Fitts D.D. - Principles of Quantum Mechanics[c] As Applied to Chemistry and Chemical Physics (1999)(en)

92 General principles of quantum theory

q1, q2, . . . may be shown by comparing equations (3.46) and (3.54) for A equal

to the coordinate vector q

…

hqi hØjqjØi Ø (q)Ø(q)q dô

…

hqi r(q)q dô

For these two expressions to be mutually consistent, we must have r(q) Ø (q)Ø(q)

Thus, this interpretation of Ø Ø follows from postulate 3 and for this reason is not included in the statement of postulate 1.

Collapse of the state function

The measurement of a physical observable A gives one of the eigenvalues ën of

the operator ^. As stated by the fourth postulate, a consequence of this

A

measurement is the sudden change in the state function of the system from its original form Ø to an eigenfunction or linear combination of eigenfunctions of

^ corresponding to ë .

A n

At a ®xed time t just before the measurement takes place, the ket jØi may

be expanded in terms of the eigenkets j ái of ^, as shown in equation (3.48). If i A

the measurement gives a non-degenerate eigenvalue ën, then immediately after the measurement the system is in state jni. The state function Ø is said to collapse to the function jni. A second measurement of A on this same system, if taken immediately after the ®rst, always yields the same result ën. If the eigenvalue ën is degenerate, then right after the measurement the state function is some linear combination of the eigenkets jnái, á 1, 2, . . . , gn. A second, immediate measurement of A still yields ën as the result.

From postulates 4 and 5, we see that the state function Ø can change with time for two different reasons. A discontinuous change in Ø occurs when some property of the system is measured. The state of the system changes suddenly from Ø to an eigenfunction or linear combination of eigenfunctions associated with the observed eigenvalue. An isolated system, on the other hand, undergoes a continuous change with time in accordance with the time-dependent SchroÈdinger equation.

Time evolution of the state function

The ®fth postulate stipulates that the time evolution of the state function Ø is determined by the time-dependent SchroÈdinger equation

	3.7 Postulates of quantum mechanics			93
		@Ø	^
	i" @t		HØ	(3:55)
where	^
	H is the Hamiltonian operator of the system and, in general, changes

with time. However, in this book we only consider systems for which the Hamiltonian operator is time-independent. To solve the time-dependent SchroÈdinger equation, we express the state function Ø(q, t) as the product of two functions

Ø(q, t) ø(q)÷(t)

(3:56)

where ø(q) depends only on the spatial variables and ÷(t) depends only on the time. In Section 2.4 we discuss the procedure for separating the partial differential equation (3.55) into two differential equations, one involving only the spatial variables and the other only the time. The state function Ø(q, t) is

then shown to be		ø(q)eÿiEt="
Ø(q, t)			(3:57)

where E is the separation constant. Since it follows from equation (3.57) that jØ(q, t)j2 jø(q)j2

the probability density is independent of the time t and Ø(q, t) is a stationary state.

The spatial differential equation, known as the time-independent SchroÈdinger equation, is

^ ø( ) ø( )

H q E q

Thus, the spatial function ø(q) is actually a set of eigenfunctions øn(q) of the

Hamiltonian operator ^ with eigenvalues . The time-independent SchroÈdin-

H En ger equation takes the form

^ ø ( ) ø ( )

H n q En n q

and the general solution of the time-dependent SchroÈdinger equation is

X

Ø(q, t) cnøn(q)eÿiEn t="

n

(3:58)

(3:59)

where cn are arbitrary complex constants.

The appearance of the Hamiltonian operator in equation (3.55) as stipulated by postulate 5 gives that operator a special status in quantum mechanics. Knowledge of the eigenfunctions and eigenvalues of the Hamiltonian operator for a given system is suf®cient to determine the stationary states of the system and the expectation values of any other dynamical variables.

