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Lecture Notes PH 411/511 ECE 598 A. La Rosa
INTRODUCTION TO QUANTUM MECHANICS ________________________________________________________________________
CHAPTER-10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS
10.1 Representations in the spatial and momentum spaces
10.1.A Representation of the wavefunction in the spatial coordinates
basis { position x }
10.2.A1 The Delta Dirac
10.2.A2 Compatibility between the physical concept of amplitude probability and the notation used for the inner product.
10.1.B Representation of the wavefunction in the momentum coordinates basis {
momentum p }
10.2.B1 Representation of the momentum p state in space-coordinates basis
{ position x }
10.2.B2 Identifying the amplitude probability momentum p as the Fourier
transform of the function ( x)
10.1.C Tensor Product of State Spaces
10.2 The Schrödinger equation as a postulate
10.2.A The Hamiltonian equations expressed in the continuum spatial coordinates. The Schrodinger Equation.
10.2.B Interpretation of the wavefunction
Einstein’s view on the granularity nature of the electromagnetic radiation.
Max Born’s probabilistic interpretation of the wavefunction.
Deterministic evolution of the wavefunction
Ensemble
10.2.C Normalization condition of the wavefunction
Hilbert space
Conservation of probability
10.2.D The Philosophy of Quantum Theory
10.3 Expectation values
10.3.A Expectation value of a particle’s position
10.3.B Expectation value of the particle’s momentum
10.3.C Expectation (average) values are calculated in an ensemble of identically prepared systems
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10.4 Operators associated to observables
10.4.A Observables, eigenvalues and eigenstates
10.4.B Definition of the quantum mechanics operator F~ to be associated with the
observable physical quantity f
10.4.C Definition of the Position Operator X~
10.4.D Definition of the Linear Momentum Operator P~
10.4.D.1 Representation of the linear momentum operator P~
in the momentum
basis { momentum p }
10.4.D.2 Representation of the linear momentum operator P~
in the spatial
coordinates basis { position x }
10.4.D3 Construction of the operators P~
, 2~P , 3~
P
10.4.E The Hamiltonian operator
10.4.E.1 Evaluation of the mean energy in terms of the Hamiltonian operator 10.4.E.2 Representation of the Hamiltonian operator in the spatial coordinate
basis
10.5 Properties of Operators
10.5.A Correspondence between bras and kets
10.5.B Adjoint operators
10.5.C Hermitian or self-adjoint operators
Properties of Hermitian (or self-adjoint) operators: - Operators associated to mean values are Hermitian (or self-adjoint) - Eigenvalues are real - Eigenvectors with different eigenvalues are orthogonal
10.5.D Observable Operators
10.5.E Operators that are not associated to mean values
10.6 The commutator 10.6.A The Heisenberg uncertainty relation
10.6.B Conjugate observables Standard deviation of two conjugate observables
10.6.C Properties of observable operators that do commute
10.6.D How to uniquely identify a basis of eigenfunctions? Complete set of commuting observables
10.7 Simultaneous measurement of observables 10.7.A Definition of compatible (or simultaneously measurable) operators
10.7.B Condition for observables A~
and B~
to be compatible
10.8 How to prepare the initial quantum states
10.8.A Knowing what can we predict about eventual outcomes from measurement?
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10.8.B After a measurement, what can we say about the state ?
References:
Feynman Lectures Vol. III; Chapter 16, 20
Claude Cohen-Tannoudji, B. Diu, F. Laloe, “Quantum Mechanics”, Wiley.
"Introduction to Quantum Mechanics" by David Griffiths; Chapter 3.
B. H. Bransden & C. J. Joachin, “Quantum Mechanics”, Prentice Hall, 2nd Ed. 2000
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CHAPTER-10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS
Quantum theory is based on two mathematical items: wavefunctions and operators.
The state of a system is represented by a wavefunction. An exact knowledge of the wavefunction is the maximum information one can have of the system: all possible information about the system can be calculated from this wavefunction.
Quantities such as position, momentum, or energy, which one measures experimentally, are called observables. In classical physics, observables are represented by ordinary variables.
In quantum mechanics observables are represented by operators; i.e. by quantities
that upon operation on a wavefunction giving a new wavefunction. This chapter presents three main sections:
The first includes a description of the
spatial-coordinates basis and
the momentum-coordinates basis, which are typically used to represent a quantum state.
The next describes how to build the quantum mechanics operator
corresponding to a given observable.
The final section addresses how to build a (mathematical) quantum state from
a given set of experimental results. The key mathematical concept used here is
the complete set of commuting operators
10.1 Representation of the wavefunctions in the spatial and momentum spaces
An arbitrary state can be expanded in terms of base states that conveniently fit the particular problem under study. For general descriptions, two bases are frequently used: the spatial coordinate basis and the linear momentum basis. These two basis are addressed in this section. 10.1.A Representation of the wavefunction in the spatial coordinates basis
{ x , x }
Chapter 9 helped to provide some clues on the proper interpretation of the wavefunction (the solutions of the Schrodinger equation.) This came through the analysis of the particular case of an electron moving across a discrete lattice: the wavefunction is pictured as a wave of amplitude probabilities (complex numbers whose magnitude is interpreted as probabilities). Notice, however, that when taking the limiting case of the lattice spacing tending to zero, one
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ends up with a situation in which the electron is propagating through a continuum line space. Thus, this limiting case takes us to the study of a particle moving in a continuum space.
In analogy to the discrete lattice, where the location of the atoms guided the selection of
the state basis { n }, in the continuum space we consider the following continuum set,
{ x , x } Continuum spatial-coordinates base (1)
In analogy to the discrete case,
x stands for a state in which a particle is located around the coordinate x.
x
For every value x along the line one conceives a corresponding state. If one includes all the points on the line, a complete basis set results as indicated in (1), which will be used to describe a general quantum state and, hence, to describe the one-dimension motion of a particle.
A given state specifies the particular way in which the amplitude probability of a particle is
distributed along a line. One way of specifying this state is by specifying all the amplitude-
probabilities that the particle will be found at each base state x; we write each of these
amplitudes as x . We must give an infinite set of amplitudes, one for each value of x. Thus,
=
x x dx Representation of the wavefunction (2)
in the spatial coordinates basis
x is the amplitude probability that a system in the state be found in the state x .
About the x notation
We could use alternative notations, like, for example, x ) ( xA Ψ
=
x x dx =
x ) ( xA Ψ dx ;
as to make this expansion to resemble the expansion of a “vector ” in terms of a base-
vectors x , and the corresponding coefficients ) ( xA Ψ playing the role of weighting-factor
coefficients.
Sometimes the following notation is preferred:x ) ( x
=
x x dx =
x (x) dx ; (2)’
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x = (x) is the amplitude probability that a system in the state be found in the state x .
Multiplying the expression (2)’ above by a particular bra x ’, we should obtain,
x ’ = ( x’ )
(This result will be justified a bit more rigorously in the next section below; see the delta Dirac section).
That is,
( x) ≡ x
Amplitude probability that the particle
initially in the state be found (immediately after a measurement) at the
state x.
(3)
Thus, we will use indistinctly the following notation
number Number
=
x x dx .=
x xdx . (4) Representation of the wave-function in the spatial coordinates basis
Caution: x does not mean
[x]* (x) dx. Working with the spatial coordinates base { x }
may constitute the only occasion in which the notation x becomes confusing with the definition of the scalar product.
Note: In Chapter 8 we used the notation (t) =n
n An(t), where the amplitudes
An(t) =n were determined by the Hamiltonian equations
dt
di n =
jjnH
j .
In this chapter, we are using instead a continuum base { x , x };
accordingly the amplitudes will be expressed as x , which is also written as x.
Soon we have to address then how the Hamiltonian equations look like when using a continuum base.
The Delta Dirac1 A subproduct of the mathematical manipulation expressing a state in a given basis is the closure relationship that the components of the basis set must comply: the Delta Dirac relationship. This is illustrated for the case of the space-coordinates basis.
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=
x’ [ x’ ] dx’
x = x
x’ [ x’ ] dx’
x =
x x’ [ x’ ] dx’
x =
x x’ [ x’ ] dx’
( x ) ( x’ )
Or, equivalently
x =
x x’ [ x’ ] dx’
On the other hand, it is accepted that for an arbitrary function f, the delta ‘function’ is defined as2
f x ≡
( x’ - x ) f x’ dx’
The last two expressions are consistent if,
x x’ = ( x’ – x ) . Thus, we have the following result:
Case of discrete states Case of continuum states
Delta Kroenecker Delta Dirac (x’-x )
n m = nm x x’ = ( x’ - x )
(5)
Compatibility between the physical concept of amplitude probability and the
notation used for the inner product
* x) x dx
We know that given a state , the amplitude probabilitiesx that appear in the
expansion =
x x dx are determined by the Hamiltonian equations.
8
Suppose we have a particle in the state and we want to know the amplitude
probability (after a given measurement process) to find the particle at the state . That is, we
want to evaluate .
There are many path way for the state to transit to state . It could do it by passing first
through any of the base-states x . Since each state x generates a path, and all these paths have the same initial and final state, then, according to the rules established in Chapter 7, the total amplitude probability will be given by,
= xall
x x
x
First, the sum over a small region of width dx would be xx multiplied by dx. Then, since x varies from to the expression above can be written as,
=
x x dx (6)
For the terms inside the inside the integral, let’s use the notation introduced in expression (3)
above, x = x. Also, since x = x * then x = x * =*(x), one obtains,
=
* x) x dx (7)
The amplitude probability is equal to the
inner product between the functions and
10.1.B Representation of the wavefunction in the momentum coordinates basis { p }
In the previous section, an arbitrary state was expressed in terms of the components
of the space-coordinates basis { x , x } . Here we present an alternative basis to
express the state , which is the continuum base of momentum coordinates. We will realize that we are already familiar with the states comprising this new basis.
Recall that in Chapter 5 we introduced the Fourier transform F= F(k) of a function
)(x . It involved the introduction of the complex harmonic functions ek, where ek (x)
= 2
1 e
i
kx for k
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( x )= 2
1
F(k) ei
kx dk
Fourier transform
Base-function ek evaluated at x
where the weight coefficients F(k), referred to as the Fourier transform of the
function , are given by,
F(k) =2
1
e' - xki
')'( dxx
However, it is convenient to express the last two expressions in terms of the variable p = k
,
( x )= 2
1
(p) ei ( p / ) x d p
Fourier transform of
(8)
where
(p) =2
1
e-i(p/ )x’
')'( dxx (9)
In expressions (8) and (9), the expression 2
1
πei ( p / x have a clear physics
interpretation,
p( x ) = 2
1
π e
i ( p / ) x According to de Broglie, p represents a plane wave of
definite linear momentum p. (10)
We ca re-write expression (8) in terms of functions,
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( x )=
( p) 2
1 ei ( p / ) x d p
Fourier transform
of ( x)
=
( p) p dp
p
Function Sum of functions
(11)
Expressions (8) and (11) are equivalent.
