Can You Help Me Couple 3 Spin 1/2 Particles?

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Discussion Overview

The discussion centers on the coupling of three spin 1/2 particles, specifically focusing on the calculation of coefficients for different states using Clebsch-Gordan (C-G) coefficients. Participants are exploring the mathematical details involved in these calculations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents their calculations for coupling three spin 1/2 particles, noting difficulties in obtaining the correct results for specific states.
  • Another participant challenges the correctness of the second C-G coefficient calculation, asserting that adding L=0 to any state results in a C-G coefficient of 1.
  • A request for further clarification and resources is made by the original poster, indicating a desire for more detailed explanations or references on the topic.
  • A link to a resource on angular momentum is provided as a suggestion for further reading.

Areas of Agreement / Disagreement

There is disagreement regarding the correctness of the second C-G coefficient calculation. Some participants assert that the original poster's calculation is incorrect, while the original poster seeks clarification and further information.

Contextual Notes

The discussion does not resolve the mathematical steps involved in the calculations, and there are limitations in the original poster's understanding of the C-G coefficients as indicated by their request for more information.

JoJoQuinoa
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Hello,

I'm trying to couple 3 spin 1/2 particles. So far, I have been able to find the coefficient for the other states but I can't get the results for ##j_{12} = 0## to ##j_3=1/2##.
Here is my attempt:
1) Using CG table ##<j_1;j_2;m_1;m_2|J;M><\frac{1}{2};\frac{1}{2};-\frac{1}{2};\frac{1}{2}|0;0> = 1/\sqrt2##
2) Using CG table again with ##\frac{1}{\sqrt2}<0;\frac{1}{2};0;\frac{1}{2}|\frac{1}{2};\frac{1}{2}> = 1/\sqrt2*1/\sqrt3 = 1/\sqrt6##

The correct answer is ##1/\sqrt2##.

Please help!

Thank you
 
Physics news on Phys.org
Your second C-G is wrong. Adding L=0 to anything has C-G=1.
 
Meir Achuz said:
Your second C-G is wrong. Adding L=0 to anything has C-G=1.
Hi Meir Achuz,

Thank you for your reply. Do you mind explaining a little bit in details or direct me to where I can find more info on that? The sources I've looked at only listed the results so I guess this is supposed to be known? I'm using Quantum Mechanics by David H. McIntyre.
 

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