Angular momentum addition and expansion in states

[Onamor]

Homework Statement

Part (e) of the attached question. Sorry for using a picture, and thanks to anyone who can help.

Homework Equations

the answer to part (d) is that the eigenvalue is
\hbar^{2}\left(l\left(l+1\right) + s\left(s+1\right)+2m_{l}m_{s}\right)
where, for this part of the question, m_{l}=l and m_{s}=s.

The Attempt at a Solution

I try to expand the state \left|1/2,1/2\right\rangle in \left|l m_{l}, s m_{s}\right\rangle but would I need six components (and therefore six coefficients)?
Because we have l=1 then m_{l}=1,0,-1 and for each of those we can have m_{s}=1/2,-1/2.
I can act with J^{2} on this expansion and use the formula for the eigenvalues above, but then I still have 6 unknown coefficients.

If I expand the state as \left|j,m_{j}\right\rangle then there are four components for m_{j}=3/2,1/2,-1/2,-3/2.
Acting with J_{+} on rids me of the first term but I still have 3 unknowns.

The last equation I haven't used yet is the normalisation of the coefficients but so far I have too many unknowns for it to be useful.

Thanks very much to any helpers, really not sure where to go with this one..

 

[vela]

For part (d), you should be able to show that the expression you have is equal to \hbar^2j(j+1) where j=l+s.

For part (e), considering just the z-component, you have J_z=L_z+S_z. Use this fact to cut down on the number of |l m_l; s m_s> states you have to consider.