Placing p in bracket notation First, taking into account expression (4),
=
x xdx,
Let’s apply it to the functionp, where p (x) = 2
1
π e
i ( p / ) x; we obtain ,
p ≡
x’ p ( x’) dx’
The above serves to define the momentum–states basis: { p , - ∞ < p < ∞ }
p ≡
x’ p ( x’) dx’
=
x’ 2
1
π e
i( p / ) x’ dx’
Representation of the
momentum state p in the space-coordinate
basis { x }
(12)
x p ≡ p ( x ) = 2
1
π e
i( p / )
x
Amplitude probability that a particle, in a state of momentum p, be found at the coordinate x.
(13)
This is the de Broglie hypothesis in the language of amplitude probabilities.
Placing expression (11) in bracket notation
11
In expression (11) we can make then the following association,
p
=
(p) p dp
Hence,
=
(p) p dp
Fourier transform ( x )
(14)
Expression (14) gives as a linear combination of the momentum states p defined in (12) above.
Expressions (8), (11) and (14) are equivalent.
The states defined in (12) constitute a base (the justification comes from the Fourier transform theory),
{ p , - ∞ < p < ∞ }
continuum base of momentum-coordinates
(15)
Exercise: Prove that p p’ = ( p – p’ )
Now we formally can justify that in expression (14): p = (p)
In effect, multiplying (14) with a bra p , one obtains,
p = p
(p’) p’ d’p
=
(p’) p p’ dp’
usingp p= ( p – p’ )
p = (p) (16)
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Summary about the momentum coordinates:
An arbitrary wavefunction can be expressed as a linear combination
of momentum-coordinates p ,
=
p (p) d p =
p p dp
where
(p) = 2
1
e-i(p/ )x’
')'( dxx p
=(p) is the Fourier transform of = (x)
p ≡ (p)
Amplitude probability that a particle
in the state can be found, upon
making a measurement, in the state p.
p 2 dp ( p) 2
dp
Probability that a particle in the state be found with a momentum within the interval ( p, p+ d p).
< p > =
p (p) 2dp =
p p 2 dp
Average linear momentum of an ensemble of systems in
the state
(17)
Expressions (4) and (17) summarizes the objective of this section, showing an expansion of the
state in two different basis,
the “continuum spatial-coordinates base” { x , x }; and
the “linear-momentum base” { p , - ∞ < p < ∞ }.
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SUMMARY
Coordinates basis Momentum basis
=
x x dx =
p p dp
=
x (x) dx ; =
p (p) d p
(p) is the Fourier transform of = (x)
p is such that,
x p ≡ p ( x ) = 2
1
π e
i( p / )
x
10.1.C Tensor Product of State Spaces
Consider the state space 1 comprised of wavefunctions describing the states of a given
system (an electron, for example). We use the index-1 to differentiate it from the state space
2 comprised of wavefunctions describing the state of another system (another electron, for
example) which is initially located far away from the system 1. The systems (the electrons) may
eventually get closer, interact, and then go far away again. How to describe the space state of
the global system ? The concept of tensor product is introduced to allows such a description
… …
1
1
2
2
1
2
Let }{ ... 3, 2, 1, i , (1)i u be a basis in the space 1ε , and
}{ ... 3, 2, 1, i , v (2)i be a basis in the space 2ε
The tensor product of of 1ε and
2ε , denoted by 21 εεε , is defined as a space whose basis
is formed by elements of the type,
} { (2)v ji (1) u
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That is, an element of 21 εε is a linear combination of the form,
(2)v ji
ji,
(1)ij uc
The tensor product is defined with the following properties,
[ (1) ] (2) [ (1) (2) ]
(1) [ (2) ] [ (1) (2) ]
[ (1)(1) ] (2) [ (1) (2) ] + [ (1) (2) ]
(1) [ (2)(2) ] [ (1) (2) ] + [ (1) (2) ]
Two important cases may arise.
Case 1: The wavefunction is the tensor product of the type,
(2) (1)
That is can be expressed as the tensor product of a state from 1ε and a state from
2ε .
(2)v jjiiji
(1)
bua
(2)v jj
ji,
(1)ii uba
Case 2: The wavefunction cannot be expressed as the tensor product between a state purely from the space
1ε and a state purely from the space 2ε .
In this case, the wavefuntion takes the form,
(2)v jj i
ji,
(1)i uc
Let’s consider, for example the case in which each space 1ε and
2ε has dimension 2.
(2)v jj i
2
j i,
(1)i uc
(2)v (2)v 22 i
2
i
11 i
2
i
(1)(1) ii uu cc
(2)v (2)v 22 111 1 (1)(1) 11 uu cc
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(2)v (2)v 22 211 2 (1)(1) 22 uu cc
Notice, cannot be expressed in the form (2) (1)
To make the case even simpler, let’s assume c11 = c22 = 0,
(2)v 22 1 (1)1 uc + (2)v 11 2 (1)2 uc entanglement
cannot be expressed in the form of a single term of the form (2) (1)
10.2 The Schrödinger Equation as a postulate
10.2.A The Hamiltonian equations expressed in the continuum spatial coordinates. The Schrödinger Equation.3
In chapter 8 we obtained the general Hamiltonian equations that describe the time evolution of
the wavefunction (t) =n
n An( t ),
jjnn AtH
dt
dAi
j )(
(18)
Chapter 9 described the particular case of an electron moving in a lattice (the latter constituted by atoms separated a distance “b”. When we took the limit b 0 the Hamiltonian equations in (25) took the form
t
tx,i
)( )()(
)(
2
2
tx,tx,Vx
tx,
m 2
2
eff
(19)
Let’s consider now an arbitrary general case.
How does the Hamiltonian equations (18) look like when expressed
in the in the continuum space coordinates { x , x }?
Let’s find out such general formal expression (one that is more general than expression (19) ).
First notice that the amplitudes jA in (18) account for the state describing the quantum
system,
= j
j Aj
Since Aj can also be written as Aj = j , Eq. (25) can also be expressed as,
dt
di n =
jjnH
j
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Let’s also recall, from Chapter 7, that the coefficients jnH
are obtained from the Hamiltonian
operator H~
(specific to the problem being solved.) That is, jnH
≡ n H~ j. In general, H
~ and
depend on t.
dt
di n =
j n H
~ j j
In the continuum space coordinates we should expect,
dt
di x = x H
~ x’ x’ d x’
(x) ( x’ )
dt
di ( x ) = H( x, x’ )( x’ ) d x’ (20)
where we have defined H( x, x’ ) ≡ x H~ x’
Quoting Feynman,4
“According to (20), the rate of change of at x would depend on the value of at all other points x’ .
x H~ x’ is the amplitude per unit time that the electron will jump from
x to x’.
It turns out in nature, however, that this amplitude is zero except for points x’ very close to x. …
… This means (as we saw in the example of the chain of atoms) that the right-hand
side of Eq. (20) can be expressed completely in terms of and the spatial derivatives
of , all evaluated at x.
The correct law of physics is
H( x, x’ ) ( x’ ) d x’ = 2
2
dx
d
m2
2 ( x ) + V ( x) ( x ) Postulate (21)
Where did we get that from? …Nowhere.
It came out of the mind of Schrodinger, invented in his struggle to find an understanding of the experimental observation of the experimental world. ”
Using (21) in (20) one obtains,
17
(22)
),(
2
2
22
txVxmt
i
Schrodinger Equation
“This equation marked a historic moment constituting the birth of the quantum mechanical description of matter. The great historical moment marking the birth of the quantum mechanical description of matter occurred when Schrodinger first wrote down his equation in 1926.
For many years the internal atomic structure of the matter had been a great mystery. No one had been able to understand what held matter together, why there was chemical binding, and especially how it could be that atoms could be stable. (Although Bohr had been able to give a description of the internal motion of an electron in a hydrogen atom which seemed to explain the observed spectrum of light emitted by this atom, the reason that electrons moved this way remained a mystery.)
Schrodinger’s discovery of the proper equations of motion for electrons on an atomic scale provided a theory from which atomic phenomena could be calculated quantitatively, accurately and in detail.” Feynman’s Lectures, Vol III, page 16-13.
Although the result (22) is kind of a postulate, we do have some clues about how to interpret it, based on the particular case of the dynamics of an electron in a crystal lattice, studied in Chapter 9.
10.2.B Interpretation of the Wavefunction
Einstein’s view on the granularity nature of the electromagnetic radiation
In Chapter 5, a harmonic function was used to describe the motion of a free particle in analogy to the existent formalism to describe electromagnetic waves,
][),(2
νtxCostxλ
πo εε electromagnetic wave
where, the electromagnetic intensity I (energy per unit time crossing a unit cross-section area
perpendicular to the direction of radiation propagation) is proportional to 2
),( txε .
Einstein (in the context of trying to explain the results from the photoelectric effect) introduced the granularity interpretation of the electromagnetic waves (later called photons), abandoning the more classical continuum interpretation.
In Einstein’s view, the intensity is interpreted as a statistical variable hNcI o
2ε .
Here N constitutes the average number of photons per second crossing a unit area
perpendicular to the direction of radiation propagation;
N~2ε
Average values are used in this interpretation because the emission process of photons by a given source is statistical in nature. The exact number of photons crossing a unit area per unit
time fluctuates around an average value N .
18
Max Born’s Probabilistic Interpretation of the wavefunction In analogy to Einstein’s view of radiation, Max Born proposed a similar view to interpret the
particle’s wave-functions. In Max Born’s view, 2
),( tx plays a role similar to 2
),( txε , 2
),( tx is a measure of the probability of
finding the particle around a given place x and at a
given time t .
This interpretation was introduced years after Schrodinger (1926) had developed a formal quantum mechanics description. More specifically,
2
)( tz,y,x, plays the role of probability density.
Pictorially, the particle is more likely to be at locations where the wavefuntion has an appreciable value.
Deterministic evolution of the wavefunction
The predictions of quantum mechanics are statistical. In order to know the state of motion of a particle, we must make a measurement But a measurement necessarily disturbs the system in a way that cannot be completely determined.
However, notice that , being the solution of a differential equation (the Schrodinger
equation), varies with time in a way that is completely deterministic. That is, if were
known at t=0, the Schrodinger equation determines precisely its form at any future time.
That is, QM makes deterministic prediction of an amplitude probability wave. However, the latter does not convey to deterministic outcomes.
