### Fine Structure

I’m teaching the undergraduate Quantum II course (“Atoms and Molecules”) this semester. We’ve come to the point where it’s time to discuss the fine structure of hydrogen. I had previously found this somewhat unsatisfactory. If one wanted to do a proper treatment, one would start with a relativistic theory and take the non-relativistic limit. But we’re not going to introduce the Dirac equation (much less QED). And, in any case, introducing the Dirac equation would get you the *leading corrections* but fail miserably to get various non-leading corrections (the Lamb shift, the anomalous magnetic moment, …).

Instead, various hand-waving arguments are invoked (“The electron has an intrinsic magnetic moment and since it’s moving in the electrostatic field of the proton, it sees a magnetic field …”) which give you the wrong answer for the spin-orbit coupling (off by a factor of two), which you then have to further correct (“Thomas precession”) and then there’s the Darwin term, with an even more hand-wavy explanation …

So I set about trying to find a better way. I want use as minimal as possible input from the relativistic theory and get the *leading* relativistic correction(s).

- For a spinless particle, the correction amounts to replacing the nonrelativistic kinetic energy by the relativistic expression $\frac{p^2}{2m} \to \sqrt{p^2 c^2 +m^2 c^4} - m c^2 = \frac{p^2}{2m} - \frac{(p^2)^2}{8m^3 c^2}+\dots$
- 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 $(\vec{\sigma}\cdot\vec{p})^2$.

- Parity and time-reversal invariance
^{1}imply only*even*powers of $\vec{\sigma}\cdot\vec{p}$ appear in the low-velocity expansion. - 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* term^{2} (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 …

^{1}On the Hilbert space $\mathcal{H}=L^2(\mathbb{R}^3)\otimes \mathbb{C}^2$, time-reversal is implemented as the anti-unitary operator $\Omega_T: \begin{pmatrix}f_1(\vec{r})\\ f_2(\vec{r})\end{pmatrix} \mapsto \begin{pmatrix}-f^*_2(\vec{r})\\ f^*_1(\vec{r})\end{pmatrix}$ and parity is implemented as the unitary operator $U_P: \begin{pmatrix}f_1(\vec{r})\\ f_2(\vec{r})\end{pmatrix} \mapsto \begin{pmatrix}f_1(-\vec{r})\\ f_2(-\vec{r})\end{pmatrix}$ These obviously satisfy $\begin{aligned} \Omega_T \vec{\sigma} \Omega_T^{-1} &= -\vec{\sigma},\quad& U_P \vec{\sigma} U_P &= \vec{\sigma}\\ \Omega_T \vec{p} \Omega_T^{-1} &= -\vec{p},\quad& U_P \vec{p} U_P &= -\vec{p}\\ \end{aligned}$

^{2}Of course, the operator $i[H_0,V]$ also appears at the same order. But it makes zero contribution to the shift of energy levels in first-order perturbation theory, so we ignore it.

## Re: Fine Structure

Just out of curiosity, what book are you using/recommending for the course (if you’re using any)?