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

In summary, the conversation is about trying to couple 3 spin 1/2 particles, with the goal of finding the coefficient for the states where ##j_{12} = 0## and ##j_3=1/2##. Using the C-G table, the person was able to find the coefficient for the other states but not for the desired one. Upon receiving help, they realized that their second C-G calculation was incorrect and that adding L=0 to anything always results in a coefficient of 1.
  • #1
JoJoQuinoa
17
0
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
 
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  • #2
Your second C-G is wrong. Adding L=0 to anything has C-G=1.
 
  • #3
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.
 

1. What is the concept of "coupling" in relation to spin 1/2 particles?

Coupling refers to the interaction between multiple spin 1/2 particles, where their individual spins combine to form a total spin state. This can result in either a higher or lower total spin value, depending on the specific coupling mechanism.

2. How is the total spin state of 3 spin 1/2 particles determined?

The total spin state of 3 spin 1/2 particles is determined by the combination of their individual spin states. For example, if all 3 particles have a spin up state, the total spin state will be 3/2. If 2 particles have a spin up state and 1 has a spin down state, the total spin state will be 1/2.

3. What are the different types of coupling for 3 spin 1/2 particles?

There are three main types of coupling for 3 spin 1/2 particles: parallel (or ferromagnetic) coupling, antiparallel (or antiferromagnetic) coupling, and mixed coupling. In parallel coupling, all 3 particles have the same spin state, while in antiparallel coupling, 2 particles have opposite spin states. Mixed coupling involves a combination of parallel and antiparallel coupling.

4. How does the coupling strength affect the total spin state of 3 spin 1/2 particles?

The coupling strength determines the difference in energy between the different spin states of the 3 particles. A stronger coupling will result in a larger energy difference and a more stable total spin state, while a weaker coupling will result in a smaller energy difference and a less stable total spin state.

5. What are some real-world applications of studying coupling in spin 1/2 particles?

Studying coupling in spin 1/2 particles has many applications in fields such as quantum computing, magnetic storage, and materials science. Understanding how spin states can be manipulated and controlled through coupling can lead to advancements in technology and materials with unique magnetic properties.

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