How Do We Calculate the Composite Spin of a System with Three or More Particles?

[Xian]

When there are two particles with spin s₁ and s₂ the composite system has possible spins s₁+s₂ all the way down to |s₂-s₁| (using integer steps). The azimuthal quantum number goes from s₁+s₂ to -s₁+s₂ (using integer steps).

Now my question is if we had three particles (or more) how would we calculate their possible composite spin?

 

[Physics Monkey]

Hi Xian,

You add the spins one at a time. For example, $$1/2 \otimes 1/2 = 0 \oplus 1$$ and $$1/2 \otimes 1/2 \otimes 1/2 = (0 \oplus 1 )\otimes 1/2 = 1/2 \oplus 1/2 \oplus 3/2$$. Basically you're doing the analog of distributing multiplication over addition. There are several different ways to group the spins in the 3 spin case, and the different choices are related by 6j symbols. Hopefully you can now figure out the generalization to N spins.

Hope this helps.

 

[SpectraCat]

When there are two particles with spin s₁ and s₂ the composite system has possible spins s₁+s₂ all the way down to |s₂-s₁| (using integer steps). The azimuthal quantum number goes from s₁+s₂ to -s₁+s₂ (using integer steps).

Now my question is if we had three particles (or more) how would we calculate their possible composite spin?

The maximum value of the spins is the same (i.e. the sum of all of the spins), and assuming , all values represented by integer steps (or half integer, if you are mixing fermions and bosons), are possible down to the minimum value. However, the minimum value (which obviously cannot be less than zero) is more complicated to determine in general. Some simple cases are easier to determine, such as the case of N particles each with the same spin s, in which case the minimum total spin will be zero for even N, and s for odd N. Where things really get complicated is with the degeneracies of the different total spin values. I know that analytical expressions exist for up to 4 spins (9-J symbols), but beyond that I am not sure.

 

[DrDu]

There exists a relatively easy way to obtain all the irreducible representations for the spin of n particles using so called "Weyl diagrams", at least for identical particles which is possibly the most important case. It makes use of the fact that the irreducible representations of the group SU(2) can be labeled by Young type diagrams pertaining to the Young type diagrams of the permutation group S_n of the n particles.