How Do You Solve Wave Functions Using Determinants?

[gazepdapi1]

[SOLVED] wave functions

I realize that I have to provide what I have done first for every problem, but this one has stumped be. It is a bonus problem provided by my teacher and it is beyond our scope. that's why its a bonus. Just helping me start it would be a big help. thank you

http://img301.imageshack.us/img301/2531/46008754yl9.jpg

thank you

 

[CompuChip]

Start by writing out \hat L_{z,\mathrm{total}} and \hat S_{z,\mathrm{total}}.

 

[gazepdapi1]

here is what I have so far

Sz(total) = Sz1 + Sz2
Sz(alpha) = +(1/2)hbar(alpha)
Sz(beta) = -(1/2)hbar(beta)

Lz(total) = Lz1 + Lz2
Lz(alpha) = + hbar(alpha)
Lz(beta) = - hbar(beta)

Is this correct?
If so, then what?

 

[J.D.]

In the case of spin, use the fact the spin functions only act on states alpha and beta. I.e. write out the determinant, which for a 2x2 matrix is ad-bc, act on \psi (1,2) with \hat{S}_{z,total} and then see if you get back an eigenvalue equation.
Also remember that S_{z,1}, S_{z,2} only acts on states 1 and 2 respectively.