## Sample Test Problems

1. A 1D harmonic oscillator is in a linear combination of the energy eigenstates

Find the expected value of .

2. Assuming represents the 1D harmonic oscillator energy eigenstate, calculate .

3. Evaluate the uncertainty'' in for the 1D HO ground state where . Similarly, evaluate the uncertainty in for the ground state. What is the product ?
Its easy to see that either from the integral or using operators. I'll use operators to compute the rest.

4. Use the commutator relation between and to derive . Now show that is the lowering operator for the harmonic oscillator energy.
5. At , a one dimensional harmonic oscillator is in the state . Calculate the expected value of as a function of time.
6. At , a harmonic oscillator is in a linear combination of the and states.

Find and as a function of time.
7. A 1D harmonic oscillator is in a linear combination of the energy eigenstates

Find .
Jim Branson 2013-04-22