(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given 3 spins, #1 and #3 are spin-1/2 and #2 is spin-1. The particles have spin operators ## \vec{S}_i, i=1,2,3 ##. The particles are fixed in space. Let ## \vec{S} = \vec{S}_1 + \vec{S}_2 + \vec{S}_3 ## be the total spin operator for the particles.

(ii) Find the eigenvalues of ## \vec{S}^2 ## and their multiplicities.

2. Relevant equations

## \vec{S}^2 \rvert s,m \rangle = s(s+1)\rvert s,m \rangle##, Spin matrices

3. The attempt at a solution

So, for each total spin ## s ## there are a number of combination of the ## s_i ## values associated with each spin, ## \{ \pm 1/2 \} ## for #1 and #3, ## \{ 0, \pm 1 \}## for #2. Then for each combination there are ## 2s_i+1 ## choices for ## m ## which yields ## (2s_1+1)(2s_2+1)(2s_3+1) ## possibilities

This comes down to eigenvalues

## 6 \hbar^2 ## with degeneracy 24

## 2 \hbar^2## with degeneracy 56

## 0 \hbar^2 ## with degeneracy 32

I am skeptical of these values and feel that there is probably a better way.

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# Homework Help: The addition of 3 spins

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