QM-3 H.W.2
1. We derived in class the propagator for particle moving in uniform electric field, use one of the methods
derived in class to derive the propagator in electro-magnetic field in 3D, take limits of either electric or
magnetic field to zero .
2. Starting for the equation of propagator (Green function) derive perturbation theory for the propagator in
terms of the potential up to a second order.
Use your results for the parabolic potential and compare with expression of exact result .
3. Using the result derived in class for parabolic potential define the density matrix
  q, q '   q e   H q ' and obtain the density matrix for the parabolic potential at finite temperature

1
. Obtain and comment on the asymptotics, taking the limit of high and low temperatures.
T
4. Consider a harmonic oscillator described by
P2 1
2
H
 m  t  x 2
2m 2
where   t   0  cos  t , and 
0 .
Assume that at t=0 the system is in the ground state. Using the perturbation theory find the transition
probability from the ground state to a final state
f
.