Energy transition in LS-coupling

[joris_pixie]

Energy transition in LS-coupling ! :)

Hello ! I have a small question I hope someone can help me :)

Lets take for example a 2 electron system with l₁,s₁ and l₂,s₂. One electron changes energy due to electric dipole.

I understand the rules for electrons in electric dipole transitions:
1) Δl = +/- 1 (parity flips)
2) Δm_l = 0 or +/- 1
3) Δm_s = 0

But I don't get a selection rule for the terms !

One of the acceptable transitions is :
ΔL = 0
But if (1) Δl can't be 0 and (2) Δm_l = 0 or +/- 1 how is this possible ? :)

Because to my understanding L = l₁ + l₂ in vector form.

 

[DrClaude]

That comes from the rules of addition of quantum angular momentum. The possible values of $L$ are
$$
L = l_1 + l_2, l_1 + l_2 - 1, \ldots, |l_1 - l_2|
$$
So even though one value of $l$ changes, it is still possible that $L$ doesn't change.