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I'm solving some practice questions and I found this one:

Consider three particles with spins $S_1 = S_2 = S_3 = 1.$

(a) Find the dimension of the Hilbert space of this system.

27, since $(2S_1+1)(2S_2+1)(2S_3+1) = 3^3 =27$

(b) What are possible values of the total angular momentum?

S is defined as $S_1+S_2+S_3$ so it would be 0, 1, 2 or 3

(c) How many linearly independent states exist for each possible value of the total angular momentum?

2 states for S=3, 6 states for S=2, 12 states for S=1 and 7 states for S=0.

(d) Find the eigenvalues of the following Hamiltonian $$H = J(S_1·S_2 + S_2·S_3 + S_3·S_1)$$ What is the degeneracy of each level?

I assume they are asking to extrapolate what I know about principal quantum numbers for an $S=1/2$ particle to particles with integer spin. But in this case, the particles are bosons so I don't have "shells to fill up" and I don't know what the equivalent of $n$, $j$, or $l$ would be. If you can point me in the right direction to understand integer spin, I would really appreciate it. Thanks!

Alex
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  • b) is wrong. $S$ is not simply additive. Also, some values of final $S$ will be repeated: see for instance this answer: https://physics.stackexchange.com/a/459411/36194 – ZeroTheHero Jun 13 '21 at 23:44

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The only relevant quantum number in this problem is the spin. Every particle is characterised by a state $|s, s_z\rangle$ in the Hilbert space $\mathcal{H}$.

The first three questions can be answered by simple tensor product decomposition while the last one requires a simple trick to put the hamitonian in a better suited form.

Davide Morgante
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