What Are the Quantum Number Combinations for Electron Angular Momentum?

[javit27]

1. Here's the problem:
Consider a system of two electrons, each with l = 1 and s = 1/2.
(A) What are the possible values of the quantum number for the total orbital angular momentum L = L₁ + L₂?
(B) What are the possible values of the quantum number for the spin orbital angular momentum S = S₁ + S₂?
(C) Using the above results, find the possible values of the quantum number for the total orbital angular momentum J = L + S.
(D) What are the possible individual quantum numbers j₁ and j₂ for the total angular momentum of each electron separately?
(E) Use the results of part (D) to compute the allowed values of j from the possible combinations of j₁ and j₂. Do your results agree with those of part (C)?

Homework Equations

L = [STRIKE]h[/STRIKE][l(l + 1)]^1/2
S = [STRIKE]h[/STRIKE][s(s + 1)]^1/2
J = [STRIKE]h[/STRIKE][j(j + 1)]^1/2

The Attempt at a Solution

(A) since l = 1,
L₁ = [STRIKE]h[/STRIKE][l(l + 1)]^1/2 = [STRIKE]h[/STRIKE](2)^1/2
I think this can be positive or negative, so
L = 2[STRIKE]h[/STRIKE](2)^1/2, or -2[STRIKE]h[/STRIKE](2)^1/2, or 0.

(B) I did this the same as (A) except with 1/2 instead of 1. I got:
S = [STRIKE]h[/STRIKE](3)^1/2, or -[STRIKE]h[/STRIKE](3)^1/2, or 0

(C) I assume just add up all the possibilities for L and S

(D) j = l + s or j = l - s
so, j = 1/2 or j = 3/2

(E) total j = 1 or 2 of 3
so J = [j (j + 1)]^1/2
so, J = [STRIKE]h[/STRIKE](2^1/2)
or [STRIKE]h[/STRIKE](6^1/2)
or 2[STRIKE]h[/STRIKE](3^1/2)

Basically I'm really unsure about (A) and (B). I have more confidence in (D) and (E), and (C) should be easy once I get the first two.

 

[_coyote_]

A) L = 2h(2)1/2, -2h(2)1/2, 0B) S = h(3)1/2, -h(3)1/2, 0C) J = 2h(2)1/2 + h(3)1/2, 2h(2)1/2 - h(3)1/2, 0D) j1 = 1/2 or j2 = 3/2E) J = h(21/2), h(61/2), 2h(31/2)