Seriously stuck 3D Quantum Harmonic Oscillator

[Xyius]

Homework Statement

The question is from Sakurai 2nd edition, problem 3.21. (See attachments)

*******
EDIT: Oops! Forgot to attach file! It should be there now..
*******

The Attempt at a Solution

Part a, I feel like I can do without too much of a problem, just re-write L as L=xp and then re-write x and p in terms of the raising and lowering operators. Part B is where I have no idea.

It says to write |01m> in TERMS of the eigenstates in cartesian coordinates. So I figure for q=0, l=1 the energy in spherical coordinates is..

$$E=\frac{5}{2}\hbar \omega$$

So, obtaining this energy in cartesian coordinates, we can have..

$$(n_x , n_y , n_z) = (1 0 0) OR (0 1 0) OR (0 0 1)$$

Since in the rectangular coordinate system

$$E=(n_x+n_y+n_z+3/2)\hbar \omega$$

So if we were to write the spherical state in terms of the cartesian degenerate states, I would assume that would mean..

$$|0 1 0>_S=A|1 0 0>_C+B|0 1 0>C + C|0 0 1>_C$$

Where subscript C = Cartesian basis and subscript S = spherical basis.

Am I on the right track here? Because In order to find the coefficients I need to compute (for example)

$$A=_S<1 0 0|0 1 0>_C$$

And I am not sure how to do that!

 

[hgrant826]

Hi, I am actually stuck on this problem at a similar state. I was curious if you were able to find a solution or any material covering this problem.

 

[vela]

Try finding the matrix that represents $\hat{L}_z$ with respect to the cartesian basis and diagonalizing it.