We next address the question as to whether equation (3.59) is actually the most general solution of the time-dependent SchroÈdinger equation. Are there other solutions that are not expressible in the form of equation (3.59)? To

94	General principles of quantum theory

answer that question, we assume that Ø(q, t) is any arbitrary solution of the parital differential equation (3.55). We suppose further that the set of functions øn(q) which satisfy the eigenvalue equation (3.58) is complete. Then we can,

in general, expand Ø(q, t) in terms of the complete set øn(q) and obtain

X

Ø(q, t)	an(t)øn(q)	(3:60)
	n
The coef®cients an(t) in the expansion are given by
an(t) høn(q)jØ(q, t)i		(3:61)

and are functions of the time t, but not of the coordinates q. We substitute the expansion (3.60) into the differential equation (3.55) to obtain

X

"

@an(t)

an(t)H^ øn(q)

X

"

@an(t)

En an(t) øn(q) 0

^

n

i

@t

n

i

@t

(3:62)

where we have also noted that the functions øn(q) are eigenfunctions of H in accordance with equation (3.58). We next multiply equation (3.62) by øk (q), the complex conjugate of one of the eigenfunctions of the orthogonal set, and

integrate over the spatial variables

X

@an(t)

n

"

En an(t) høk (q)jøn(q)i

i

@t

X

"

@ak (t)

n

"

@an(t)

En an(t) äkn

Ek ak (t) 0

i

@t

i

@t

Replacing the dummy index k by n, we obtain the result

an(t) cneÿiEn t="

(3:63)

where cn is a constant independent of both q and t. Substitution of equation (3.63) into (3.60) gives equation (3.59), showing that equation (3.59) is indeed the most general form for a solution of the time-dependent SchroÈdinger equation. All solutions may be expressed as the sum over stationary states.

3.8 Parity operator
^
The parity operator Ð is de®ned by the relation
^	(3:64)
Ðø(q) ø(ÿq)

Thus, the parity operator reverses the sign of each cartesian coordinate. This operator is equivalent to an inversion of the coordinate system through the origin. In one and three dimensions, equation (3.64) takes the form

Ð^ ø(x) ø(ÿx), Ð^ ø(r) Ð^ ø(x, y, z) Ð^ ø(ÿx, ÿy, ÿz) ø(ÿr) The operator Ð^ 2 is equal to unity since

		3.8		Parity operator			95
^ 2		^	^		^		ø(q)
Ð	ø(q) Ð(Ðø(q)) Ðø(ÿq)
Further, we see that
	^ n	ø(q) ø(q),				n even
	Ð
or				ø(ÿq),		n odd
		^ n		1,	n even
		Ð
				^	n odd
				Ð,

^
The operator Ð is linear and hermitian. In the one-dimensional case, the
^
hermiticity of Ð is demonstrated as follows			ö (ÿx9)ø(x9) dx9
höjÐ^ jøi	…ÿ1ö (x)ø(ÿx) dx ÿ…1
	1	ÿ1
	…ÿ1ø(x9)Ð^	ö (x9) dx9 hÐ^	öjøi
	1

where x in the second integral is replaced by x9 ÿx to obtain the third integral. By applying the same procedure to each coordinate, we can show that

^
Ð is hermitian with respect to multi-dimensional functions.
					^
The eigenvalues ë of the parity operator Ð are given by
					^
					Ðøë(q) ëøë	(q)	^	(3:65)
where øë(q) are the corresponding eigenfunctions. If we apply Ð to both sides
of equation (3.65), we obtain
	^	2			^	2
^	Ð		øë(q) ëÐøë(q) ë øë(q)
	2			2	1 and that the eigenvalues		ë, which must be
Since Ð	1, we see that ë
	^					either 1 or	ÿ1.	To ®nd the
real because Ð is hermitian, are equal to

eigenfunctions øë(q), we note that equation (3.65) now becomes øë(ÿq) øë(q)

For ë 1, the eigenfunctions of Ð^ are even functions of q, while for ë ÿ1, they are odd functions of q. An even function of q is said to be of even parity, while odd parity refers to an odd function of q. Thus, the eigenfunctions of Ð^ are any well-behaved functions that are either of even or odd parity in their cartesian variables.

