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

72	General principles of quantum theory

hermitian operator involve several variables and that the weighting function must be used. The functions are, therefore, orthogonal with respect to the weighting function w(q1, q2, . . .).

If the weighting function is real and positive, then we can de®ne ö1 and ö2

as		ö1 pwø1,	ö2 pwø2
The functions ö1	and ö2 are ••••		••••		with respect	to a
		then mutually orthogonal
weighting function	of	unity. Moreover,	if the operator	^	is hermitian	with
				A
respect to ø1 and ø2		with a weighting function w, then		^
				A is hermitian with

respect to ö1 and ö2 with a weighting function equal to unity.

If two or more linearly independent eigenfunctions have the same eigenvalue, so that the eigenvalue is degenerate, the orthogonality theorem does not apply. However, it is possible to construct eigenfunctions that are mutually orthogonal. Suppose there are two independent eigenfunctions ø1 and ø2 of

^	with	the same eigenvalue á. Any		linear combination
the operator A
c1ø1 c2ø2, where c1		and c2 are any pair of complex numbers, is also an
eigenfunction of	^
	A with the same eigenvalue, so that
^	ø1 c2	^	^	c2ø2)
A(c1		ø2) c1Aø1	c2Aø2 á(c1ø1

From any pair ø1, ø2 which initially are not orthogonal, we can construct by selecting appropriate values for c1 and c2 a new pair which are orthogonal. By selecting different sets of values for c1, c2, we may obtain in®nitely many new pairs of eigenfunctions which are mutually orthogonal.

As an illustration, suppose the members of a set of functions ø1, ø2, . . . , øn are not orthogonal. We de®ne a new set of functions ö1, ö2, . . . , ön by the relations

ö1 ø1

ö2 aö1 ø2

ö3 b1ö1 b2ö2 ø3

...

If we require that ö2 be orthogonal to ö1 by setting hö1 j ö2i 0, then the constant a is given by

a ÿhø1 j ø2i=hø1 j ø1i ÿhö1 j ø2i=hö1 j ö1i

and ö2 is determined. We next require ö3 to be orthogonal to ö1 and to ö2, which gives

3.3 Hermitian operators

73

b1 ÿhö1 j ø3i=hö1 j ö1i

b2 ÿhö2 j ø3i=hö2 j ö2i

In general, we have

Xsÿ1

ös øs ksiöi i 1

ksi ÿhöi j øsi=höi j öii

This construction is known as the Schmidt orthogonalization procedure. Since the initial selection for ö1 can be any of the original functions øi or any linear combination of them, an in®nite number of orthogonal sets öi can be obtained by the Schmidt procedure.

We conclude that all eigenfunctions of a hermitian operator are either mutually orthogonal or, if belonging to a degenerate eigenvalue, can be chosen to be mutually orthogonal. Throughout the remainder of this book, we treat all the eigenfunctions of a hermitian operator as an orthogonal set.

Extended orthogonality theorem

The orthogonality theorem can also be extended to cover a somewhat more general form of the eigenvalue equation. For the sake of convenience, we present in detail the case of a single variable, although the treatment can be generalized to any number of variables. Suppose that instead of the eigenvalue

^
equation (3.5), we have for a hermitian operator A of one variable
^	(3:18)
Aøi(x) ái w(x)øi(x)

where the function w(x) is real, positive, and the same for all values of i. Therefore, equation (3.18) can also be written as

	A^ øj (x) áj w(x)øj (x)		(3:19)
Multiplication of equation (3.18) by øj (x) and integration over x give
…	øj (x)A^øi(x) dx ái …	øj (x)øi(x)w(x) dx	(3:20)
Now, the operator	^	respect to the functions øi	with a
	A is hermitian with

weighting function equaling unity, so that the integral on the left-hand side of

equation (3.20) becomes

… … …

ø ( ) ^ø ( ) d ø ( ) ^ ø ( ) d á ø ( )ø ( ) ( ) d j x A i x x i x A j x x j j x i x w x x

where equation (3.19) has been used as well. Accordingly, equation (3.20) becomes