There is one further point to consider. How to determine the wavefunction at t=0? How
do the experimental measurements lead to the reconstruction of the wavefunction? Or, how to prepare a system in a definitely unique state? If we could not reconstruct a wavefunction, what would be the benefit of having a theoretical formulation that describes a deterministic evolution of something we do not know? As it turns out, despite the fact that measurement in QM in general affect the state of a system, such a reconstruction is possible in some cases (think of system that are in stationary sates.) But in a larger context, to benefit of the QM deterministic formulation what we need is to prepare a system (or many systems) in a definite state; for the theory could then be used to make predictions about the evolution of that particular state. We will address this issue in the next chapters, after the introduction of observables and eigenstates. We will see that finding eigenstates common to different observables narrows the selection pool of states in which the system can be found. This procedure leads to the concept of “ensemble of system” constituted by (in this way) “equally prepared systems”, which constitutes the “laboratory” in which the QM concept are develop. (A the end,
19
systems cannot be determined with absolute certainty simply because a set of
measurements at t=0 at most may lead to the determination of but not to uniquely
define ).
Let’s explain the statistical interpretation a bit further in the context of an ensemble of identically prepared systems.
Ensemble5
Imagine a very large number of identically prepared independent system (assumed to be all of them in the same state), each of them consisting of a single particle moving under the influence of a given external force.
All these systems are identically prepared.
The whole ensemble is assumed to be described by a complex-variable single wavefunction )( tz,y,x, , which contains all the information that can be obtained about them.
Ensemble .
.
.
1
2
N
describes the whole ensemble
is used to make a probabilistic prediction on what may happen in a particular member of the ensemble.
It is postulated that:
If measurement of the particle’s position are made on each of the
N member of the ensemble, the fraction of times the particle will
be found within the volume element d
3r =dx dy dz around the
position )( tz,y,x,r at the time t is given by
),,,(),,,( * tzyxtzyx d
3r
where * stands for the complex conjugate number.
(23)
Notice that this is nothing but the language of probability; in this case, position probability density P.
2* )( )( )( )( tz,y,x,tz,y,x,tz,y,x,tz,y,x,P
20
Caution: For convenience, we shall often speak of “the wavefunction of a particular system”,
BUT it must always be understood that this is shorthand for
“the wavefunction associated with an ensemble of identical and identically prepared systems”,
as required by the statistical nature of the theory.6
10.2.C Normalization condition for the wavefunction
The probabilistic interpretation of the wavefunction implies, therefore, the following requirement:
1)(*)( spaceAll
r3dtz,y,x,tz,y,x, (24)
because given a particle, the likelihood to find it anywhere should be one. Inherent to this requirement is that,
0),(
2
rr t (25)
Notice that if is a solution of the Schrodinger equation, the function c (c being a constant)
is also a solution. The multiplicative factor c therefore has to be chosen such that the function c satisfies the condition (24). This process is called normalizing the wavefunction.
In general, there will be solutions to the Schrodinger equation (22) whose solution tend to an infinite value. This means they are non-normalizable and therefore can not represent a particle probability density. Such functions must be rejected on the grounds of Born’s probability interpretation.
Quantum mechanics states are represented by square- integrable functions that satisfy the Schrodinger equation.
The particular subset of square integrable functions form a vector space called the Hilbert space.
QUESTION: Suppose that is normalized at 0t . As the time evolves, will change. How do
we know if it will remain normalized?
Here we show that the Schrodinger equation has the remarkable property that it automatically preserves the normalization of the wavefunction:
21
If satisfies the Schrodinger equation
Then if the potential is real
dxtx, dt
d 2 )( = 0 (26)
Proof: Let’s start with
dxtx,t
dxtx,dt
d
22
)( )(
(27)
We provide below a graphic justification of (27).
fx, t +)dx -
fx, t +)
1
t
fx, t ) dx
1
t [fx, t + ) - fx, t) ]
dx
=
fx, t )
x x
On the other hand,
tttt
)2
(28)
We use the Schrodinger equation (29) to calculate the time derivatives,
)(2 2
22
tx,Vxm
i
t
)(2 2
2
tx,Vi
xm
i
)(
2
2
2
tx,Vi
xm
i
t
Taking the complex conjugate, and assuming that the potential is real,
)(Vi
m2
i2
2
tx,xt
Adding the last two expressions,
2
2
2
2
m2
i
xxtt
22
xxx2m
i
(29)
Replacing (29) in (28) we obtain,
2
t
xxx m2
i
Accordingly,
dxtx,t
dxtx,dt
d
22 ) (
)(
dxxxx
m2
i
xx
m2
i
The expression on the right is zero because 0 xx
10.2.D The Philosophy of Quantum Theory
There has been a controversy over the Quantum Theory’s philosophic foundations.
Neils Bohr has been the principal architect of what is known as the Copenhagen interpretation (statistical interpretation)
Einstein was the principal critic of Bohr ‘s interpretation.
His statement “God does not play dice with the universe”, refers to the abandonment of strict causality and individual events by quantum theory.
Heisenberg counteracts arguing: “We have not assumed that the quantum theory (as opposed to the classical theory) is a statistical theory, in the sense that only statistical conclusions can be drawn from exact data.
In the formulation of the causal law, namely, ’if we know the present exactly, we can predict the future’ it is not the conclusion, but rather the premise which is false. We cannot know, as a matter of principle, the present in all its details.
Louis de Broglie, on the other hand, argues that that limited knowledge of the present may be rather a limitation of the current measurement methods being used.
He recognizes that
23
a) it is certain that the methods of measurement do not allow us to determine simultaneously all the magnitude which would be necessary to obtain a picture of the classical type, and that
b) perturbations introduced by the measurement, which are impossible to eliminate, prevent us in general from predicting precisely the results which it will produce and allow only statistical predictions.
The construction of purely probabilistic formulae was thus completely justified.
But, the assertion that i) The uncertain and incomplete character of the knowledge that experiment at its
present stage gives us about what really happens in microphysics, is the result of ii) a real indeterminacy of the physical states and of their evolution,
constitutes an extrapolation that does not appear in any way to be justified.
De Broglie considers possible that looking into the future we will be able to interpret the laws of probability and quantum physics as being the statistical results of the development of completely determined values of variables which are at present hidden from us.
Louis de Broglie’s view given above highlights the objection to quantum mechanics’ philosophic indeterminism.
According to Einstein:
“The belief of an external world independent of the perceiving subject is the basis of all natural science.”
Quantum mechanics, however,
regards the interaction between object and observer as the ultimate reality;
rejects as meaningless and useless the notion that behind the universe of our perception there lies a hidden objective world ruled by causality;
confines itself to the description of the relations among perceptions7
Physics has given up on the problem of trying predicting exactly what will happen in a definite circumstance.
10.3 Expectation values
10.3.A Expectation (or mean) value of a particle’s position
Let’s assume we have a system consisting of a box containing a single particle, which is
(we assume) in a state . Since 2
)( x is interpreted as probability, the expectation value of
the particle’s position is defined by,
dxxxx
-
2)( (30)
But what does this integral exactly mean?
24
It is worth to emphasize first what type of interpretation should be avoided.8
Expression (30) does not imply that if you measure the position of the particle over and over
again then dxxx
-
2)( would be the average of the results.
In fact, if repeated measurements were to be made on the same particle, the first measurement (whose outcome is unpredictable) will make the wavefunction to collapse to a state of corresponding particle’s position x (let’s say x0); subsequent measurements (if they are performed quickly) will simply repeat that same result x0.
On the contrary, dxxxx
-
2)( means the average obtained from measurements
performed on many systems, all in the same state. That is,
An ensemble of systems is prepared, each in the same state, and measurement of the position is performed in all of them. x is the average from such a measurement.
Ensemble .
.
.
1
2
N
describes the whole ensemble
is used to make a probabilistic prediction on what may happen in a particular member of the ensemble.
Fig. 10.1 Ensemble of identically prepared systems. When we say that a system is in the
state , we are actually referring to an ensemble of systems all of them in the same
state. Thus, represents the whole ensemble.
10.3.B Calculation of dt
xdm
As time goes on, the expectation value x may change with time, since the wavefunction evolves with time. Let’s calculate its rate of change.
dxt
xdxtxt
xdxtxxdt
d
dt
xd
22
),(),(
25
dxtt
x
For the case where the potential is real, we obtained in expression (29) that,
2
2
2
2
2 xxm
i
tt
2
2
2
2
2 xxm
ix
dt
xd
xxxm
ix
2
After integrating by parts, it gives
xxm
i
dt
xd
2
Integrating one more time by parts (just the second term on the right side,)
dx
xm
i
dt
xd
(31)
We would be tempted to postulate that the expectation (or mean) value of the linear
momentum is equal to dt
xdmp
. In that case, expression (31) would give,
dt
xdmp
dx
xi
=
xi
(32)
However, look at the functions inside the integral. How to understand that a term like x
would lead to the average value of the linear momentum? This appears a bit strange, to say the least. (This result will make more sense in the sections below, when the concept of quantum mechanics operators is introduced).
10.3.C Expectation (average) values are calculated in an ensemble of identically prepared systems
In general, the mean-value of a given physical property f (namely, energy, linear momentum, position, etc.), more generically called observable, is obtained by making measurement in each of the N equally prepared systems of an ensemble (and not by averaging
26
repeated measurements on a single system.) The whole ensemble is assumed to be described by a complex-variable single wavefunction )( tz,y,x, .
When making measurements on each of the N identically prepared systems of the
ensemble (see left-side of the figure below) let’s assume we get a series of results (see right-side of the figure below) like this:
N1 systems are found to have a value of f equal to f 1,
from which we deduce that the particular system collapsed to the
state f1 right after the measurement N2 systems are found to have a value of f equal to f 2,
from which we deduce that the particular system collapsed to the
state f2 right after the measurement
etc.
Accordingly, the average value of f is calculated as follows,
N
NN
Nn
n
nn
n
n fff av
1
(33)
Usual procedure to calculate average values
Notice that when the total number of measurement N is a very large number, the ratio N
Nn
is nothing but the probability of finding the system in the particular state fn .
QM postulates that the value of 2
ψfn should be interpreted as the probability to obtain
the value fn,
2
ψfn = NNn
Quantum Mechanics postulate (34)
Representing the
ensemble ψ
Hence, expression (33) can be written as,
2
av
ψfff nnn
(35)
27
Ensemble .
.
.
1
2
N
.
.
.
1
2
N
Before the measurements
The value of f in each system is unknown.
After the measurements
The value of f has been measured in each system; afterwards we proceed to
calculate the average value < f >.
10.4 Operators associated to Observables
Quantities such as position, momentum, or energy (which are measured experimentally) are called observables.
In classical physics, observables are represented by ordinary variables (E, p, for example).
In quantum mechanics observables are represented by operators (quantities that
operate on a function to give a new function.)
When a system in a state enters some apparatus, like, for example, a magnetic field in the Stern Gerlach experiment, or a maser resonant cavity, it may leave in a different
state .
That is, as a result of its interaction with the apparatus, the state of the system is modified.
Symbolically, the apparatus can be represented by a corresponding operator F~ such
that,
= F~ (36)
Note: We will distinguish the operators (from other quantities) by putting a small
hat ~ on top of its corresponding symbol.
