This question, posed in a problem sheet that I have been asked to do, has stumped me. I really don't know what to do here. Any help would be greatly appreciated.
I know that the magnetic moment of an electron orbiting a nucleus is given by $$\mu=-\frac e{2m}L.$$
I also know that the angular momentum quantum number is zero so therefore so is $m_l$. I don't know which one is more relevant since $J_z=\hbar m_l$. How would one incorporate spin into this?
When you have orbital angular momentum and spin angular momentum for an electron then the resulting magnetic moment $\mu_j$ is $$\vec \mu_j=\vec \mu_l+\vec \mu_s= -\frac{e}{2m_e}(\vec l+g\vec s)=-\frac{\mu_b}{\hbar}(\vec l+g\vec s),$$ where $\mu_b=\frac{e\hbar}{2m_e}$ is Bohr's magneton, $l$ denotes the orbital angular momentum, $s$ the spin angular momentum, $e$ the electron charge and $m_e$ the electron mass. $g$ is the Landé factor which can be measured in experiment and calculated. The value of the Landé factor for an electron is $g\approx2$.
In your question the electron is in an s-state, so $\vec l$ is zero and the total magnetic moment is
$$\vec \mu_j=-g\frac{\mu_b}{\hbar}\vec s=-\frac{e}{m_e}\vec s$$
You can calculate the z-component and the magnetude of $\vec\mu_j$ by considering the z-component and the magnitude of $\vec s$:
$$|\vec s|=\sqrt{3/4}\hbar, \quad s_z=\pm1/2\hbar$$
Anticipating the next question from your sheet:
What if $l\neq0$, is there a nice, clean expression for $\mu_j$?
The problem is that this magnetic momentum vector $\vec\mu$ is not parallel to the total angular momentum vector $$\vec j=\vec l+\vec s.$$ Due to an interaction between the electron spin and it's orbital motion the vector $\vec s$ is not fixed in space. This then means that $\vec\mu_j$ is also not fixed in space. However, you can calculate the average value of $\vec\mu_j$ by projecting it on the vector $\vec{j}$, this is why it says "averaged according to..." in your problem sheet. If you do this then you will find that the average value is
$$\langle\mu_j \rangle = g_j\frac{\mu_b}{\hbar}|\vec j|=g_j\frac{\mu_b}{\hbar}\sqrt{j(j+1).} $$
$g_j$ is the Landé factor from your problem sheet. Since you consider an electron, the gyromagnetic factor for it's motion ($g_l$) is one (for the neutron it's zero) and $g_s$ is two, so you need the second formula.
In the formula for $g_j$ and $\mu_l$ you have the values for s,l and j. When you calculate $g_j$ you have always to consider those values for a specific state. So for a 2p electron $l=1$, $s=1/2$ and $$j=l\pm s=1/2 \quad \text{or} \quad 3/2.$$ So, for each of those set of values (1,1/2,3/2 and 1,1/2,1/2) you calculate $\langle \mu_j\rangle$.
I hope this helps!