74	General principles of quantum theory
	(ái ÿ áj ) …øj (x)øi(x)w(x) dx 0	(3:21)

When j i, the integral in equation (3.21) cannot vanish because the product øi øi and the function w(x) are always positive. Therefore, we have ái ái and the eigenvalues ái are real. For the situation where i 6j and

ái 6áj , the integral in equation (3.21) must vanish,

…

øj (x)øi(x)w(x) dx 0

(3:22)

Thus, the set of functions øi(x) for non-degenerate eigenvalues are mutually orthogonal when integrated with a weighting function w(x). Eigenfunctions corresponding to degenerate eigenvalues can be made orthogonal as discussed earlier.

The discussion above may be generalized to more than one variable. In the general case, equation (3.18) is replaced by

^	, . . .) ái w(q1	, q2,	. . .)øi(q1, q2, . . .)	(3:23)
Aøi(q1, q2
and equation (3.22) by
…
øj (q1, q2, . . .)øi(q1, q2, . . .)w(q1, q2, . . .) dq1 dq2 . . . 0				(3:24)

Equation (3.18) can also be transformed into the more usual form, equation (3.5). We ®rst de®ne a set of functions öi(x) as

öi(x) [w(x)]1=2øi(x) øi(x)=u(x)			(3:25)
where
u(x)		[w(x)]ÿ1=2	(3:26)

The function u(x) is real because w(x) is always positive and u(x) is positive because we take the positive square root. If w(x) approaches in®nity at any

point within the range of hermiticity of	^	(as x	approaches in®nity, for
	A

example), then øi(x) must approach zero such that the ratio öi(x) approaches

zero. Equation (3.18) is now multiplied by			u(x) and øi(x) is replaced by
u(x)öi(x)
	^		2	öi(x)
u(x)Au(x)öi(x) ái w(x)[u(x)]
If we de®ne an operator	^	^		^
	B by the relation B		u(x)Au(x) and apply equation

(3.26), we obtain

^ö ( ) á ö ( )

B i x i i x which has the form of equation (3.5). We observe that

3.4 Eigenfunction expansions

75

… … …

ø ^ø d ö ^ ö d ö ^ö d j A i x j uAu i x j B i x

… … …

( ^ø ) ø d ( ^ ö ) ö d ( ^ö ) ö d

A j i x Au j u i x B j i x

Since ^ is hermitian with respect to the ø s, the two integrals on the left of

A i

each equation equal each other, from which it follows that

……

ö ^ö d ( ^ö ) ö d

j B i x B j i x

and ^ is therefore hermitian with respect to the ö s.

B i

3.4 Eigenfunction expansions

Consider a set of orthonormal eigenfunctions øi of a hermitian operator. Any arbitrary function f of the same variables as øi de®ned over the same range of these variables may be expanded in terms of the members of set øi

X

f aiøi	(3:27)
i

where the ais are constants. The summation in equation (3.27) converges to the function f if the set of eigenfunctions is complete. By complete we mean that no other function g exists with the property that hg j øii 0 for any value of i, where g and øi are functions of the same variables and are de®ned over the same variable range. As a general rule, the eigenfunctions of a hermitian operator are not only orthogonal, but are also complete. A mathematical criterion for completeness is presented at the end of this section.

