15.4 The Quantum Harmonic Oscillator
681
where E is the energy of the system, ψ is a wave function, and
H is the Hamiltonian
operator. The time-dependent Schrödinger equation is:
Hψ i ¯
h
∂Ψ
∂t
By assuming that the wave function Ψ is a product of a coordinate factor ψ and a
time factor η, the coordinate factor is found to obey the time-independent Schrödinger
equation.
The solutions to the time-independent Schrödinger equation for three example systems were presented: the particle in a box (in one dimension and in three dimensions),
the free particle, and the harmonic oscillator. Sets of energy eigenfunctions and energy
eigenvalues were obtained, and in the cases of the particle in a box and the harmonic
oscillator, we found a discrete spectrum of energies, corresponding to energy quantization. Two new phenomena occurred. First, the particle in a box and harmonic oscillator
exhibited a zero-point energy. Second, the harmonic oscillator has a nonzero wave
function in regions where classical mechanics predicts that the particle cannot enter.
A D D I T I O N A L P R O B L E M S
15.29 Describe and discuss the conditions that a
one-dimensional wave function must obey at a point
where
a. V is a continuous function.
b. V is a discontinuous function with a finite step
discontinuity.
c. V is a discontinuous function with an infinite step
discontinuity.
15.30 Consider an automobile with a coil spring at each wheel.
If a mass of 100 kg is suspended from one such spring,
the spring lengthens by 0.020 m. The “unsprung weight”
(the effective mass of the wheel and suspension
components) of one wheel is equal to 25 kg. The mass of
the part of the automobile supported by the springs is
1400 kg.
a. Find the force constant for each spring.
b. Assuming that all four springs are identical and that
one-fourth of the supported mass is supported at each
wheel, find the distance that each spring is compressed
from its equilibrium length when the automobile is
resting on its wheels.
c. Find the potential energy of each spring when the
automobile is resting on its wheels.
d. Find the period and the frequency of oscillation of a
wheel when it is hanging freely.
e. If the automobile is suddenly lifted off its wheels, find
the speed of the wheel when the spring passes through
its equilibrium length if no shock absorber is present to
slow it down.
f. Find the energy of a quantum of energy of an
oscillating wheel according to quantum mechanics.
g. Find the value of the quantum number when the energy
of the oscillating wheel is equal to the energy of part c.
h. Find the wavelength of the electromagnetic radiation
whose photons have energy equal to hν, where ν is the
frequency of oscillation of part d.
15.31 Calculate the de Broglie wavelength of an electron
moving with the kinetic energy corresponding to the
n 5 state of a hydrogen atom according to the Bohr
theory. Show that this wavelength is equal to 1/5 of the
circumference of the fifth Bohr orbit.
15.32 Assume that the motion of the earth around the sun is
described by the Bohr hydrogen atom theory. The
electrostatic attraction is replaced by the gravitational
attraction, given by the formula
F −Gm 1 m 2 /r
2
where G is the gravitational constant, equal to
6.673 × 10 −11 m 3 s −2 kg −1 and m 1 and m 2 are the
masses of the two objects. The mass of the earth is
5.983 × 10 24 kg, and the mass of the sun is larger by a
681
where E is the energy of the system, ψ is a wave function, and
H is the Hamiltonian
operator. The time-dependent Schrödinger equation is:
Hψ i ¯
h
∂Ψ
∂t
By assuming that the wave function Ψ is a product of a coordinate factor ψ and a
time factor η, the coordinate factor is found to obey the time-independent Schrödinger
equation.
The solutions to the time-independent Schrödinger equation for three example systems were presented: the particle in a box (in one dimension and in three dimensions),
the free particle, and the harmonic oscillator. Sets of energy eigenfunctions and energy
eigenvalues were obtained, and in the cases of the particle in a box and the harmonic
oscillator, we found a discrete spectrum of energies, corresponding to energy quantization. Two new phenomena occurred. First, the particle in a box and harmonic oscillator
exhibited a zero-point energy. Second, the harmonic oscillator has a nonzero wave
function in regions where classical mechanics predicts that the particle cannot enter.
A D D I T I O N A L P R O B L E M S
15.29 Describe and discuss the conditions that a
one-dimensional wave function must obey at a point
where
a. V is a continuous function.
b. V is a discontinuous function with a finite step
discontinuity.
c. V is a discontinuous function with an infinite step
discontinuity.
15.30 Consider an automobile with a coil spring at each wheel.
If a mass of 100 kg is suspended from one such spring,
the spring lengthens by 0.020 m. The “unsprung weight”
(the effective mass of the wheel and suspension
components) of one wheel is equal to 25 kg. The mass of
the part of the automobile supported by the springs is
1400 kg.
a. Find the force constant for each spring.
b. Assuming that all four springs are identical and that
one-fourth of the supported mass is supported at each
wheel, find the distance that each spring is compressed
from its equilibrium length when the automobile is
resting on its wheels.
c. Find the potential energy of each spring when the
automobile is resting on its wheels.
d. Find the period and the frequency of oscillation of a
wheel when it is hanging freely.
e. If the automobile is suddenly lifted off its wheels, find
the speed of the wheel when the spring passes through
its equilibrium length if no shock absorber is present to
slow it down.
f. Find the energy of a quantum of energy of an
oscillating wheel according to quantum mechanics.
g. Find the value of the quantum number when the energy
of the oscillating wheel is equal to the energy of part c.
h. Find the wavelength of the electromagnetic radiation
whose photons have energy equal to hν, where ν is the
frequency of oscillation of part d.
15.31 Calculate the de Broglie wavelength of an electron
moving with the kinetic energy corresponding to the
n 5 state of a hydrogen atom according to the Bohr
theory. Show that this wavelength is equal to 1/5 of the
circumference of the fifth Bohr orbit.
15.32 Assume that the motion of the earth around the sun is
described by the Bohr hydrogen atom theory. The
electrostatic attraction is replaced by the gravitational
attraction, given by the formula
F −Gm 1 m 2 /r
2
where G is the gravitational constant, equal to
6.673 × 10 −11 m 3 s −2 kg −1 and m 1 and m 2 are the
masses of the two objects. The mass of the earth is
5.983 × 10 24 kg, and the mass of the sun is larger by a