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 wavefunctionin the space- 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 pstate in space-coordinates basis {
position x }
10.2.B2 Identifyingthe amplitude probabilitymomentum pas 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
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 in the
momentum basis { momentum p } 10.4.D.2 Representation of the linear momentum operator 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?
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.
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 (x, p, E).
In quantum mechanics, observables are represented by operators ( X~
, P~
, 𝐻 );
An operator�̃� acting on a wavefunction gives another wavefunction �̃� =.
In bracket notation:�̃� = .
Notice also that by definition �̃� =�̃�
In this chapter we address three main subjects:
Description of the spatial-coordinates basis and the momentum-coordinates basis. (A base may be more convenient than another base to represent the same quantum
state.)
How to build the quantum mechanics operator corresponding to a given observable.
How to build a quantum state from a given set of experimental results. The key 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 (the latter are complex numbers whose magnitude is interpreted as probabilities).
Notice, however, that when taking the limiting case of the lattice spacing tending to zero, one 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)
xstands 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). Such a base will be used to describe a general quantum state.
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 then an infinite set of amplitudes, one for each value of x. Thus,
=
-
x xdx Representation of the wavefunction (2)
in the spatial coordinates basis
x Amplitude probability that the particle
initially in the state be found (immediately after a position measurement)
at the state x.
Sometimes the following notation is preferred:
x (x) (3) Summary,
=
x xdx =
x (x) dx
Number
Representation of the wave-function in the spatial coordinates basis
(4)
Number
Note: Multiplying the expression (4) 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 section on “the delta Dirac”).
Caution: xdoes not mean [x]*(x) dx.
Working with the spatial coordinates base { x}may constitute the only occasion in which the notation xbecomes confusing with the definition of the scalar product.1
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
j
nj jHndt
d
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 Dirac2
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.
=
-
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 as3
fx ≡
-
( x’ - x ) fx’dx’
The last two expressions are consistent if,
x x’ = ( x’ – x ) . (5)
Thus, we have the following result:
Compatibility between the physical concept of amplitude probability and
the notation used for the inner product *x)xdx
We know that given a state , the amplitude probabilitiesx that appear in the
expansion =
-
x xdx are determined by the Hamiltonian equations.
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 and then arriving at . The
amplitude probability assigned to this path is xx
Since each of these paths have the same initial state and final state , then, according to the rules established in Chapter 7, the total amplitude probability will be given by,
= x
xx
First, the sum over a small region of width dx would be xxmultiplied by dx. Then,
since x varies from to the expression above can be written as,
=
xx dx
For the terms inside the inside the integral, let’s use the notation introduced in expression (3)
above, x(x). Also, since 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 } .
=
-
x xdx
Here we present an alternative basis to express the same state , but in the continuum
base of momentum coordinates { p , p }
p (p) dp p p dp
p (p)
(We will see below that we are already familiar with this new base.)
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 xki ee xk
2
1)(
for k
)(x
F(k) 2
1 eikx dk
Fourier Base-function ek
Transform evaluated at x
where the weight coefficients F(k), referred to as the Fourier transform of the function ,
are given by,
dx xkF xki- e-
)(2
1)(
Exercise. Demonstrate that 2
1
-
[e-i(k-k’) x ] dx = ( k - k’) (8)
Hint: In the expression for dx xkF xki- e-
)(2
1)(
replace )(x
F(k’ ) 2
1 eik’x dk’
It is convenient to express the last two expressions in terms of the variable p = k
)(x
(p) 2
1 ei(p/
ћ) x
dp (9)
Fourier transform of where
dx xpxp/i- e
-
)()(
2
1)(
(10)
We can re-write expression (9) in terms of functions,
)(x
(p) 2
1 ei(p/
ћ) x
dp
Function Function p
evaluated at x evaluated at x Fourier transform
of evaluated at p
(p) p dp (11)
Function Sum of functions
Expressions (9) and (11) are equivalent.
