---
title: The Dirac Equation for Hydrogen
module: Fine Structure and the Dirac Atom
moduleNumber: 3
lessonNumber: 5
order: 305
summary: >
  The fine-structure formula was assembled from three perturbations; the Dirac
  equation produces it in one stroke and exactly. A first-order relativistic wave
  equation forces a four-component spinor, from which spin s=½, the g-factor of 2,
  the spin-orbit term, and antiparticles all emerge automatically. Its exact
  Coulomb spectrum depends only on n and j, and expanding in Zα reproduces the
  perturbative result, including the 2S₁/₂–2P₁/₂ degeneracy that sets up the Lamb
  shift.
topics: [Fine Structure and the Dirac Atom]
sources:
  - book: Bransden & Joachain
    ref: "Ch. 5 §5.2–5.3; Ch. 15 — The Dirac Equation and One-Electron Atoms"
  - book: Griffiths & Schroeter
    ref: "Afterword — The Dirac Equation; relativistic quantum mechanics"
  - book: Foot
    ref: "Ch. 2 — The Hydrogen Atom; Appendix on the Dirac result"
draft: false
---

The [fine-structure formula](/atomic-physics/fine-structure-and-the-dirac-atom/darwin-term-fine-structure-formula)
required three separate perturbations, each of order $(Z\alpha)^4 mc^2$, and a
factor of $\tfrac12$ from [Thomas precession](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-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 $E^2 = p^2c^2 + m^2c^4$
directly, giving the **Klein–Gordon equation**

$$
\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 + \frac{m^2c^2}{\hbar^2}\right)\phi = 0.
$$

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

$$
i\hbar\frac{\partial\psi}{\partial t}
= \left(c\,\vec\alpha\cdot\vec p + \beta mc^2\right)\psi
\equiv H_D\psi,
$$

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

$$
\alpha_i\alpha_j + \alpha_j\alpha_i = 2\delta_{ij},
\qquad
\alpha_i\beta + \beta\alpha_i = 0,
\qquad
\alpha_i^2 = \beta^2 = 1.
$$

No numbers satisfy these anticommutation relations; the smallest objects that do
are $4\times4$ 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.[^bj-15]

> **Definition (Dirac equation).** The first-order relativistic wave equation
> $i\hbar\,\partial_t\psi = (c\,\vec\alpha\cdot\vec p + \beta mc^2)\psi$ for a
> four-component spinor $\psi$, where the Hermitian matrices $\vec\alpha$, $\beta$
> obey $\{\alpha_i,\alpha_j\}=2\delta_{ij}$, $\{\alpha_i,\beta\}=0$,
> $\alpha_i^2=\beta^2=1$. In covariant form $(i\hbar\gamma^\mu\partial_\mu - mc)\psi
> = 0$ with $\gamma^0=\beta$, $\gamma^i=\beta\alpha_i$.

## The four components

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

$$
\psi = \begin{pmatrix}\varphi\\[2pt]\chi\end{pmatrix},
$$

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

$$
\chi \approx \frac{\vec\sigma\cdot\vec p}{2mc}\,\varphi,
$$

where $\vec\sigma$ are the Pauli matrices. The two components of $\varphi$ are the
spin-up and spin-down amplitudes of the electron; spin $s=\tfrac12$ is built into
the equation, not assumed. Substituting $\chi$ back into the equation for
$\varphi$ and keeping terms through order $(v/c)^2$ reproduces, in order:

- the non-relativistic kinetic energy $p^2/2m$;
- the [mass-velocity term](/atomic-physics/fine-structure-and-the-dirac-atom/relativistic-kinetic-correction) $-p^4/8m^3c^2$;
- the [spin-orbit term](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession) with its Thomas $\tfrac12$ automatic;
- the [Darwin term](/atomic-physics/fine-structure-and-the-dirac-atom/darwin-term-fine-structure-formula) $(\hbar^2/8m^2c^2)\nabla^2 V$.