We show below how to associate a quantum mechanics operator F~ to a given physical
quantity f.
10.4.A Observables, eigenvalues and eigen-states
28
Let’s consider a physical quantity or observable f ( f could be the particle’s angular
momentum for example) that characterizes the state of a quantum system.
In quantum mechanics, the different values that a given physical quantity f (observable) can take,
f1, f2 , f3 , … (37)
are called its eigenvalues;
The set of these quantum eigenvalues is referred to as the spectrum of eigenvalues of the corresponding quantity f. For simplicity, let’ assume for the moment that the spectrum of eigenvalues is discrete.
n will denote the state where the quantity f has the value fn ;
These states will be called eigenstates (38)
We will assume that these eigensates satisfy,
n m =
n( x) m
( x) dx = nm (39)
Let’s assume also that the eigenstates associated to the observable f constitute a basis,
{ n ; n = 1, 2, 3, …} Basis of eigenstates
An arbitrary state can then be represented by the expansion,
= n
n An = n
n n . (40)
where An = n =
n( x) ( x) dx.
Since the state must be a normalized state, then
n
An 2 =
n
n 2 = 1
According to expression (35), the mean value of f , when the system is in the state , is given by,
av
f = n
f n An 2 =
n
f n n 2 (41)
Expression (41) is still very general, since we do not know yet the eigenstates n.
We describe next how to figure out those states.
29
10.4.B Definition of the QM operator F~ to be associated with the
observable physical quantity f. 9,10
The operator F~ to be associated with the observable f is such that, when acting on an
arbitrary state , satisfies the following:
Definition of the Operator F~
associated to the observable f
F~ ≡ av
f (42)
(the average value on the right side is calculated over the ensemble represented by the state )
But, how to obtain an explicit expression for such an operator F~ ?
If the observable quantity f were, for example, the linear momentum, how to build its corresponding quantum mechanics operator?
Before building the operators explicitly, we first we derive in this section a general self-consistent expression that shows how an operator, associated to a given observable, should look like (see expression (47) below). Subsequently, sections 10.4.C and 10.4.D will provide a specific procedure on how to construct the position and momentum operators.
Deriving a self-consistent expression of a QM operator F~
If the system is in the state ,
= n
n An = n
n n ,
the average value of the quantityc f is given by (41),
av
f = n
fn An 2 =
n
fn n 2
The requirement to build the operator F~ is,
F~ ≡ av
f
= n
f n An 2 =
n
f n n 2
= n
f n n * n
= n
f n n n (43)
30
Notice, we can obtain a more compact expression for the operator F~ if we use the notation,
n proj,P~
≡ n n Projection operator (44)
Expression (44) describes an operator that when acting on a state gives the projection of
that state along the state n ; that is,
n proj,P~
= n n (45)
Accordingly, (43) can be expressed as,
F~ = [ n
f n n n ]
number state
operator
(46)
where n denote the state where the quantity f has the value fn ;
This notation trick allows us to express the operator in a more compact form,
F~ =
n
f n n n
eigenvalues Projection operator n proj,P~
built
out of eigenstates associated
to the physical quantity f.
(47)
This is a self-consistent expression for the quantum
operator F~ that will be associated with the classical
quantity f. It is an expression that is compatible with the
requirement that av
~fΨΨ F .
Note: The operator defined in (47) should remain the same if we changed the basis to express
the n. This claim is supported by the fact that the value of
av f , which intervenes
in the definition of F~ , should be independent of the basis chosen.
Notice (47) is self-consistent with its definition. By applying the operator F~ to one of the
eigenstates, it gives,
F~ j = jf j (48)
31
That is, once the operator F~ (associated to the physical quantity f ) is known, the
eigenfunctions of that given physical quantity are the solutions of the equation F~ =
where is a constant.
Still, notice that (47) and (48) give just self-consistent expressions for the operator F~ . It is an
expression that is compatible with the requirement that F~ ≡ av
f , but it is defined
in terms of eigenstates j that, for a given physics quantity f, we do not know yet but would
like to find out. Accordingly, F~ is not completely known yet. But, paraphrasing Landau,
“Although the operator F~ is defined by (48), which itself contains the
eigenfunctions n, no further conclusions can be drawn from the results we have obtained. However, as we shall see below, the form of the operators for various physical quantities can be determined from direct physical considerations, which subsequently, using the above properties of the operators, will enable us to find
the eigenfunctions and eigenvalues by solving the equation F~ = ” 11
Expressions (42) and (47) will guide us to build the quantum operators.
10.4.C Definition of the Position Operator X~
We are looking for operator X~
such that X~
gives us the
mean value of the position when the system is in the state ,
X~
≡ dxxxx
-
2)( , X
~ =? (49)
*( x) [ ( X~ )( x) ] dx ≡ x
[ *( x) ] x [ ( x) ] dx
which requires,
( X~
) ( x ) = x ( x ) (50)
Notice, we cannot say X~
= x; that would be incorrect.
(For instance, what value of x would you choose to make the expression X~
= x valid?).
More appropriate is first to define the identity function I, which satisfies,
I(x) = x,
Then (50) can be written as,
( X~
) ( x) = (I ) ( x)
32
because that would give [X~]( x) = [I ] ( x) = I( x) ( x) = x ( x) .
That is,
X~
= I Answer (51)
In summary,
The Position Operator X
~
X~
is the operator associated to x
X~
= I where I is the identity function; I (x) = x
( X~) x= x x (52)
X~
=
[ *x ] [ X~x ] dx =
[ *x ] x [x]
≡ x
Notice, it is straightforward to realize that2~
X is the operator associated to 2 x , and that
3~X is the operator associated to 3x , etc. In general,
nX
~ is the operator associated to nx (53)
Average value Quantum mechanics
of physical quantity operator
av x X
~
av
2
x ~2X
…
av
nx
~nX
More general,
if h(x) is a polynomial or a convergent series, (54)
then h(X~
) will be the operator associated to )( xh
33
For example: If a physical quantity changes as,
h(x) = 3 x2 - 27 x3
then the corresponding operator will be
3 2X~
- 27 3X~
,
that is, the last expression is obtained by evaluating h(X~
).
Average value Quantum mechanics
of physical quantity operator
av )(xh h(X
~)
For example, if )(xh were a classical potential that depends only on position x, then h(X~
)
would be then the corresponding quantum mechanics potential operator. Take the case of the harmonic oscillator, where V(x) = (1/2)k x
2
the corresponding QM operator is
V~ = (1/2)k 2~
X .
10.4.D Definition of the Linear Momentum Operator
Here we construct the quantum Linear Momentum Operator, and give its representation in both the momentum-coordinates and spatial-coordinates basis. This example will help to illustrate that operators are general mathematical concepts independent of the base used, but whose representation depends on the particular base states being selected.
10.4.D1 The Linear Momentum Operator P~
expressed in the momentum states basis { p }
In this case, the observable f refers to the linear momentum p.
Before building the “linear momentum operator” let’s summarize what we know about the momentum states. According to de Broglie, for a given p we associate a momentum state
p given by expression (12) above ,
p ≡
x’ p ( x’ ) dx’ =
x’ 2
1
π e
i( p / ) x’ dx’ (55)
Representation of the momentum state p in the
space-coordinate basis { x }
x p ≡ p ( x ) = 2
1
π e
i( p / ) x
34
Sometimes, for convenience, we will use the notation momentum p or p instead of
p .
From the Fourier transform theory we learned that an arbitrary state can be expressed as a linear combination of the momentum states,
=
p p dp =
p (p) d p (56)
where
(p) = p =2
1
e-i(p/ )x’
')'( dxx
Now we want to build the Linear Momentum Operator that will be associated to the
physical quantity p.
We are looking for the operator P~
such that P~
gives us the
mean value of the momentum when the system is in the state .
P~ ≡ x , P
~=?
But first, if a system is in the state =
p p dp =
p (p) d p ,
What is the average momentum p ?
Answer: p =
p (p) 2 d p
Hence,
We are looking for the operator P~
such that
P~
= p =
p (p) 2 d p; P~
= ?
p =
p (p) 2 d p
=
(d p) p (p ) (p)
=
(d p)(p ) p(p) .
35
=
(d p) (p )
p’ (p’) ( p’ – p ) dp’
Using p’ p = ( p’ – p ) = ( p – p’ ) = p p’
=
(d p) (p )
p’ (p’) p p’ dp’
= [
p (p) d p] [
p’ (p’) p’ dp’ ]
p = [
p’ (p’) p’ dp’ ] (57)
With the hint taken from expression (47), we realize the following perator,
P~
=
p p p dp Linear Momentum Operator (58)
would do the job
In effect, applying P~
to an arbitrary state gives,
P~ = [
p p p dp] ,
=
p p p dp,
According to (56)p =(p), which gives
P~ =
p p (p) dp (59)
Replacing (59) in (57) one obtains,
p = P~ (60)
This confirms the oprator defined in (58) is indeed the momentum operator.
36
Matrix representation of the Linear Momentum Operator P~
in the momentum coordinates basis { p , - ∞ < p < ∞ }
Notice in (57) that applying P~
to a particular state p’ gives,
P~ p’ = p’
p’ (61)
p P~ p’ = p’
p p’
= p’ ( p – p’ )
'
~pp
P = p P~
p’ = p’ ( p’ – p ) (62)
Elements of the matrix representation of the operator in the momentum basis
Summary
The Momentum Operator P~
P~
=
p p p dp
=
p p dp =
p (p) d p
where = (p) is the Fourier transform of (x)
P~ =
p p p dp =
p (p) p dp
P~ =
p (p) 2 d p
= p
10.4.D2 The linear momentum operator P~
expressed in the spatial coordinates basis { x , x }
Expression (59) indicates that if the system were in the state and we wanted to
evaluate P~ then we have to first expand in the momentum basis =
p (p)dp
37
That way, once all the (p) values are known, then we would be able to evaluate P~ =
pp (p) dp
What about if the expansion of in the momentum basis { p , - ∞ < p < ∞ } is not handy available, and, instead, its expansion in the spatial coordinate is available?
(i.e. (x) is known for every value of x, but its Fourier transform (p) is not ready available).
What to do to find P~
(without having to go through the trouble of expressing in terms of the momentum basis as expression (59) requires)?
Here an alternative to (59) is presented.