In expressions (9) and (11), the function p has a clear physics interpretation,
p (x) =
2
1 ei(p/ ћ) x (12)
p in bracket notation
Recall, for a given function the corresponding bracket notation in the space base is=
xxdx. Let’s apply this procedure to the functionp,
p ≡
-
x’ p ( x’ ) dx’ (13)
x p ≡
-
x x’ p ( x’ ) dx’
Using expression (5), x x’ = ( x’ – x ) .
=
-
( x’ – x ) p ( x’ ) dx’
= p ( x )
= ei(p/ ћ) x
x p = 2
1 ei(p/ ћ) x (14)
Placing expression (11) in bracket notation In expression (11) we make then the following association,
(p) p dp
p
(p) p dp (15)
Fourier transform
of (x)
2
1
According to de Brogliep represents a
plane wave of definite linear momentum p.
as a linear combination of the
momentum states p
Representation of momentum state p
In the space-coordinates basis { x}.
Amplitude probability that a particle, in a state
p of momentum p, be found (immediately
after a position measurement) at the state x.
This expansion is an alternative to =
-
x xdx =
-
x xdx
The states defined in (13) constitute a base (the justification comes from the Fourier transform theory),
{ p, where - ∞ < p < ∞ } Momentum-coordinates basis (16)
Exercise: Prove that p p’= ( p – p’ ) (17)
Answer:
p’ ≡ x p’ ( x ) dx
pp’ ≡ p x p’ ( x ) dx
= p x p’ ( x ) dx
= p ( x )]* p’ ( x ) dx
= 2
1 ei(p/ ћ) x ]*
2
1 ei(p’/ ћ) x
dx
=2
1 ei(p/ ћ)
x ]* ei(p’/ ћ)
x
dx
=2
1 e-i(p-p’) x/ ћ ]
dx
=2
1 e-i(p-p’) u ]
du
Using (8)
= (p-p’)
10.1.C Tensor Product of State Spaces
Consider the state space 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 comprised of wavefunctions describing the state of another system (another electron, for example) which is initially located far away from the system 1.
These two 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 allow such a description.
Let { ui(1) , i =1, 2, 3, ...} Let { vi (2) , i =1, 2, 3, ...}
be a basis in the space ε1 be a basis in the space ε2
The tensor product of ε1 and ε2 , denoted by ε ε1 ε2 , is defined as a space whose basis
is formed by elements of the type,
{ ui(1) vj (2) , i,j = 1, 2, 3, … } (18)
That is, an element of ε1 ε2 is a linear combination of the form,
j i,
cij ui(1) vj (2) (19)
The tensor product is defined with the following properties,
[ a(1) ] (2) a[ (1) (2) ]
(1)[ c(2) ] c[ (1) (2) ]
[a(1) b(1) ] (2) a[ (1) (2)] + b[ (1) (2) ]
(1) [ c(2) + d (2)] c[ (1) (2) ] + d[ (1) (2) ] In (19), two important cases arise.
Case 1: The wavefunction is the tensor product of the type,
(1) (2) (20a)
That is, can be expressed as the tensor product of a state from the space ε1 and
a state from the space ε2.
ji,
ai bj ui(1) vj (2) (20b)
{ i
ai ui(1) } { j
bjvj (2)}
Belongs to Belongs to space ε1 space ε2
Case 2: The wavefunction cannot be expressed as the tensor product between a state purely from the space ε1 and a state purely from the space ε2.
As an example, let’s consider first the simple case
cu1(1) v2 (2)cu2(1) v1(2) (21)
Notice, cannot be expressed in the form of a single term of the
form (1) (2)
In cases like this, it is said that the spaces ε1 and ε2 are entangled
More general,
ji,
cij ui(1) vj (2) (22)
In general, expressions like (22) serve to describe entangled states.
Exercise: Consider spaces ε1 and ε2 have dimension 2.
Write down explicitly the state ji,
cij ui(1) vj (2)
Show that if all the coefficient cij are equal then the state can be expressed as the tensor product of a state purely from the space ε1 and a state purely from the space ε2.
Show that if c11 = c22 =0, then describes an entangled state.
