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

82 General principles of quantum theory

cjøii, the corresponding bra is c høij. We can also write cjøii as jcøii, in

which case the corresponding bra is hcøij, so that hcøij c høij

For every linear operator ^ that transforms jø i in ket space into jö i j ^ø i,

A i i A i

there is a corresponding linear operator ^y in bra space which transforms hø j

A i

into hö j h ^ø j. This operator ^y is called the of ^. In bra space the i A i A adjoint A

transformation is expressed as

h ^ø j hø j ^y

A i i A

Thus, for bras the operator acts on the vector to its left, whereas for kets the operator acts on the vector to its right.

To ®nd the relationship between ^ and its adjoint ^y, we take the scalar

A A

product of h ^ø j and jø i

A j i

hA^øjjøii højjA^yjøii	(3:33a)
or in integral notation …(A^øj) øi dx …øj A^yøi dx	(3:33b)
^
A comparison with equation (3.8) shows that if A is hermitian, then we have
A^y A^ and A^ is said to be self-adjoint. The two terms,	hermitian and self-

adjoint, are synonymous. To ®nd the adjoint of a non-hermitian operator, we

apply equations (3.33). For example, we see from equation (3.10) that the
adjoint of the operator d=dx is ÿd=dx.
Since the scalar product højöi is equal to höjøi , we see that
hA^øjjøii høijA^jøji	(3:34)
Combining equations (3.33a) and (3.34) gives
højjA^yjøii høijA^jøji	(3:35)
If we replace A^ in equation (3.35) by the operator A^y, we obtain
højj(A^y)yjøii høijA^yjøji	(3:36)

where ( ^y)y is the adjoint of the operator ^y. Equation (3.35) may be rewritten

A A

as

høijA^yjøji højjA^jøii
and when compared with (3.36), we see that
højj(A^y)yjøii højjA^jøii
We conclude that
(A^y)y A^	(3:37)
From equation (3.35) we can also show that

	3.6	Hilbert space and Dirac notation			83
		(cA^)y c A^y			(3:38)
where c is any complex constant, and that
		(A^ B^)y A^y B^y			(3:39)
	To obtain the adjoint of the product		^^
			AB of two operators, we apply equation
	^^	^	^
(3.33a), ®rst to AB, then to A, and ®nally to B
	højj(AB^^)yjøii hAB^^øjjøii hB^øjjA^yjøii højjB^yA^yjøii
Thus, we have the relation
		(AB^^)y B^yA^y			(3:40)
If A^ and B^ are hermitian (self-adjoint), then we have (AB^^)y					BA^ ^ and further, if
^	^	^^

A and B commute, then the product AB is hermitian or self-adjoint.

The outer product of a bra höj and a ket jøi is jøihöj and behaves as an operator. If we let this outer product operate on another ket j÷i, we obtain the expression jøihöj÷i, which can be regarded in two ways. The scalar product höj÷i is a complex number multiplying the ket jøi, so that the complete expression is a ket parallel to jøi. Alternatively, the operator jøihöj acts on the ket j÷i and transfroms j÷i into a ket proportional to jøi.

To ®nd the adjoint of the outer product j÷ihöj of the ket j÷i and the bra höj,

we let ^ in equation (3.35) be equal to j÷ihöj and obtain

A

højj(j÷ihöj)yjøii høij(j÷ihöj)jøji høij÷i höjøji

h÷jøiihøjjöi højjöih÷jøii højj(jöih÷j)jøii

Setting equal the operators in the left-most and right-most integrals, we ®nd that

(j÷ihöj)y jöih÷j

(3:41)

Projection operator

^
We de®ne the operator Pi as the outer product of jøii and its corresponding bra
^	(3:42)
Pi jøiihøij jiihij
^
and apply Pi to an arbitrary ket jöi
^
Pijöi jiihijöi

					^
Thus, the result of Pi acting on jöi is a ket proportional to jii, the proportion-
ality constant being the scalar product høijöi. The operator								^
								Pi, then, projects
jöi	onto	j	2	ii	and for that reason is known as the		projection operator. The
			ø
operator		^		is given by
		Pi
					^2	^ ^	^
					Pi	Pi Pi jiihijiihij jiihij Pi

where we have noted that the kets j i are normalized. Likewise, the operator ^ n i Pi

84		General principles of quantum theory
for	n . 2 also equals	^
		Pi. This property is consistent with the interpretation of

^ as a projection operator since the result of projecting jöi onto j i should be

Pi i

the same whether the projection is carried out once, twice, or multiple times.

^
The operator Pi is hermitian, so that the projection of jöi on jøii is equal to
the projection of jøii on jöi. To show	that	^	is hermitian, we let j÷i
		Pi
^y	^
jöi jii in equation (3.41) and obtain P i	Pi.

