# Spin orbital coupling and total angular momentum

The Sodium emission spectrum has 2 bright yellow lines called the Na-D lines. Superficially I understand the cause; there's spin-orbit coupling and the 3p orbital is split into 3p(3/2) and 3p(1/2).

But I don't understand why those numbers (3/2 and 1/2)?

Are these the total angular momentum values or the orbital angular momentum values?

And how does one work them out?

J is the total angular momentum. L is the orbital angular momentum and S is the intrinsic total electron spin angular momentum. J = L + S

L = $$\sqrt{L\left(L+1\right)}\frac{h}{2\pi }$$ --> L can be n-1 where n is the Principal quantum number

S = $$\sqrt{S\left(S+1\right)}\frac{h}{2\pi }$$ --> S is an integer or half an odd integer, depending on whether the number of electrons is even or odd

J = $$\sqrt{J\left(J+1\right)}\frac{h}{2\pi }$$

J can be L+S ... L-S

For the Na-D lines the J = 3/2 is caused by L=1 and S=1/2 for the 3/2 value and L=1 and S=-1/2

The values for $$J,L,S$$ have a range of values e.g. $$-X, -X+1,\cdots X-1, X$$, e.g. if $$L=5/2$$ the range is $$-5/2,-3/2,3/2,5/2$$