Solution
If all the cij are equal to c
cu1(1) v1(2)v2(2)
+ cu2(1) v1(2)v2(2)
cu1(1) u2(1) v1(2)v2(2)
which has the form
c (1) (2)
If instead c11 = c22 =0
c12u1(1) v2(2)
+ c21u2(1) v1(2)
which cannot be put in the form
(1) (2)
10.2 The Schrödinger Equation as a postulate 10.2.A The Hamiltonian equations expressed in the continuum spatial coordinates. The
Schrödinger Equation4 In chapter 8 we obtained the general Hamiltonian equations that describe the time
evolution of the wavefunction (t)= n
nAn(t), where the amplitudes An(t) =n were
determined by the Hamiltonian equations,
jj
nj AHAdt
di n (23)
Chapter 9 described the particular case of an electron moving in a lattice (the latter
constituted by atoms separated a distance “b”) and under the effects of a potential )( t,,xV n .
When we took the limit b 0 the Hamiltonian equations took the form,
)()()(
2
)(2
22
t,x,At,x,Vx
t,x,A
mt
t,x,Ai
eff
(24)
Let’s consider now an arbitrary general case.
How does the Hamiltonian equations 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 (24) ).
First, since Aj can also be written as Aj = j , the Hamiltonian Eqs. (23) can also be expressed as,
(t) = n
n An(t) = n
n n ,
j
nj jHndt
di
Let’s also recall, from Chapter 7, that the coefficients Hnj are obtained from the Hamiltonian
operator H~
(one specific to the problem being solved.) That is, Hnj ≡ n H~j. In general, H
~
and depend on t.
j
jjHnndt
di
~ (25)
Extrapolating this expression to the continuum space coordinates, we should expect,
dx'x'x'Hxxdt
di
-
~
(x) ( x’ )
dx'x'x'Hxdt
di
-
)( ~
dx'x'xx,Hdt
di )( )'(
-
(26)
where we have defined H( x, x’ ) ≡ x H~x’
Quoting Feynman,5
“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,
dx'x'xx,H )( )'(-
)( )()(
2 2
22
t,x,t,x,Vx
t,x,
m
Postulate (27)
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 (27) in (26) one obtains,
dt
di )( )(
)(
2 2
22
t,x,t,x,Vx
t,x,
m
(28)
“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.
Schrodinger Equation
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 (28) is a postulate, we have gained 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
10.2.Ba 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,
]22
[)( t-xcosEtx,E 0
electromagnetic wave
where, the electromagnetic intensity I (that is, the energy per unit time crossing a unit cross-
section area perpendicular to the direction of radiation propagation) is proportional to E(x,t) .
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 I = cE2N . Here N constitutes the average
number of photons per second crossing a unit area perpendicular to the direction of radiation propagation;
E2N
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.
10.2.Bb 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, (x,t) plays a role similar to E(x,t).
(x, t) 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,
(x, t) plays the role of probability density. (See expression (29) below.)
Pictorially, the particle is more likely to be at locations where the wavefuntion has an appreciable value.
10.2.Bc 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, 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 sections, after the introduction of observables and eigenstates. Ahead of time, let’s mention that 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.” This ensemble constitutes the “laboratory” in which the QM concepts are develop. (It should be clear, however, that 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.
Ensemble6
Imagine a very large number N of identically prepared independent systems (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
(x, y, z,t) , which contains all the information that can be obtained about them.
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 dzdy dx d 3 r around the position )( z y, x, r at the time t is given by,
r3dz y, x,z y, x, )( )( * (29)
where * stands for the complex conjugate number.
Notice that this is nothing but the language of probability; in this case, position probability density P.
2* )( )( )( )( z y, x,z y, x,z y, x,z y, x,P (30)
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.7 10.2.C Normalization condition for the wavefunction The probabilistic interpretation of the wavefunction implies, therefore, the following requirement:
1 )( )( * dz y, x,z y, x, spaceAll
3 r (31)
because given a particle, the likelihood to find it anywhere should be one. Inherent to this requirement is that,
0)( 2
r
r
(32)
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 (31). This process is called normalization condition of the
wavefunction. In general, there will be solutions to the Schrodinger equation (28) whose solution tend to an infinite value when r
. This means they are non-normalizable and therefore cannot
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 (with a well-defined scalar product between any two of them) form a vector space called the Hilbert space.8 Preservation of the normalization condition
QUESTION: Suppose that is normalized at t 0. 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.
