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## Homework Statement

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?

## Homework Equations

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