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Microstates of Spin-Orbit Levels in Atomic Physics

Physics Asked on March 4, 2021

I have searched for this answer in many atomic physics books and didn’t find the answer. I have a question on how to assign the microstates to their associated level. For example, I worked out the microstates of the first three terms for the Sb atom: $rm^4S^o$, $rm^2D^o$, and $rm^2P^o$. For the term $rm^2P^o$, we have two levels $rm^2P^o_{1/2}$ and $rm^2P^o_{3/2}$.

The obtained microstates in the $rm(M_L, M_S)$ representation are as follows: $(+1,-frac{1}{2})$, $(0,-frac{1}{2})$, $(-1,-frac{1}{2})$, $(+1,+frac{1}{2})$, $(0,+frac{1}{2})$, and $(-1,+frac{1}{2})$.

Now, $rm^2P^o_{1/2}$ and $rm^2P^o_{3/2}$ have 2 and 4 microstates, respectively. So 2 of these microstates belong to $rm^2P^o_{1/2}$ and 4 of them belong to $rm^2P^o_{3/2}$. My question is how can I know which 2 microstates belong to $rm^2P^o_{1/2}$ and which 4 microstates belong to $rm^2P^o_{3/2}$?

Thank you so much for your help. Much Appreciated!

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