Values of the Total Angular Momentum Quantum Number $j$ I'm having some trouble understanding how values are determined for the total angular momentum quantum number $j$. In a single electron system we've been taught that:
$$j = l \pm s$$
Where $s = \frac 12$ is the spin. Is this equivalent to saying:
$$j = l + m_s$$
With $m_s = \pm \frac 12$ the spin orientation? So theoretically if one had the values $l=1$ and $m_s=+ \frac 12$ could you say that $j = \frac 32$?
 A: Just because the total spin quantum number $s$ and the $z$-component spin quantum number $m_s$ can take the same value ($1/2$), they are not the same thing. As a consequence, the short answer to your question is: no, you cannot say that $j = 3/2$ if $l = 1$ and $m_s = 1/2$. For the long answer, please read on.
Assume we have two angular momentum operators $J_1$ and $J_2$ with quantum numbers $j_1,m_1,j_2,m_2$ and we are considering a problem where the Hamiltonian depends on $J := J_1 + J_2$. An example for this would be a term in the fine structure of the hydrogen atom which reads (up to some constants) $J^2/r^3$ where $J$ is the sum of the spin $S$ and the orbital momentum $L$, and $r$ is the radial coordinate. Now the good quantum numbers are $j$ and $m_j$ with
$$
J^2 |j,m_j\rangle = \hbar^2 j (j+1) |j,m_j\rangle~, \qquad J_z |j,m_j\rangle = \hbar m_j |j,m_j\rangle~.
$$
The total angular momentum number $j$ can take any value in $\{j_1 + j_2, j_1 + j_2 - 1,\dots,|j_1 - j_2|\}$ and for a given $j$ it holds $m_j \in \{j,j-1,\dots,-j+1,-j\}$. The only other thing one can say at this point, is, that due to angular momentum conservation in the $z$-direction, it holds $m_j = m_1+m_2$. In particular, the value of $j$ is not determined, even if you know  $j_1,j_2,m_1,m_2$.
Now what exactly does "not determined" mean? To illustrate this, let's assume we indeed know $j_1,j_2,m_1,m_2$ at some point in time, meaning that the system is in the state $|j_1,j_2;m_1,m_2\rangle$. Please note, that because these are not the good quantum numbers, this state may change over time. However, it is possible to expand it in eigenstates of the total angular momentum, using the good quantum numbers:
$$
|j_1,j_2;m_1,m_2\rangle = \sum_{j \in \{j_1 + j_2,\dots,|j_1-j_2|\}} \langle j,m_j|j_1,j_2;m_1,m_2\rangle |j,m_j\rangle = \sum_{j \in \{j_1 + j_2,\dots,|j_1-j_2|\}} \langle j,m_1+m_2|j_1,j_2;m_1,m_2\rangle |j,m_1+m_2\rangle~.
$$
The numbers $\langle j,m_j|j_1,j_2;m_1,m_2\rangle$ are called Clebsch-Gordan-coefficients and they are a measure of how probable it is to find a certain $j$ in the state $|j_1,j_2;m_1,m_2\rangle$. If you do not want to calculate them yourself, tabulated values can be found for example here
Remark:
The above expansion using the Clebsch-Gordan-coefficients is actually simpler than it looks, because we know right away that the coefficients will be zero if $|m_1 + m_2| > j$, because it holds $m_j \in \{j,\dots,-j\}$, as mentioned above.