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

$$
% caption: The Dirac spinor has four components: two large (electron spin up and
% down) and two small, suppressed by v/c ~ Zα. In the non-relativistic limit the
% small pair is slaved to the large pair by σ·p/2mc.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth,
  cell/.style={draw, minimum width=13mm, minimum height=8mm, align=center}]
\definecolor{acc}{HTML}{4A6FA5}
\node[cell, draw=acc, text=acc, thick] (u1) at (0,1.5) {large up};
\node[cell, draw=acc, text=acc, thick] (u2) at (0,0.5) {large down};
\node[cell, draw=black, text=black] (l1) at (0,-0.7) {small up};
\node[cell, draw=black, text=black] (l2) at (0,-1.7) {small down};
% bracket
\draw[black] (-1.05,2.0) -- (-1.25,2.0) -- (-1.25,-2.2) -- (-1.05,-2.2);
\node[black, anchor=east] at (-1.35,-0.1) {spinor};
% relation
\draw[black, ->] (1.1,1.0) .. controls (2.4,0.6) and (2.4,-0.6) .. (1.1,-1.2);
\node[black, anchor=west, align=left] at (2.4,-0.1) {small pair\\suppressed};
\end{tikzpicture}
$$

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 $V = -Ze^2/4\pi\epsilon_0 r$ 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 energies[^bj-53][^foot-2app]

$$
E_{nj} = mc^2\left[1 + \left(\frac{Z\alpha}
{\,n - (j+\tfrac12) + \sqrt{(j+\tfrac12)^2 - (Z\alpha)^2}\,}\right)^2\right]^{-1/2}.
$$

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, $(j+\tfrac12)^2 - (Z\alpha)^2$, must stay positive: for $j=\tfrac12$
this fails at $Z\alpha = \tfrac12$, 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.

> **Theorem (Dirac spectrum of hydrogen).** The exact bound-state energies of the
> Dirac–Coulomb problem are
> $$
> E_{nj} = mc^2\left[1 + \frac{(Z\alpha)^2}{\big(n - (j+\tfrac12) + \sqrt{(j+\tfrac12)^2 - (Z\alpha)^2}\big)^2}\right]^{-1/2},
> $$
> functions of $n$ and $j$ alone, with $n \ge 1$ and $j = \tfrac12,\tfrac32,\ldots,
> n-\tfrac12$. States of equal $n$ and $j$ but different $\ell$ are exactly
> degenerate.

## 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

$$
\kappa = \mp\left(j + \tfrac12\right)
\quad\text{for}\quad j = \ell \pm \tfrac12,
\qquad
\kappa = \begin{cases}-(\ell+1), & j = \ell+\tfrac12,\\[2pt] +\ell, & j=\ell-\tfrac12,\end{cases}
$$

so $|\kappa| = j+\tfrac12$ fixes $j$ while the sign selects which of $\ell = j\pm\tfrac12$
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 $\ell' =
\ell\pm1$, 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 $G(r) = r\,g(r)$ and $F(r) = r\,f(r)$, with $V(r)$
the potential energy and $E$ the total energy (rest mass included),

$$
\frac{\d G}{\d r} + \frac{\kappa}{r}G
= \frac{1}{\hbar c}\big(E - V + mc^2\big)F,
\qquad
\frac{\d F}{\d r} - \frac{\kappa}{r}F
= -\frac{1}{\hbar c}\big(E - V - mc^2\big)G,
$$

whose normalizable solutions exist only at the quantized energies $E_{nj}$ above.
The structure parallels the non-relativistic
[radial equation](/atomic-physics/quantum-hydrogen-atom/radial-equation-in-full),
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
$k = j+\tfrac12$ and expand the square root and the outer power:

$$
E_{nj} = mc^2\left[1 - \frac{(Z\alpha)^2}{2n^2}
- \frac{(Z\alpha)^4}{2n^4}\left(\frac{n}{j+\tfrac12} - \frac34\right)
+ \mathcal{O}\big((Z\alpha)^6\big)\right].
$$

