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.