Physical Chemistry Third Edition

722 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

it was established that if the state just before a measurement ofAcorresponds to an

eigenfunction ofÂ, the only possible outcome of the measurement is the eigenvalue

corresponding to that eigenfunction. This is called the predictable case. If the wave

function is not an eigenfunction of̂A, the standard deviation gives a measure of the

spread of the distribution of results. This is called the statistical case.

The fifth postulate states that in a measurement ofA, if the result isaj, one of

the eigenvalues ofÂ, then the state of the system immediately after the measurement

corresponds to a wave function that is a linear combination only of those eigenfunctions

whose eigenvalues equalaj.

The measurement on the same system of a complete set of commuting observables

suffices to put the system into a state that is completely known, even though only partial

information is available about the state of the system prior to the measurements.

ADDITIONAL PROBLEMS

16.43 a.Show that the operatorL̂zis hermitian. Use spherical
polar coordinates.
b.Repeat the proof using Cartesian coordinates.
16.44a. Obtain a formula for the uncertainty productσxσpxfor
each of the first three energy eigenfunctions of a
particle in a one-dimensional box. Comment on any
trend that you see in these values.
b. Evaluate the uncertainty product for each of the first
three energy eigenfunctions of an electron in a box of
length 10.0 Å (1.00× 10 −^9 m).
16.45A free particle that can move only in thexdirection has a
wave function given by

ψ

√

2

3

eiax+

√

1

3

e−iax

whereais a real constant.

a.Find〈px〉andσpx.

b.Find the possible values that could occur in a single

measurement.

c.For each of the possible outcomes in part b, tell what

the wave function would be immediately after the

measurement.

d.For each of the values in part b, tell what the outcome

of a second measurement immediately after the first

measurement would be.

16.46a. Obtain a formula for the uncertaintyσxfor a particle

in a one-dimensional box of lengthafor a state

corresponding to a general energy eigenfunctionψn.

b.Find the limit of the formula of part a asn→∞.

c.Obtain a formula for the uncertaintyσpxfor a particle

in a one-dimensional box of lengthafor a state

corresponding to a general energy eigenfunctionψn.

d.Obtain a formula for the uncertainty productσxσpx

corresponding to a general energy eigenfunctionψn

for a particle in a one-dimensional box of lengtha.

e.For the baseball in Problem 15.13, find the value of

the uncertaintyσx.

f.For the baseball in Problem 15.13, find the value of

the uncertaintyσpx.

g.For the baseball in Problem 15.13, find the value of

the uncertainty productσxσpx.

16.47Label each statement as either true or false. If a statement

is true only under certain circumstances, label it as

false.

a.Every correct nonrelativistic wave function of an

isolated system satisfies the time-independent

Schrödinger equation.

b.Every correct wave function satisfies the

time-dependent Schrödinger equation.

c.Knowledge of the time-independent wave function

provides all available information about mechanical

variables of a system.

d.Knowledge of the time-dependent wave function

provides all available information about mechanical

variables of a system.