Let be a solution of the Schrodinger equation that satisfies the normalization condition at a given time. If the potential is real, then
0)(2
dxt x,dt
d
-
(33)
Proof: Let’s start with
--
dxt x,t
dxt x,dt
d 22)()( (34)
We provide below a graphic justification of (34).
On the other hand,
**
*2)(
tttt
(35)
We use the Schrodinger equation (28) to calculate the time derivatives,
)(2
)(2 2
2
2
22
t,x,Vi
xm
it,x,V
xm
i
t
**
2
2* )(
2
t,x,V
i
xm
i
t
Taking the complex conjugate, and assuming that the potential is real,
*
2
*2*
)( 2
t,x,Vi
xm
i-
t
Adding the last two expressions,
2
*2*
2
2*
*
2 xxm
i
tt
xxxm
i **
2
(36)
Replacing (35) in (36) we obtain,
xxxm
i
t
**2
2
Accordingly,
--
dxt x,t
dxt x,dt
d 22)()(
xxxm
i **
2
xxm
i **
2
The expression on the right is zero because 0)( 2 x x
So this guy ∫ |𝜓(𝑥, 𝑡)|2∞
−∞𝑑𝑥 does not change with time.
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, 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 perceptions.9
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,
dxx x x
2
)( (37)
But what does this integral exactly mean?
First, it is worth to emphasize what type of interpretation should be avoided.10
Expression (37) does not imply that if you measure the position of the particle over and
over again then dxx x
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, dxx x x
2
)( means the average obtained from measurements
performed on many systems, all in the same initial 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. dxx x x
2
)( is the average from such a
measurement.
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.
Just for fun, let’s find out what our math give us for an expression like dt
xdm
. Whatever
expression we end up with, it should represent the liner momentum of the ensemble, right? Let’s find out.
10.3.B Calculation of dt
xdm
As time goes on, the expectation value x may change with time because the wavefunction evolves with time. Let’s calculate its rate of change.
dx t
xdxx t
xdxx x dt
d
dt
xd
)( )( *2
2
dx t t
x
*
*
For the case where the potential is real, we obtained in expression (36) that,
xxxm
i
tt
***
*
2
dx xxxm
i x
dt
xd
**
2
After integrating by parts, it gives,
dx xxm
i
dt
xd
**
2
Integrating one more time by parts (just the second term on the right side,)
dx x
m
i
dt
xd
*
(38)
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 (38) would give,
xi
dx xi
dt
xdmp
* (39)
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
repeated measurements on a single system.) The whole ensemble is assumed to be described by a complex-variable single wavefunction )( z 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,
NN
NN
n
nnn
nn ff
f
1 av (40)
Usual procedure to calculate average values
Notice that when the total number of measurement N is a very large number, the ratio NN n is
nothing but the probability of finding the system in the particular state fn.
QM postulates that the value of 2
nf should be interpreted as the probability to obtain the
value fn,
NN n
nf 2
Quantum Mechanics postulate (41)
Representing the
ensemble Hence, expression (40) can be written as,
n
nn fff2
av (42)
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, L, 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~ (43)
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
Let’s consider a physical quantity or observable f (for example f could be the particle’s angular momentum) 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 , … (44)
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 ; (45)
These states will be called eigenstates.
We will assume that these eigensates satisfy,
dxxx )()(*
mnmn
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
nAn = n
nn
where An = n = dxxxn )( )(*
Since the state must be a normalized state, then
n
An 2 = n
n 2 = 1
According to expression (42), when the system is in the state , the mean value of f is given by,
av f = n
fn n2 = n
fn n 2 (46)
10.4.B Definition of the QM operator F~ to be associated with the observable physical quantity f.
