The Darwin Term and the Fine-Structure Formula

Two of the three fine-structure corrections are now in hand: the relativistic kinetic term, which acts on every state, and the spin-orbit term, which acts only on $\ell \ge 1$ states and left an unresolved $0\times \infty$ for $\ell =0$. The third correction, the Darwin term, fills exactly that gap: it acts only on $\ell =0$ states, where $|\psi (0){|}^2\ne 0$. It has no classical analogue, and its origin is only made fully transparent by the Dirac equation; the non-relativistic account describes it as a smearing of the electron's position over a length of order its Compton wavelength. Once all three shifts are summed, a striking simplification occurs: the separate $\ell$-dependences cancel, and the total fine-structure shift depends only on $n$ and the total angular momentum $j$.

The contact interaction

The reduction of the Dirac equation to a two-component non-relativistic theory (the Foldy–Wouthuysen transformation) produces, beyond the mass-velocity and spin-orbit terms, a third correction of the same order:¹²

For the Coulomb potential $V=-Z{e}^2/4\pi {ϵ}_{0}r$, the Laplacian of $1/r$ is a delta function,

so the Darwin term is a contact interaction, nonzero only at the origin:

A delta function at the origin sees only the value of the wave function there. The first-order shift is

Only $\ell =0$ states have nonzero amplitude at the nucleus. Every $\ell \ge 1$ wave function carries a factor $r^{\ell }$ that vanishes at the origin, so $|\psi (0){|}^2=0$ and the Darwin term contributes nothing there, precisely complementing the spin-orbit term's domain.

Zitterbewegung: why the electron is smeared

The physical picture behind the contact term is that the electron does not sit at a sharp point but jitters over a region of order its reduced Compton wavelength $\hslash /mc$. In the Dirac theory this trembling motion is called zitterbewegung, an interference between the positive- and negative-energy components of the spinor at the frequency $2m{c}^2/\hslash$. Averaged over the jitter, the electron samples the potential not at $\vec{r}$ but over a small neighborhood, and the potential it feels is the smeared average

With $⟨(\delta r{)}^2⟩$ of order $(\hslash /mc{)}^2$, the correction is $\sim (\hslash /mc{)}^2{\nabla }^{2}V$, matching the Darwin form up to the numerical factor that the careful reduction fixes at ${\hslash }^2/8m^2{c}^2$. Where the potential is smooth (away from the origin) the smearing does nothing; where it is sharply curved (at the Coulomb singularity) the average differs from the point value, and only s-states, which visit the origin, register the difference.

Evaluating the Darwin shift

The s-state density at the origin follows from the hydrogenic wave functions. For $\ell =0$ the spherical harmonic is $Y_0^0=1/\sqrt{4\pi }$ and the radial function at the origin gives

Substituting into $E_D^{(1)}$,

Reducing the constants against $E_n$ and $a_0$ (using $a_0={\hslash }^2/mb$ with $b=Z{e}^2/4\pi {ϵ}_0$, so $m{a}_{0}b={\hslash }^2$) gives the compact form¹

The structural role of this value appears when it is combined with the relativistic shift at $\ell =0$. Adding $E_r^{(1)}(\ell =0)=-(E_n^2/2m{c}^2)(8n-3)$ to $E_D^{(1)}=(E_n^2/2m{c}^2)(4n)$ gives $(E_n^2/2m{c}^2)(3-4n)$, reproducing the fine-structure formula evaluated at $j=\frac{1}{2}$. The Darwin term is precisely what makes the combined shift, once written in terms of $j$, extend continuously to the $\ell =0$ end where the spin-orbit expression alone is indeterminate.

Summing the three corrections

For $\ell \ge 1$ the fine-structure shift is $E_r^{(1)}+E_{\text{so}}^{(1)}$ (no Darwin), and for $\ell =0$ it is $E_r^{(1)}+E_D^{(1)}$ (no spin-orbit). Both cases reduce to the same expression. Take $\ell \ge 1$ and add:

The algebra is a matter of putting both terms over the common denominator $\ell (\ell +\frac{1}{2})(\ell +1)$ and using $j(j+1)=\ell (\ell +1)+\frac{3}{4}\pm (\ell +\frac{1}{2})$ for $j=\ell \pm \frac{1}{2}$. Every explicit $\ell$ cancels, leaving a result that depends on $\ell$ only through $j$:¹

The $\ell =0$ case, with $E_r^{(1)}$ and $E_D^{(1)}$ and $j=\frac{1}{2}$, reproduces the same formula. The cancellation of $\ell$ is not an accident of the perturbative bookkeeping; it reflects the deeper structure the Dirac equation makes exact, in which $j$ and not $\ell$ is the natural label.

The n=2 shell

The fine-structure formula turns the fourfold-degenerate (eightfold with spin) $n=2$ level into a small ladder. The allowed states are $2s$ ($\ell =0$, $j=\frac{1}{2}$), $2p$ ($\ell =1$, $j=\frac{1}{2}$ or $\frac{3}{2}$). In spectroscopic notation $n{}^{2s+1}{L}_j$ these are $2S_{1/2}$, $2P_{1/2}$, and $2P_{3/2}$. The formula assigns

So $2S_{1/2}$ and $2P_{1/2}$ share the lower energy (both $j=\frac{1}{2}$), and $2P_{3/2}$ sits above them by $4E_2^2/2m{c}^2=2E_2^2/m{c}^2$. Numerically the $2P_{3/2}$–$2P_{1/2}$ separation is about $4.5\times 10^{-5}$ eV, corresponding to $10.9$ GHz, the fine-structure splitting of the Balmer-$\alpha$ line.

The level ordering and the surviving degeneracy

The fine structure lowers all levels (the bracket $n/(j+\frac{1}{2})-\frac{3}{4}$ is positive for every allowed state) and orders them by $j$: at fixed $n$, larger $j$ means a smaller downward shift, so the levels rise with $j$. The dependence on $j$ alone, not $\ell$, is the signature feature.

The degeneracy of $2S_{1/2}$ and $2P_{1/2}$ is the sharpest prediction of the formula and the most consequential. Perturbation theory gives it, and the exact Dirac equation gives it too: states of equal $n$ and $j$ are degenerate to all orders in the Dirac Coulomb problem. Experiment disagrees. In 1947 Lamb and Retherford measured a splitting of about $1$ GHz between $2S_{1/2}$ and $2P_{1/2}$, the Lamb shift, which no theory built on a single-particle wave equation can produce. Its explanation requires the quantized electromagnetic field, and it marks the boundary where atomic fine structure meets quantum electrodynamics. The fine-structure formula is exact within its domain; the domain simply does not include the vacuum fluctuations that lift the last degeneracy.