Subtracting the rest energy $mc^2$ isolates the binding energy:

$$
E_{nj} - mc^2 = -\frac{mc^2(Z\alpha)^2}{2n^2}
\left[1 + \frac{(Z\alpha)^2}{n^2}\left(\frac{n}{j+\tfrac12} - \frac34\right)
+ \cdots\right].
$$

The first term is the Bohr energy $E_n = -mc^2(Z\alpha)^2/2n^2 = -Z^2\,\mathrm{Ry}/n^2$.
The second reproduces the
[fine-structure formula](/atomic-physics/fine-structure-and-the-dirac-atom/darwin-term-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.

$$
% caption: The exact Dirac binding energy (solid) and its α²-truncated expansion
% (dashed) as functions of Zα; they agree at small Zα and diverge as Zα approaches
% ½, where the j=½ square root reaches zero.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.2,0) node[right, black!70] {$Z$ alpha};
\draw[->, black] (0,0.3) -- (0,-3.2) node[below, black!70] {binding energy};
\node[black, anchor=north] at (5.6,0) {$\tfrac12$};
% exact: dives steeply near Za=1/2
\draw[acc, very thick] (0,-0.05) .. controls (1.6,-0.25) and (3.2,-0.7) .. (4.4,-1.5)
  .. controls (5.0,-2.1) and (5.4,-2.8) .. (5.6,-3.1);
\node[acc, anchor=west] at (3.3,-0.9) {exact Dirac};
% expansion: gentler, only two terms
\draw[black, very thick, dashed] (0,-0.05) .. controls (1.6,-0.22) and (3.2,-0.6) .. (4.4,-1.15)
  .. controls (5.0,-1.5) and (5.4,-1.85) .. (5.6,-2.05);
\node[black, anchor=north west] at (4.4,-1.3) {expansion};
\end{tikzpicture}
$$

## 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 $j = \ell\pm\tfrac12$. For $n=2$ the
$j=\tfrac12$ level contains both $2S_{1/2}$ ($\ell=0$) and $2P_{1/2}$ ($\ell=1$),
exactly degenerate; $2P_{3/2}$ ($j=\tfrac32$) sits above.

$$
% caption: The Dirac fine structure of n=1 and n=2. Levels are labelled by (n, j);
% those of equal n and j but different ℓ coincide, so 2S₁/₂ and 2P₁/₂ share a
% level.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% n=1
\draw[black, very thick] (0,-2.4) -- (2.0,-2.4);
\node[black, anchor=east] at (-0.1,-2.4) {$1S_{\frac12}$};
\node[black, anchor=west, font=\scriptsize] at (2.1,-2.4) {$n{=}1$, $j{=}\tfrac12$};
% n=2, j=1/2
\draw[acc, very thick] (0,1.0) -- (2.0,1.0);
\node[acc, anchor=east] at (-0.1,1.0) {$2S_{\frac12}{=}2P_{\frac12}$};
\node[black, anchor=west, font=\scriptsize] at (2.1,1.0) {$n{=}2$, $j{=}\tfrac12$};
% n=2, j=3/2
\draw[black, very thick] (0,1.7) -- (2.0,1.7);
\node[black, anchor=east] at (-0.1,1.7) {$2P_{\frac32}$};
\node[black, anchor=west, font=\scriptsize] at (2.1,1.7) {$n{=}2$, $j{=}\tfrac32$};
% brace for n=2 group
\draw[black] (3.9,0.9) -- (4.1,0.9) -- (4.1,1.8) -- (3.9,1.8);
\node[black, anchor=west] at (4.2,1.35) {$n=2$};
\end{tikzpicture}
$$

The degeneracy structure is summarized compactly.

| $n$ | $j$ | states (spectroscopic) | $\ell$ values | degeneracy $2j+1$ |
| --- | --- | --- | --- | --- |
| $1$ | $\tfrac12$ | $1S_{1/2}$ | $0$ | $2$ |
| $2$ | $\tfrac12$ | $2S_{1/2}, 2P_{1/2}$ | $0,1$ | $2$ each |
| $2$ | $\tfrac32$ | $2P_{3/2}$ | $1$ | $4$ |
| $3$ | $\tfrac12$ | $3S_{1/2}, 3P_{1/2}$ | $0,1$ | $2$ each |
| $3$ | $\tfrac32$ | $3P_{3/2}, 3D_{3/2}$ | $1,2$ | $4$ each |
| $3$ | $\tfrac52$ | $3D_{5/2}$ | $2$ | $6$ |

