Physical Chemistry Third Edition

16.6 Postulate 5. Measurements and the Determination of the State of a System 721

PROBLEMS

Section 16.6: Postulate 5. Measurements and the
Determination of the State of a System

16.40a.Find〈E〉for the coordinate wave function

ψ(x)

√

1

3

ψ 0 +

√

1

3

ψ 1 +

√

1

3

ψ 2

whereψ 0 ,ψ 1 , andψ 2 are the three lowest-energy

harmonic oscillator energy eigenfunctions.

b.Show that the function is normalized.

c.FindσEfor the wave function in part a.

d.Tell what values would occur in a set of many

measurements ofE, given that the system is in the

state corresponding to the wave function of part a

immediately before each measurement. Give the

probability of each value.

16.41a.A measurement of the energy of a particle in a
three-dimensional cubic box gives a value

14 h^2 /(8ma^2 ). Tell what eigenfunctions are included in

the linear combination representation of the wave

function after the measurement.

b.How could the particle be put into a known state?

16.42The energy of a particle in a one-dimensional box of

lengthais measured repeatedly with the particle restored

to a specific but unknown state before each measurement.

The results are summarized as follows:

Value Probability

h^2 /(8ma^2 ) 0.25

4 h^2 /(8ma^2 ) 0.375

9 h^2 /(8ma^2 ) 0.125

16 h^2 /(8ma^2 ) 0.25

What can you say about the state prior to the

measurement?

Summary of the Chapter

In this chapter we have presented five postulates that are the theoretical basis of quantum

mechanics. The first two postulates establish a one-to-one correspondence between the

mechanical state of a system and a wave function and establish the time-dependent

Schrödinger equation, which governs the wave functions.

The third postulate asserts that there is a hermitian mathematical operator for each

mechanical variable. To write the operator for a given variable: (1) write the classical

expression for the variable in terms of Cartesian coordinates and momentum compo-

nents, (2) replace each momentum component by the relation

px

h ̄

i

∂

∂x

and its analogues.

The first part of the fourth postulate asserts that the only possible outcomes of a

measurement of a variable are the eigenvalues of the operator corresponding to that

variable. The second part of the fourth postulate asserts that the expectation value of

the variableAis given by

〈A〉

∫

Ψ∗ÂΨdq

∫

Ψ∗Ψdq

By study of the standard deviation ofA, given by

σA[〈A^2 〉−〈A〉^2 ]^1 /^2