11.2 Classical Harmonic Oscillator
11.1.Convert 3.558 mdyn/Å into units of N/m.
11.2.A swinging pendulum has a frequency of 0.277 Hz and
a mass of 500.0 kg. Calculate the force constant for this har-
monic oscillator.
11.3.An object having mass mat some height above the
ground hhas a gravitational potential energy of mgh, where
gis the acceleration due to gravity (9.8 m/s^2 ). Explain why
objects moving back and forth under the influence of gravity
(like a clock’s pendulum) can be treated as harmonic oscilla-
tors. (Hint:see equation 11.1.)
11.3 Quantum-Mechanical Harmonic Oscillator
11.4.In equation 11.6, in order to properly subtract the two
terms in parentheses on the left, they must have the same
units overall. Verify that 2mE/^2 and ^2 x^2 have the same units.
Use standard SI units for x(position/distance). Do the same for
the two terms in parentheses in equation 11.11.
11.5.Verify that the three substitutions mentioned in the text
yield equation 11.6.
11.6.Verify that the second derivative of given by equa-
tion 11.8 gives equation 11.9.
11.7.Derive equation 11.16 from the equation immediately
preceding it.
11.8.Show that the energy separation between anytwo ad-
jacent energy levels for an ideal harmonic oscillator is h ,
where is the classical frequency of the oscillator.
11.9. (a)For a pendulum having a classical frequency of
1.00 s^1 , what is the energy difference in J between quantized
energy levels? (b)Calculate the wavelength of light that must
be absorbed in order for the pendulum to go from one level
to another. (c)Can you determine in what region of the elec-
tromagnetic spectrum such a wavelength belongs? (d)
Comment on your results for parts a and b based on your
knowledge of the state of science in the early twentieth cen-
tury. Why wasn’t the quantum mechanical behavior of nature
noticed?
11.10. (a)A hydrogen atom bonded to a surface is acting as
a harmonic oscillator with a classical frequency of 6.000
1013 s^1. What is the energy difference in J between quantized
energy levels? (b)Calculate the wavelength of light that must
be absorbed in order for the hydrogen atom to go from one
level to another. (c)Can you determine in what region of
the electromagnetic spectrum such a wavelength belongs?
(d) Comment on your results for parts a and b based on
your knowledge of the state of science in the early twentieth
century.
11.11.The O–H bond in water vibrates at a frequency of
3650 cm^1. What wavelength and frequency (in s^1 ) of light
would be required to change the vibrational quantum num-
ber from n0 to n4, assuming O–H acts as a harmonic
oscillator?
11.4 Harmonic Oscillator Wavefunctions
11.12.Show that 2 and 3 for the harmonic oscillator are
orthogonal.
11.13.Substitute 1 into the complete expression for the
Hamiltonian operator of an ideal harmonic oscillator and show
that E^32 h.
11.14.Calculate pxfor 0 and 1 for a harmonic oscillator.
Do the values you calculate make sense?
11.15.Use the expression for 1 in equations 11.17 and nor-
malize the wavefunction. Use the integral defined for the
Hermite polynomials in Table 11.2. Compare your answer with
the wavefunction defined by equation 11.19.
11.16.Simply using arguments based on odd or even func-
tions, determine whether the following integrals involving har-
monic oscillator wavefunctions are identically zero, are not
identically zero, or are indeterminate. If indeterminate, state
why.
(a)1 * 2 dx (b)
1 * xˆ 1 dx
(c)
1 *xˆ^2 1 dx, where xˆ^2 xˆ xˆ
(d)
1 3 dx (e) 3 3 dx
(f)
1 *Vˆ 1 dx, where Vˆis some undefined potential energy
function.11.17.Determine the value(s) of xfor the classical turning
point of a harmonic oscillator in terms of kand n. There may
be other constants in the expression you derive.11.5 Reduced Mass
11.18.Compare the mass of the electron, me, with (a)the
reduced mass of a hydrogen atom; (b)the reduced mass of a
deuterium atom (deuterium ^2 H); (c)the reduced mass of
a carbon-12 atom having a 5 charge, that is, C^5 . Suggest
a conclusion to the trend presented by parts a–c.
11.19.Reduced mass is not reserved only for atomic systems.
A solar system or a planet/satellite system, for example, can
have its behavior described by first determining its reduced
mass. If the mass of Earth is 2.435
1024 kg and that of the
moon is 2.995
1022 kg, what is the reduced mass of the
Earth-moon system? (This is not to imply any support of a
planetary model for atoms!)
11.20. (a)Calculate the expected harmonic-oscillator fre-
quency of vibration for carbon monoxide, CO, if the force
constant is 1902 N/m. (b)What is the expected frequency of(^13) CO, assuming the force constant remains the same?
11.21.An O–H bond has a frequency of 3650 cm^1. Using
equation 11.27 twice, set up a ratio and determine the ex-
pected frequency of an O–D bond, without calculating the
Exercises for Chapter 11 367
EXERCISES FOR CHAPTER 11