Physical Chemistry Third Edition

16.4 Postulate 4 and Expectation Values 709

When we multiply out the integrand, there are four terms. Sinceψ 0 andψ 1 are orthogonal

to each other, the terms containing bothψ 0 andψ 1 vanish after integration.

〈E〉

1

2

∫

ψ∗ 0 (q)eiE^0 t/ ̄hE 0 ψ 0 (q)e−iE^0 t/h ̄dq

+

1

2

∫

ψ∗ 1 (q)eiE^1 t/ ̄hE 1 ψ 1 (q)e−iE^1 t/h ̄)dq

We can factor theEvalues out of the integrals. The complex time factors cancel, and

sinceψ 0 andψ 1 are normalized

〈E〉

1

2

(E 0 +E 1 )

b.To show that〈E^2 〉is time-independent, we carry out exactly the same steps as in part a

except that we operate twice withĤ, obtaining equations exactly as in part a except that

we haveE^2 values instead ofEvalues. The result is that

〈E^2 〉

1

2

(E^20 +E^21 )

so thatσEis time-independent and has the same value as in the previous example.

Since the eigenfunctions of any hermitian operator form a complete set, we can

express a wave function at a fixed time as the following linear combination

ψ

∑∞

k 1

ckfk∗ (16.4-29)

wheref 1 ,f 2 , and so on are eigenfunctions of a quantum-mechanical operatorÂ.

We substitute the expansion of Eq. (16.4-29) into the expression for the expectation

value〈A〉:

〈A〉

∫

ψ∗Aψdq̂ 

∑∞

j 1

∑∞

k 1

c∗jckak

∫

fj∗fkdq (16.4-30)

whereakis the eigenvalue ofÂcorresponding tofk. We have used the eigenfunction

property, have factored the constants out of the integrals, and have exchanged the order

of integrating and summing. This exchange is acceptable if the sums and integrals are

uniformly convergent, which means that the sum and the integral converge with at least

a certain rate for all values of the variables on which they depend after summation or

integration.

Since the functionsf 1 ,f 2 ,...are orthogonal to each other, those integrals in which

jkwill vanish, and since the functions are normalized the integrals withjkwill

equal unity. We write

∫

fj∗fkdqδjk

{

1ifjk

0ifjk

(16.4-31)

This equation defines the quantityδjk, which equals unity when its two indices are

equal to each other and equals zero otherwise. It is called theKronecker delta. When

the sum overkis performed, only thejkterm will be nonzero. The sum overk

collapses to a single term withkj: