force constant. D deuterium (^2 H). Assume that the force
constant remains the same.
11.6 2-D Rotations
11.22.Why can’t the quantized values of the 2-D angular
momentum be used to determine the mass of a rotating sys-
tem, like classical angular momentum can?
11.23.Show that 3 of 2-D rotational motion has the same
normalization constant as 13 by normalizing both wavefunc-
tions.
11.24.What are the energies and angular momenta of the
first five energy levels of benzene in the 2-D rotational motion
approximation? Use the mass of the electron and a radius of
1.51 Å to determine I.
11.25.A 25-kg child is on a merry-go-round/calliope, going
around and around in a large circle that has a radius of 8 me-
ters. The child has an angular momentum of 600. kg m^2 /s. (a)
From these facts, estimate the approximate quantum number
for the angular momentum the child has. (b)Estimate the
quantized amount of energy the child has in this situation.
How does this compare to the child’s classical energy? What
principle does this illustrate?
11.26.Using Euler’s identity, rewrite the first four 2-D rota-
tional wavefunctions in terms of sine and cosine.
11.27. (a)Using the expression for the energy of a 2-D rigid
rotor, construct the expression for the energy difference be-
tween two adjacent levels, E(m1) E(m). (b)For HCl, E(1)
E(0) 20.7 cm^1. Calculate E(2) E(1), assuming HCl acts
as a 2-D rigid rotor. (c)This energy difference is determined
experimentally as 41.4 cm^1. How good would you say a 2-D
model is for this system?
11.28.Derive equation 11.35 from 11.34.
11.7 & 11.8 3-D Rotations
11.29.Use trigonometry to verify the relationships between
the Cartesian and spherical polar coordinates as given in equa-
tion 11.40.
11.30.Why can’t the square root of equation 11.45 be taken
analytically? (Hint:consider how you would have to take the
square root of the right side of the equation. Can it be done?)
11.31.For both 2-D and 3-D rotations, the radius of the par-
ticle’s motion is kept constant. Consider a nonzero, constant
potential energy acting on the particle. Show that the form of
the Schrödinger equation in equation 11.46 would be equiv-
alent to its form if Vwere identically zero. (Hint:use the idea
that EnewEV.)
11.32.Can you evaluate rfor the spherical harmonic Y^2 2?
Why or why not?
11.33.Using the complete form of 3, 2 (where 3 and
m2) for 3-D rotations (get the Legendre polynomial from
Table 11.3) and the complete forms of the operators, evaluate
the eigenvalues of (a)L^2 , (b)Lz, (c)E. Do not use the ana-
lytic expressions for the observables. Instead, operate on 3, 2
with the appropriate operators and see that you do get the
proper eigenvalue equation. From the eigenvalue equation,
determine the value of the observable.
11.34.A 3-D rotational wavefunction has the quantum num-
ber equal to 2 and a moment of inertia of 4.445
10 ^47
kg m^2. What are the possible numerical values of (a)the en-
ergy; (b)the total angular momentum; (c)the zcomponent
of the total angular momentum?
11.35. (a)Using the expression for the energy of a 3-D rigid
rotor, construct the expression for the energy difference be-
tween two adjacent levels, E(1) E().
(b)For HCl, E(1) E(0) 20.7 cm^1. Calculate E(2) E(1),
assuming HCl acts as a 3-D rigid rotor.
(c)This energy difference is determined experimentally as
41.4 cm^1. How good would you say a 3-D model is for this
system?
11.36.See Example 11.17, regarding the “spherical” C 60
molecule. Assuming the electrons in this molecule are experi-
encing 3-D rotations, calculate the wavelength of light neces-
sary to cause a transition from state 5 to 6 and from
7 to 8. Compare your answers with experimentally
measured absorptions at wavelengths of 328 and 256 nm.
How good is this model for describing C 60 ’s electronic
absorptions?
11.37.In exercise 11.36 regarding C 60 , what are the numer-
ical values of the total angular momenta of the electron for
each state having quantum number ? What are the zcom-
ponents of the angular momentum for each state?
11.38.Draw graphical representations (see Figure 11.15) of
the possible values for and mfor the first four energy levels
of the 3-D rigid rotor. What are the degeneracies of each state?
11.39.What is a physical explanation of the difference be-
tween a particle having the 3-D rotational wavefunction 3,2
and an identical particle having the wavefunction 3, 2?11.9, 11.10, & 11.11 Hydrogen-Like Atoms
11.40.List the charges on hydrogen-like atoms whose nuclei
are of the following elements. (a)lithium, (b)carbon, (c)
iron, (d)samarium, (e)xenon, (f)francium, (g)uranium,
(h)seaborgium
11.41.Calculate the electrostatic potential energy Vbetween
an electron and a proton if the electron is at a distance of 1
Bohr radius (0.529 Å) from the proton. Be careful that the cor-
rect units are used!
11.42.Using Newton’s law of gravity and the relationship be-
tween force and potential energy, the gravitational potential
energy can be written asVGm^1 rm^2Use the masses of the electron and the proton and the gravi-
tational constant G6.673
10 ^11 N m^2 /kg^2 to show that
the gravitational potential energy is negligible compared to the
electrostatic potential energy at a distance of 1 Bohr radius.
11.43.Show that for constant rand V0, equation 11.56
becomes equation 11.46. (Hint:you will have to apply the
chain rule of differentiation to the derivatives in the second
term of equation 11.56.)368 Exercises for Chapter 11