Can I Couple Angular Momenta Into One Big One?

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Discussion Overview

The discussion revolves around the coupling of three angular momenta, denoted as l_1, l_2, and l_3, into a single angular momentum L. Participants explore the methods of coupling these momenta and whether different coupling sequences yield the same result.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant proposes to couple l_1 and l_2 into an intermediate angular momentum L' and then couple L' with l_3 to obtain L.
  • Another participant agrees with this method and references historical work by Wigner and Racah, suggesting that this approach is established in the literature.
  • A third participant confirms that both coupling sequences (l_1 with l_2 first or l_2 with l_3 first) are valid and notes that the resulting states are related through Wigner's 6J coefficients.
  • A fourth participant expresses gratitude for the confirmation of their understanding, emphasizing the value of asking questions.

Areas of Agreement / Disagreement

Participants generally agree that the proposed methods of coupling angular momenta are valid, but there is no explicit consensus on the implications or applications of these methods.

Contextual Notes

Participants reference established literature and concepts such as Wigner's 6J coefficients, but the discussion does not resolve any potential complexities or nuances in the coupling process.

Who May Find This Useful

This discussion may be useful for students and researchers interested in angular momentum coupling in quantum mechanics, particularly those exploring group theory applications.

suyver
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I have three angular momenta l_1,l_2,l_3 which I want to couple into one big one:

L\equiv l_1+l_2+l_3.

Can I just do this by coupling l_1,l_2 into L' and then couple L',l_3 into L?

I would guess that I could equally couple l_2,l_3 into L' and then couple l_1,L' into L and this would give the same result. Correct?
My reason to assume this: the different l_i work on different parts of the system.
 
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That is the correct way to do it, couple two of the angular momenta then couple the third. If memory serves me correctly, this was done by Wigner and also by Racah...Look in Wigners book or Tinkhams book on Group Theory and Quantum Mechanics, it is all there.
 
I've been doing a lot of reading on this subject lately. You are completley correct in stating that you can first add j1 and j2, and then add this to j3. Likewise you can first add j2 and j3 and add this to j1. These product eigenstates are related through the wigner 6J coefficients.
 
Thanks for answering, all! I was quite sure that I was right, but I thought it never hurts to ask. After all, there are no stupid questions (only stupid people )
 

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