The coef®cients ai are evaluated by multiplying (3.27) by the complex conjugate øj of one of the eigenfunctions, integrating over the range of the variables, and noting that the øis are orthonormal

* X + X

høj j f i øj aiøi aihøj j øii aj

ii

Replacing the dummy index j by i, we have
ai høi j f i		(3:28)
Substitution of equation (3.28) back into (3.27) gives
f Xi	høi j f iøi	(3:29)

øi (x9)øi(x) ä(x ÿ x9)

76 General principles of quantum theory

Completeness				and f are expanded as in equation (3.27),
We now evaluate hf j f i in which f
with the two independent summations given different dummy indices								jaij2
hf j f i * j	ajøj	i	aiøi	+	j i	aj ajhøj j øii	i
X		X			XX		X

Without loss of generality we may assume that the function f is normalized, so

that hf j f i 1 and

X

jaij2 1 (3:30)

i

Equation (3.30) may be used as a criterion for completeness. If an eigenfunction øn with a non-vanishing coef®cient an were missing from the summation in equation (3.27), then the series would still converge, but it would be incomplete and would therefore not converge to f . The corresponding coef®- cient an would be missing from the left-hand side of equation (3.30). Since each term in the summation in equation (3.30) is positive, the sum without an would be less than unity. Only if the expansion set øi in equation (3.27) is complete will (3.30) be satis®ed.

The completeness criterion can also be expressed in another form. For this purpose we need to introduce the variables explicitly. For simplicity we assume

®rst that f is a function of only one variable x. In this case, equation (3.29) is
f (x)	i	…øi (x9) f (x9) dx9 øi(x)
	X

where x9 is the dummy variable of integration. Interchanging the order of summation and integration gives

…	"	X	øi (x9)øi(x)#f (x9) dx9
f (x)		i

Thus, the summation is equal to the Dirac delta function (see Appendix C)

X

(3:31)

i

This expression, known as the completeness relation and sometimes as the closure relation, is valid only if the set of eigenfunctions is complete, and may be used as a mathematical test for completeness. Notice that the completeness relation (3.31) is not related to the choice of the arbitrary function f , whereas the criterion (3.30) is related.

The completeness relation for the multi-variable case is slightly more complex. When expressed explicitly in terms of its variables, equation (3.29) is

		3.5 Simultaneous eigenfunctions			77
	X
f (q1, q2, . . .)	i	…øi (q91, q92, . . .) f (q91, q92, . . .)w(q91, q92, . . .) dq91, dq92, . . .
	3 øi(q1, q2, . . .)
Interchanging the order of summation and integration gives
		…	X	øi (q91, q92, . . .)øi(q1, q2, . . .)#
f (q1, q2, . . .)			" i
		3 f (q91, q92, . . .)w(q91, q92, . . .) dq1 dq2 . . .
so that the completeness relation takes the form
w(q91, q92, . . .) Xi		øi (q91, q92, . . .)øi(q1, q2, . . .) ä(q1 ÿ q91)ä(q2 ÿ q92) . . .
					(3:32)

3.5 Simultaneous eigenfunctions

Suppose the members of a complete set of functions øi are simultaneously

		^	^
eigenfunctions of two hermitian operators A and B with eigenvalues ái and âi,
respectively
	^
	Aøi áiøi
	^
	Bøi âiøi			^
			^
If we operate on the ®rst eigenvalue equation with B and on the second with A,
we obtain
^ ^		^	áiâiøi
BAøi		áiBøi
^^		^	áiâiøi
ABøi		âiAøi
from which it follows that
^^	^ ^
(AB	ÿ BA)øi [A, B]øi 0

Thus, the functions ø are eigenfunctions of the commutator [ ^, ^] with i A B

eigenvalues equal to zero. An operator that gives zero when applied to any

			^	^
member of a complete set of functions is itself zero, so that A and B commute.
We have just shown that		^	^
	if the operators A and B have a complete set of
	^	^
simultaneous eigenfunctions, then A and B commute.

We now prove the converse, namely, that eigenfunctions of commuting

operators can always be		constructed	to	be simultaneous		eigenfunctions.
Suppose that	^	^	^		^	^
	Aøi áiøi	and that [A,	B] 0. Since		A and	B commute, we

have

78	General principles of quantum theory
^^		^ ^	^	^
ABøi BAøi			B(áiøi) áiBøi
Therefore, the function	^			^
	Bøi is an eigenfunction of A with eigenvalue ái.
There are now two	possibilities;		the	eigenvalue ái of	^	is either non-
					A

degenerate or degenerate. If ái is non-degenerate, then it corresponds to only

	^	is proportional to
	one independent eigenfunction øi, so that the function Bøi
	øi
	^
	Bøi âiøi		^
	where âi is the proportionality constant and therefore the	eigenvalue of
			B

corresponding to øi. Thus, the function øi is a simultaneous eigenfunction of

both ^ and ^.