## 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](/atomic-physics/qed-corrections-and-hyperfine-structure/lamb-shift-qed).
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.

$$
% caption: The n=2 level of hydrogen across three theories. Bohr and Schrödinger
% give one level; Dirac fine structure splits it into two, with 2S₁/₂ and 2P₁/₂
% degenerate; QED (dashed) lifts 2S₁/₂ above 2P₁/₂ by the Lamb shift.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% Bohr/Schrodinger single level
\draw[black, very thick] (0,1.4) -- (1.6,1.4);
\node[black, anchor=south, font=\scriptsize] at (0.8,1.45) {one level};
\node[anchor=north] at (0.8,-0.4) {Bohr};
% Bohr -> Dirac connectors (open space)
\draw[black, dotted] (1.6,1.4) -- (2.8,1.0);
\draw[black, dotted] (1.6,1.4) -- (2.8,2.1);
% Dirac: two levels, labels above the lines
\draw[black, very thick] (2.8,2.1) -- (4.4,2.1);
\node[black, anchor=south, font=\scriptsize] at (3.6,2.15) {$2P_{\frac32}$};
\draw[black, very thick] (2.8,1.0) -- (4.4,1.0);
\node[black, anchor=south, font=\scriptsize] at (3.6,1.05) {$2S_{\frac12}{=}2P_{\frac12}$};
\node[anchor=north] at (3.6,-0.4) {Dirac};
% QED: split the j=1/2 pair, labels to the right (open)
\draw[black, very thick, dashed] (6.8,2.1) -- (8.2,2.1);
\node[black, anchor=west, font=\scriptsize] at (8.3,2.1) {$2P_{\frac32}$};
\draw[black, very thick, dashed] (6.8,1.3) -- (8.2,1.3);
\node[black, anchor=west, font=\scriptsize] at (8.3,1.3) {$2S_{\frac12}$};
\draw[black, very thick, dashed] (6.8,0.7) -- (8.2,0.7);
\node[black, anchor=west, font=\scriptsize] at (8.3,0.7) {$2P_{\frac12}$};
\node[anchor=north] at (7.5,-0.4) {QED};
% Lamb shift bracket, label to the left in open space
\draw[acc, <->] (6.65,1.3) -- (6.65,0.7);
\node[acc, anchor=east, font=\scriptsize] at (6.55,1.0) {Lamb};
\end{tikzpicture}
$$

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](/atomic-physics/qed-corrections-and-hyperfine-structure/lamb-shift-qed)
that live in that gap.

[^bj-15]: **Bransden & Joachain**, _Physics of Atoms and Molecules_, 2nd ed., Ch. 15 — the Dirac equation, the anticommuting $4\times4$ matrices forced by first-order covariance, four-component spinors, large and small components, and the negative-energy solutions reinterpreted as antiparticles. <https://www.pearson.com/en-gb/subject-catalog/p/physics-of-atoms-and-molecules/P200000005386>
[^bj-53]: **Bransden & Joachain**, _Physics of Atoms and Molecules_, 2nd ed., §5.2–5.3 — the exact Dirac–Coulomb energy levels, their dependence on $n$ and $j$ alone, and the $Z\alpha$ expansion reproducing the fine-structure formula and the $2S_{1/2}$–$2P_{1/2}$ degeneracy.
[^foot-2app]: **Foot**, _Atomic Physics_, Ch. 2 Appendix — the Dirac energy formula quoted for hydrogen, its reduction to the fine-structure formula at order $\alpha^2$, and the connection to the Lamb shift beyond it. <https://global.oup.com/academic/product/atomic-physics-9780198506959>