## Sample Test Problems

1. Derive the commutators and . Now show that , that is, raises the eigenvalue but does not change the eigenvalue.

Since commutes with both and ,

We have the commutators. Now we apply them to a .

So, is also an eigenfunction of with the same eigenvalue. does not change .

So, raises the eigenvalue of .
2. Write the (normalized) state which is an eigenstate of with eigenvalue and also an eigenstate of with eigenvalue in terms of the usual .
An eigenvalue of implies . We will need a linear combination of the to get the eigenstate of .

Since this is true for all and , each term must be zero.

The state is

The trivial solution that is just a zero state, not normalizable to 1.
3. Write the (normalized) state which is an eigenstate of with eigenvalue and also an eigenstate of with eigenvalue in terms of the usual .
4. Calculate the commutators and .
5. Derive the relation .
6. A particle is in a state and is known to have angular momentum in the direction equal to . That is . Since we know , must have the form . Find the coefficients and for normalized.
7. Calculate the following commutators: , , .
8. Prove that, if the Hamiltonian is symmetric under rotations, then .
9. In 3 dimensions, a particle is in the state:

where is some arbitrary radial wave function normalized such that

a)
Find the value of C that will normalize this wave function.
b)
If a measurement of is made, what are the possible measured values and what are probabilities for each.
c)
Find the expected value of in the above state.
10. Two (different) atoms of masses and are bound together into the ground state of a diatomic molecule. The binding is such that radial excitations can be neglected at low energy and that the atoms can be assumed to be a constant distance apart. (We will ignore the small spread around .)
a)
What is the energy spectrum due to rotations of the molecule?
b)
Assuming that is given, write down the energy eigenfunctions for the ground state and the first excited state.
c)
Assuming that both masses are about 1000 MeV, how does the excitation energy of the first excited state compare to thermal energies at K.

Jim Branson 2013-04-22