# I Why is association rule in angular momentum sum not valid?

1. Jul 21, 2016

### goodphy

Hello.

The quantum mechanics textbook shows the relation of J1 + J2 + J3 ≠ J1 + (J2 + J3). I believe Ji is total angular momentum operator for ith group of electrons (but actually I have not seen J1 operator while I have seen J12 operator so far).

I don't know how to prove J1 + J2 + J3 ≠ J1 + (J2 + J3). I think this play essential role of explaining why there are many coupling schemes.

2. Jul 21, 2016

### blue_leaf77

That's strange, which book did you find this from?

3. Jul 21, 2016

### goodphy

This is actually Korea book so you may not know this. Not famous book but it shows this relation with some proof which is...difficult for me to understand. Have you ever seen this relation? I think essence is this; total angular momentum quantum number J resulted from LS coupling is different to that of being from jj coupling.

Can you at least tell me it is right?

4. Jul 21, 2016

### blue_leaf77

The thing about LS and jj coupling schemes is that which effect is larger than which. In LS coupling, the electron-electron repulsion effect is far stronger than the spin-orbit effect and thus the latter is treated as a perturbation. In this case, the zeroth order state is determined by performing the sum $\mathbf L = \sum_{i=1}^N \mathbf L_i$ and $\mathbf S = \sum_{i=1}^N \mathbf S_i$ to obtain $L$ and $S$. When the spin-orbit perturbation is taken into account, there might be states/terms which are degenerate, so one must switch into using the total angular momentum $\mathbf J = \mathbf L+\mathbf S$. In JJ coupling case where spin-orbit is much larger than electron-electron repulsion, one calculate $\mathbf J_i = \mathbf L_i + \mathbf S_i$ first, then calculate $\mathbf J = \sum_{i=1}^N\mathbf J_i$.

My point is, I think writing $(\mathbf J_1 + \mathbf J_2) + \mathbf J_3 \neq \mathbf J_1 + (\mathbf J_2 + \mathbf J_3)$ is not mathematically correct. The reason why the final state in LS and jj coupling differs lies in the question of which effect is taken as perturbation.

Last edited: Jul 21, 2016