Applying (58), P~
=
pp p dp , to the state function =
p’ (p’) dp’
one obtains,
P~ = [
pp p dp] =
pp p dp
=
p p(p) dp
x P~ =
p (p) x p dp,
[P~x=
p (p)p x dp
Using x p ≡ p ( x ) = 2
1
π e
i( p / ) x
[P~x=
p (p)
2
1
π e
i ( p / ) x dp,
[P~x =
(p) p
2
1
π e
i ( p / ) x dp,
38
=
(p) dx
d
i
2
1
π e
i ( p / ) x dp
= dx
d
i
(p)
2
1
π e
i (p / ) x dp
= dx
d
i
(p) p xdp,
= dx
d
i
x
Hence,
P~=
dx
d
i
(63)
The Momentum Operator P~
(in the spatial-coordinates space)
P~
= dx
d
i
(64)
(alternative to expression (58))
P~ =
dx
d
i
which leads to
av p = P
~ = dx
dx
d
i
In passing, notice that the last expression is identical to (32) where we calculated the rate of
change of the average position of a particle dt
xdm
. At that time, the presence of a spatial-
derivative dx
d
i
appeared to make no sense in being involved in what could be interpreted as
the velocity of a particle. Now we see that such spatial derivative appears because we are using the representation of the momentum operator in the spatial coordinates (that spatial derivative does not appear when working in the momentum basis).
39
10.4.D3 Systematic way to express the operators P~
, 2~P , 3~
P , etc., in the spatial-coordinates basis
Construction of the operator associated to av
2 p
By definition:
av
2
p =
p (p) 2 d p (65)
=
p p 2 p dp
The systematic procedure consists in evaluating first the factor p2 p inside the
integral.
p2 p = p2
2
1
e-i(p/ )
x’ ')'( dxx
= 2
1
p2 e
-i(p/ )
x’ ')'( dxx
Continuing with a similar procedure as above (integrating by parts), we obtain,
p2 p = p
2
i
2
2
dx
d (66)
Replacing (66) in (65) leads to,
av
2 p =
p p 2
i
2
2
dx
d dp
Factoring out
=
p p 2
i
2
2
dx
d dp
The integral term is the expansion of the ket 2
i
2
2
dx
d in the momentum base
av
2
p = 2
dx
d
i
40
=
*x ,t
2
dx
d
i
x ,t ] dx
If we define the operator
2 ~
P = 2
dx
d
i
(67)
The last expression becomes,
av
2
p =
*x ,t 2 ~
P x ,t ] dx (68)
That is, 2 ~
P = 2
dx
d
i
satisfies the requirement for being the quantum operator to be
associated with p2.
More general, for a positive integer n,
av
np =
*x, t
n
dx
d
i
x ,t ] dx (69)
That is,
n ~
P = n
dx
d
i
is the quantum mechanics operator associated to pn
Average value Quantum mechanics
of physical quantity operator
av p
dx
d
i
av
2
p 2
dx
d
i
…
av
np
n
dx
d
i
If g(p, t) is a polynomial, or an absolutely convergent series, in the classical momentum variable p, one obtains,
41
av )( t p,g =
*x, t
)( t , dx
d
i g
x ,t ] dx (70)
Average value Quantum mechanics
of physical quantity operator
av )( t p,g )( t ,
dx
d
i g
An example of (70) constitutes the QM operator associated to the kinetic energy.
2
av
1p
2m 2~1
P2m
= 2
1
dx
d
i2m
= dx
d
2m 2
22
10.4.E The Hamiltonian operator
10.4.E.1 Mean energy in terms of the Hamiltonian operator One of the virtues of using QM operators is that mean values can be expressed independent
of a particular basis. For example, recall that < x > = X~
and <p> = P~ . We
are going to do something similar here for the average energy <E>.
In Chapter 8, the Hamiltonian Operator H~
was recognized through its matrix representation [H] , the latter being interpreted as an energy matrix due to the fact that, when working with stationary states, the components of the matrix were the energy of the corresponding stationary states. In the language of eigenvalues used in this chapter we can write,
H~ En = En En (71)
where
{ En , n=1, 2, 3, …} is the basis constituted by energy stationary states
For a system in an arbitrary state let’s calculate the expectation (or mean) value of energy.
= n
En En (72)
Its mean energy is given by,
2
n
n
nav ψE E E
= n
En * En En
42
= n
En En En
This factor can be expressed in terms of the Hamiltonian operator
From (71): H~ En = En En
= n
H~ En En
= H~
n
En En
=
av
E = H~ (73)
Thus, we have found again (as we did for the linear momentum and the position operators) an elegant way to express a mean value (in this case for the energy) that does not make reference to the particular base states.
When explicit notation of the basis states is required to calculate 2
avE , we may use the
stationary states basis { En , n=1, 2, 3, …}
2
n
n
nav ψE E E (74)
But, in some cases it may be convenient to use a different base. For a general basis
{ j for j =1,2,3,… } we will have,
av
E = H~
Expressing and in the { j } basis
= [ i
i* i ] H~
[ j
j j ]
= j i,
i i H~
j j (75)
10.4.E.2 Representation of the Hamiltonian Operator in the spatial coordinate basis Similar to the case of the linear momentum operator addressed in the previous sections,
the Hamiltonian operator can adopt different “shapes” depending on whether we are working in the spatial-coordinates basis or in the momentum coordinate bases. Here we address the representation of the Hamiltonian operator in the spatial-coordinates
43
In (73), let’s take the case of spatial coordinate basis
avE = H
~
= [
-
x * x dx ] H
~ [
-
x’ x’ dx’ ]
= [
-
x * x dx ] [
-
H~ x’ x’ dx’ ]
avE =
-
x *
-
x H~
x’ x’ dx’ dx ]
Defining
H( x, x’ ) ≡ x H~ x’ (76)
where x H~ x’ is the amplitude per unit time that the electron will jump
from x to x’.
(see also expression (20) above).
avE =
-
x *
-
H( x, x’ ) x’ dx’ dx
=
-
( x)*
-
H( x, x’ ) ( x’ ) dx’ dx
But Schrodinger established that (see expression (21) above)
-
H( x, x’ ) ( x’ ) d x’ = 2
2
dx
d
m2
2 ( x ) + V ( x)( x )
Accordingly
avE =
-
( x)* [
2
2
dx
d
m2
2 ( x ) + V ( x)( x )] dx
If we define the potential operator V~
,
Ψ VΨ V ~
(77)
avE =
-
( x)* [
2
2
dx
d
m2
2 + V
~ ] (x) dx
H
~
44
avE = H
~
Representation of the Hamiltonian Operator in the spatial coordinates basis
2
2
dx
d
m2
2~ H + V
~ (78)
Notice, according to expression (67) 2~1P
2m =
dx
d
2m 2
22 , therefore we can write,
V2m
~
~1 2~
PH
Summary ______________________________________________________________________________ Observable Mean Operator Operator in the spatial value coordinates representation ______________________________________________________________________________
X~
X~
position x ____________________________________________________________
x = X~
X~
= x ______________________________________________________________________________
P~
dx
d
i
momentum p ____________________________________________________________
p = P~ P
~ =
dx
d
i
______________________________________________________________________________
~
H 2
2
dx
d
m2
2 + V
~
Energy E ____________________________________________________________
E = H~
_______________________________________________________________________ ______________________________________________________________________________
10.5 Properties of Operators
10.5.A Correspondence between bras and kets
Bra Ket
χχ
45
*
m
m
m
m
m
m
b
b
m
m
m
m
m
m
b
b
(79)
b b
b b
* Φ Φ
ΦΦ
*
**
a a
aa
Φ Φ
ΦΦ
Until now we have defined the action of a linear operator A~
on kets χ . We want to define
the action of an operator on bras. 10.5.B The Adjoint operator
Consider an operator A~
. We picture an operator A~
that when acting on an arbitrary
wavefunction the result is, for example, 3 ; or dx
d; or I , etc. For a given A
~, we
will define its correspondent adjoint operator A~
. Through the definition we will find out what
does A~
do on a wavefunction ; that is is,
if A~
= 3 then we will find out ~A =?; or
if A~
= dx
d then we will find out
~A =?
… etc.
In addition, we would like to know also how to express the action of operator onto wavefunctions in the language of bras and kets.
For an operator acting on a ket, what does A~
χ mean ?
For an operator acting on a bra, what does A~
mean ?
Let A~
be an operator.
A
~ is the adjoint operator of A
~
46
if it which fulfills the following inner product for any arbitrary wavefunctions and ,
~ , ,
~ AA
(80)
Interpretation of the adjoint operator in bracket notation. First, a straight forward translation of expression (80) is to write is as,
AA~~
(81)
Taking the conjugate values,
AA~~
We would like to know also how to take the operators out of the bra or the ket.
First,
A~
A~
~
~
That is, we define
A~ A
~ (82)
Second, let’s compare the actions of an operator on the bras and kets
A~
A~
A~
A~
That is, A
~ is defined through the
following expression
A~
A~ (83)
In other words,
A~
applied on the ket gives the ket A~
A
~applied on the bra gives the bra A
~
Example: Show that )(A
~ = A
~
Applying the definition of adjoint operator to A~
,
A~ = A
~
47
AA~~
Applying the definition of adjoint operator to A
~,
AA~
)~
(
*~
A
Using the definition of adjoint
*~
A
)~
(A A~
This implies,
)
~(A A
~ (84)
Notice,
A~
A
~
A
~
)
~(A
That is,
)~
(A A~
(85)
Using (84)
A~
A~
(86)
Summary
A~ A
~ .
A~
A~ . (87)
A~
A~
Question: How to interpret the following quantity A~ ?
48
Does it mean ( A~
) or ( A~ ) ?
On one hand,
A~
= ( A~
)
A~
= ( A~
) (88)
On the other hand,
A~
= A~
= ( A~ ) (89)
From (88) and (89)
( A~
) = ( A~ ) (90)
That is, it in unnecessary to place the parenthesis. That is, in A~ , one obtains the same
result by making A~
to act first on the bra (and then multiply by ), or
by making A~
to act first on the ket (and then multiply by ).
In short,
A~
= ( A~
) = ( A~ ) = A
~
A~
= A~
= A~ (91)
Exercise: Show that the adjoint operator A~
is linear.
Example: Show that )(B
~ = B
~
49
Let’s apply the definition (81), AA~~
, to the operator A~
= B~
BB~~
)(
**]
~[ B
*]
~[ B
*]
~[ B
B~
Hence,
)(B~
= B~
Exercise. Given the operatorx d
dD ~
, what is the operator D~
?
( D~
, ) ≡ D~ =
* x) [xd
d
]x dx
=
* x)xd
ψd
x dx
Integrating by parts
= -
xd
φd
x) x dx =
[ D~
- x) ]* x dx
= ( - D~ , )
This implies,
D~
= - D~
Matrix Representation of the adjoint operator
( n ,
~m ) = ( m ,
~ n )
*
Hence their matrix representation are related through,
[ ~
nm = [ ~
mn* (92)
50
(Notice the order of the indexes is reversed)
Exercise: Show that
)
~
~( BA = B
~ A~
(93)
Example: What is the adjoint operator of the position operator X~
?
( X~
, ) ≡ ( , X~
) =
* x) [ X~ ]x dx
=
* x) x x dx =
x * x) x dx
=
[x x) ] * x dx =
[ X~ x) ] * x dx
( X~
, ) = ( X~ )
Since this expression is valid for any arbitrary states and , then
X~
= X~
(94)
That is, the adjoint of the position operator is itself.
