Summary: Physics 606, Quantum Mechanics, Exam 1
Please show all your work.
(You are graded on your work, with partial credit where it is deserved.)
All problems are, of course, nonrelativistic.
1. In one dimension, a free particle with mass m is perturbed by a position-independent force F t( ).
(a) (5) Write down the Hamiltonian for this particle, in terms of the momentum operator p , the position
operator x , and the time-dependent force F t( ).
(b) (5) Using the Heisenberg equation of motion for the Heisenberg operator p t( ), obtain dp t( )/ dt , and
integrate to find p t( ) in terms of F t( ) and the initial value p 0( ).
[Please show all your work here and elsewhere.]
(c) (5) Using the Heisenberg equation of motion for x t( ), obtain dx t( )/ dt , and integrate to find x t( ) in
terms of F t( ) and the initial values x 0( ) and p 0( ).
(d) (5) Calculate the momentum imparted by the force after it acts for time t (i.e., the change in the
expectation value of the momentum between time = 0 and time = t ).
(e) (5) Calculate the change in the energy of the particle during this time.
(f) (5) Calculate the time-dependent uncertainly in momentum, !p t( ), in terms of the uncertainty !p 0( ) at
time = 0.
2. (30) For a particle with charge q in an electromagnetic field, the Hamiltonian is