674
15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation
∞
−∞
ψ 2 (x)ψ 3 (x)dx 0
b. Work out the integral and show that ψ 2 and ψ 3 are
orthogonal. Remember that ψ 0 outside of
the box.
15.13 Think 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
number n corresponding to this value of E. 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.
15.14 Assume 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 absorbed in a transition between the two levels of part a.
Do you think that such a transition would actually be
observed spectroscopically?
15.15 An electron is confined in a cubical box
10.0 Å(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?
15.16 Assume 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 3k B T /2 where k B is
Boltzmann’s constant and where T is the absolute
temperature, assumed to be equal to 298.15 K. Find the
value of n x , n y , and n z , assumed equal each other.
15.17 Consider 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 six π electrons, 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.
15.18 Consider a model representing the pi electrons of a
benzene molecule as electrons moving around a ring of
circumference L equal 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 function ψ is a function of x, the distance
around the ring from an initial point. The boundary
condition is that ψ(0) ψ(L).
a. Show that an acceptable wave function is
Asin(2nπx/L) + Bcos(2nπx/L) where n is an
integer.
b. Show that another acceptable wave function is
Cexp(2nπix/L).
c. Show that the energy eigenvalue is
E n (h 2 /2mL 2 )n 2 .
d. The molecule has six pi electrons. Assume that two
occupy the n 0 state, two occupy the n 1 state,
and that two occupy the n −1 state. Find the
wavelength of the photon absorbed when an electron
makes a transition from n 2 to n 3. 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 oscillator. We now solve the time-independent Schrödinger equation for this model system.
The Hamiltonian operator must contain the potential energy expression from
15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation
∞
−∞
ψ 2 (x)ψ 3 (x)dx 0
b. Work out the integral and show that ψ 2 and ψ 3 are
orthogonal. Remember that ψ 0 outside of
the box.
15.13 Think 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
number n corresponding to this value of E. 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.
15.14 Assume 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 absorbed in a transition between the two levels of part a.
Do you think that such a transition would actually be
observed spectroscopically?
15.15 An electron is confined in a cubical box
10.0 Å(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?
15.16 Assume 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 3k B T /2 where k B is
Boltzmann’s constant and where T is the absolute
temperature, assumed to be equal to 298.15 K. Find the
value of n x , n y , and n z , assumed equal each other.
15.17 Consider 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 six π electrons, 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.
15.18 Consider a model representing the pi electrons of a
benzene molecule as electrons moving around a ring of
circumference L equal 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 function ψ is a function of x, the distance
around the ring from an initial point. The boundary
condition is that ψ(0) ψ(L).
a. Show that an acceptable wave function is
Asin(2nπx/L) + Bcos(2nπx/L) where n is an
integer.
b. Show that another acceptable wave function is
Cexp(2nπix/L).
c. Show that the energy eigenvalue is
E n (h 2 /2mL 2 )n 2 .
d. The molecule has six pi electrons. Assume that two
occupy the n 0 state, two occupy the n 1 state,
and that two occupy the n −1 state. Find the
wavelength of the photon absorbed when an electron
makes a transition from n 2 to n 3. 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 oscillator. We now solve the time-independent Schrödinger equation for this model system.
The Hamiltonian operator must contain the potential energy expression from