Why is association rule in angular momentum sum not valid?

[goodphy]

Hello.

The quantum mechanics textbook shows the relation of J₁ + J₂ + J₃ ≠ J₁ + (J₂ + J₃). I believe J_i is total angular momentum operator for i^th 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.

Please help me to do this:)

 

[blue_leaf77]

The quantum mechanics textbook shows the relation of J1 + J2 + J3 ≠ J1 + (J2 + J3).

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

 

[goodphy]

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

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?

 

[blue_leaf77]

I think essence is this; total angular momentum quantum number J resulted from LS coupling is different to that of being from jj coupling.

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.