The Dirac Equation for Hydrogen

The fine-structure formula required three separate perturbations, each of order $(Z\alpha {)}^{4}m{c}^2$, and a factor of $\frac{1}{2}$ from Thomas precession inserted by hand. The Dirac equation delivers the same spectrum in one calculation, exact to all orders in $Z\alpha$, and with every ad hoc ingredient replaced by something forced by the structure of the equation. Spin, the $g$-factor of $2$, the Thomas factor, and the spin-orbit coupling are not added; they are consequences of demanding a relativistic wave equation that is first order in time. This lesson states the equation, reads off what its four-component structure means, quotes the exact Coulomb spectrum, and expands it to recover the fine-structure formula and the degeneracy that experiment then breaks.

Why a first-order equation

The Schrödinger equation is first order in time but second order in space, an asymmetry incompatible with relativity, which treats time and space on equal footing. The obvious relativistic patch is to quantize $E^2=p^2{c}^2+m^2{c}^4$ directly, giving the Klein–Gordon equation

Being second order in time, it requires both $\varphi$ and $\partial \varphi /\partial t$ as initial data, and the natural conserved density is not positive-definite, so $\varphi$ cannot be a single-particle probability amplitude in the Schrödinger sense. Dirac's demand was an equation first order in both time and space,

with $\vec{\alpha }$ and $\beta$ constant coefficients. Requiring that iterating $H_D$ reproduce $E^2=p^2{c}^2+m^2{c}^4$ forces the coefficients to satisfy

No numbers satisfy these anticommutation relations; the smallest objects that do are $4\times 4$ matrices. The wave function $\psi$ must therefore have four components. That single algebraic fact, forced by first-order relativistic covariance, is the origin of spin and antimatter.¹

The four components

In the standard (Dirac) representation the four-component spinor splits into two two-component pieces,

with $\phi$ the large components and $\chi$ the small components. For a particle at rest, or in the non-relativistic limit, $\chi$ is smaller than $\phi$ by a factor of order $v/c\sim Z\alpha$: the coupled equations give

where $\vec{\sigma }$ are the Pauli matrices. The two components of $\phi$ are the spin-up and spin-down amplitudes of the electron; spin $s=\frac{1}{2}$ is built into the equation, not assumed. Substituting $\chi$ back into the equation for $\phi$ and keeping terms through order $(v/c{)}^2$ reproduces, in order:

• the non-relativistic kinetic energy $p^2/2m$;
• the mass-velocity term $-p^4/8m^3{c}^2$;
• the spin-orbit term with its Thomas $\frac{1}{2}$ automatic;
• the Darwin term $({\hslash }^2/8m^2{c}^2){\nabla }^{2}V$.

This reduction, the Foldy–Wouthuysen transformation, is where the three fine-structure corrections come from as a package, with no factor inserted by hand.

The two other solutions, with the roles of large and small reversed, are negative-energy states. Dirac reinterpreted the unfilled negative-energy sea as antiparticles, predicting the positron before its 1932 discovery. For the atomic bound-state problem the negative-energy sector matters only as the source of the small components and the zitterbewegung behind the Darwin term.

The exact Coulomb spectrum

For a Coulomb potential $V=-Z{e}^2/4\pi {ϵ}_{0}r$ the Dirac equation is solved exactly, in closed form, by separating angular and radial parts much as in the non-relativistic problem but with the spinor structure carried through. The bound states have energies²³

The energy depends on $n$ and $j$ only, never on $\ell$ separately, and the dependence is exact in $Z\alpha$, not a truncated expansion. The quantity under the square root, $(j+\frac{1}{2}{)}^2-(Z\alpha {)}^2$, must stay positive: for $j=\frac{1}{2}$ this fails at $Z\alpha =\frac{1}{2}$, i.e. $Z\approx 68$, and beyond it the point-nucleus Dirac equation has no real ground-state energy, the first sign that very heavy nuclei need finite-size and QED treatment.

The κ quantum number

The angular part of the Dirac–Coulomb solution is not labelled by $\ell$ but by a single integer $\kappa$ that packages the orbital and total angular momenta together. It is defined by

so $|\kappa |=j+\frac{1}{2}$ fixes $j$ while the sign selects which of $\ell =j\pm \frac{1}{2}$ appears in the large component. The exact energy depends on $\kappa$ only through $|\kappa |$, which is why it depends on $j$ and not on the sign that distinguishes the two $\ell$ values, the exact statement of the $\ell$-degeneracy. The large and small components carry orbital angular momenta differing by one unit, $\ell$ and ${\ell }^{\prime }=\ell \pm 1$, so the spinor mixes two parities' worth of orbital character; the pair is what a non-relativistic observer resolves into a single $\ell$.

The radial problem reduces to two coupled first-order equations. Writing the large and small radial components as $G(r)=rg(r)$ and $F(r)=rf(r)$, with $V(r)$ the potential energy and $E$ the total energy (rest mass included),

whose normalizable solutions exist only at the quantized energies $E_{nj}$ above. The structure parallels the non-relativistic radial equation, now a coupled pair rather than a single second-order equation, with $\kappa$ playing the role $\ell$ played there.

Recovering the fine-structure formula

Expanding the exact energy in powers of $Z\alpha$ shows how it contains the non-relativistic and fine-structure results as its first two terms. Write $k=j+\frac{1}{2}$ and expand the square root and the outer power:

Subtracting the rest energy $m{c}^2$ isolates the binding energy:

The first term is the Bohr energy $E_n=-m{c}^2(Z\alpha {)}^2/2n^2=-Z^2\mathrm{Ry}/n^2$. The second reproduces the fine-structure formula built term by term in the previous lessons. The Dirac theory thus confirms the perturbative assembly and supplies all higher orders in $Z\alpha$ for free.

The Dirac level structure and its degeneracy

Ordering the levels by $n$ and $j$ produces the same map as the fine-structure formula, now exact. Within each $n$ the levels group by $j$, and each $(n,j)$ level holds all the $\ell$ values compatible with $j=\ell \pm \frac{1}{2}$. For $n=2$ the $j=\frac{1}{2}$ level contains both $2S_{1/2}$ ($\ell =0$) and $2P_{1/2}$ ($\ell =1$), exactly degenerate; $2P_{3/2}$ ($j=\frac{3}{2}$) sits above.

The degeneracy structure is summarized compactly.

The boundary with QED

The $2S_{1/2}$–$2P_{1/2}$ degeneracy is the sharpest testable prediction of the Dirac theory: two states of different orbital character, held at exactly the same energy by the $\ell$-independence of the Dirac spectrum. The prediction is wrong. The 1947 Lamb–Retherford experiment found $2S_{1/2}$ lying about $1058$ MHz above $2P_{1/2}$, a splitting no single-particle relativistic wave equation can produce. Its origin is the interaction of the electron with the quantized electromagnetic field: self-energy, vacuum polarization, and the vacuum fluctuations that smear the electron's position, all treated in quantum electrodynamics. The same field theory produces the anomalous magnetic moment $g=2(1+\alpha /2\pi +\cdots )$, correcting the Dirac value $g=2$ that the four-component structure gave exactly.

The Dirac equation is therefore the last word in single-particle atomic theory and the first place it visibly fails. It captures the fine structure exactly, predicts antimatter, and forces spin from covariance alone; the residual $1$ GHz gap it cannot close is the entry point to a field-theoretic description, where hydrogen becomes the most precisely calculated and measured system in physics. The next module takes up the QED corrections and hyperfine structure that live in that gap.