					^
We show next that the parity operator Ð commutes with the Hamiltonian
operator	^
	H if the potential energy V(q) is an even function of q. The kinetic
energy term in the Hamiltonian operator is given by
	ÿ 2m =2		ÿ 2m		@q12		@q22 !
	"2		"2		@2		@2

96 General principles of quantum theory

and is an even function of each qk . If the potential energy V (q) is also an even

			^	^
function of each qk , then we have H(q) H(ÿq) and
^ ^	^	^	^ ^		^	^
[H, Ð] f (q)	H(q)Ð f (q) ÿ ÐH(q) f (q) H(q) f (ÿq) ÿ H(ÿq) f (ÿq) 0
Since the function					^	^
		f (q) is arbitrary, the commutator of H and Ð vanishes.
Thus, these operators have simultaneous					eigenfunctions	for systems with
V (q) V (ÿq).

If the potential energy of a system is an even function of the coordinates and if ø(q) is a solution of the time-independent SchroÈdinger equation, then the function ø(ÿq) is also a solution. When the eigenvalues of the Hamiltonian operator are non-degenerate, these two solutions are not independent of each other, but are proportional

ø(ÿq) cø(q)

These eigenfunctions are also eigenfunctions of the parity operator, leading to the conclusion that c 1. Consequently, some eigenfunctions will be of even parity while all the others will be of odd parity.

For a degenerate energy eigenvalue, the several corresponding eigenfunc-

tions of ^ may not initially have a de®nite parity. However, each eigenfunction

H

may be written as the sum of an even part øe(q) and an odd part øo(q) ø(q) øe(q) øo(q)

where

øe(q) 12[ø(q) ø(ÿq)] øe(ÿq) øo(q) 12[ø(q) ÿ ø(ÿq)] ÿøo(ÿq)

Since any linear combination of ø(q) and ø(ÿq) satis®es SchroÈdinger's equa-

tion, the functions ø ( ) and ø ( ) are eigenfunctions of ^ . Furthermore, the

e q o q H

functions øe(q) and øo(q) are also eigenfunctions of the parity operator Ð^ , the ®rst with eigenvalue 1 and the second with eigenvalue ÿ1.

3.9 Hellmann±Feynman theorem

A useful expression for evaluating expectation values is known as the Hellmann±Feynman theorem. This theorem is based on the observation that the Hamiltonian operator for a system depends on at least one parameter ë, which can be considered for mathematical purposes to be a continuous variable. For example, depending on the particular system, this parameter ë may be the mass of an electron or a nucleus, the electronic charge, the nuclear charge parameter Z, a constant in the potential energy, a quantum number, or even Planck's

constant. The eigenfunctions and eigenvalues of ^ (ë) also depend on this

H

3.10 Time dependence of the expectation value

97

parameter, so that the time-independent SchroÈdinger equation (3.58) may be written as

^	(3:66)
H(ë)øn(ë) En(ë)øn(ë)
^
The expectation value of H(ë) is, then
^	(3:67)
En(ë) høn(ë)jH(ë)jøn(ë)i
where we assume that øn(ë) is normalized
høn(ë)jøn(ë)i 1	(3:68)

To obtain the Hellmann±Feynman theorem, we differentiate equation (3.67)

with respect to ë

ddë En(ë) *øn(ë) ddë H^ (ë) øn(ë)+

d

*øn(ë) H^

d

øn(ë)+

(3:69)

*dë

øn(ë) H^

(ë) øn(ë)+

(ë) dë

ë

Applying the hermitian property

of H^

(

) to the third

integral

on the right-hand

side of equation (3.69) and then applying (3.66) to the second and third terms,

we obtain

*øn(ë) dë H^ (ë) øn(ë)+

dë En(ë)

d

d

d

d

øn

(ë)+#

(3:70)

En(ë)"*dë øn(ë) øn(ë)+

*øn(ë) dë

respect to

ë

is

The derivative of equation (3.68) with

*ddë øn(ë)

øn(ë)+ *øn(ë)

ddë øn(ë)+ 0

showing that the last term on

the right-hand side

of (3.70) vanishes. We thereby

obtain the Hellmann±Feynman theorem

*øn(ë)

øn(ë)+

ddë En(ë)

d

(3:71)

dë H^ (ë)

3.10 Time dependence of the expectation value

The expectation value hAi of the dynamical quantity or observable A is, in general, a function of the time t. To determine how hAi changes with time, we take the time derivative of equation (3.46)

