---
title: Fine Structure and the Real Hydrogen Atom
module: Approximation Methods for Bound States
moduleNumber: 10
lessonNumber: 2
order: 1002
summary: >
  The Bohr spectrum is only the leading term. Two relativistic corrections of
  order alpha-squared — the relativistic kinetic-energy correction and
  spin–orbit coupling, joined by the Darwin term for s states — split the
  hydrogen levels into fine structure that depends on the total angular momentum
  j. We derive each shift as a first-order perturbation, combine them into a
  formula depending only on n and j, and continue down the energy ladder to the
  Lamb shift and the hyperfine 21 cm line.
topics: [Approximation Methods for Bound States]
sources:
  - book: Griffiths & Schroeter
    ref: "Ch. 7; §7.3 Fine Structure of Hydrogen, §7.4 (Darwin term), §7.5 Hyperfine Splitting"
  - book: Sakurai & Napolitano
    ref: "Ch. 5; §5.3 Hydrogenlike Atoms: Fine Structure and the Zeeman Effect"
  - book: Griffiths & Schroeter
    ref: "CODATA fine-structure constant alpha — https://physics.nist.gov/cuu/Constants/"
draft: false
---

The [hydrogen spectrum](/quantum-mechanics/central-potentials/the-hydrogen-atom)
$E_n = -13.6\,\text{eV}/n^2$ comes from a nonrelativistic electron in a pure
Coulomb potential with no spin. High-resolution spectroscopy shows each of those
levels is split into closely spaced components, the **fine structure**, smaller
than the gross structure by a factor of order $\alpha^2 \approx 5\times10^{-5}$,
where $\alpha = e^2/4\pi\epsilon_0\hbar c \approx 1/137.036$ is the fine-structure
constant.[^codata] The splittings are the first corrections to a model that
treated the electron as slow, spinless, and pointlike. Two of them, the
relativistic kinetic correction and spin–orbit coupling, are both of order
$\alpha^2 E_n$; a third, the Darwin term, matters only for $s$ states. Treated
together as a perturbation on the Bohr Hamiltonian they combine into a single
formula depending on $n$ and the total angular momentum $j$.

## The hierarchy of corrections

The corrections form a graded sequence, each smaller than the last by a power of
$\alpha$ or the electron-to-proton mass ratio. Reading the Bohr energy as
$E_n \sim \alpha^2 mc^2/n^2$ fixes the scale of everything below it.

$$
% caption: The energy corrections of hydrogen fall in a strict hierarchy. The
% Bohr spectrum ($\sim\alpha^2 mc^2$, of order $13.6\ \text{eV}/n^2$) is followed
% by fine structure ($\sim\alpha^4 mc^2$, $\sim 10^{-4}\ \text{eV}$), the Lamb
% shift ($\sim\alpha^5 mc^2$), and hyperfine splitting ($\sim (m_e/m_p)\alpha^4 mc^2$).
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  box/.style={draw, minimum width=26mm, minimum height=8mm, align=center}]
\definecolor{acc}{HTML}{4A6FA5}
\node[box, draw=acc, text=acc] (b) at (0,0) {Bohr spectrum};
\node[box] (f) at (0,-1.1) {$E_{fs}$};
\node[box] (l) at (0,-2.2) {Lamb shift};
\node[box] (h) at (0,-3.3) {$E_{hf}$};
\draw[->, acc, thick] (b) -- (f);
\draw[->, acc, thick] (f) -- (l);
\draw[->, acc, thick] (l) -- (h);
\node[anchor=west, black!70] at (1.7,0) {gross structure};
\node[anchor=west, black!70] at (1.7,-1.1) {relativistic};
\node[anchor=west, black!70] at (1.7,-2.2) {vacuum (QED)};
\node[anchor=west, black!70] at (1.7,-3.3) {nuclear spin};
\end{tikzpicture}
$$

Because the perturbations act on the degenerate hydrogen levels (each $n$ level
is $n^2$-fold degenerate in $\ell$ and $m_\ell$, and doubly so in spin), the
[degenerate theory](/quantum-mechanics/approximation-methods/time-independent-perturbation-theory)
governs. The saving grace is that both fine-structure operators commute with
$L^2$, $J^2$, and $J_z$, so the coupled basis $\ket{n,\ell,j,m_j}$ is the good
basis and each shift is a diagonal expectation value.