10.5.C Hermitian or self-adjoint operators
Many important operators of quantum mechanics have the special property that when you take the Hermitian adjoint you get back the same operator.
~
= ~
. (95)
Such operators are called the “self-adjoint” or “Hermitian” operators.
Example: The position operator X~
is a self adjoint operator because X~
= X~
, as shown in the example in the previous section (see expression 91).
Example: Let’s see if the linear momentum operator P~
is self adjoint
( P~
, ) ≡ ( ,P~ ) =
* x) [ i
xd
d
]x dx
= i
* x)xd
ψd
x dx
Integrating by parts
51
= -i
xd
φd
x) x dx
=
[i
xd
φd
x) ]* x dx
= ( P~ , )
That is,
P~
= P~
(96)
Properties of Hermitian (or self-adjoint) operators
Operators associated to mean values are Hermitian (or self-adjoint) In section 10.4 above we defined quantum mechanics operators associated to a classical
observable quantity. The definition involved the calculation of mean values of observables. From the fact that mean values are real, we can draw some conclusions concerning the properties of those operators.
av
f = F~
Since this quantity is real, it will be equal t its complex conjugate
= F~ *
= F~
That is,
F~ = F~
Using definition of adjoint operator
F~ = F~
which implies,
F~ = F~ Operators corresponding to observables (97)
(i.e. operators obtained through the requirement
av f = F~ )
must be hermitians (self-adjoint).
The eigenvalues of a Hermitian (self-adjoint) operator are real.
Let j be and eigenvalue of F~ and j the corresponding eigenvector
F~ j = jj (98)
52
Since F~ is Hermitian (self-adjoint), F~ = F~ , for any state we will have,
F~ ) = F~ ,
In particular, for =j,
F~ j j = jF~j
Using (98)
jj j = j jj
jj j = j j j
which implies
*j = j ( for eigenvalues of Hermitian operators) (99)
Eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal
Let,
F~ j = j j and F~k = k k
where j ≠ k
k F~ j = k , j j and F~k j = k k j
Since F~ = F~ , and j as well as k are real, we obtain,
F~k j = j k j and F~
k j = k k j
This implies,
j k j = k k , j
( j - k ) k j = 0
Thus,
j ≠ k implies k j = 0 ( for Hermitian operators) (100)
10.5.D Observable Operators When working in a space of finite dimension, it can be demonstrated that it is always
possible to form a basis with the eigen-vectors of a Hermitian operator. But, when the space is infinite dimensional, this is not necessarily the case. That is the reason why it is useful to introduce the concept of an observable operator.
An Hermitian (or self-adjoint) operator ̂ is an observable (101) operator if its orthonormal eigen-states form a basis. . .
53
10.5.E Operators that are not associated to mean values In the previous section we addressed operators associated to i) classical physical quantities
f(x) that depends on the position x, and ii) physical quantities g(p) that depend on the momentum turn out to be hermitian (self-adjoint) operators.
Here we explore building operators associated to classical physical quantities like xp.
Let’s calculate
*( x ) [ PX~
~
] ( x ) dx
Naively, let’s call that quantity av xp
i. e. one would expect that the calculation of the integral will give a positive number and thus reflect a quantity associated to a classical measurement of xp. We will see below that this assumption is incorrect. As a matter of fact we will
realize that PX~
~
is not Hermitian.
av xp ≡
*( x ) [ P~
~X ] ( x ) dx
≡
*(x) [ x )(xdx
d
i
] dx
Rearranging the order of the terms,
≡
x *(x) [ )(xdx
d
i
]dx
dv
Integrating by parts
≡ -
dx
d[x *(x) ] } [ )(x
i
]dx
≡ -
[ *(x) ] } [ )(xi
]dx
-
[x dx
d *(x) ] } [ )(x
i
]dx
54
≡ i -
[x dx
d *(x)] [ )(x
i
]dx
≡ i +
[x *
i
dx
d*(x) ] )(x dx
≡ i +
[x
i
dx
d (x)]* )(x dx
≡ i +
[ P~
~X ]* (x) )(x dx
≡ i + [
)(* x [ P~
~X ] (x) dx ]*
av xp ≡ i +
*
av xp (102)
This result indicates that av xp is not a real quantity.
Let’s put the result (102) more explicitly in terms of operators.
Expression (102) can be written more explicitly,
*(x) [ P~
~X ] ( x )dx = i + [
)(* x [ P~
~X ] (x) dx ]*
P~
~X = i + P
~
~X ]*
P~
~X = i + P
~
~X
P~
~X = i +
)~
~
( PX
Although this has been demonstrated for identical bra and ket, the procedure with different bra and ket would give a similar result. That is,
PX~
~
= i + )
~
~( PX (103)
which shows more explicitly that the operator PX~
~
is not hermitian
55
Also, since )~
~
( BA = B~ A
~
PX~
~
= i + ~~XP
PX~
~
= i + XP~~
(104)
More properly, this result should be expressed as,
PX~
~
= i I~
+ XP~~
where I~
is the identity operator; 1~
What to do then if a quantity like xp is present in the classical Hamiltonian? How to build the corresponding quantum Operator?
The result in (104), that PX~
~
XP ~~
, indicates that different quantum mechanics operators may correspond to equivalent classical quantities
For example x p, p x, (1/2)( x p + p x ) are equivalent classically.
Still, three different operators PX~
~
, XP~~
, (1/2)( PX~
~
+ XP~~
) can be associated to the same classical quantity.
The guidance to select the proper operator is to manipulate the classical quantity such that the resulting operator result in a Hermitian operator.
In this particular case of x p, it turns out that,
~
(1/2) ( PX~
~
+ XP~~
) (105)
fulfills ~
~
10.6 The commutator
In general, two operator do not commute; that is, BA~~
is different than AB~~
(that is the case
of X~
and P~
, for example).
The commutator between two operators is defined as,
, A BBA BA~~
~~
]~~
[ (106)
According to (104),
i ]~
, ~
[ PX (107)
10.6.A The Heisenberg uncertainty relation
In this section we will be using terms like 1~
~
22 )( A AA . What does this
term mean? To get familiar first with that terminology, let’s work out explicitly a couple of terms. (You may skip the following paragraph and go directly to expression (109)).
56
The meaning of ( P~
- < p > )2
First, let’s expand in the based formed by its eigenstates
=
p p dp (i)
(Note: To indicate the momentum state p sometimes I am using the notation p . That is p = p . )
P ~
=
( P ~
p ) p dp
P ~
=
( p p ) p dp (ii)
Since < p > is a constant, then
< p > =
< p > p p dp (iii)
(ii) – (iii)
P ~
- < p > =
( p - < p > ) p p dp
[ P~
- < p > ] =
( p - < p > ) p p dp
[ P~
- < p > ]2 =
( p - < p > ) 2
p p dp
We realize this is the procedure to calculate standard deviations. In effect,
[ P~
- < p > I~
]2 =
( p - < p > ) 2
p p dp
=
( p - < p > ) 2
p p dp
=
( p - < p > ) 2 p 2 dp (108)
Probability with which the term ( p - < p > )
2 occurs
For two given observable operators A~
and B~
, let’s define the corresponding standard
deviation (the statistics is taken from an ensemble characterized by the wavefunction ,
1~
~
22 )( A AA (109)
57
1~
~
2 2)( B BB
To simplify the notation, let’s work with the Hermitian operators a~ and b~
defined as,
1~~
~ A Aa and 1~~
~
B Bb (110)
Notice,
] ~~
[ ]~~[ B ,Ab ,a (111)
2
A and 2
B can then be expressed as,
ψa ψψa ψa 22
Aσ~~~ (112)
ψb ψψb ψb 22
Bσ~~~
Consider the not Hermitian operator C~
,
bλia C~
~~ where is a real constant (113)
Notice: bλia C~
~~ , and
0~~~~
CCC C (114)
0~~
~~ ) () ( ψ bλi abλiaψ
0 ) ]~~ [
~~ ( 222 b,ai b a
Using (112),
0 ~~
222
][ B ,Ai BσσA (115)
Notice that the term ]~~
[ B,A must be a
purely imaginary number.
The function f = f () defined as,
)(f ~~
222
][ B ,Ai BσσA (116)
satisfies, according to (115),
0 )( f
In addition 0)(2 B" f . Therefore the value of at which 0)( 'f is a minimum; such
a value is,
58
ψ B,A ψ2
i2
Bσ]
~~[min
The value of f at min is ,
2
2
2
2
2
min ]~~
[ 2
1 ]
~
~ [
4
1 )( B,AB,Af
BσσBA
2
2
2 ]
~
~ [
4
1
B,A
BA
According to (116) this value must be greater or equal to zero.
0 ]~~
[ 4
1 2
2
2
B ,A
BA
222
]~
,~
[ 4
1 BABA Generalized uncertainty principle (117)
where ]~
,~
[ BA , according to (115), is a purely imaginary number.
10.6.B Conjugate observables Standard deviation of two conjugate observables
Two observable operators A~
and B~
are called conjugate observables if their commutator
is equal to i .
iBA ]~
, ~
[ definition of conjugate observables (115)
The result (114) then gives for this type of pair operators the following requirement,
2
BA uncertainty principle for conjugate observables (116)
In particular, the result (104) indicates that X~
and P~
constitutes a pair of conjugate observables. Hence,
2
px (117)
10.6.C Properties of observable operators that do commute
Some of the theorems listed below are valid even if the operators are not observables. But we will assume the latter, since the main objective of this section is to show that it is possible to build a basis out of eigenfunctions common to two observables.
59
Let’s start with the assumption that the eigenvalues { a1, a2, ,… } and eigenfunctions { 1a ,
2a , … } of the operator A~
are known.
Theorem-1:
Let A~
and B~
two observable operator such that 0]~
, ~
[ BA
If a is an eigenfunction of A~
then aB ~
is also an eigenfunction of A~
, with the same eigenvalue.
Proof:
A~ a = a a
B~
A~ a = a B
~ a
Since A~
and B~
commute
A~
( B~ a ) = a ( B
~ a ) (119)
Thus, the theorem is proven.
1
~a ,A
2a
~
1a
~
2a is the space
generated by the eigenfunctions of
A~
that have
eigenvalue2a
2
~a ,A
1~ a
B
2a
~
2a B
1a is the space
generated by the eigenfunctions of
A~
that have
eigenvalue1a
2a
1a
2a
The operator B
~ acting on an eigenfunction in the space a of A
~ gives a
state that remains in the space a .
Two cases arise:
i) The eigenvalue a is non-degenerate.
i.e. the eigenvalue-space a associated to a is one dimensional.
B~ a is therefore proportional to a ; that is,
60
B~ a = b a (120)
In this case a is also an eigenfunction of B~
.
ii) The eigenvalue a is ag -fold degenerate.
That is, there exists ag independent eigenvectors associated to the same eigenvalue a.