The expansion of a function f (x) in terms of the orthonormal set øi(x), as shown in equation (3.27), may be expressed in terms of kets as

XX

jf i aijøii aijii

ii

where jf i is regarded as a vector in ket space. The constants ai are the projections of jf i on the `unit ket vectors' jii and are given by equation (3.28)

ai hijf i

Combining these two equations gives equation (3.29), which when expressed in Dirac notation is

X

jf i jiihijf i

i

Since f (x) is an arbitrary function of x, the operator Pijiihij must equal the identity operator, so that

X

jiihij 1

(3:43)

i

From the de®nition of ^ in equation (3.42), we see that

Pi

X^ 1

Pi

i

Since the operator Pijiihij equals unity, it may be inserted at any point in an equation. Accordingly, we insert it between the bra and the ket in the scalar product of jf i with itself

hf jf i *f			i	jiihij! f + 1
			X
where we have assumed jf i is normalized.					This expression		may be written as
hf jf i Xi	hf jiihijf i Xi				jhijf ij2 Xi	jaij2 1

Thus, the expression (3.43) is related to the completeness criterion (3.30) and is called, therefore, the completeness relation.

3.7 Postulates of quantum mechanics

85

3.7 Postulates of quantum mechanics

In this section we state the postulates of quantum mechanics in terms of the properties of linear operators. By way of an introduction to quantum theory, the basic principles have already been presented in Chapters 1 and 2. The purpose of that introduction is to provide a rationale for the quantum concepts by showing how the particle±wave duality leads to the postulate of a wave function based on the properties of a wave packet. Although this approach, based in part on historical development, helps to explain why certain quantum concepts were proposed, the basic principles of quantum mechanics cannot be obtained by any process of deduction. They must be stated as postulates to be accepted because the conclusions drawn from them agree with experiment without exception.

We ®rst state the postulates succinctly and then elaborate on each of them with particular regard to the mathematical properties of linear operators. The postulates are as follows.

1.	The state of a physical system is de®ned by a normalized function Ø of the spatial
	coordinates and the time. This function contains all the information that exists on
	the state of the system.
2.	^
	Every physical observable A is represented by a linear hermitian operator A.
3.	Every individual measurement of a physical observable A yields an eigenvalue of
	^
	the corresponding operator A. The average value or expectation value hAi from a
	series of measurements of A for systems, each of which is in the exact same state
	Ø, is given by hAi hØjAjØi.
4.	If a measurement of a physical observable A for a system in state Ø gives the
	^
	eigenvalue ën of A, then the state of the system immediately after the measurement

is the eigenfunction (if ën is non-degenerate) or a linear combination of eigenfunctions (if ën is degenerate) corresponding to ën.

5.The time dependence of the state function Ø is determined by the time-dependent SchroÈdinger differential equation

i" @Ø ^ Ø

@t

H

where ^ is the Hamiltonian operator for the system.

H

This list of postulates is not complete in that two quantum concepts are not covered, spin and identical particles. In Section 1.7 we mentioned in passing that an electron has an intrinsic angular momentum called spin. Other particles also possess spin. The quantum-mechanical treatment of spin is postponed until Chapter 7. Moreover, the state function for a system of two or more identical and therefore indistinguishable particles requires special consideration and is discussed in Chapter 8.

86 General principles of quantum theory

State function

According to the ®rst postulate, the state of a physical system is completely described by a state function Ø(q, t) or ket jØi, which depends on spatial coordinates q and the time t. This function is sometimes also called a state vector or a wave function. The coordinate vector q has components q1, q2, . . . , so that the state function may also be written as Ø(q1, q2, . . . , t). For a particle or system that moves in only one dimension (say along the x-axis), the vector q has only one component and the state vector Ø is a function of x and t: Ø(x, t). For a particle or system in three dimensions, the components of q are x, y, z and Ø is a function of the position vector r and t: Ø(r, t). The state function is single-valued, a continuous function of each of its variables, and square or quadratically integrable.

For a one-dimensional system, the quantity Ø (x, t)Ø(x, t) is the probability density for ®nding the system at position x at time t. In three dimensions, the quantity Ø (r, t)Ø(r, t) is the probability density for ®nding the system at point r at time t. For a multi-variable system, the product Ø (q1, q2,

. . . , t)Ø(q1, q2, . . . , t) is the probability density that the system has coordinates q1, q2, . . . at time t. We show below that this interpretation of Ø Ø follows from postulate 3. We usually assume that the state function is normalized …

Ø (q1, q2, . . . , t)Ø(q1, q2, . . . , t)w(q1, q2, . . .) dq1 dq2 . . . 1

or in Dirac notation

hØjØi 1

where the limits of integration are over all allowed values of q1, q2, . . .

