674 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation∫∞−∞ψ 2 (x)ψ 3 (x)dx 0b.Work out the integral and show thatψ 2 andψ 3 are
orthogonal. Remember thatψ0 outside of
the box.15.13Think of a baseball on its way from the pitcher’s mound
to home plate as being a particle in a box of length
60 feet. Assume that the baseball has a mass of 5.1
ounces. If the baseball has a speed of 95 miles per hour,
find its kinetic energy and the value of the quantum
numberncorresponding to this value ofE. Find the
number of nodes in the wave function and find the
wavelength corresponding to this many nodes in a length
of 60 feet. Compare this wavelength with the de Broglie
wavelength in Example 15.1.
15.14Assume that a sample of helium has been trapped in a
matrix of solid argon at 84 K, and that each helium atom
is confined in a cavity such that the center of mass of the
helium atom can move (“translate”) in a cubical region
5. 00 × 10 −^10 m on a side.
a.Find the energies and degeneracies of the two
lowest-energy translational levels.
b.Find the frequency and wavelength of a photon absor-
bed in a transition between the two levels of part a.
Do you think that such a transition would actually be
observed spectroscopically?15.15An electron is confined in a cubical box
10 .0Å(1. 00 × 10 −^9 m) in length, width, and height.
a.Find the energy eigenvalues of the first two energy
levels.
b.Find the degeneracies of the first two energy levels.
c.Find the frequency and wavelength of the light
absorbed if the electron makes a transition from the
lowest energy level to the next lowest energy level.
d.How do this frequency and this wavelength compare
with the corresponding quantities if the particle were
in a one-dimensional box of length 10.0?15.16Assume that an argon atom can be represented as a
particle in a three-dimensional box with height, width,
and length all equal to 0.100 m. Assume that the argon
atom has an energy equal to 3kBT/2 wherekBis
Boltzmann’s constant and whereTis the absolute
temperature, assumed to be equal to 298.15 K. Find the
value ofnx,ny, andnz, assumed equal each other.15.17Consider a crude model representing a benzene molecule
as a three-dimensional rectangular box with dimensions
3 .5 Å by 3.5 Å by 1.25 Å (350 pm by 350 pm by 125 pm).
Include only the sixπelectrons, which are assumed to
move in the entire box, and let them occupy the three
lowest energy states with paired spins. Find the
wavelength of the photon absorbed when an electron
makes a transition from the highest occupied state to the
lowest unoccupied state. Compare it with the wavelength
of the actual transition, 180 nm.15.18Consider a model representing the pi electrons of a
benzene molecule as electrons moving around a ring of
circumferenceLequal to 8.35 Å (8. 35 × 10 −^10 m) with a
constant potential energy set equal to zero. The
Schrödinger equation is like that of a particle in a box,
except the wave functionψis a function ofx, the distance
around the ring from an initial point. The boundary
condition is thatψ(0)ψ(L).a.Show that an acceptable wave function is
Asin(2nπx/L)+Bcos(2nπx/L) wherenis an
integer.b.Show that another acceptable wave function is
Cexp(2nπix/L).c.Show that the energy eigenvalue is
En(h^2 / 2 mL^2 )n^2.d.The molecule has six pi electrons. Assume that two
occupy then0 state, two occupy then1 state,
and that two occupy then−1 state. Find the
wavelength of the photon absorbed when an electron
makes a transition fromn2ton3. Compare it
with the wavelength of the actual transition,
180 nm.15.4 The Quantum Harmonic Oscillator
In Chapter 14, we solved the classical equation of motion for a harmonic oscilla-
tor. We now solve the time-independent Schrödinger equation for this model system.
The Hamiltonian operator must contain the potential energy expression from