Physics 401 Homework1---Due September 16
1.
Using the conservation of energy and momentum and
assuming a two-body collision between a photon and an electron initially at
rest, derive the 
where the subscripts i and s indicate the wavelength of the
incident and scattered radiation respectively. (Hint: You may want to use
relativity to first solve the problem in the center of mass frame and then
boost. If you do not know relativeity well enough to
do this uses brute force.).
2.
In class we discussed the Bohr atom. The derivation of
the allowed energies was based on the electrostatic force, the quantization of
angular momentum into integral units of Ñ and
the assumption of circular orbits. Repeat this derivation supposing that the
force holding a particle in its orbit is that of a three dimensional harmonic
oscillator with a force given by 
with
where k is
the spring constant and of the oscillator and m is the mass of the
orbiting particle. Show the allowable radii for the circular orbits and the
allowable energies. How do these allowable energies compare with those of
Planck?
3. Consider the following problem. A bead is constrained to move (without friction) along a wire. The wire is bent into a circular loop of radius R. Imposing the condition that the angular momentum is quantized into integral units of Ñ find the allowable energies.
4.
Let us try to gain
some insights into the quantization of L in problem 3. Suppose,
following de Broglie, that there is a wave function y which describes
the system. This wave function can be
described in terms of q, the angular variable describing motion around the ring. By
analogy with de Broglie’s description of linear
motion one would expect the wave function for a state of good angular momentum
to be 
. In deriving this we simply replace the ordinary position,
x, by the angular position, q, and the ordinary momentum, p, by the angular momentum,
L. In order that the wave function is well defined in the sense of being
single-valued it must satisfy the condition y (q + 2p )
=y (q )
. Show that this condition implies that L
= n Ñ
with integral n.