Physical quantities or observables

The second postulate states that a physical quantity or observable is represented in quantum mechanics by a hermitian operator. To every classically de®ned function A(r, p) of position and momentum there corresponds a quantum-

mechanical linear hermitian operator ^( , ("=i)=). Thus, to obtain the quan-

A r

tum-mechanical operator, the momentum p in the classical function is replaced by the operator p^

"

:

p i =

or, in terms of components

^

(3 44)

" @

" @

" @

^px

,

^py

,

^pz

i

@x

i

@ y

i

@z

3.7 Postulates of quantum mechanics

87

For multi-particle systems with cartesian coordinates r1, r2, . . . , the classical

function			A(r1, r2, . . . , p1, p2, . . .) possesses	the corresponding operator
^	, r2	,	. . . , ("=i)=1, ("=i)=2, . . .) where =k	is the gradient with respect to
A(r1

rk . For non-cartesian coordinates, the construction of the quantum-mechanical

operator ^ is more complex and is not presented here.

A

The classical function A is an observable, meaning that it is a physically measurable property of the system. For example, for a one-particle system the

^
Hamiltonian operator H corresponding to the classical Hamiltonian function
			p2
H(r, p)				V (r)
		2m
where p2 p : p p2x p2y p2z , is
^	"2		2
H ÿ 2m = V (r)

The linear operator ^ is easily shown to be hermitian.

H

Measurement of observable properties

The third postulate relates to the measurement of observable properties. Every individual measurement of a physical observable A yields an eigenvalue ëi of

	^
	the operator A. The eigenvalues are given by
	^			(3:45)
	Ajii ëijii	^	^
	where jii are the orthonormal eigenkets of			is hermitian, the
		A. Since	A

eigenvalues are all real. It is essential for the theory that ^ is hermitian because

A

any measured quantity must, of course, be a real number. If the spectrum of ^

A is discrete, then the eigenvalues ëi are discrete and the measurements of A are quantized. If, on the other hand, the eigenfunctions jii form a continuous, in®nite set, then the eigenvalues ëi are continuous and the measured values of

are not quantized. The set of eigenkets j i of the dynamical operator ^ are

A i A assumed to be complete. In some cases it is possible to show explicitly that jii forms a complete set, but in other cases we must assume that property.

The expectation value or mean value hAi of the physical observable A at

time t for a system in a normalized state Ø is given by
			^	(3:46)
hAi hØjAjØi
If Ø is not normalized, then the appropriate expression is
			^
h	A	i	hØjAjØi
			hØjØi

Some examples of expectation values are as follows

88

General principles of quantum theory

hxi hØjxjØi

Ø+

hpxi

*Ø

"i @@x

r

Ø r

Ø

h i h j j

i

hpi

*Ø

"i

= Ø+

2

E hHi

"

2

V(r) Ø+

*Ø

ÿ 2m =

The expectation value hAi is not the

result of a single

measurement of the

property A, but rather the average of a large number (in the limit, an in®nite number) of measurements of A on systems, each of which is in the same state Ø. Each individual measurement yields one of the eigenvalues ëi, and hAi is then the average of the observed array of eigenvalues. For example, if the eigenvalue ë1 is observed four times, the eigenvalue ë2 three times, the eigenvalue ë3 once, and no other eigenvalues are observed, then the expectation value hAi is given by

hAi 4ë1 3ë2 ë3

8

In practice, many more than eight observations would be required to obtain a reliable value for hAi.

In general, the expectation value hAi of the observable A may be written for

a discrete set of eigenfunctions as
hAi Xi	Piëi	(3:47)

where Pi is the probability of obtaining the value ëi. If the state function Ø for a system happens to coincide with one of the eigenstates jii, then only the eigenvalue ëi would be observed each time a measurement of A is made and therefore the expectation value hAi would equal ëi

h i h j ^j i h jë j i ë

A i A i i i i i

It is important not to confuse the expectation value hAi with the time average of A for a single system.

For an arbitrary state Ø at a ®xed time t, the ket jØi may be expanded in

terms of the complete set of eigenkets of ^. In order to make the following

A

discussion clearer, we now introduce a slightly more complicated notation. Each eigenvalue ëi will now be distinct, so that ëi 6 ëj for i 6 j. We let gi be

3.7 Postulates of quantum mechanics

89

the degeneracy of the eigenvalue ëi and let jiái, á 1, 2, . . . , gi, be the

orthonormal eigenkets of ^. We assume that the subset of kets corresponding

A

to each eigenvalue ëi has been made orthogonal by the Schmidt procedure outlined in Section 3.3.

