# Spin wavefunction confusion

My lecturer writes:

The spin wavefunctions are symmetric on exchange of spins for the spin 3/2 states. These states include:

$$|\uparrow \uparrow \uparrow \rangle$$

and $$|\uparrow \uparrow \downarrow \rangle + |\uparrow \downarrow \uparrow \rangle + |\downarrow \uparrow \uparrow \rangle$$

How is the second wavefunction a state for a spin 3/2 particle? I thought the spin is 1/2 + 1/2 - 1/2 = 1, so the measured spin can be 1, 0 or -1?

Start with the spin 1 states that you get from adding two spin 1/2 particles and then add the third, standard excersice in QM

|++>

|+-> + |-+>

|-->

are the three spin 1 states you can build from adding two spin 1/2 particles-

The second state you wrote is the |S, S_z> = |3/2, 1> state

How are they spin 1 states though? How do you figure that out from those states?

How are they spin 1 states though? How do you figure that out from those states?
Have you done adding angular momenta in your QM class yet? yes or no?

We did it briefly, just in terms of quantum numbers though, so S = s1 + s2...|s1-s2|, we didn't relate it to the spin wavefunctions like the ones you have mentioned.

ok, there are three spin-1 states - do you agree?

do you also agree that |+-> + |-+> has S_z = 0?

and total spin

S^2 = (S_1 + S_2)^2 on that state gives s(s+1) = 1(1+1) = 2

as eigenvalue.

S^2 on |+-> + |-+> gives 0, right?