Modern physics, quantum numbers, subshells

[fluidistic]

Homework Statement

Find all the terms of an atom whose last subshell is np³.

Homework Equations

$M_L=\sum _i m_{l _i}$
$M_S=\sum _i m_{s _i}$

The Attempt at a Solution

My professor did the same exercise but with np². Basically he wrotes all the possible quantum numbers for the atom:
1)$m_l=1$, $m_s=1/2$. 2)$m_l=0$, $m_s=1/2$. 3)$m_l=-1$, $m_s=1/2$
1')$m_l=1$, $m_s=-1/2$. 2')$m_l=0$, $m_s=-1/2$. 3')$m_l=-1$, $m_s=-1/2$
Then he calculated all the possible values for $M_L$ and $M_S$. There are 15 values in total.
After this, there is an obscure step to me (he counted I don't really know what) and went to the conclusion that the solution to the problem is $^1 D ^3 P ^1 S$. Where the upper script is worth $2S+1$.
So I did the same method as him for np³ (I guess this notation means that there are 3 electrons on the subshell p or an arbitrary n?). I got 20 values for $M_L$, $M_S$. I'm stuck at doing the obscure step now. I have all possible values for $M_L$ and $M_S$.
Can someone explain me what I should do next?
Another question is... since n seems arbitrary, can I for example take $n=1$, so that $l=0$ and $m_l=0$. My professor seems to have taken n=2 for some misterious reason to me. Does someone understand why?
Edit: since there are at least 3 electrons I guess I cannot take n=1, since at least n=2. Ah... n must equal 2... ok that's what I considered, good. So I'm stuck where I pointed out.

 

[Nate810]

A:It looks like you are trying to label an atomic orbital. This can be done with the following equations:$M_L = \sum_i m_{l_i}$, and$M_S = \sum_i m_{s_i}$where there is one $m_{l_i}$ and one $m_{s_i}$ for each electron in the orbital.The terms of a np3 orbital are the same as for a np2 orbital, namely: $M_L = 0,\pm 1$$M_S = \frac{1}{2}, -\frac{1}{2}$You have listed these correctly.Now what does this mean? There are two possible values for the total angular momentum quantum number, $J$. The maximum value for $J$ is obtained when $M_L = M_S$, and the minimum value for $J$ is obtained when $M_L = -M_S$. Thus the possible values of $J$ are $J=M_L + M_S$ and $J=|M_L - M_S|$.For example, if $M_L = 1$ and $M_S = \frac{1}{2}$, then $J= \frac{3}{2}$ and $J = \frac{1}{2}$. So the terms of a np3 orbital are (in order of increasing energy):$^1D_2$, $^3P_2$, $^3P_1$, $^3P_0$, $^1S_0$