If the eigenkets jiái constitute a discrete set, we may expand the state vector jØi as

jØi	gi
	ciájiái	(3:48)
i	á 1
XX
where the expansion coef®cients ciá are
ciá hiájØi		(3:49)
The expansion of the bra vector hØj is, therefore, given by
	g j
hØj XX c jâhjâj		(3:50)
j	â 1

where the dummy indices i and á have been replaced by j and â.

	^
The expectation value of A is obtained by substituting equations (3.48) and
(3.50) into (3.46)
g j	gi	g j	gi
XXXX		XXXX
hAi	c jâciáhjâjA^jiái	j â 1 i	c jâciáëihjâjiái
j â 1 i	á 1		á 1
gi
XXjciáj2ëi			(3:51)

i á 1

where we have noted that the kets jiái are orthonormal, so that hjâjiái äijäáâ

A comparison of equations (3.47) and (3.51) relates the probability Pi to the

expansion coef®cients ciá	X	X
Pi	gi	gi
	jciáj2	jhiájØij2	(3:52)
	á 1	á 1

where equation (3.49) has also been introduced. For the case where ëi is nondegenerate, the index á is not needed and equation (3.52) reduces to

Pi jcij2 jhijØij2

For a continuous spectrum of eigenkets with non-degenerate eigenvalues, it is more convenient to write the eigenvalue equation (3.45) in the form

^jëi ëjëi

A

where ë is now a continuous variable and jëi is the eigenfunction whose eigenvalue is ë. The expansion of the state vector Ø becomes

90	General principles of quantum theory
where	jØi …c(ë)jëi dë
	c(ë) hëjØi
and the expectation value of A takes the form
	…
	hAi jc(ë)j2 ë dë	(3:53)

If dPë is the probability of obtaining a value of A between ë and ë dë, then

equation (3.47) is replaced by

…

hAi ë dPë

and we see that

dPë jc(ë)j2 dë jhëjØij2 dë The probability dPë is often written in the form

dPë r(ë) dë

where r(ë) is the probability density of obtaining the result ë and is given by

	r(ë) jc(ë)j2 jhëjØij2
	In terms of the probability density, equation (3.53) becomes
	hAi	…ër(ë) dë		^	(3:54)
	In some applications to physical	systems, the eigenkets	of		possess a
				A

partially discrete and a partially continuous spectrum, in which case equations (3.51) and (3.53) must be combined.

The scalar product hØjØi may be evaluated from equations (3.48) and

(3.50) as		XX
XXXX
g j	gi	gi
hØjØi		c jâciáhjâjiái jciáj2
j â 1 i á 1		i á 1
XX		X
gi	jhiájØij2 Pi
i á 1		i

Since the state vector Ø is normalized, this expression gives

X

Pi 1

i

Thus, the sum of the probabilities Pi equals unity as it must from the de®nition of probability. For a continuous set of eigenkets, this relationship is replaced by

3.7 Postulates of quantum mechanics

91

……

dPë r(ë) dë 1

As an example, we consider a particle in a one-dimensional box as discussed in Section 2.5. Suppose that the state function Ø(x) for this particle is time-

independent and is given by
	ðx
Ø(x) C sin5			, 0	< x < a
		a

where C is a constant which normalizes Ø(x). The eigenfunctions jni and

							^
eigenvalues En of the Hamiltonian operator H are
j i r•••							n2 h2
2			nðx
n	a	sin		,	En		8ma	2	, n 1, 2, . . .
			a
Obviously, the state function Ø(x) is not an eigenfunction of											^
											H. Following the

general procedure described above, we expand Ø(x) in terms of the eigenfunctions jni. This expansion is the same as an expansion in a Fourier series, as described in Appendix B. As a shortcut we may use equations (A.39) and (A.40) to obtain the identity

sin5

è

1

(10 sin è ÿ 5 sin 3è sin 5è)

16

so that the expansion of Ø(x) is

C

5ðx

Ø(x)

"10 sin

ð

ÿ 5 sin

sin

#

16

a

a

a

16 r•••

j i ÿ

j i j i

C

a

(10

1

5 3

5 )

2

A measurement of the energy of a particle in state Ø(x) yields one of three values and no other value. The values and their probabilities are

			h2					102				100		0:794
E1					,	P1
			8ma2					102 52		12		126
			9h2					52	0:198
E3				,		P3
			8ma2					126
			25h2					12	0:008
E5				,		P5
			8ma2					126

The sum of the probabilities is unity,

P1 P3 P5 0:794 0:198 0:008 1

The interpretation that the quantity Ø (q1, q2, . . . , t)Ø(q1, q2, . . . , t) is the probability density that the coordinates of the system at time t are