# Quantization axis

1. Jul 14, 2012

### Niles

Hi

I am looking at a bunch of atoms in a homogeneous magnetic field, irridiated by a monochromatic EM wave. I am trying to figure out how to intensity pattern of the emitted light by the atoms looks.

Case 1) I have attached a picture of the situation called "case_1.jpg". It is very clear that only π-transitions are being driven, i.e. Δm=0 transitions.

Case 2) I have attached a picture of the situation again. The quantization axis points along the magnetic field, but the polarization is orthogonal to it. So somehow I need to decompose the polarization into something in the same plane as the B-field. How can I do that?

I would be very happy to recieve some feedback.

Best,
Niles.

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• ###### case_2.jpg
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2. Jul 14, 2012

### Niles

Ah, ok. I think I figured it out entirely by myself. I can of course always decompose it into circularly polarized light along k. So they will drive the Δm=+1 and Δm=-1 transition. But then what happens when B is perpendicular to both k and E? Then my "trick" doesn't work anymore.

Best,
Niles.

Last edited: Jul 14, 2012
3. Jul 14, 2012

### Darwin123

Maybe that is a forbidden transition.
Or maybe you better look at quadropole moments, or magnetic dipole moments.
I thought that the m=0 transition is dipole forbidden, anyway.

4. Jul 14, 2012

### Niles

I'm pretty sure having E perp. to k perp. to B will still yield a signal. I just don't see how I can ever decompose E into something along B, but I know it is possible.

5. Jul 17, 2012

### M Quack

You can still decompose the linear polarization into two circular ones. What changes wrt case_1 is the relative phase between the two circular waves.

Therefore you should get the same spectrum as in case_1, i.e. delta-m=0.

6. Jul 18, 2012

### Niles

Hi

Thanks for replying! However I have to disagree. So B is perp. to k, which is perp. to E: If the electron is oscillating circularly along B, then looking "edge on", it looks linear. And it is exactly this motion that the E-field excites. So the transitions being driven are delta-m = +/- 1.

Does this sound reasonable to you?

7. Jul 21, 2012

### M Quack

When I find a moment I will work this out in detail.

You can write the dipole operator ε.r as Ʃ_m |r| ε_m Y_1,m
where m=-1,0,1 and Y_1,m is a spherical harmonic.

The matrix element then reduces to an amplitude prefactor and some
Clebsch-Gordans. If you know the initial and final angular momenta
this is easy to write down exactly.