58.Gutzwiller, M. (1990) Chaos in Classical and

Quantum Mechanics, Springer-Verlag, NewYork.

Hasegawa, Y., Loidl, R., Badurek, G., Baron, M.,and Rauch, H. (2006) Phys. Rev. Lett., 97,230401.

Heisenberg, W. (1927) Z. Phys., 43, 172–98.Hellmuth, T., Walter, H., Zajonc, A. and

Schleich, W. (1987) Phys. Rev. A, 35,2532–41.

Hylleraas, E. (1929) Z. Phys., 54, 347–66.Lighthill, M.J. (1958) Introduction to Fourier

Analysis and Generalized Functions,Cambridge University Press, Cambridge.

Majumder, P.K., Venema, B.J., Lamoreaux, S.K.,Heckel, B.R., and Fortson, E.N. (1990) Phys.Rev. Lett., 65, 2931–34.

Marcus, C.M., Rimberg, A.J., Westervelt, R.M.,Hopkins, P.F., and Gossard, A.C. (1992) Phys.Rev. Lett., 69, 506–9.

Noid, D. and Marcus, R. (1977) J. Chem. Phys.,67, 559–67.

(a) Pekeris, C.L. (1958) Phys. Rev., 112, 1649–58;(b) Pekeris C.L. (1959) Phys. Rev., 115,1216–21.

Rutherford, E. (1911) Philos. Mag., 21, 669–83.Schwartz, L. (1965) Methodes Mathematiques pour

Les Sciences Physiques, Hermann, Paris.Shor, P.W. (1994) Proceedings of the 35th

annual symposium on foundations ofcomputer science; revised version availableat: (1997). SIAM J. Comput., 26, 1484–509.

(a) Stern, O. and Gerlach, W. (1921) Z. Phys., 7,249–53; (b) Stern, O. and Gerlach, W. (1921)8, 110–111; (c) Stern, O. and Gerlach, W.(1922) 9, 349.

Ursin, R. et al. (2007) Nature Physics, 3, 481–86.Van Vleck, J. (1928) Proc. Natl. Acad. Sci. U.S.A.,

14, 178–88.Weihs, G., Jennewein, T., Simon, C., Weinfurter,

H., and Zeilinger, A. (1998) Phys. Rev. Lett.,81, 5039–43.

Wootters, W.K. and Zurek, W.H. (1982) Nature,299, 802–3.

Further Reading

Bell, J.S. (1987) Speakable and Unspeakable inQuantum Mechanics, Cambridge UniversityPress, Cambridge.

Bjorken, J.D. and Drell, S.D. (1965) RelativisticQuantum Mechanics and Relativistic FieldTheory, McGraw-Hill, New York.

Bransden, B.H. and Joachain, C.J. (1983) Physicsof Atoms and Molecules, Longman, New York.

Burgdorfer, J., Yang, X., and Muller, J. (1995)Chaos Solitons Fractals, 5, 1235–73.

Cohen-Tannoudji, C., Diu, B. and Laloe, F.(1977) Quantum Mechanics, Vols. I and II,John Wiley & Sons, New York.

Courant, R. and Hilbert, D. (1953) Methods ofMathematical Physics, Wiley Interscience,New York.

Eisberg, R. and Resnick, R. (1985) QuantumPhysics of Atoms, Molecules, Solids, Nuclei andParticles, 2nd edn, John Wiley & Sons, NewYork.

Feynman, R.P. and Hibbs, A.R. (1965) QuantumMechanics and Path Integrals, McGraw-Hill,New York.

Feynman R.P., Leighton, R.B., Sands, M. (1965)The Feynman Lectures on Physics, Vol. III,Quantum Mechanics, Addison-Wesley,Reading, MA.

Krane, K. (1983) Modern Physics, John Wiley &Sons, New York.

Landau, L. and Lifshitz, E. (1965) QuantumMechanics, Non-Relativistic Theory, Pergamon,Oxford.

Merzbacher, E. (1970) Quantum Mechanics, JohnWiley & Sons, New York.

Messiah, A. (1961) Quantum Mechanics,North-Holland, Amsterdam.

von Neumann, J. (1955) MathematicalFoundations of Quantum Mechanics,Princeton University Press, Princeton, NJ.

Schiff, L. (1968) Quantum Mechanics,McGraw-Hill, Singapore.


Sommerfeld, A. (1950) Atombau undSpektrallinien, Vieweg, Braunschweig;Reprint (1978) Verlag Harri Deutsch, Thun,Switzerland.

Wichmann E.H. (1971) Berkeley Physics Course,Quantum Physics, Vol. 4, McGraw-Hill, NewYork.

Documents

1 Quantum Mechanics - Wiley-VCH · Quantum mechanics (QM), also called wave mechanics, is the modern theory of matter, of atoms, molecules, solids, and of the interaction of electromagnetic