Finding Eigenstates of J_z and the Harmonic Oscillator Operators

[ausdreamer]

Homework Statement

http://img191.imageshack.us/i/questionyw.png/

Homework Equations

Given in problem

The Attempt at a Solution

a) I've been able to find expressions of operators x, p_x, y and p_y in terms of the creation/annihilation operators and hence been able to express the angular momentum operator as:

http://img232.imageshack.us/i/solutionq.png/

However, I'm having trouble just writing the operators x, p_x, y and p_y in terms of the "new" creation/annihilation operators and so I can proceed to find J_z in terms of the new operators.

b) I'm guessing I need to formulate an eigenequation involving J_z and H to find the eigenstates of J_z? I'm pretty stumped on this question to be honest.

 

[ausdreamer]

anyone?