A~
a
j
= a a
j
for j = 1, 2, … , ag
The result (119) indicates that the following wavefunctions
B~ a
j
for j = 1, 2, … , ag
are also eigenfunctions of A~
and with the same eigenvalue a.
But we cannot state, in general, that the states a
j
are also eigenfunctions of B~
.
All we can say at the moment is that B~ a
j
belong to the space a ,
For a
j
a we also have B~
a
j
a , for j = 1, 2, … , ag (121)
Or more general,
For any a we also have B~ a .
and the space a is said to be globally invariant under the action of B~
.
Hence, theorem-1 can also be stated as,
If two operators A~
and B~
commute, every subspace of A~
(122)
is globally invariant under the action of B~
.
Theorem-2:
If two observables A~
and B~
commute, and if 1 and 2 are two eigenvectors with
different eigenvalues a1 and a2 respectively, then the matrix element 21 ~ B = 0.
Proof: Since 2 ~B
2a and 1
1a then they are orthogonal.
(Eigenvector of hermitian operators that have different eigenvalue are orthogonal; see expression 100 above).
Theorem-3:
If two observables A~
and B~
commute, one can construct an orthonormal basis with
eigenvectors common to A~
and B~
.
Proof:
Let A~ i
n
= an i
n
; n = 1, 2, 3, …
61
i = 1, 2, … , gn Here gn is the degeneracy of the eigenvalue an. That is, gn is the dimension of the space
na .
Also, nn'ii'i
ni'
n'
How does the matrix representing B~
in the basis { i
n
} look like?
Let’s arrange the basis { i
n
} in the following order
1
1 , 2
1 , … , 11
g ; 1
2 , 2
2 , … , 22
g ; …
We know: For n'n we know that 0~ i
ni'
n' B (see theorem 2 above).
For n'n we can say nothing a priori about in
i'n' B ~
(see below).
The matrix representing B~
is the a “block-diagonal matrix
1
000
000
000
000
0000
0000
0000
0000
0000
0000
0000
0000
000
0000
0000
000
1 2 3 4 5
1
2
3
4
5
Two cases arise: i) The eigenvalue an is non-degenerate.
Then the block n is a 11 matrix
ii) The eigenvalue an is ng -fold degenerate.
Then B~
i
n
n , but i
n
is not necessarily eigenstate of B~
The block n is the nn gg matrix
Notice, however, the matrix representation of A~
in the sub-base { i
n
; I = 1, 2, …, ng } is
diagonal. It will remain diagonal if the elements of the sub-base are replaced by any linear
62
combination of its elements. On the other hand, particular linear combination can be work
out in n to obtain eigenvectors of B~
; that is, we can always diagonalize the block-matrix
Let’s work out explicitly the simple case of ng =2.
a
1 and a
2 are known.
B~ a
1 and B~ a
2 are known.
Let’s express B~ a
1 and B~ a
2 as linear combinations of a
1 and a
2
B~ a
1 = 11 a
1 + 12 a
2 (123)
B~ a
2 = 21 a
1 + 22 a
2
The coefficients jk are known.
We have to find p, q and such that,
b = p a
1 + q a
2 ( p and q are unknown) (124)
is an eigenstate of the operator B~
B~
b = b ( is unknown) (125)
Note: we cannot ensure whether one or two different eigenvalues will be found. That is, we
cannot anticipate whether the degeneracy ng will be reduced or not after the following
procedure).
Using (124) in (125),
B~
( p a
1 + q a
2 ) = ( p a
1 + q a
2 ) (126)
Since B~
is a linear operator,
p B~
( p a
1 + q B~
a
2 = p a
1 + q a
2
p(11 a
1 + 12 a
2 ) +
+ q ( 21 a
1 + 22 a
2 ) = p a
1 + q a
2
( p11 + q 21 ) a
1 +
+ ( p12 + q 22 ) a
2 = p a
1 + q a
2
This implies,
11 p + 21 q = p (127) 12 p + 22 q = q
63
which can be rewritten as,
q
p
q
p
2212
2111
(128)
The coefficients jk are known
This is an eigenvalue problem. We expect then to obtain a couple of independent solutions,
(p1 q1) with eigenvalue and (129) (p2 q2) with eigenvalue
Using these solutions in expressions (124) and (125) above, gives a couple of eigensates for
the operator B~
.
1 1
b = p1 a
1 + q1 a
2
Eigenstates of B~
(130)
2 2
b = p2 a
1 + q2 a
2
It is straightforward to verify that these two states are also eigenstates of A~
.
We have thus shown a procedure to obtain eigenstates common to A~
and B~
. That is,
there exist in n eigenvectors of B~
.
Therefore, for the degenerate case, it is possible to find (131)
in n common eigenvectors to A~
and B~
.
In general, the eigenvalues andwill be different; but it may also occur that they are equal.
Let’s denote by i n,s
the eigenvectors common to A~
and B~
.
A~ i
n,s = an i
n,s (132)
B~ i
n,s = bs
i n,s
n,s specify the eigenvalues an and bs of A~
and B~
.
i = 1,2, …, nsg allows to distinguish the different basis eigenvectors corresponding to the
same eigenvalues an and bs of A~
and B~
. The reciprocal of theorem-3 is also valid.
If there is a base composed of eigenvectors common to A~
and ~B , (133)
64
then 0]~
, ~
[ BA
Proof:
For any state =
nn
n
u c
A~ =
~
nn
n
uA c =
n nn
n
ca u ; ~B =
~
nn
n
uB c =
nnn
n
b u c
AB~~ =
~n nn
n
a uB c =
n nnn
n
ab u c ; ~~BA =
~
nnn
n
b uA c =
nnnn
n
ba u c
That is,
AB~
~
= ~~BA , for any state
Hence, 0]~
, ~
[ BA
Let’s put theorem-3 and its reciprocal in just one statement;
Theorem-4:
Two observables A~
and B~
satisfy 0]~
, ~
[ BA .
If and only if there exists a basis composed of states (134)
that are mutual eigenfunctions of A~
and ~B . .
10.6.D How to uniquely identify a basis of eigenfunctions? Complete set of commuting observables
Let A~
be an observable. Its eigenfunctions { in , i = 1, 2, … , ng / for n=1, 2, 3, … … }
constitute a basis in space . Here ng specifies the degeneracy of the eigenvalue an.
If all ng =1: Then specifying an determines in a unique
way the corresponding eigenfunction n .
That is, there exists only only one basis formed by the eigenfunctions of A~
.
If a particular ng >1:
Specifying an is not sufficient to characterize a basis eigenfunction. Several set
of ng vectors in the sub-set n can be chosen as a base.
That is, the base of eigenfunctions of A~
is not unique.
Let’s choose then another observable B~
that commutes with A~
.
Let’s construct a basis comprising eigenfunctions common to A~
and ~B ,
{ i n,s
, i = 1, 2, … , nsg / for n=1, 2, 3, … }.
If all nsg =1: Then specifying an and bs determines in a unique
way the corresponding eigenfunction i n,s
.
65
That is, there exists only only one basis formed by the eigenfunctions of A~
and ~B .
If a particular nsg >1:
Specifying an and bs do not characterize a basis eigenfunction uniquely. The
diagonalization process of B~
in the subspace n may have rendered an
eigenvalue with multiplicity (degeneracy) nsg greater than 1. Hence several set
of nsg vectors in the sub-set ns can be chosen as a base.
That is, the base of eigenfunctions of A~
and B~
is not unique.
Let’s add then another observable C~
that commutes with A~
and B~
. Then, we repeat the
procedure of diagonalization of C~
in the sub-space ns , … , . And so on.
We repeat adding more operators that commute with the previous ones, until we obtain all
subspaces with degeneracy equal to 1. Thus, we have a set of commuting operators { A~
, B~
, … ,
Q~
} such that to any set of eigenvalues { na , sb , … , zq } there exists exactly one eigenfunction.
We call such an observation maximal or complete, and the corresponding set of observables a complete set of commuting observables.
A set of observables { A~
, B~
, C~
, … } is a complete set of commuting observables if there exists a unique orthonormal basis of common (135)
eigenfunctions.
Example. For the case of a particle in a central field, H~
, 2L~
and zL~
constitute a complete set of
commuting observables.
Notice, after an observation with a complete set of commuting observables { A~
, B~
, … ,Q~
} we
do not know the values of all possible observables.
- We cannot ascribe a definite value to an observable W~
that does not commute with
all the observables in the set { A~
, B~
, … ,Q~
}
- Only if W~
commutes with { A~
, B~
, … ,Q~
} can we ascribe a definite value to it.
However, after an observation with a complete set of commuting observables
{ A~
, B~
, … , Q~
} we do of course know the exact wavefunction of the system. Hence,
- We can calculate the statistics (mean values, probability distributions, etc.) of any
observable W~
.
It should be clear that there is not one unique complete set of commuting observables.
- If we had taken an observable W~
for the first measurement, and W~
does not commute
with every operator of the set { A~
, B~
, … ,Q~
}, we could still obtain a different complete
set of commuting observables.
Example. For the case of a particle in a central field, { H~
, 2L~
, zL~
} form a complete set of
commuting observables.
66
Since xL~
nor yL~
commute with zL~
, we cannot determine their values simultaneously
with zL~
.
But { H~
, 2L~
, xL~
} or { H~
, 2L~
, yL~
} would be equally acceptable complete set of commuting
observables. 10.7 Simultaneous measurement of observables
According to classical mechanics:
The state of a system with N degrees of freedom can be determined, at the time t, by
measuring the 2N position and momentum coordinates )(tqi , )(tpi ; i= 1, 2, …, N .
We can think of these measurements as performed simultaneously, or as carried out in very rapid succession so that the coordinates have not varied appreciable (on accounts of their development according to the equations of motion) during the time required for all the measurement.
But it is inherent in the concepts of classical physics that,
The measurements do not affect the values of the variables describing the system (i.e. the measurements do not affect the state of the system).
That the order of measuring the 2N coordinates is immaterial.
Once the 2N coordinates are known, all conceivable observables can be calculated.
In quantum mechanics the situation is entirely different.
It is no longer possible to specify the simultaneous values of all observables of a system.
In general, observables are mutually incompatible. Measuring one of them partly or completely destroys any knowledge about the others. Such an incompatibility of operators is related to the uncertainty principle (expression (117) above.)
Below we explore the maximum information (maximum number of observables) one can obtain from a system
10.7.A Definition of compatible (or simultaneously measurable) operators
If A~
and B~
are measured on a system (in quick succession) with the
results a and b, and an immediate re-measurement of A~
or B~
(136) necessarily reproduces these results, whatever the initial state of the
system, then we say that A~
and B~
are compatible or simultaneously measurable.
