Adding Spin and Orbital Momentum

From two spins to L and S

Everything we derived for two spins applies to any pair of angular momenta, because the construction only used the angular-momentum algebra. The most important application is coupling a particle's orbital angular momentum $\mathbf{L}$ to its spin $\mathbf{S}$ into a total

Here $j_1 = \ell$ (an integer: $0, 1, 2, \ldots$ for $s, p, d, \ldots$ states) and $j_2 = s = \tfrac12$ for a single electron.

The allowed total $j$

The triangle rule gives the possible totals at once. With $s = \tfrac12$ there are just two, provided $\ell \ge 1$:

For an $s$-state ($\ell = 0$) the only possibility is $j = \tfrac12$. As a dimension check, the two multiplets for $\ell \ge 1$ have sizes $2(\ell+\tfrac12)+1 = 2\ell+2$ and $2(\ell-\tfrac12)+1 = 2\ell$, summing to $4\ell+2 = (2\ell+1)\cdot 2$, exactly the uncoupled count of orbital times spin states.

A concrete case: the $p$-electron

Take $\ell = 1$ (a $p$-electron). The allowed totals are

The $j=\tfrac32$ multiplet holds $2\cdot\tfrac32+1 = 4$ states; the $j=\tfrac12$ multiplet holds $2$. Together they account for all $(2\cdot1+1)\cdot2 = 6$ uncoupled states $|m_\ell, m_s\rangle$. In spectroscopic notation these coupled levels are written $^2P_{3/2}$ and $^2P_{1/2}$, where the superscript $2 = 2s+1$ is the spin multiplicity, the letter $P$ stands for $\ell = 1$, and the subscript is $j$.

The coupled basis $|j, m_j\rangle$

A coupled state $|j, m_j\rangle$ is the Clebsch–Gordan combination of uncoupled states $|\ell, m_\ell\rangle|s, m_s\rangle$ with $m_\ell + m_s = m_j$. For the $p$-electron stretched state, for example,

and lowering with $J_- = L_- + S_-$ builds the rest of the $j=\tfrac32$ ladder, exactly as the two-spin case but with the orbital ladder operator now changing $m_\ell$.

Looking ahead

The coupled basis is not just convenient bookkeeping: it diagonalizes the spin–orbit interaction, which is why the energy levels of real atoms split into the multiplets we just enumerated. The next lesson motivates that interaction and shows how $\mathbf{L}\cdot\mathbf{S}$ is expressed through $\mathbf{J}^2$, $\mathbf{L}^2$, and $\mathbf{S}^2$.

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