The operator F~ to be associated with the observable f is such that, when acting on an arbitrary
state , satisfies the following requirement: 11,12
F~ ≡ av f (47)
(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 (52) 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~
From (47),
F~ ≡ av f
= n
fn n2 = n
fn n 2
= n
fn n n = n
fn nn (48)
For convenience, we introduce the “Projection operator” n proj,P~
,
n proj.P~
≡ nn (49)
Notice,
n proj,P~
nn
nn (50)
scalar
Requirement for the Operator F~
associated to the observable f
That is, n proj,P~
gives the projection of along the state nor
the component of along the state n
Using (49) in (48)
F~ = n
fn n proj,P~
Factoring out
= n
fn n proj,P~
This notation trick allows us to express the operator in a more compact form,
F~ = n
fn n proj.P~ (51)
= n
fn nn
eigenvalues Projection operator n proj.P~
built
out of eigenstates associated to the physical quantity f.
Self-consistent expression for the quantum operator
F~ associated with the observable f.
Notice (51) is self-consistent with its definition. By applying the operator F~ to one of the
eigenstates j gives,
F~ j = f j j (52)
That is, if the operator F~ associated to the physical quantity f were known, the quantum
eigenfunctions of that physical quantity are the solutions of the equation, F~=
where is a constant.
Expression (51) is a self-consistent definition for the operator F~ and compatible with the
requirement that F~ ≡av f . But F~ is defined in terms of eigenstates j that, for
a given physics quantity f, we do not know yet. Therefore, F~ is not completely known yet.
But, paraphrasing Landau,
“Although the operator F~ is defined by (51), 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~= "
Expressions (47) and (51) will guide us to build the quantum operators.
10.4.C Definition of the Position Operator X~
In the expressions below X~
() “the operator X~
evaluated at "
will be written, for simplicity, as well as 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~≡ av x
-
x (x)2 dx, X~?
-
(x) X~(x) dx
-
x (x)(x) dx,
-
(x) [ X~ ()] (x) dx
-
(x) x(x) dx,
This implies,
[ X~ ()](x) x ( x)
For convenience, let’s use the identity function I, which satisfies I(x) = x,
[ X~ ()](x) I(x) ( x )
[ I ] ( x )
where stands for the multiplication of functions
Thus,
X~ () I Answer
Note: Frequently, X~ () “the operator X
~ evaluated at the state “ is written as X
~
The Position Operator X~
X~
is the operator associated to av x
X~ ()I where
I is the identity function; I (x) = x
and stands for the multiplication of functions
[ X~ ()] (x) x ( x ) (53)
X~
-
[(x)] [ X~(x) ] dx
-
[(x ) ] x(x) ] dx
-
x (x)2 dx av x
Exercise: Demonstrate that 2X~
is the operator associated to av 2x
Answer:
X~ () I
X~
( X~) X
~ (I )I I I 2
X~
( X~)
means 2X~
2X~
I 2
( 2X~
) (x) (I 2 )(x)
I 2(x) (x)
x2 (x)
Note: I 2(x) does not mean the composition operation I ( I(x) ); the latter
would give I ( I(x) ) I ( x ) x.
Instead I 2 means I I , which gives I 2(x) = I (x) I(x) .
Now let’s evaluate 2X~
2X~
-
[(x)] [ 2X~
(x) ] dx
-
[(x ) ] [ x2(x) ] dx
-
x2 (x)2 dx
av
2 x
Exercise: Demonstrate that 3~X is the operator associated to av
3 x .
In general,
nX~
is the operator associated to av nx . (54)
Even more general,
if h(x) is a polynomial or a convergent series,
then h( X~
) will be the operator associated to )( av xh
For example: If a physical quantity changes as,
3 x2 - 27 x3 ≡ h(x)
then the corresponding operator will be
3 2X
~ - 27 3~
X
Notice, the last expression can be obtained by evaluating h( X~
). Average value Quantum mechanics of physical quantity operator
)( av xh h( X~
) (55)
For example, if h(x) were a classical potential that depends only on position x, then h( X~
) would then be the corresponding quantum mechanics potential operator. Take the case of the harmonic oscillator, where
V(x) = (1/2) k x2
the corresponding QM operator is
V~
(x) = (1/2) k 2X~
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, where - ∞ < p < ∞ }
In this case, the observable f refers to the linear momentum p. Sometimes, for convenience,
we will use the notation momentum p or p instead of p.
