9 This problem refers to the analysis of the ammonia molecule
in sec. 14.7.2, p. 992. (a) The bond lengths in this molecule are on
the order of 0.1 nm. Use this fact to estimate the moment of inertia
for rotation about the symmetry axis, and verify that states with
Lz>0 are likely to be populated at room temperature.
(b) The original 1955 paper by Townes and Schawlow on the mi-
crowave spectroscopy of ammonia detected about 55 lines lying be-
tween 17 and 29 GHz. Each of these corresponds to a certain value
ofLandLz. Since there are many lines crowded together in this
region of the spectrum, the issue arises of whether the resolution of
the experiment will be sufficient to distinguish them. One of the
factors limiting the resolution is that the molecules of ammonia gas
have velocities that are random and randomly oriented, and this
causes random Doppler shifts in the lines. Estimate the Doppler
shifts at room temperature and determine whether or not they are
likely to cause problems.10 This problem refers to the analysis of the ammonia molecule
in sec. 14.7.2, p. 992. (a) The text constructs the ground state|g.s.〉,
which has energy−|f|=f. Use the same method to find the excited
state, which has energy +|f|=−f.
(b) Verify that these two states are orthogonal.
(c) Find normalized versions of the two states.11 Consider the wavefunctions Ψ 1 = and Ψ 2 = for a par-
ticle in a one-dimensional box. Suppose we have the superposition
Ψ =A(2Ψ 1 + Ψ 2 ).
(a) If Ψ is to be properly normalized, what is|A|?√(b) Sketch the wavefunction.
(c) Suppose you can measure the position of the particle very ac-
curately. What is the probability that the particle will be found in
the left half of the box?√(d) Instead of measuring position, suppose you measure the energy
of the state. What is the probability that you’ll measure the ground
state energy?√(e) Suppose that the wavefunction had been Φ =A(2Ψ 1 −Ψ 2 ).
Which of your answers to parts a-d would remain the same, and
which would change? (You need not redo the work for the ones that
would change. Just give your reasoning as to whether they would
or would not.) [Problem by B. Shotwell.]1010 Chapter 14 Additional Topics in Quantum Physics