14.4 Operators in Hilbert space

An operator acts on a function (or vector) to transform it into another funct-

ion (or vector) in the same space. We restrict ourselves to linear operators

which transform a function into a linear combination of other functions and

ˆ

denote operators by a hat, as A:

ˆ

ψ = Aψ. (14.9)

An operator can be represented by a matrix on a given orthonormal basis

set {φn }, transforming the representation c of ψ into c of ψ by an ordinary

matrix multiplication

c = Ac, (14.10)

where

φ— Aφm d„.

ˆ ˆ

Anm = φn |A|φm = (14.11)

n

Proof Expanding ψ on an orthonormal basis set φm and applying (14.9)

we have:

ˆ ˆ

ψ= cm φm = Aψ = cm Aφm .

m m

Now left-multiply by φ— and integrate over coordinates to form the scalar

n

products

ˆ

cm φn |φm = cm φn |A|φm = Anm cm ,

m m m

or

cn = (Ac)n .

With the superscript † we denote the hermitian conjugate, which is the transpose of the complex

1

conjugate: (A† )nm = A— . This is the usual notation in physics and chemistry, but in

mn

mathematical texts the hermitian conjugate is often denoted by —.

14.4 Operators in Hilbert space 383

ˆ

The eigenvalue equation for an operator A:

ˆ

Aψ = »ψ, (14.12)

now becomes on an orthonormal basis set an eigenvalue equation for the

matrix A:

Ac = »c. (14.13)

Solutions are eigenvectors c and eigenvalues ». If the basis set is not or-

thonormal, the equation becomes

Ac = »Sc. (14.14)

Hermitian operators form an important subclass of operators. An

ˆ

operator A is hermitian if

f |Ag = g|Af — ,

ˆ ˆ (14.15)

or

f — Ag d„ = (A— f — )g d„.

ˆ ˆ (14.16)

Hermitian operators have real expectation values (f = g = ψ) and real

eigenvalues (f = g; Af = »f ’ » = »— ). The operators of physically mean-

ˆ

ingful observables are hermitian. The matrix representation of a hermitian

operator is a hermitian matrix A = A† (f = φn , g = φm ).

Not only do hermitian operators have real eigenvalues, they also have

orthogonal eigenfunctions for non-degenerate (di¬erent) eigenvalues. The

eigenfunctions within the subspace corresponding to a set of degenerate

eigenvalues can be chosen to be orthogonal as well,2 and all eigenfunctions

may be normalized: The eigenfunctions of a hermitian operator (can be

chosen to) form an orthonormal set.

ˆ

Proof Let »n , ψn be eigenvalues and eigenfunctions of A:

ˆ

Aψn = »n ψn .

Then

ψ — Aψm d„ = »m —

ˆ ψn ψm d„,

2 If φ1 and φ2 are two eigenfunctions of the same (degenerate) eigenvalue », then any linear

combination of φ1 and φ2 is also an eigenfunction.

384 Vectors, operators and vector spaces

and

ψm Aψn d„ )— = »—

—ˆ —

( ψm ψn d„.

n

When A = A† then for n = m : »n = »— ’ » is real; for m = n and

n

— ψ d„ = 0.

»m = »n : ψ n m

ˆˆ

The commutator [A, B] of two operators is de¬ned as

ˆˆ ˆˆ ˆˆ

[A, B] = AB ’ B, A, (14.17)

ˆ ˆ

and we say that A and B commute if their commutator is zero. If two

operators commute, they have the same set of eigenvectors.

14.5 Transformations of the basis set

It is important to clearly distinguish operators that act on functions (vec-

tors) in Hilbert space, changing the vector itself, from coordinate transfor-

mations which are operators acting on the basis functions, thus changing

the representation of a vector, without touching the vector itself.

Consider a linear coordinate transformation Q changing a basis set {φn }

into a new basis set {φn }:

φn = Qin φi . (14.18)

i

ˆ

Let A be the representation of an operator A on {φn } and A its represen-

tation on {φn }. Then A and A relate as

A = Q† AQ. (14.19)

Proof Consider one element of A and insert (14.18):

Q— Qjm Aij = (Q† AQ)nm .

ˆ

(A )nm = φn |A|φm = in

ij

If both basis sets are orthonormal, then the transformation Q is unitary.3

Proof Orthonormality implies that

φn |φm = δnm .

A transformation (matrix) U is unitary if U† = U’1 .

3

14.6 Exponential operators and matrices 385