A B

On the other hand, suppose the eigenvalue ái is degenerate. For simplicity, we consider the case of a doubly degenerate eigenvalue ái; the extension to n- fold degeneracy is straightforward. The function øi is then any linear combina-

tion of two linearly independent, orthonormal eigenfunctions øi1 and øi2 of ^

A

corresponding to the eigenvalue ái

øi c1øi1 c2øi2

		^
We need to determine the coef®cients c1, c2 such that Bøi âiøi, that is
^	^	âi(c1øi1 c2øi2)
c1Bøi1	c2Bøi2

If we take the scalar product of this equation ®rst with øi1 and then with øi2, we obtain

c1(B11 ÿ âi) c2 B12 0 c1 B21 c2(B22 ÿ âi) 0

where we have introduced the simpli®ed notation

jk høij j ^øik i

B

B

These simultaneous linear homogeneous equations determine c1 and c2 and have a non-trivial solution if the determinant of the coef®cients of c1, c2

vanishes			ÿ
		B11 ÿ âi
or			B12			0
		B21	B22	âi

â2i ÿ (B11 B22)âi B11 B22 ÿ B12 B21 0

â(1) â(2)

This quadratic equation has two roots i and i , which lead to two corresponding sets of constants c(1)1 , c(1)2 and c(2)1 , c(2)2 . Thus, there are two

ø(1) ø(2)

distinct functions i and i

3.5 Simultaneous eigenfunctions				79
ø(1)i c1(1)øi1		c2(1)øi2
ø(2)i c1(2)øi1		c2(2)øi2
which satisfy the relations
^ (1)		(1)	(1)
Bøi	âi		øi
^ (2)		(2)	(2)
Bøi	âi		øi

and are, therefore, simultaneous eigenfunctions of the commuting operators ^

A

and ^.

B

	This analysis can be extended to three or more operators. If three operators
^	^	^
A,	B, and	C have a complete set of simultaneous eigenfunctions, then the
		^	^	^	^	^	^
argument above shows that A and B commute, B and					C commute, and A and C
commute. Furthermore, the converse is also true. If					^		^
					A commutes with both B
	^	^	^
and C, and B commutes with			C, then the three operators possess simultaneous

eigenfunctions. To show this, suppose that the three operators commute with

one another. We know that since ^ and ^ commute, they possess simultaneous

A B

eigenfunctions øi such that

		^
		Aøi áiøi
		^
		Bøi âiøi
			^
We next operate on each of these expressions with C, giving
^ ^		^ ^	^	^
CAøi		A(Cøi) C(áiøi) ái(Cøi)
^^		^ ^	^	^
CBøi		B(Cøi) C(âiøi) âi(Cøi)
^			^	^
Thus, the function Cøi	is an eigenfunction of both A and B with eigenvalues ái
and âi, respectively. If ái and âi			are non-degenerate, then there is only one
eigenfunction øi corresponding to them and the function					^
					Cøi is proportional
to øi
		^
		Cøi ãiøi
^	^	^	possess simultaneous		eigenfunctions. For
and, consequently, A,	B,	and C

degenerate eigenvalues ái and/or âi, simultaneous eigenfunctions may be constructed using a procedure parallel to the one described above for the

doubly degenerate two-operator case.
	^			^	^		^	^
	We note here that if A commutes with B and B commutes with C, but A does
	^	^	^					^
not commute with C, then A and B possess simultaneous eigenfunctions, B and
^	possess simultaneous	eigenfunctions, but			^	^	do not. The set of
C					A and	C
			^	^			^	^

simultaneous eigenfunctions of A and B will differ from the set for B and C.

