The spin and orbital angular momenta are coupled giving rise to a total angular momentum that we represent by J, $\vec{J}=\vec{L}+\vec{S}$. The total angular momentum quantum number is obtained by summing the total orbital and spin angular momentum quantum numbers: \begin{equation} J=L\otimes S= L+S, L+S-1, L+S-2 ,......,|LS| \end{equation} The operators compatible with the fine structure Hamiltonian are $J^2$ and $J_z$, whose eigenvalues are $\hbar^2 J(J+1)$ and $M_J\hbar$, respectively. A fine structure level is given by: $^{2S+1}L_J$ Fine structure states have the form: $\left|^{2S+1}L_J\;M_J\right\rangle$. The degeneracy of each fine structure level is $2J+1$ The spin-orbit coupling produces the splitting of the RS terms into fine structure levels, which have a slightly different energy. For example, carbon in its ground state $1s^22s^22p^2$, has three RS terms $^1S,^3P$ and $^1D$. Considering the spin-orbit coupling, the term $^3P$ is split into three levels $^3P_2, ^3P_1$ and $^3P_0$.

# Polyelectronic atoms : Quantum mechanics

## Fine structure of the atom. Spin-orbit interaction.

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- Written by: Germán Fernández
- Category: Polyelectron atoms | Quantum mechanics
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