# How does one determine the number of eigenstates of a system with a given spin? [closed]

I have had a true/false question in a practice exam stating:

For a spin 3/2 system (S=3/2), there are only four spin eigenstates.

which is true. (solutions)

I do not understand how one can determine how many eigenstates exist for a given spin system.

All I know is a s=1/2 system has two eigenstates.

## closed as off-topic by Thomas Fritsch, Jon Custer, ZeroTheHero, GiorgioP, tpg2114♦Jul 5 at 23:49

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• A general formula for a number of the spin eigenstates is $2S+1$ – Gec Jul 5 at 7:36

The states may take any value between $$S$$ and $$-S$$ in steps of 1. I.e. for $$S = \tfrac{3}{2}$$ valid states would be $$\lbrace\tfrac{3}{2},\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{3}{2}\rbrace$$, for $$S = \tfrac{1}{2}$$ we get $$\lbrace\tfrac{1}{2},-\tfrac{1}{2}\rbrace$$ and for $$S = 2$$ we get the five states $$\lbrace2,1,0,-1,-2\rbrace$$.
Notice that then nececarrily follows that $$S$$ may take only full or half integer-values.