## The relativistic kinetic correction

The kinetic energy $T = p^2/2m$ is the nonrelativistic limit of the relativistic
expression $T = \sqrt{p^2c^2 + m^2c^4} - mc^2$. Expanding in $p/mc$,

$$
T = \frac{p^2}{2m} - \frac{p^4}{8m^3c^2} + \cdots,
$$

identifies the leading correction as the perturbation
$\hat H_r' = -\dfrac{\hat p^4}{8m^3c^2}$. Its first-order shift needs
$\braket{\hat p^4}$ in the unperturbed state. Rather than compute a quartic
momentum integral, use the unperturbed equation $\hat p^2/2m = E_n - V$ with
$V = -e^2/4\pi\epsilon_0 r$ to write $\hat p^2\psi = 2m(E_n - V)\psi$, so

$$
E_r^1 = -\frac{1}{2mc^2}\braket{(E_n - V)^2}
= -\frac{1}{2mc^2}\left[E_n^2 - 2E_n\braket{V} + \braket{V^2}\right].
$$

The needed radial averages are standard results for hydrogen,
$\braket{1/r} = 1/(n^2 a_0)$ and
$\braket{1/r^2} = 1/[(\ell+\tfrac12)n^3 a_0^2]$. Substituting and using
$E_n = -e^2/8\pi\epsilon_0 a_0 n^2$ to eliminate $a_0$ gives a compact result.

> **Theorem (Relativistic correction).** The first-order shift from the leading
> relativistic kinetic term is
> $$E_r^1 = -\frac{(E_n)^2}{2mc^2}\left[\frac{4n}{\ell+\tfrac12} - 3\right].$$

It is negative (the true kinetic energy is less than the nonrelativistic
estimate at fixed momentum spread) and grows with $n$ and with decreasing
$\ell$, because low-$\ell$ orbits penetrate closer to the nucleus where the
electron moves fastest.

## Spin–orbit coupling

In the electron's instantaneous rest frame the proton circulates and produces a
magnetic field $\vec B$; the electron's intrinsic magnetic moment
$\vec\mu = -(e/m)\vec S$ has energy $-\vec\mu\cdot\vec B$ in that field. Transforming
the proton's Coulomb field to the electron frame and including the **Thomas
precession** factor of $\tfrac12$, which corrects for the non-inertial rotation
of the electron frame, gives the spin–orbit Hamiltonian

$$
\hat H_{so}' = \frac{1}{2m^2c^2}\frac{1}{r}\frac{\d V}{\d r}\,\vec S\cdot\vec L
= \left(\frac{e^2}{8\pi\epsilon_0}\right)\frac{1}{m^2c^2}\frac{1}{r^3}\,\vec S\cdot\vec L .
$$

The operator $\vec S\cdot\vec L$ is not diagonal in the uncoupled basis
$\ket{\ell, m_\ell}\ket{s, m_s}$, since it mixes $m_\ell$ and $m_s$ while
conserving their sum. It is diagonal in the coupled basis, where the total
angular momentum $\vec J = \vec L + \vec S$ has definite $j$. Squaring
$\vec J = \vec L + \vec S$,

$$
\vec S\cdot\vec L = \tfrac12\left(J^2 - L^2 - S^2\right)
= \frac{\hbar^2}{2}\left[\,j(j+1) - \ell(\ell+1) - \tfrac34\,\right],
$$

with $s = \tfrac12$. This is the algebraic reason the coupled basis is good: the
[addition of angular momenta](/quantum-mechanics/angular-momentum/addition-of-angular-momenta-and-clebsch-gordan)
diagonalizes the perturbation.

$$
% caption: Spin–orbit coupling adds the orbital and spin angular momenta into a
% conserved total J; the interaction energy depends on the relative orientation
% through the projection of S on L, diagonal only in the coupled basis of
% definite j.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% L vector
\draw[->, acc, very thick] (0,0) -- (2.6,1.4) node[anchor=south east] {$L$};
% S vector added at tip of L
\draw[->, black, very thick, dashed] (2.6,1.4) -- (3.2,3.0) node[anchor=south] {$S$};
% J = L + S
\draw[->, black, very thick] (0,0) -- (3.2,3.0) node[anchor=west] {$J$};
\node[anchor=north, black!70] at (1.3,0.0) {$J = L + S$};
\end{tikzpicture}
$$

The first-order shift uses $\braket{1/r^3} = 1/[\ell(\ell+\tfrac12)(\ell+1)n^3 a_0^3]$,
valid for $\ell \ge 1$.

