Physical Chemistry Third Edition

16.4 Postulate 4 and Expectation Values 707

The predictable case is always identifiable by calculating the standard deviation of

the measurement. Here is an important fact:If a system is in a state corresponding

to an eigenfunction of the variable to be measured, the measurement belongs to the

predictable case.

EXAMPLE16.18

For a particle in a one-dimensional box, find〈E〉andσEfor then1 stationary state.

Solution

σE

[

〈E^2 〉−〈E〉^2

] 1 / 2

〈E〉

∫

ψ∗ 1 Hψdx̂ 

∫

ψ 1 ∗E 1 ψ 1 dxE

∫

ψ 1 ∗ψ 1 dxE 1 

h^2

8 ma^2

〈E^2 〉

∫

ψ∗ 1 Ĥ^2 ψ 1 dx

∫

ψ∗ 1 E 12 ψ 1 dxE^21

∫

ψ 1 ∗ψ 1 dxE^21

where we have used the fact thatĤ^2 ĤĤ. The standard deviation vanishes:

σE(E^21 −E^21 )^1 /^2  0

The outcome of the energy measurement is completely predictable if the particle is in this

stationary state, which corresponds to an energy eigenfunction.

Exercise 16.10

a.For a general system whose wave functionψjis an eigenfunction of the operator̂Awith

eigenvalueaj, show that〈A〉ajand thatσA0.

b.For a one-dimensional harmonic oscillator, find〈E〉andσEfor the state corresponding to

thev1 energy eigenfunction.

Wave Functions That Are Not Energy Eigenfunctions

The wave function of a system at a given instant can be any function that obeys

the proper boundary conditions. There is no requirement that it must obey the time-

independent Schrödinger equation. However, since the energy eigenfunctions are a

complete set, any wave function at a fixed time can be written as a linear combination

of energy eigenfunctions, as in Eqs. (15.3-20) and (16.3-35) withtset equal to zero.

EXAMPLE16.19

For a one-dimensional harmonic oscillator, find〈E〉andσEif the state just prior to the

measurement corresponds to the normalized wave function

ψ

√

1

2

(ψ 0 +ψ 1 )

whereψ 0 andψ 1 are the first two energy eigenfunctions, given in Eqs. (15.4-10) and (15.4-11).