Intensity pattern of the emitted light

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Discussion Overview

The discussion revolves around the intensity pattern of light emitted by atoms in a homogeneous magnetic field when irradiated by a monochromatic electromagnetic wave. Participants explore different cases of atomic transitions, particularly focusing on the effects of polarization and the orientation of the magnetic field.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes a scenario where only π-transitions (Δm=0) are driven by the electromagnetic wave in a magnetic field.
  • Another participant suggests that circularly polarized light can be used to drive Δm=+1 and Δm=-1 transitions, but questions the situation when the magnetic field is perpendicular to both the wave vector and the electric field.
  • Concerns are raised about whether certain transitions might be forbidden, particularly the m=0 transition being dipole forbidden.
  • There is a proposal that even with the electric field perpendicular to the wave vector and magnetic field, a signal can still be obtained, although the decomposition of the electric field into components along the magnetic field remains unclear.
  • One participant asserts that the electron's circular oscillation along the magnetic field leads to linear motion when viewed edge-on, suggesting that this motion excites transitions of Δm=±1.
  • A later reply discusses the mathematical representation of the dipole operator and the role of Clebsch-Gordan coefficients in determining transition amplitudes based on angular momentum states.

Areas of Agreement / Disagreement

Participants express differing views on the nature of transitions driven by the electric field in the presence of a magnetic field. There is no consensus on whether certain transitions are forbidden or how to effectively decompose the electric field in the given configurations.

Contextual Notes

Participants mention various assumptions regarding the nature of transitions and the mathematical treatment of the dipole operator, but these assumptions remain unresolved within the discussion.

Niles
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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 receive some feedback.


Niles.
 

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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.

Niles.
 
Last edited:
Niles said:
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.




Niles.
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.
 
Darwin123 said:
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.

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.
 
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.
 
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?
 
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.
 

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