In LS coupling, the individual orbital angular momenta $ \vec{L}_i $ combine to form the total orbital angular momentum $ \vec{L} $, and the individual spin angular momenta $ \vec{S}_i $ combine to form the total spin angular momentum $ \vec{S} $. Then, $ \vec{L} $ and $ \vec{S} $ are coupled to produce the total angular momentum $ \vec{J} $.
In JJ coupling, each particle’s orbital and spin angular momenta ($ \vec{L}_i $ and $ \vec{S}_i $) are first coupled individually to form the total angular momentum for each particle, $ \vec{J}_i = \vec{L}_i + \vec{S}_i $. These individual $ \vec{J}_i $ values are then combined to yield the total angular momentum $ \vec{J} $ for the system.
While both LS and JJ coupling ultimately lead to the same total $ \vec{J} $, they differ in the sequence of combining angular momenta. Since vector addition is associative, I wonder: $$ \mathbf J = (\mathbf L_1 + \mathbf L_2) + (\mathbf S_1+\mathbf S_2) = (\mathbf L_1+\mathbf S_1)+(\mathbf L_2+\mathbf S_2) $$
How does the order of coupling affect the eigenstates of the system if the end result is total angular momenta?
In LS coupling, why is the eigenbasis constructed from $ \vec{L} $ and $ \vec{S} $ even though $ \vec{L}_i $, $ \vec{S}_i $, and $ \vec{J}_i $ are also constants of motion and commute with the Hamiltonian? Does this choice of basis relate to which quantities span the irreducible representations (irreps) of the symmetry group?
One possible resolution, as suggested in this post, is that $$[L^2,J_{i}]\ne0$$ This implies that the choice of $L$ and $S$ must be made sacrificing $J_i$ because it forms the good basis for $L\cdot S$. Additionally, I’m curious about how these coupling schemes affect degeneracies and the role of irreps:
- In LS coupling, the spin-orbit interaction lifts the degeneracy of states with the same $ L $ and $ S $ by coupling them into $ J $ eigenstates, splitting energy levels based on $ \vec{L} \cdot \vec{S} $.
- In JJ coupling, the strong spin-orbit interaction ensures $ \vec{J}_i $ is a good quantum number for each electron, and the residual Coulomb interaction then breaks degeneracy within states of the same $J$ but differing distributions of $ j_i $ values.
Finally, regarding the symmetry group and irreps:
- Do $L$, $S$, and $J_i$ always form irreducible representations of the rotation group $SO(3)$ in these schemes? For example, in LS coupling, $\vec{L}$ and $\vec{S}$ clearly span irreps of $SO(3)$, but $J_i$ seems less relevant. Why?
- In JJ coupling, $ J_i$ becomes central due to the spin-orbit interaction. Do $\vec{L}$ and $\vec{S}$ retain any physical significance, or are they merely mathematical constructs?
Could someone clarify how degeneracy breaking, choice of basis, and irreps play out in LS versus JJ coupling, and why certain angular momenta are prioritized in different regimes?
