Physical Chemistry Third Edition

710 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

〈A〉

∑∞

j 1

∑∞

k 1

c∗jckakδjk

∑∞

j 1

c∗jcjaj

∑∞

j 1

|cj|^2 aj (16.4-32)

Comparison of Eq. (16.4-32) with Eq. (16.4-13) shows that the probability that the

eigenvalueajwill occur is

Probability ofajpj|cj|^2 (16.4-33)

Exercise 16.11

Find the probability of each of the eigenvalues in Example 16.19 and in Example 16.20.

PROBLEMS

Section 16.4: Postulate 4 and Expectation Values

16.20Show that the momentum operator (h/i ̄ )∂/∂xgives the
correct sign for〈px〉for a traveling wave given by

Ψeiκxe−iEt/h ̄

16.21a.Find the eigenfunctionsΦ(φ) of the operator for
thezcomponent of the angular momentum,
L̂z−ih ̄(∂/∂φ).
b.Sinceφ0 andφ 2 πrefer to the same location,
impose the boundary condition

Φ(0)Φ(2π)

and find the eigenvalues of̂Lz.

16.22Carry out the integration to show that the harmonic
oscillator coordinate wave function in Eq. (15.4-10) is
normalized.

16.23Obtain a formula for the expectation value of the
potential energy of a harmonic oscillator in thev 1
state. How does this relate to the total energy of the
harmonic oscillator in this state?

16.24a.For a particle in a hard box of lengtha, find the
expectation value of the quantityp^4 xfor then1 state.
b.Find the standard deviation ofp^2 x. Compare it with the
square of the standard deviation ofpx. Explain your
result.

16.25a.Draw sketches of the first two energy eigenfunctions
of a particle in a one-dimensional box of lengtha.
Without doing the integral explicitly, argue from the
graphs that the two functions are orthogonal.

b.Draw sketches of the first two energy eigenfunctions

of a harmonic oscillator. Without doing the integral

explicitly, argue from the graphs that the two functions

are orthogonal.

16.26a.Show for a harmonic oscillator in thev0 state that

〈V〉〈K〉whereVis the potential energy andKis

the kinetic energy.Hint:One way to proceed is to

calculate〈K〉and use the fact that〈K〉+〈V〉E.

b.Do you think that this will also be true for the other

energy eigenfunctions? Check it out forv1.

16.27a.Calculate〈p^2 x〉for each of the first three energy

eigenfunctions for the particle in a one-dimensional

box.Hint:Use the energy eigenvalues to avoid

detailed calculations.

b.Obtain a formula (a function ofn) for〈p^2 x〉for a

general energy eigenfunction of a particle in a

one-dimensional box.

c.Find the limit of the expression of part b asn→∞.

16.28From inspection of Figure 16.2, estimate the probability

of finding the particle in the left one-third of the box for

then1 state. After making this estimate, make a

calculation of the probability.

16.29a.Find a formula representing the turning point for a

classical harmonic oscillator that has the same energy

as thev1 quantum-mechanical energy.

b.Construct an accurate graph of the square of thev 1

wave function for a harmonic oscillator as a function

of

√

az.