98

General principles of quantum theory

*Ø @t

Ø+

hdt i dt hØjA^jØi *@@t

A^ Ø+ *Ø A^ @@t +

Ø

Ø

^

d A d

@A

Equation

(3.55) may be substituted

for

the time derivatives

of the wave

function

to give

j i * @t +

dt " h j j i ÿ " h j

dhAi

^

i H^ Ø A^ Ø

i Ø A^H^ Ø Ø

@A

Ø

*Ø

Ø+

" hØjH^ A^jØi ÿ " hØjA^H^ jØi

@t

i

i

@A^

i

*Ø

@A^

Ø+

" hØj[H^ , A^]jØi

@t

i

^

@A

h[H^ , A^]i

"

@t

where the hermiticity of ^ and the de®nition (equation (3.3)) of the commu-

H

					^
tator have been used. If the operator A is not an explicit function of time, then
the last term on the right-hand side vanishes and we have
			dhAi			i		[ H^		, A^]	(3:72)
			dt
					" h					i
^										^	^
If we set A equal to unity, then the commutator [H,											A] vanishes and equation
(3.72) becomes
				dhAi					0
					dt
or
	d	^						d
	dt	hØjAjØi						dt	hØjØi 0

We thereby obtain the result in Section 2.2 that if Ø is normalized, it remains

normalized as time progresses.
If the operator	^	in equation			(3.72)		is set			equal to	^
	A										H, then again the
commutator vanishes and we have
			dhAi			dhHi		dE		0
			dt			dt	dt

Thus, the energy E of the system, which is equal to the expectation value of the Hamiltonian, is conserved if the Hamiltonian does not depend explicitly on time.

By setting the operator ^ in equation (3.72) equal ®rst to the position

A

variable x, then the variable y, and ®nally the variable z, we can show that

3.11

Heisenberg uncertainty principle

99

m

dhxi

h

p

xi

,

m

dhyi

h

p

,

m

dhzi

h

p

dt

dt

yi

dt

zi

or, in vector notation

m

dhri

h

p

i

dt

which is one of the Ehrenfest theorems discussed in Section 2.3. The other Ehrenfest theorem,

dhpi	ÿh	=V (r)
dt		i
		^	^px,
may be obtained from equation (3.72) by setting A successively equal to
^py, and ^pz.

3.11 Heisenberg uncertainty principle

We have shown in Section 3.5 that commuting hermitian operators have simultaneous eigenfunctions and, therefore, that the physical quantities associated with those operators can be observed simultaneously. On the other hand,

if the hermitian operators	^	and	^	do not commute, then the physical
	A		B

observables A and B cannot both be precisely determined at the same time. We

begin by demonstrating this conclusion.
^	^	^
Suppose that A and B do not commute. Let ái and âi be the eigenvalues of A
^
and B, respectively, with corresponding eigenstates jáii and jâii
	^	(3:73a)
	Ajáii áijáii
	^	(3:73b)
	Bjâii âijâii

Some or all of the eigenvalues may be degenerate, but each eigenfunction has a unique index i. Suppose further that the system is in state jáji, one of the

eigenstates of ^. If we measure the physical observable , we obtain the result

A A

áj. What happens if we simultaneously measure the physical observable B? To answer this question we need to calculate the expectation value hBi for this system

^	(3:74)
hBi hájjBjáji

If we expand the state function jáji in terms of the complete, orthonormal set

jâii

X

jáji cij âii

i

where ci are the expansion coef®cients, and substitute the expansion into equation (3.74), we obtain

100		General principles of quantum theory
hBi Xi	Xk	ck cihâk jB^jâii Xi	Xk	ck ciâiäki Xi	jcij2 âi

where (3.73b) has been used. Thus, a measurement of B yields one of the many values âi with a probability jcij2. There is no way to predict which of the values âi will be obtained and, therefore, the observables A and B cannot both be determined concurrently.

For a system in an arbitrary state Ø, neither of the physical observables A

and can be precisely determined simultaneously if ^ and ^ do not commute.