We are looking for an operator P~
such that P~ gives the
mean value of the momentum when the system is in the state .
P~≡ av p P
~=?
First, from expression (15) let’s recall that,
pp dp p (p)
dp
where
dx xpxp/i- e
-
)()(
2
1)(
is the Fourier transform of (x).
For a system in that state the average momentum will be given by,
av p p (p)2 dp
Hence,
we are looking for an operator P~
such that
P~≡ p (p)2 dp , P
~=?
pp p
dp
p p p
dp
[ pp p
dp] [ pp
p
dp ]
We realize,
P~
= pp p
dp Linear Momentum Operator (56)
Notice,
P~p’ [ pp
p
dp ]p’
pp p p’ dp
Using the result (17) p p’= ( p – p’ )
pp ( p – p’ )
dp
P~p’ p’p’ (57)
Notice also, using (56)
P~[ pp
p
dp]
pp p
dp
pp (p)
dp
P~ p p
(p)
dp
Or simply (58)
P~ p p
(p)
dp Calculation of P
~in the momentum basis
Matrix representation of the Linear Momentum Operator P~
in the momentum-coordinates basis { p , where - ∞ < p < ∞ } From (57),
P~p’ p’p’
p P~p’ p’p p’
p’( p – p’ )
The left side defines the components of the matrix representation of the operator P~
,
' ~
ppP p P~p’ p’( p – p’ )
(59)
Elements of the matrix representation of the operator in the momentum basis
Summary
10.4.D2 The Linear Momentum Operator P~
expressed in the spatial coordinates basis { x, where - ∞ < x < ∞ }
Expression (59) indicates that if the system were in the state and we wanted to evaluate P~
what we have to do is,
to first expand in the momentum basis p (p) dp
(that is, we have to evaluate the Fourier transform (p) )
and only then be able to evaluate P~ p p(p)
dp
What about if the expansion of in the momentum basis 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 finding the Fourier transform
(p) as (58) requires)?
Here we show an alternative to (58),
P~ p p
(p)
dp
x P~ x p p
(p)
dp
Using (14), x p = ei(p/ ћ) x
x P~ (p )
p ei(p/ ћ)
x
dp
(p) dx
d
i
ei(p/ ћ) x
dp
dx
d
i
(p ) ei(p/ ћ)
x
dp
x P~
dx
d
i
(p ) x p dp
We use x P~x P
~(P
~)(x)
and x p p (x)
x P~
dx
d
i
( p) p(x) dp
(P~)(x)
dx
d
i
( p) p (x) dp
But p (p) dp, or (x) = p (x)(p)
dp. Hence,
2
1
2
1
2
1
2
1
P~(x)
dx
d
i
(x)
P~
dx
d
i
Calculation of P
~in the position basis (60)
Expression (60) is an alternative to expression (58),
P~ p p
(p)
dp Calculation of P
~in the momentum basis
In passing, notice that the last expression is identical to (39) 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 xi
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).
10.4.D3 The operator associated to av 2p
Consider a system is in the state
pp dp p (p)
dp
where dx xpxp/i- e
-
)()(
2
1)(
= p is the Fourier transform of (x).
By definition,
av 2p ≡ p2 (p)2 dp (61)
Hence, we are looking for an operator F~ such that,
F~ av 2p = p2 (p)2 dp
p2 (p)(p)dp
p2 p p
dp
p2 p p
dp
p2 p p
dp
[ p2 p p
dp ]
Hence,
F~ p2 p p
dp Operator associated to av 2p (62)
On the other hand,
notice, P~
= p’p’ p’
dp’
from (58)
P~ p p
(p)
dp
Let’s apply the composition operation of operators.
