Physical Chemistry Third Edition

680 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation

d.Express the value ofψ^22 in terms of the parameteraat

each relative maximum.

15.24From the recursion relation of Eq. (15.4-6),

a.Show that ifc 2 is nonzero andc 4 vanishes, the energy

eigenvalue forv2 results.

b.Show that ifb 2 mE 2 /h ̄and ifc 0 −1, then

c 2  2 a, verifying the formula given forψ 2 in Eq.

(15.4-12).

c.Show that the formula in Eq. (F-41) produces

coefficients in the same ratio for H 2 (

√

ax) as given in

Eq. (15.4-12).

d.Generate the Hermite polynomial H 4 (

√

ax) from

Eq. (F-41) and write the formula forψ 4 for the

harmonic oscillator. Do not include the constant in

front, which will be discussed later.

15.25A two-dimensional harmonic oscillator has the potential
energy function

VV(x,y)

k

2

(x^2 +y^2 )

a.Write the time-independent Schrödinger equation and

find its solutions by separation of variables, using the

one-dimensional harmonic oscillator solutions.

b.Find the energy eigenvalues and degeneracies for the

first 10 energy levels.

15.26A three-dimensional harmonic oscillator has the potential
energy function

VV(x,y)

k

2

(x^2 +y^2 +z^2 )

a.Write the time-independent Schrödinger equation and

find its solutions by separation of variables, using the

one-dimensional harmonic oscillator solutions.

b.Find the energy eigenvalues and degeneracies for the

first 5 energy levels.

15.27A diatomic molecule vibrates like a harmonic oscillator

with mass equal to the reduced mass of the nuclei of the

molecule.

a.Calculate the reduced mass of the nuclei of an HBr

molecule. Calculate its ratio to the mass of a hydrogen

atom.

b.The vibrational frequency of the HBr molecule is

ν 7. 944 × 1013 s−^1. Find the force constantk.

15.28A harmonic oscillator potential energy function is

modified so that

V

{

kx^2 /2if|x|<x′

∞ if|x|>x′

wherex′is some positive constant that is greater than the

classical turning point for the energies that we will

consider.

a.Tell qualitatively how this will affect the classical

solution.

b.Tell qualitatively how this will affect the

quantum-mechanical solution.

c.Will tunneling occur? In what region?

d.Draw a rough sketch of the first two wave functions.

Summary of the Chapter

De Broglie sought a physical justification for Bohr’s assumption of quantization, and

hypothesized that all particles move with a wave-like character with a wavelength

given by

λ

h

mv



h

p

wherehis Planck’s constant,mis the mass of the particle, andvis its speed. According

to the concept of wave–particle duality, electrons and other objects have some of the

properties of classical waves and some of the properties of classical particles.

Schrödinger discovered a wave equation for de Broglie waves. The time-independent

Schrödinger equation is an eigenvalue equation

Hψ̂ Eψ