In this problem, you will analyze the motion of a bouncing ball of mass m subject to the gravitational force mg. You are now able to treat this problem at a level far beyond that you used in Quiz I. You will now investigate the consequences of the de Broglie relations (i.e., Bohr-Sommerfeld quantization) on the motion of the ball. A sketch of the potential experienced by the ball is given in Figure 2.
The bouncing of a ball is a periodic motion; we shall call the period of the motion T. The Bohr-Sommerfeld quantization condition
restricts the allowed energies of the bouncing ball to a discrete set
of energies 
indexed by the quantum number n. Your task again
is to compute the spectrum of allowed energies 
, and to verify that
the ground state gives results consistent with the expectations of the
uncertainty principle and that the large n limit is consistent
with the expectations of classical physics.
a) Without performing an analysis, how to you expect the period of the ball's bouncing to vary (increase or decrease) as the energy of its motion increases? What does this then tell you about the spacing of the energy levels for large n?
b) Compute the period of the ball's motion T in terms of its total energy E.
c) If a small test charge q is placed on the mass m, what frequency radiation would this bouncing ball emit?
d) Use the Bohr-Sommerfeld quantization condition to find the allowed energy levels of the ball. Sketch the energy spectrum on an energy-level diagram. Show that the spacings for large n behave as you expected in a).
e) Check your result for d) for the case n=1
by using the uncertainty principle to to estimate the ground
state energy of the ball, 
.
f) Show explicity that, in the classical limit, the period of the motion of the ball T is exactly consistent with the energy-level spacings.
