- #1

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There's something that I don't understand about the spin-orbit interaction.

First of all I know that

[itex][\hat{S} \cdot \hat{L}, \hat{L_z}] \ne 0[/itex]

[itex][\hat{S} \cdot \hat{L}, \hat{S_z}] \ne 0[/itex]

so this means that [itex] \hat{S} \cdot \hat{L} [/itex] doesn't share a common set of eigenstates with [itex] \hat{S_z} [/itex] and [itex] \hat{L_z} [/itex].

I know that [itex]| nlm_lsm_s>[/itex] is a common eigenstate for [itex] \hat{S_z} [/itex] and [itex] \hat{L_z} [/itex],

so that would mean it is not an eigenstate for [itex] \hat{S} \cdot \hat{L} [/itex].

However, I've read that [itex]<nlm_l'sm_s'|\hat{S}\cdot\hat{J}|nlm_lsm_s>\ne0[/itex] for all [itex] m_l \ne m_l', m_s\ne m_s' [/itex] i.e. the diagonal elements are non-zero. Surely if [itex]| nlm_lsm_s>[/itex] is not an eigenstate of [itex] \hat{S} \cdot \hat{L} [/itex], then the matrix element cannot be evaluated?

Thank you in advance!