Finding Eigenstates of J_z and the Harmonic Oscillator Operators

ausdreamer
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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.
 
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This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
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The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
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