Phys621: Homework Assignment 6

==> Due on Thursday 11/30 in class <==

^* Note: The operator J² should have been defined: J²=J_z² + (J₊J_- + J_-J₊)/2.

Cont'd:

We now relate these results to the 2-D Harmonic oscillator problem we discussed in class. It turns out that the 2-D Harmonic oscillator can be used as a model to understand angular momentum, which you will see as we walk through the steps below. Let the `+' and `-' oscillators in this problem be the x- and y- dimensions, respectively, of the 2-D problem.

1. Write down the Hamiltonian H of the 2-D Harmonic oscillator in terms of the operator N. Argue that H commutes with both J² and J_z.
2. Denote the eigenvalues of the operator N by n. What are the energy eigenvalues of the 2-D Harmonic oscillator? Express the corresponding eigenvalues of the operator J² in terms of j=n/2.
3. As we know, the eigenkets of the 2-D SHO are given by |n₊; n_- >. What is the degeneracy of each energy level? Show that within each energy level, the degenerate states are eigenstates of the operator J_z, with different eigenvalues. Express these eigenvalues as (hbar)*m.
4. We can therefore uniquely specify each eigenket |n₊; n_- > with |j;m>, where j and m are determined in parts (2) and (3), respectively. For example, the ground-state has j=m=0 and is thus |0;0>. Write down the collection of degenerate eigenstates in this notation for each of the next three energy levels.
5. What is the outcome of applying the operator J₊ on the state |j;m>? What about J_-? J₊ and J_- are referred to as ladder operators.

(2) Pauli matrices and rotation --- in pdf file.

(3) Sakurai Problem 3.16.