## October 17, 2022

### 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).

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 term (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}
Posted by distler at October 17, 2022 9:00 PM

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### Re: Fine Structure

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

Posted by: Dave on October 17, 2022 11:52 PM | Permalink | Reply to this

### Textbook

Quantum I and II are (roughly) the first and second half of Griffiths’ book.

Posted by: Jacques Distler on October 18, 2022 10:40 AM | Permalink | PGP Sig | Reply to this

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