P~ [P
~]P~ p p
(p)
dp P
~ p p
(p)
dp
p p p (p)
dp p p2 (p)
dp
p p2 (p) dp p p2 p
dp
2~
P [ p p2 p dp]
hence
2~
P p2 p p dp (63)
which, according to (62), it is the operator associated to av 2p
Thus, from (62) and (63),
Physical quantity Quantum Operator
av 2p 2~
P = p2 p p
dp (64)
Generalizing,
Physical quantity Quantum Operator
av np nP
~ = pn p
p
dp (65)
2~
P in spatial coordinates
2~
P p p2 (p) dp
x 2~P x p p2 (p)
dp
Using (14)x p = ei(p/ ћ) x
x 2~P (p )
p2 ei(p/ ћ) x
dp
(p ) 2
22
dx
d
i
ei(p/ ћ) x
dp
2
1
2
1
2
1
2
22
dx
d
i
(p ) ei(p/ ћ)
x
dp
x 2~P
2
22
dx
d
i
(p ) x p dp
We will use x 2~P ≡x
2~P ≡( 2~
P )(x) and x p ≡p (x)
( 2~P )(x)
2
22
dx
d
i
(p ) p (x) dp
But p (p) dp, or (x) = p (x)(p)
dp. Hence,
( 2~P )(x)
2
22
dx
d
i
(x)
2~
P 2
22
dx
d
i
(66)
Thus,
Physical quantity Quantum operator In the momentum base In the spatial coordinates base
av 2p 2~
P = p2 p p
dp
2~P =
2
22
dx
d
i
(67)
Generalizing
av np nP
~ = pn p
p
dp
nP~
= (ℏ
𝑖)𝑛
𝑑𝑛
𝑑𝑥𝑛
Kinetic energy operator
K = 1
2𝑚 av 2p 𝐾 =
1
2𝑚
2~P =
1
2𝑚 p2 p
p
dp 𝐾 =
1
2𝑚
2
22
dx
d
i
Physical property 𝐾 in the momentum base 𝐾 in the spatial coordinates base
2
1
Expressing the Schrodinger Eq. in terms of operators
−ℏ
𝑖
𝜕𝜓𝜕𝑡
= − ℏ2
2𝑚 𝜕2𝜓
𝜕𝑥2 + V(x,t)
Since −ℏ2
2𝑚 𝜕2𝜓
𝜕𝑥2 =
1
2𝑚 (
ℏ
𝑖)2 𝜕2𝜓
𝜕𝑥2
= 1
2𝑚 (
ℏ
𝑖
𝜕
𝜕𝑥)2
= 1
2𝑚 2~P
Thus, we realize that
−ℏ
𝑖
𝜕𝜕𝑡
𝜓 = 1
2𝑚 2~P + V(x,t)
1 Here we explore a possible procedure to use whenever we deal with scalar products
involving the position states x}: We could exploit the use of the Delta-Dirac.
Since xmust be equal to (x),
and
the delta-Dirac has the property that [(x’-x)](x’) dx’ = (x),
then
x= [(x’-x)_](x’) dx’ = (x) .
The latter suggests to make the following guess:
x’ x x x’is equal to (x’-x)
because
[(x’-x)_](x’) dx’ = x x’(x’) dx’
= x x’(x’) dx’
= x
(x)
So, our guess that x x’ (x’-x) let us arrive to a well-known result.
2 See Section 16-4 in Feynman Lectures: Vol III (See Fig 16-2 in this reference for an illustration
of the Delta Dirac). 3 See Feynman Lectures, Vol III, page 16-9 and 16-10. 4 See Feynman Lectures, Vol III, Section 16-5. 5 See Feynman Lectures, Vol III, page 16-12 6 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 Yet Still Giant,” Science 316, 63 (2007).
7 B. H. Brasden and C. J. Joachain, “Quantum Mechanics,” 2nd Edition, page 56. 8 It should be mentioned that in scattering theory one deals with functions that are not
normalizable. One deals with fluxes of incident and scattered particles, which must be well defined.
9 Eisberg, Resnick, “Quantum Physics,” page 80, 2nd Edition Wiley (1985). 10 D. Griffiths, Introduction to Quantum Mechanics,” Second Edition, Pearson Prentice Hall
(2005), page 15. 11 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.
12 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.
