Spin-orbit interaction and two inconsistent values

[hokhani]

TL;DR

Two inconsistent values for spin-orbit interaction in different approaches

For an electron in the orbital characterized by $l=0$ we have $j=0\pm1/2$ and so $J^2=j(j+1)$ gives $J^2=3/4$ and $-1/4$ (normalized to $\hbar^2$). Finally, $L.S=1/2(J^2-L^2-S^2)$ results in $L.S=0$ and $-1$. However, according to $L.S=l_xs_x+l_ys_y+l_zs_z$ we find $L.S=0$ for $l=0$. I don't know where is my mistake?

 

[PeroK]

The quantum number $j$ is the characteristic AM of the system. We have $j = 0, \frac 1 2, 1, \frac 3 2 \dots$

We never have $j = - \frac 1 2$. This is not a valid quantum number for $j$.

You are confusing this with the quantum number $m$. For $j = \frac 1 2$, we have $m = \pm \frac 1 2$.