10.7.B Condition for observables A~
and B~
to be compatible It is rare for two arbitrary observables to be compatible. It would be interesting to establish
the condition under which two observables are compatible. One clue comes from the fact that
if two observables A~
and B~
possess a mutual eigenfunction (i.e A~ i
n,s = a i
n,s and B
~
i n,s
= b i n,s
) then, successive measurements with A~
and B~
would give the results a and b
67
and the system will remain in the same state i n,s
. It is straightforward to infer then that if A~
and B~
were to have a common basis of mutual eigenfunctions, then they would display their compatibility when measuring any state (not only the eigenfunctions). Indeed, we have a theorem of fundamental importance:
A necessary and sufficient condition for A~
and B~
to be compatible is for them to possess a complete orthonormal set of simultaneous (137) eigenfunctions.
Proof: i) Necessary condition
If A~
and B~
are compatible then B~
A~ = A
~B~ ; that is ][
~
~B ,A = 0. From theorem-4
(expression (134) above) a basis of common eigenfunctions exists.
ii) Sufficient condition
Let { i n,s
, i = 1, 2, … , nsg / for n=1, 2, 3, … } be a complete set of simultaneous
eigenfunctions of A~
and B~
.
Let’s assume the system is initially in a state .
= is,n,
ss Ψi n,
i n,
A~
= is,n,
ss ΨA i n,
i n,
~
= is,n,
ss Ψa i n,
i n, n
B~
A~
= is,n,
ss ΨaB i n,
i n, n
~
= is,n,
ss Ψab i n,
i n, ns
(138)
Similarly,
A~
B~
= is,n,
ss Ψba i n,
i n, sn
(139)
From (138) and (139).
][~
~
B ,A = 0.
Therefore B~
A~ = A
~B~ ; and A
~and B
~are compatible.
Putting together the results (134) and (137):
Theorem: Any one of these conditions imply the other one;
(1) A~
and B~
are compatible
(2) A~
and B~
possess a basis composed of simultaneous eigenvalues. (140)
(3) A~
and B~
commute
68
Summary
When two observables A~
and B~
are compatible, the measurement of B~
does not cause
any loss of information previously obtained from a measurement of A~
(and vice versa) but, on the contrary, adds to it. Moreover, the order of measuring the two observables is of no importance.
On the other hand, if A~
and B~
do not commute, the preceding arguments are no longer valid.
10.8 How to prepare the initial quantum states12
In the previous sections we have introduced i) wave functions as mathematical entities that represent states, and ii) operators as representing the physical quantities of systems. We have described how to predict the probable outcomes of a measurement once a wave function is known. We have also seen the association between operators and the average values of the physical quantities.
What has not been addressed yet is how to prepare quantum state ensembles upon which measurement can be made. We raised this question in in Section 6.4 of Lecture 6, but we postponed its discussion until we were armed with better mathematical tools and expanded concepts (like QM operators, observables, and, in particular, commutation properties). It is time then to retake and explain those questions.
Let’s start mentioning,
A system, classically described as one of n degrees of freedom, is completely specified quantum mechanically by a normalized wave
function ( q1, q2, … , qn ) which contains an arbitrary factor of modulus 1. All possible information about the system can be derived from this wave function.
(141)
How to build a QM wave function? How to use the classical measurements of the physical properties of a
system to build a QM wave function?
It turns out, “the state of a system, specified by a wave function in the Hilbert space, represents a theoretical abstraction. One cannot measure the state directly in any way. What one does do is to measure certain physical quantities such as energies, momenta, etc., which Dirac referred to as observables. From these observations one then has to infer the state of the system.”
We have introduced wavefunctions as representing states and operators as representing the physical observables of the system. We must now see how these quantities are related to actual measurements.
Given a system in a definite state, described by a given wavefuntion, how can we predict its physical properties, which are to be compared with experiments?
69
How to predict
energy linear momentum
angular momentum
of the system
?
Given
Conversely, how do we incorporate information about a system, gained as a result of measurement, into the wavefunction? How, for example, do we determine the initial state
0 a system upon which we can predict, using the Schrodinger, the state at a later time t?
Having measured Energies Linear momenta Angular momentum
?
How to infer the state of the system?
10.7.A Knowing , what can we predict about eventual outcomes from the measurement?
Consider F~ to be an observable operator associated to the observable f, which has a complete
orthonormal set of eigenfunctions { 1, 2, … } and corresponding eigenvalues { f1, f2, … }
If a system is in an eigenstate n of F~ , then measuring f gives a definite result f n.
In general, a system will be in a state , which is not an eigenstate of F~ . But it can be
expanded in the form = m
m cm . Measuring f on a system in a state no longer
leads to a definite result; rather there exists a probability distribution to find anyone of the values f1, f2, …
P ( f n ) = 2
nc Postulate (142)
The most important implication of this postulate is that, in contrast to classical mechanics, a
system with the maximal specification of its state (namely with a given wavefunction ) still shows dispersion; measuring f does not necessarily leads to one definite result.
This means, if we measured f on a large number of systems (the ensemble characterized by
) then the different possible results f1, f2, … would occur with a frequency given by (142).
10.7.B After a measurement, what can we say about the state of a system ? We come now to the converse question of how information obtained about a system
through measurements is incorporated in the wavefunction.
Let’s assume the observable f is measured on a system in the (143)
state , and the value f n (an eigenvalue of F~ ) is observed.
If this statement is to mean anything, it must imply that,
If one repeats the measurement of f sufficiently quickly one (144) necessarily again finds the same value f n.
70
[In general, this only holds for a sufficiently short interval of time between the two measurements (for if the interval is too long the system will have changed appreciably, according to the Schrodinger equation).]
If the second measurement is to have the definite result f n,
then the wavefunction ' (representing the system immediately after the first
measurement) must be an eigenfunction of F~ belonging to the eigenvalue f n.
Let’s make some predictions about ':
Case: The state before the first measurement is unknown
If f n is non-degenerate, then the state after the measurements ' is n .
If fn is gn-fold degenerate (i.e. r n for r =1,2, … gn have the same eigenvalue
fn), then we only know that ' lies in a gn -dimensional sub-space,
n
nnn
n
g
gr
r r c
c 1
1
1
2 r
r
' ( rnc unknown) (145)
The amplitudes rnc are in general unknown.
Case: The state before the first measurement is known
We calculate
rrnnc ( r
nc known) (146)
and postulate that the wavefunction ' is given by (140) as well.
That is, before the measurement, we have
n
nnn
n
k
grr
grnk
k c
c
c1
1
1
2 r
r
(147)
n
nnn
n
grr
grnk
kk
c1
1
1
2 r
r
If after a measurement on that measurement renders the eigenvalue fn, then:
n
nnn
n
grr
gr
c
c1
1
1 '
2 r
r
(148)
71
n
nnn
n
grr
grc
1
1
1
2 r
r
In the particular case that before the measurement we knew that is an eigenfunction belonging to the eigenvalue f n then, the above simply means that,
' = (149)
That is, the wavefunction is unchanged by the measurement.
We have just described how to incorporate information about a system, obtained as a result of experiments, into the wavefunction describing that system.
In general, before the measurement, there may be a variety of possible results. But once the experiments have been carried out and one particular result obtained, we can “discard” most of the wavefunction (corresponding to all the results which were not observed) and retain only that part which on immediate measurement would lead to the same result.
We are thus partially able to determine the state of a quantum system. This determination is complete only if the eigenvalue measured is non-degenerate. But in general it will be degenerate and further measurements are required to determine the state completely. The latter is achieved with simultaneous measurements of several observables: With a complete set of commuting operators.
For the state of the system after a measurement to be completely defined uniquely by the result obtained, this measurement must (150) be made on a complete set of commuting observables (CSCO).
Note: The measurement of a CSCO enables us to prepare only states comprising the unique basis associated with this CSCO. However, changing the set of observables allows us to obtain other states of the system
72
Appendix-1
Accordingly, each state B~ a
j
can be expanded in terms of the base { a
n
}
B~ a
j
= a
n
n
jn
j
g
(122)
Notice, this gives the matrix representation [ ] of the operator B~
in the { a
n
} basis.
Since B~
is an observable, we can claim that the wavefunctions B~ a
j
are linearly
independent. That is to say, the matrix obtained from (122) should be of rank- ag
[ Indeed, since B~
is an observable, we should be able to express any eigenfunction from a
in terms of its eigenfunctions
ju , j= 1, 2, … ag , where
B~
ju =
jb
ju .
There will be a transformation of coordinates from the { a
n
} to the {
ju } base:
a
j
= uT n
n
jn
j
1
g
which gives,
B~ a
j
=
j
n
jnT
g
1
B~
nu =
j
n
njn bT
g
1
nu
We can thus identify kv ≡ B~ a
k as elements of a base. It relates to {
ju } through,
kv =
j
n
njn bT
g
1
nu
73
{ kv } is a base because the transformation of coordinates matrix [T][b] satisfies,
det [T][b] = det [T] det [b] ≠ 0 .
(Here [b] stands for the diagonal matrix that represents B~
in its eigenstates basis)
Hence, the vectors kv ≡ B
~ a
k in (124) are linearly independent ]
Since the vectors B~ a
j
in (122) are linearly independent, then one can implement a
process of diagonalization. That is, there exist in a eigenvectors of B~
.
Therefore, for the degenerate case, it is also possible to find (123)
in a eigenvectors that are common to A~
and B~
REFERENCES
Mandl, “Quantum Mechanics”, Butterworths Scientific Publications
1 See Section 16-4 in Feynman Lectures: Vol III (See Fig 16-2 in this reference for an illustration of the
Delta Dirac). 2 See Feynman Lectures, Vol III, page 16-9 and 16-10. 3 See Feynman Lectures, Vol III, Section 16-5. 4 See Feynman Lectures, Vol III, page 16-12 5 It is interesting to observe new strategies beyond the ensemble approach: B. L. Altshuler, JETP Lett.
41, 648 (1985). P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985). See also the introduction article by Igor V. Lerner, “So Small Yey Still Giant,” Science 316, 63 (2007).
6 B. H. Brasden and C. J. Joachain, “Quantum Mechanics,” 2nd Edition, page 56. 7 Eisberg, Resnick, “Quantum Physics,” page 80, 2nd Edition Wiley (1985). 8 D. Griffiths, Introduction to Quantum Mechanics,” Second Edition,; Pearson Prentice Hall (2005),
page 15. 9 L. D. Landau and E.M. Lifshitz, “Quantum Mechanics, Non-Relativistic Theory,” Pergamon Press,
1965; Chapter 1, Section 3. He introduces the concept of Quantum operators in the context of finding mean values of observables.
10 Feynman Lectures, Vol III, Section 20-4 and Section 20-5. Feynman also introduces the concept of Quantum operators in the context of finding mean values of observables.
11 L. D. Landau and E. M. Lifshitz, “Quantum Mechanics, Non-Relativistic Theory,” Pergamon Press, 1965; Chapter 1, Section 3.
12 This section follows closely the book by Mandl, “Quantum mechanics”
