1) Consider a quantum
mechanics problem with a particle in a three-dimensional box with V = 0 inside
the box and V = infinite outside.
Now, suppose there is a perturbing potential
a) Find the new
energies for what was the lowest degenerate state.
b) What is the new
state for the lowest energy
2) Consider a driven harmonic oscillator.
a) Using the matching
method, find the Green’s function G(t,t’) which can be used to solve this
equation for x(t).
b) If f(t) = 0 for t’
<0
=
1 for 0
< t < to
(this is a pulse of width to)
=
0 for t > to
what is the form for
x(t)?
c) Discuss (carefully and intelligently) whether you could
use other methods to find
the Green’s function for this problem.
3) Show the following
a) In the calculus of
variations show that the usual Euler-Lagrange condition
where f = f(y, yx,
x) and yx = dy/dx is equivalent to
The second form is often useful if f is not an explicit
function of x.
b) Find the
Euler-Lagrange Equations if f depends
on f(y, yx, yxx,
x) and we want
to minimize the
integral
here 
You may assume
that our “arbitrary” function h(x)
vanishes at the endpoint, and that
its derivative
also vanishes at the endpoints.
4) Consider the vibration of a circular drumhead
of radius a. The differential
equation that governs the amplitude u is given by
The boundary conditions are that the drum head is pinned
(u=0) at the edge where r=a.
This is an ugly problem that can be solved by Bessel
functions, but here we will solve it in a much easier way.
The lowest eigenvalue can be estimated by
where the integration
is over the surface area of the drumhead and 
is only
two-dimensional.
Guess a sensible form
for u(r) and estimate k. (You do not
necessarily have to have
a variational
parameter in your guess.) The correct
answer is k2 = 5.78/a2.
Hints:
Problem 1:
I shouldn’t have to tell you this, but the wavefunctions for
the 3D box are products of the
1D wavefunctions
where a,b,c are integers.
The energy without the perturbation is given by
The ground state (a=b=c=1) is not degenerate, but the next
state is 3-fold degenerate.
Problem 2:
This problem is different from similar ones we have done
with strings because time only goes forward.
Because of this
G(t,t’) = 0
for t < t’ we don’t want a
response which starts before the stimulus.
Problem 3:
Be careful using the chain rule and keeping track of partial
and total derivatives
Problem 4:
Make sure you look up in polar coordinates
