Problem Set 1---Due February 4

1) Consider
a quantum system composed of two subsystems. Each subsystem is expressed in
terms of a basis of orthonormal states which are eigenstates of some hermitian
operator (associated with some physical properties). A basis for the combined
system can be given by states of the form _{} where the first label denotes system
1 and the second system 2. Suppose the state of the system is given by _{}.

a) Show that the state is properly normalized.

b) Suppose
both system 1 and 2 are measured. What is the probability that we will find
the system in the state _{}?

c) Suppose
system 2 is measured and found to be in state _{}. Given this, what is the probability
that system 1 will be in state _{}?

d) Suppose
system 2 is measured and found to be in state _{}. Given this, what is the probability
that system 1 will be in state _{}?

2) In
class we considered the two dimensional isotropic harmonic oscillator (where
isotropic means the spring constant is the same in all directions. We showed
the energy spectrum was given _{} with the *N*th level having
degeneracy of *(N+1)*. In this problem, generalize the analysis to three
dimensions: show that for a three dimensional isotropic harmonic oscillator the
showed the energy spectrum was given _{} with the *N*th level having
degeneracy of _{}.

3) Consider
two distinct systems with two degrees of freedom. The first one is a system
of two particles (of masses _{}and _{}) confined to a one-dimensional box of
length *L*. The second is a single particle (of mass* m*) confined
to a two-dimensional box with sides _{}and _{}. Show that the two systems have
exactly the same energy spectrum if _{} and _{}.

4) In
class we discussed the separation of variables for the three dimensional
time-independent Schrödinger equation with a spherically symmetric potential.
In this problem I want you to work thorough the equivalent for a two dimension
system with an axially symmetric potential. In particular consider the equation
_{}in polar
coordinates. Show that the it has solutions of the form _{} with _{} (integer *m*) and *R
*satisfying _{}.