Musings

1. For a spinless particle, the correction amounts to replacing the nonrelativistic kinetic energy by the relativistic expression $$\frac{p^2}{2m} \to \sqrt{p^2 c^2 +m^2 c^4} - m c^2 = \frac{p^2}{2m} - \frac{(p^2)^2}{8m^3 c^2}+\dots$$
2. For a spin-1/2 particle, “$\vec{p}$” only appears dotted into the Pauli matrices, $\vec{\sigma}\cdot\vec{p}$.
   • In particular, this tells us how the spin couples to external magnetic fields $\vec{\sigma}\cdot\vec{p} \to \vec{\sigma}\cdot(\vec{p}-q \vec{A}/c)$.
   • What we previously wrote as “$p^2$” could just as well have been written as $(\vec{\sigma}\cdot\vec{p})^2$.
3. Parity and time-reversal invariance¹ imply only even powers of $\vec{\sigma}\cdot\vec{p}$ appear in the low-velocity expansion.
4. Shifting the potential energy, $V(\vec{r})\to V(\vec{r})+\text{const}$, should shift $H\to H+\text{const}$.

With those ingredients, at $O(\vec{v}^2/c^2)$ there is a unique term² (in addition to the correction to the kinetic energy that we found for a spinless particle) that can be written down for spin-1/2 particle. $$H = \frac{p^2}{2m} +V(\vec{r}) - \frac{(p^2)^2}{8m^3 c^2} - \frac{c_1}{m^2 c^2} [\vec{\sigma}\cdot\vec{p},[\vec{\sigma}\cdot\vec{p},V(\vec{r})]]$$ Expanding this out a bit, $$[\vec{\sigma}\cdot\vec{p},[\vec{\sigma}\cdot\vec{p},V]] = (p^2 V + V p^2) - 2 \vec{\sigma}\cdot\vec{p} V \vec{\sigma}\cdot\vec{p}$$ Both terms are separately Hermitian, but condition (4) fixes their relative coefficient.

Expanding this out, yet further (and letting $\vec{S}=\tfrac{\hbar}{2}\vec{\sigma}$) $$-\frac{c_1}{m^2 c^2} [\vec{\sigma}\cdot\vec{p},[\vec{\sigma}\cdot\vec{p},V]]= \frac{4c_1}{m^2 c^2} (\vec{\nabla}(V)\times \vec{p})\cdot\vec{S} + \frac{c_1\hbar^2}{m^2 c^2} \nabla^2(V)$$

For a central force potential, $\vec{\nabla}(V)= \vec{r}\frac{1}{r}\frac{d V}{d r}$ and the first term is the spin-orbit coupling, $\frac{4c_1}{m^2 c^2} \frac{1}{r}\frac{d V}{d r}\vec{L}\cdot\vec{S}$. The second term is the Darwin term. While I haven’t fixed the overall coefficient ($c_1=1/8$), I got the form of the spin-orbit coupling and of the Darwin term correct and I fixed their relative coefficient (correctly!).

No hand-wavy hocus-pocus was required.

And I did learn something that I never knew before, namely that this correction can be succinctly written as a double-commutator $[\vec{\sigma}\cdot\vec{p},[\vec{\sigma}\cdot\vec{p},V]]$. I don’t think I’ve ever seen that before …