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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 p 22mp 2c 2+m 2c 4mc 2=p 22m(p 2) 28m 3c 2+ \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, “p\vec{p}” only appears dotted into the Pauli matrices, σp\vec{\sigma}\cdot\vec{p}.
    • In particular, this tells us how the spin couples to external magnetic fields σpσ(pqA/c)\vec{\sigma}\cdot\vec{p} \to \vec{\sigma}\cdot(\vec{p}-q \vec{A}/c).
    • What we previously wrote as “p 2p^2” could just as well have been written as (σp) 2(\vec{\sigma}\cdot\vec{p})^2.
  3. Parity and time-reversal invariance1 imply only even powers of σp\vec{\sigma}\cdot\vec{p} appear in the low-velocity expansion.
  4. Shifting the potential energy, V(r)V(r)+constV(\vec{r})\to V(\vec{r})+\text{const}, should shift HH+constH\to H+\text{const}.

With those ingredients, at O(v 2/c 2)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=p 22m+V(r)(p 2) 28m 3c 2c 1m 2c 2[σp,[σp,V(r)]] 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, [σp,[σp,V]]=(p 2V+Vp 2)2σpVσp [\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 S=2σ\vec{S}=\tfrac{\hbar}{2}\vec{\sigma}) c 1m 2c 2[σp,[σp,V]]=4c 1m 2c 2((V)×p)S+c 1 2m 2c 2 2(V) -\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, (V)=r1rdVdr\vec{\nabla}(V)= \vec{r}\frac{1}{r}\frac{d V}{d r} and the first term is the spin-orbit coupling, 4c 1m 2c 21rdVdrLS\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=1/8c_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 [σp,[σp,V]][\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 =L 2( 3) 2\mathcal{H}=L^2(\mathbb{R}^3)\otimes \mathbb{C}^2, time-reversal is implemented as the anti-unitary operator Ω T:(f 1(r) f 2(r))(f 2 *(r) f 1 *(r)) \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:(f 1(r) f 2(r))(f 1(r) f 2(r)) 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 Ω TσΩ T 1 =σ, U PσU P =σ Ω TpΩ T 1 =p, U PpU P =p \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]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.
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


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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