Physical Chemistry Third Edition

16.4 Postulate 4 and Expectation Values 697

The expectation value is the predicted mean of a set of many measurements of the

variable, given that the system is in the state corresponding to the wave functionΨat

the time of each measurement. As is the case with all quantum mechanical integrations,

the integrals in Eq. (16.4-1) extend over all values of the coordinates, which are abbrevi-

ated byq. We will see that Eq. (16.4-1) leads to the famous fact that quantum mechanics

often provides only statistical information.

Consider first the case of a wave functionΨthat is a product of a coordinate wave

function and a time-dependent factor. The complex conjugate of the time-dependent

factor can be obtained by changing the sign in front of theisymbol in the exponent

(see Appendix B):

(

e−iEt/h ̄

)∗

eiEt/h ̄ (16.4-2)

The expectation value ofAis given by

〈A〉

∫

ψ∗eiEt/h ̄̂Aψe−iEt/h ̄dq

∫

ψ∗eiEt/h ̄ψe−iEt/h ̄dq

(16.4-3)

Since the operatorÂcontains no time dependence, the time-dependent factors cancel:

〈A〉

∫

ψ∗Aψdq̂

∫

ψ∗ψdq

(16.4-4)

The expectation value can be obtained from the coordinate wave function and is time-

independent if the wave function is the product of a coordinate factor and a time

factor. A state corresponding to such a wave function is calleda stationary state, and

corresponds to a standing wave.

Normalization

There is a conventional way to simplify the formula for an expectation value. We use

the following fact, established in Problem 15.5:If any wave function that satisfies

the Schrödinger equation is multiplied by an arbitrary constant it will still satisfy the

Schrödinger equation and will give the same value for any expectation value.

EXAMPLE16.9

Show that the formula in Eq. (16.4-1) for the expectation value is unchanged ifΨis replaced

byCΨ, whereCis any constant.

Solution

〈A〉

∫

C∗Ψ∗AĈΨdq

∫

C∗Ψ∗CΨdq



C∗C

∫

Ψ∗̂AΨdq

C∗C

∫

Ψ∗Ψdq



∫

Ψ∗̂AΨdq

∫

Ψ∗Ψdq