Total Angular Momentum of Many Particles

[andrewm]

Hi,

How does one find the s=0 state for the addition of the spin of 4 (for example) electrons?

More generally, how does one obtain the total spin of 4 electrons?

I understand that for 2 electrons one can read the s=1 and s=0 states from a table of Clebsch-Gordan coefficients.

 

[Vanadium 50]

To add 4 angular momenta, you can use something called Wigner 9j symbols. (Clebsch-Gordon coefficients are related to what are called 3j symbols; adding 3 angular momenta takes 6j symbols.)

 

[denian]

Hello,

To find the s=0 state for the addition of spin of 4 electrons, one would use the same method as for 2 electrons, but with a larger table of Clebsch-Gordan coefficients. The total spin of 4 electrons can be obtained by adding the individual spin values of each electron. For example, if each electron has a spin of 1/2, the total spin would be 4 * 1/2 = 2. This can also be visualized using the vector model of spin, where each electron's spin is represented by a vector and the total spin is the vector sum of all the individual spins.
It is important to note that the total spin can only take on certain values, as determined by the Pauli exclusion principle. For 4 electrons, the possible total spin values are 0, 1, 2, and 3. The s=0 state corresponds to a total spin of 0, meaning all the individual spins are cancelled out and the system has no net spin.

In general, the total spin of a system of particles can be obtained by using the Clebsch-Gordan coefficients to combine the individual spins. The Clebsch-Gordan coefficients represent the probability amplitudes for each possible combination of individual spin states. By adding these probabilities, one can determine the overall probability of a given total spin state.

I hope this helps to clarify the process of finding the total spin of multiple particles. Let me know if you have any further questions.