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

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In summary, the conversation is about evaluating the dipole transition matrix for hydrogen wavefunctions in order to understand how radiation is polarized. The calculation involves resolving 'r' into two components and using a spherical harmonic equal to sine theta times e to the power of i*varphi. The expectation for x+iy is equal to Y(l=1, m=1) and the expectation for z is equal to r times Y(l=1, m=0). Consultation of a table of 3-j symbols may be necessary for more complex cases.
  • #1
can anyone give me any ideas on how to evaluate this:


(for say hydrogen wavefunctions). Similarly


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:


so I am simply resolving 'r' into two components (in the xy plane and z axis).
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  • #2
The only hard part in the calculation is knowing [tex]\left\langle r \right\rangle[/tex],
from which
[tex]x = r \sin\theta \cos\varphi[/tex]
[tex]y = r \sin\theta \sin\varphi[/tex].
Can you think of a spherical harmonic equal to [tex]\sin\theta e^{i\varphi}[/tex]? Then you need to know how to compute [tex]\left\langle Y^\ell_m\right\rangle[/tex], 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.
  • #3
aah, that's clever. thanks Ibrits.

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

what about the expectation of z?
  • #4
I assume you know the representation of [tex]z[/tex] in spherical coordinates. I also assume you have a table of spherical harmonics handy. It shouldn't be hard to figure the rest out =)
  • #5
oh i see, i was being really stupid (as per usual)... z=rcos(theta), so its just <r><Y1,0>. thanks:)

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