An example of this situation is discussed in Chapter 5.

80	General principles of quantum theory

In some of the derivations presented in this section, operators need not be hermitian. However, we are only interested in the properties of hermitian operators because quantum mechanics requires them. Therefore, we have implicitly assumed that all the operators are hermitian and we have not bothered to comment on the parts where hermiticity is not required.

3.6 Hilbert space and Dirac notation

This section introduces the basic mathematics of linear vector spaces as an alternative conceptual scheme for quantum-mechanical wave functions. The concept of vector spaces was developed before quantum mechanics, but Dirac applied it to wave functions and introduced a particularly useful and widely accepted notation. Much of the literature on quantum mechanics uses Dirac's ideas and notation.

A set of complete orthonormal functions øi(x) of a single variable x may be regarded as the basis vectors of a linear vector space of either ®nite or in®nite dimensions, depending on whether the complete set contains a ®nite or in®nite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an in®nite number of members and, therefore, are usually concerned with linear vector spaces of in®nite dimensionality. Such a linear vector space is called a Hilbert space. The functions øi(x) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of in®nite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol jøii or sometimes simply by jii. These ket vectors determine a ket space.

When a ket jøii is multiplied by a constant c, the result c jøii jcøii is a

ket in the same direction as jøii; only		the magnitude of the ket vector is
changed. However, when an operator	^
	A acts on a ket jøii, the result is another
ket jöii
^		^

jöii Ajøii jAøii

In general, the ket jöii is not in the same direction as jøii nor in the same direction as any other ket jøji, but rather has projections along several or all

basis kets. If an operator	^	acts	on all kets jøii of the basis set, and the
	A
		^	are orthonormal, then the net result of the
resulting set of kets jöii jAøii

3.6 Hilbert space and Dirac notation

81

operation is a rotation of the basis set jøii about the origin to a new basis set

	^
jöii. In the situation where A acting on jøii gives a constant times jøii (cf.
equation (3.5))
^	^
Ajøii jAøii áijøii
^		jøii and the kets jøii are said to be
the ket jAøii is along the direction of
^
eigenkets of the operator A.
	^	^
Although the expressions Ajøii and		jAøii are completely equivalent, there
		^	^

is a subtle distinction between them. The ®rst, Ajøii, indicates the operator A

being applied to the ket jø i. The quantity j ^ø i is the ket which results from

i A i

that application.

Bra vectors

The functions øi(x) are, in general, complex functions. As a consequence, ket space is a complex vector space, making it mathematically necessary to introduce a corresponding set of vectors which are the adjoints of the ket vectors. The adjoint (sometimes also called the complex conjugate transpose) of a complex vector is the generalization of the complex conjugate of a complex number. In Dirac notation these adjoint vectors are called bra vectors or bras and are denoted by høij or hij. Thus, the bra høij is the adjoint jøiiy of the ket jøii and, conversely, the ket jøii is the adjoint høijy of the bra høij

jøiiy høij

høijy jøii

These bra vectors determine a bra space, just as the kets determine ket space. The scalar product or inner product of a bra höj and a ket jøi is written in

Dirac notation as höjøi and is de®ned as

…

höjøi ö (x)ø(x) dx

The bracket (bra-c-ket) in höjøi provides the names for the component vectors. This notation was introduced in Section 3.2 as a shorthand for the scalar product integral. The scalar product of a ket jøi with its corresponding bra høj gives a real, positive number and is the analog of multiplying a complex number by its complex conjugate. The scalar product of a bra højj

^	^		^
and the ket jAøii is expressed in Dirac notation as højjAjøii or as			hjjAjii.
These scalar products are also known as the	matrix elements of	^	and are
		A

sometimes denoted by Aij.

To every ket in ket space, there corresponds a bra in bra space. For the ket