Modified Spectroscopic Notation (Shankar)

Question

In Shankar's Chapter on addition of angular momentum in his Principles of Quantum Mechanics (Chapter 15 of the 2nd edition), he includes the section attached after describing the basic strategy for obtaining the CG coefficients.

What I'm afraid I don't follow is what the inclusion of the superscript tells us. In particular, if we are saying that the electron is in an eigenstate of total angular momentum (j=3/2 in the example), then it is not a spin projection (quantum number $m_s$) eigenstate. So what is the superscript telling us? I suppose there is a one-to-one correspondence between 2s+1 and s (the spin angular momentum quantum number), so does the 2s+1 just tell us about the statistics of the particle? I have a feeling that this interpretation is almost certainly wrong.

Answer 1

Let's take the same example as given in the book $${}^{2S+1}L_J={}^{2}P_{3/2}$$ This implies $j=3/2$ and letter $P$ denote that $l=1$ from here it's trivial to see $s=j-l=1/2$ which can be seen from the fact that $2s+1=2\Rightarrow s=1/2$.

The $S$ is the total spin which might not be a statistic of the particles for a multiparticle system. For example the second example $${}^{2S+1}L_J={}^{1}S_{0}$$

$S\Rightarrow l=0$ (total orbital angular momentum) and $j=0$ total angular momentum and $2s+1=1\rightarrow s=0$ that is total spin angular momentum which doesn't denote the statistics of the particles.

Answer 2

$2S+1$ is the multiplicity, meaning how many degenerate states there are for a given $S$ since $-S \leq M_s \leq S$ (where $M_s$ is the spin projection number). For $S=1$, $M_s = -1, 0, 1$. So the superscript is $2(1) + 1 = 3$ for the $3$ degenerate states in spin.