B A B

Let ÄA and ÄB represent the width of the spread of values for A and B, respectively. We de®ne the variance (ÄA)2 by the relation

2	^	2	i	(3:75)
(ÄA)	h(A	ÿ hAi)

that is, as the expectation value of the square of the deviation of A from its mean value. The positive square root ÄA is the standard deviation and is called

the uncertainty in A. Noting				2that hAi is a real number, we can obtain an
alternative expression for (ÄA)				as follows:
	(ÄA)	2	^	2		^2		^	2
			h(A ÿ hAi)		i hA		ÿ 2hAiA hAi i
			^2				2	^2	i ÿ hAi	2	(3:76)
			hA i ÿ 2hAihAi hAi					hA
Expressions analogous to equations (3.75) and (3.76) apply for (ÄB)2.
^	^								^
Since A and B do not commute, we de®ne the operator C by the relation
			^	^	^^	^ ^		^			(3:77)
			[A,	B] AB		ÿ BA iC

	^
The operator C is hermitian as discussed in Section 3.3, so that its expectation
value hCi is real. The commutator of				^	^	ÿ hBi may be expanded
				A ÿ hAi and	B
as follows
^	^	^	^	^		^
[A	ÿ hAi, B	ÿ hBi] (A ÿ hAi)(B		ÿ hBi) ÿ (B	ÿ hBi)(A ÿ hAi)
		^^	^ ^	^		(3:78)
		AB	ÿ BA iC

where the cross terms cancel since hAi and hBi are numbers and commute with

^	^
the operators A and B. We use equation (3.78) later in this section.
We now introduce the operator
	^	^	ÿ hBi)
	A	ÿ hAi ië(B

where ë is a real constant, and let this operator act on the state function Ø

^	^	ÿ hBi)]Ø
[A	ÿ hAi ië(B

The scalar product of the resulting function with itself is, of course, always positive, so that

^	^	^	^	ÿ hBi)]Øi > 0	(3:79)
h[A	ÿ hAi ië(B	ÿ hBi)]Øj[A	ÿ hAi ië(B

Expansion of this expression gives

3.11

Heisenberg uncertainty principle

101

^

^

2

^

^

h(A ÿ hAi)Øj(A ÿ hAi)Øi ë

h(B ÿ hBi)Øj(B ÿ hBi)Øi

^

^

^

^

ÿ hAi)Øi > 0

iëh(A

ÿ hAi)Øj(B ÿ hBi)Øi ÿ iëh(B

ÿ hBi)Øj(A

or, since

^

^

A and B are hermitian

^

2

2

^

2

jØi

hØj(A

ÿ hAi) jØi ë

hØj(B ÿ hBi)

^

^

> 0

iëhØj[A ÿ hAi, B ÿ hBi]jØi

Applying equations (3.75) and (3.78), we have

(ÄA)2 ë2(ÄB)2 ÿ ëhCi > 0

If we complete the square of the terms involving ë, we obtain

ÿ 2(hÄB)2

ÿ

4(ÄB)2

Ci

2

hCi2

(ÄA)2

(ÄB)2 ë

> 0

Since ë is arbitrary, we select its value so as to eliminate the second term

ë

hCi

(3:80)

2(ÄB)2

thereby giving

(ÄA)2(ÄB)2 > 41hCi2

or, upon taking the positive square root,

ÄAÄB > 21jhCij

Substituting equation (3.77) into this result yields

ÄAÄB >

1

^

^

(3:81)

2jh[A,

B]ij

This general expression relates the uncertainties in the simultaneous measure-

							^	^
ments of A and B to the commutator of the corresponding operators A and B
and is a general statement of the Heisenberg uncertainty principle.
Position±momentum uncertainty principle
We now consider the special case for which				^			x) and B
				A is the variable x (A
^				^ ^
is the momentum px (B ÿi" d=dx). The commutator [A, B] may be evaluated
by letting it operate on Ø					dx i"Ø
[A^, B^]Ø ÿi" x dx ÿ
dØ				dxØ
^ ^
so that jh[A, B]ij " and equation (3.81) gives
ÄxÄ px >			"					(3:82)
2

The Heisenberg position±momentum uncertainty principle (3.82) agrees with equation (2.26), which was derived by a different, but mathematically