Spin-Orbit Coupling

An electron has orbital angular momentum $l = 1$ and spin $s = 1/2$.

a) If the projection of the electron spin along its orbital angular momentum $\overrightarrow{L}*\overrightarrow{S}$ is measured, what are the possible outcomes?

b) Suppose it is known that $S_z = -\hbar/2$ and $L_z= -\hbar$. What are the possible outcomes if the total angular momentum quantum number $J$ is measured?

For a particular choice of the z axis (z axis parallel to spin) the Spin projections on it have just 2 values   ($S_z =m_s*\hbar$)
$S_z =+(1/2)*\hbar$ or $S_z =-(1/2)*\hbar$

For the same choice of z axis the Angular momentum projections on it have the following possibilities ($Lz =m_l*\hbar$):

$L_z = -\hbar$, $L_z =0$ or $L_z =+ \hbar$

If we make the product $\overrightarrow{S}*\overrightarrow{L}=\overrightarrow{L}*\overrightarrow{S}$  then the possible outcomes are:

$\overrightarrow{L}*\overrightarrow{S} =-(1/2)*\hbar^2$

$\overrightarrow{L}*\overrightarrow{S} =0$

$\overrightarrow{L}*\overrightarrow{S} =+(1/2)*\hbar^2$

b)

For the given values in text of $S_z$ and $L_z$ we have $m_s =-1/2$ and $m_l =-1$

so that $s=1/2$ (spin quantum number is unique) and $l =1$ (orbital quantum number)

(always s and l are positive!)

Total angular momentum quantum number is defined as $j = l + s (=1+1/2 =3/2)$

possible values of $j$ are $-|l+s| < j < |l+s|$   with integer steps so that $j=-1,0,1$

In absolute value $J = \sqrt {(j*(j+1)} *\hbar = 0$ or $\sqrt{2}*\hbar$