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

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 . - Write the (normalized) state which is an eigenstate of
with eigenvalue
and also
an
**eigenstate of**with eigenvalue in terms of the usual .**Answer**

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. - Write the (normalized) state which is an eigenstate of
with eigenvalue
and also
an
**eigenstate of**with eigenvalue in terms of the usual . - Calculate the
**commutators**and . **Derive**the relation .- 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.
- Calculate the following commutators: , , .
- Prove that, if the Hamiltonian is symmetric under rotations, then .
- 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.

- 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