Can I Couple Angular Momenta Into One Big One?

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SUMMARY

The discussion centers on the coupling of three angular momenta, denoted as l_1, l_2, and l_3, into a single total angular momentum L, defined as L ≡ l_1 + l_2 + l_3. Participants confirm that one can couple l_1 and l_2 into an intermediate angular momentum L' and then couple L' with l_3 to obtain L. Alternatively, coupling l_2 and l_3 first, followed by l_1 and L', yields the same result. This method aligns with established principles in quantum mechanics, as referenced in Wigner's and Tinkham's texts on Group Theory and Quantum Mechanics, particularly involving Wigner 6J coefficients.

PREREQUISITES
  • Understanding of angular momentum in quantum mechanics
  • Familiarity with Wigner 6J coefficients
  • Knowledge of coupling schemes in quantum mechanics
  • Basic concepts of Group Theory as applied to quantum systems
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  • Study Wigner's book on Group Theory and Quantum Mechanics for detailed methodologies
  • Explore Tinkham's work for applications of angular momentum coupling
  • Research the mathematical formulation of Wigner 6J coefficients
  • Investigate other coupling schemes in quantum mechanics, such as Clebsch-Gordan coefficients
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Quantum physicists, students of quantum mechanics, and researchers focusing on angular momentum coupling and group theory applications in physics.

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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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