The expectation of 'z' and 'x+iy'

[vertices]

can anyone give me any ideas on how to evaluate this:

<z>=<\Phi1|z|\Phi2>

(for say hydrogen wavefunctions). Similarly

<x+iy>=<\Phi1|x+iy|\Phi2>

FYI, I'm trying to understand how radiation is polarised (an external B field polarises radiation, so we must consider the dipole transition matrix thus:

<r>=<\Phi1|r|\Phi2>

so I am simply resolving 'r' into two components (in the xy plane and z axis).

 

[lbrits]

The only hard part in the calculation is knowing \left\langle r \right\rangle,
from which
x = r \sin\theta \cos\varphi
y = r \sin\theta \sin\varphi.
Can you think of a spherical harmonic equal to \sin\theta e^{i\varphi}? Then you need to know how to compute \left\langle Y^\ell_m\right\rangle, which is easy enough for simple cases, but if you want you can consult a table of 3-j symbols.

That should be all that you need.

Good luck.

 

[vertices]

aah, that's clever. thanks Ibrits.

for hydrogen like wavefunctions, x+iy=\sin\theta e^{i\varphi}=Y(l=1,m=1) so yeah its just the expectation of that.

what about the expectation of z?

 

[lbrits]

I assume you know the representation of z in spherical coordinates. I also assume you have a table of spherical harmonics handy. It shouldn't be hard to figure the rest out =)

 

[vertices]

oh i see, i was being really stupid (as per usual)... z=rcos(theta), so its just <r><Y1,0>. thanks:)