Physical Chemistry Third Edition

15.4 The Quantum Harmonic Oscillator 681

whereEis the energy of the system,ψis a wave function, andĤis the Hamiltonian

operator. The time-dependent Schrödinger equation is:

Hψ̂ ih ̄∂Ψ

∂t

By assuming that the wave functionΨis a product of a coordinate factorψand a

time factorη, the coordinate factor is found to obey the time-independent Schrödinger

equation.

The solutions to the time-independent Schrödinger equation for three example sys-

tems 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 quantiza-

tion. 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.

ADDITIONAL PROBLEMS

15.29Describe and discuss the conditions that a
one-dimensional wave function must obey at a point
where

a.Vis a continuous function.

b.Vis a discontinuous function with a finite step

discontinuity.

c.Vis a discontinuous function with an infinite step

discontinuity.

15.30Consider 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 tohν, whereνis the

frequency of oscillation of part d.

15.31Calculate the de Broglie wavelength of an electron

moving with the kinetic energy corresponding to the

n5 state of a hydrogen atom according to the Bohr

theory. Show that this wavelength is equal to 1/5ofthe

circumference of the fifth Bohr orbit.

15.32Assume 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

whereGis the gravitational constant, equal to

6. 673 × 10 −^11 m^3 s−^2 kg−^1 andm 1 andm 2 are the

masses of the two objects. The mass of the earth is

5. 983 × 1024 kg, and the mass of the sun is larger by a