The total angular momentum of a particle with orbital angular momentum l (vector) and spin angular momentum s (vector) is j = l + s (vectors). The eigenvalues of j^2, l^2 and s^2 (vectors) are j(j + 1)ħ^2, l(l + 1)ħ^2 and s(s + 1)ħ^2 respectively. State the possible values of j for l = 1, s = 1/2
What is the general rule which tells you how many values of j to expect for arbitrary l, s?
What basic information about the fine structure of hydrogen does all this tell us?
j = l + s
The Attempt at a Solution
If l = 1 then l (vector) may take values (+ or -) ħ (2)^0.5
If s = 0.5 then s (vector) may take values (+ or -) ħ (3/4)^0.5
=> j = (+ or -) ħ (2)^0.5 (+ or -) ħ (3/4)^0.5
How do I know that the directions of these vectors (l and s) will line up, so I can just add their magnitudes? Also, surely the expression I have found won't necessarily satisfy
j^2 = j(j + 1)ħ^2.
Any help would be massively appreciated. Thanks