## Sample Test Problems

1. Calculate the commutator where and . State the uncertainty principle for and .

2. * A particle of mass is in a 1 dimensional potential . Calculate the rate of change of the expected values of and , ( and ). Your answer will obviously depend on the state of the particle and on the potential.

3. Compute the commutators and for the 1D harmonic oscillator.

4. * Assume that the states are the eigenstates of the Hamiltonian with eigenvalues , ( ).
a)
Prove that for an arbitrary linear operator .
b)
For a particle of mass moving in 1-dimension, the Hamiltonian is given by . Compute the commutator [H,X] where is the position operator.
c)
Compute the mean momentum in the state .
5. * At , a particle of mass is in the Harmonic Oscillator state . Use the Heisenberg picture to find the expected value of as a function of time.

Jim Branson 2013-04-22