# Spin wavefunction confusion

1. May 22, 2010

### barnflakes

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?

2. May 22, 2010

### ansgar

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

3. May 23, 2010

### barnflakes

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

4. May 23, 2010

### ansgar

Have you done adding angular momenta in your QM class yet? yes or no?

5. May 23, 2010

### barnflakes

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.

6. May 23, 2010

### ansgar

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?