Quantum - Projection Probability - Projection amplitudes for SHO states.

[metgt4]

Quantum - Projection Probability - "Projection amplitudes for SHO states."

Given the two normalized 2D SHO wave functions <x,y|m<sub>x[/SUB ],ny> for the second energy level n = nx + ny = 1 in the mx[/SUB ],ny representation:

<x,y|1,0> = (2/pi)^1/2xexp[-(x²+y²)/2]
<x,y|0,1> = (2/pi)^1/2yexp[-(x²+y²)/2]

and in the alternative n, m representation <x,y|n,m>

<x,y|1,+1> = (1/pi)^1/2(x+iy)exp[-(x²+y²)/2]

<x,y|1,-1> = (1/pi)^1/2(x-iy)exp[-(x²+y²)/2]

(a) Construct a table of projection amplitudes between these two similar to Table 7-1 (I'll include this table in the attachments) for photon polarization states.

The question continues, but I would like to work the rest out myself. I just don't seem to understand how to change from one base to the other for this question!Thanks for any hints or help you can give me!

Andrew</sub>

 

[metgt4]

Am I at all on the right track here? I have a relation between the two bases, but this is as far as I've gotten all week! What I've got is attached.

Thanks!
Andrew