Physical Chemistry Third Edition

694 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

coordinates, but the wave function vanishes outside of the box, and we can omit that part of

the integration:

∫∞

−∞

ψ 1 (x)ψ 2 (x)dxC^2

∫a

0

sin

(

πx

a

)

sin

(

2 πx

a

)

dxC^2

a

π

∫π

0

sin(y) sin(2y)dy 0

where we have looked the integral up in Appendix C. One can also make a rough graph of

the integrand and argue that the positive and negative contributions to the integral cancel

each other.

Property 6, that two commuting hermitian operators can have a set of common

eigenfunctions, means the following: If ̂AandB̂are two hermitian operators that

commute with each other, then a set of functionsfjk(q) can be found such that

Af̂ jk(q)ajfjk(q) (16.3-32)

Bf̂ jk(q)bkfjk(q) (16.3-33)

whereajandbkare constant eigenvalues. Two indices are needed on the functions in

the set, because two functions can have the same eigenvalue forÂbut have different

eigenvalues forB̂. An example of simultaneous eigenfunctions is found in the elec-

tronic wave functions of the hydrogen atom, which are simultaneous eigenfunctions of

the Hamiltonian operator and two angular momentum operators. We discuss these in

Chapter 17.

The completeness property specified in Property 7 means that an arbitrary function

ψthat obeys the same boundary conditions as the set of eigenfunctions of a hermi-

tian operator̂Acan be exactly represented as alinear combination(sum of functions

multiplied by constant coefficients) of all of the eigenfunctions ofÂ.

ψ

∑∞

j 1

cjfj (16.3-34)

wheref 1 ,f 2 ,f 3 ,...are the set of eigenfunctions. This set of functions is called the

basis setor the set ofbasis functions. The functionψis said to beexpandedin terms

of the basis functions. The coefficientsc 1 ,c 2 ,c 3 ,...are calledexpansion coefficients.

There are generally infinitely many eigenfunctions of a given operator, and all of them

must be included in the sum of Eq. (16.3-34) for the representation to be exact. There

is apparently no general proof that the eigenfunctions of a hermitian operator form

a complete set. However, there are no known counterexamples, and this assertion is

generally accepted.

We can represent a time-dependent wave function in terms of time-independent

basis functions if the expansion coefficients have the correct time dependence. For

example, in Eq. (15.3-20) we chose the energy eigenfunctions as the basis functions

and wrote

Ψ(x,t)

∑∞

n 1

Ane−iEnt/h ̄ψn(x) (16.3-35)

where the coefficientsA 1 ,A 2 ,...are constants. It was shown in Example 15.3 that

this function satisfies the time-dependent Schrödinger equation.