680
15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation
d. Express the value of ψ 2
2 in terms of the parameter a at
each relative maximum.
15.24 From the recursion relation of Eq. (15.4-6),
a. Show that if c 2 is nonzero and c 4 vanishes, the energy
eigenvalue for v 2 results.
b. Show that if b 2mE 2 / ¯
h and if c 0 −1, then
c 2 2a, verifying the formula given for ψ 2 in Eq.
(15.4-12).
c. Show that the formula in Eq. (F-41) produces
coefficients in the same ratio for H 2 (
√
a x) as given in
Eq. (15.4-12).
d. Generate the Hermite polynomial H 4 (
√
a x) from
Eq. (F-41) and write the formula for ψ 4 for the
harmonic oscillator. Do not include the constant in
front, which will be discussed later.
15.25 A two-dimensional harmonic oscillator has the potential
energy function
V V (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.
15.26 A three-dimensional harmonic oscillator has the potential
energy function
V V (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.
15.27 A 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 × 10 13 s −1 . Find the force constant k.
15.28 A harmonic oscillator potential energy function is
modified so that
V
kx 2 /2 if |x| < x
∞
if |x| > x
where x 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
where h is Planck’s constant, m is the mass of the particle, and v is 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
Hψ Eψ
15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation
d. Express the value of ψ 2
2 in terms of the parameter a at
each relative maximum.
15.24 From the recursion relation of Eq. (15.4-6),
a. Show that if c 2 is nonzero and c 4 vanishes, the energy
eigenvalue for v 2 results.
b. Show that if b 2mE 2 / ¯
h and if c 0 −1, then
c 2 2a, verifying the formula given for ψ 2 in Eq.
(15.4-12).
c. Show that the formula in Eq. (F-41) produces
coefficients in the same ratio for H 2 (
√
a x) as given in
Eq. (15.4-12).
d. Generate the Hermite polynomial H 4 (
√
a x) from
Eq. (F-41) and write the formula for ψ 4 for the
harmonic oscillator. Do not include the constant in
front, which will be discussed later.
15.25 A two-dimensional harmonic oscillator has the potential
energy function
V V (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.
15.26 A three-dimensional harmonic oscillator has the potential
energy function
V V (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.
15.27 A 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 × 10 13 s −1 . Find the force constant k.
15.28 A harmonic oscillator potential energy function is
modified so that
V
kx 2 /2 if |x| < x
∞
if |x| > x
where x 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
where h is Planck’s constant, m is the mass of the particle, and v is 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
Hψ Eψ