> **Theorem (Spin–orbit shift).** For $\ell \ge 1$,
> $$E_{so}^1 = \frac{(E_n)^2}{mc^2}\,\frac{n\left[\,j(j+1) - \ell(\ell+1) - \tfrac34\,\right]}{\ell(\ell+\tfrac12)(\ell+1)}.$$

A given $\ell \ge 1$ couples with spin to $j = \ell \pm \tfrac12$; the
$j = \ell + \tfrac12$ level (spin aligned with orbit) is pushed up and
$j = \ell - \tfrac12$ down, splitting each $\ell$ shell into a doublet.

## The combined fine-structure formula

The relativistic and spin–orbit shifts have different-looking $\ell$ and $j$
dependence, yet their sum collapses. Adding the two theorems and using
$j = \ell \pm \tfrac12$ to trade $\ell$ for $j$ (each fixed $j$ receives the
$\ell = j - \tfrac12$ or $\ell = j + \tfrac12$ contribution), the $\ell$
dependence cancels exactly.

> **Theorem (Fine structure of hydrogen).** The combined first-order shift
> depends on $n$ and $j$ alone,
> $$E_{fs}^1 = \frac{(E_n)^2}{2mc^2}\left(3 - \frac{4n}{j + \tfrac12}\right),$$
> so the corrected level is
> $$E_{nj} = -\frac{13.6\,\text{eV}}{n^2}\left[\,1 + \frac{\alpha^2}{n^2}\left(\frac{n}{j + \tfrac12} - \frac34\right)\right].$$

The cancellation of $\ell$ is not an accident: it reflects the exact $SO(4)$
symmetry of the Coulomb problem carried into the relativistic correction, and
the same $n,j$-only formula follows from the exact Dirac equation expanded to
order $\alpha^2$. Levels of equal $n$ and $j$ but different $\ell$, such as
$2S_{\frac12}$ and $2P_{\frac12}$, remain degenerate at this order.

$$
% caption: The n = 2 shell splits by fine structure into a lower j = 1/2 pair
% and a raised j = 3/2 level; the two j = 1/2 states of different orbital angular
% momentum stay degenerate until the Lamb shift separates them.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% unperturbed n=2 level
\draw[black, line width=2pt] (0,1.4) -- (2.0,1.4);
\node[anchor=east, black!70] at (0,1.4) {$n = 2$};
% split levels
\draw[acc, very thick] (4.0,2.4) -- (6.0,2.4);
\draw[acc, very thick] (4.0,0.6) -- (6.0,0.6);
\node[anchor=west, acc] at (6.1,2.4) {$2P_{\frac{3}{2}}$};
\node[anchor=west, acc] at (6.1,0.6) {$2S_{\frac{1}{2}},\ 2P_{\frac{1}{2}}$};
% connectors
\draw[black, dashed] (2.0,1.4) -- (4.0,2.4);
\draw[black, dashed] (2.0,1.4) -- (4.0,0.6);
\draw[acc, <->] (3.0,0.8) -- (3.0,2.2);
\node[anchor=east, black!70] at (2.95,1.5) {splitting};
\end{tikzpicture}
$$

> **Worked example.** The $n = 2$ fine-structure splitting between $2P_{3/2}$ and
> $2P_{1/2}$ follows from the formula. With $E_2 = -3.4\ \text{eV}$ and the prefactor
> $(E_2)^2/2mc^2 = (3.4)^2 / (2\cdot 0.511\times10^6)\ \text{eV} = 1.13\times10^{-5}\ \text{eV}$,
> $$
> E_{3/2}^{fs} - E_{1/2}^{fs}
> = \frac{(E_2)^2}{2mc^2}\left[\left(3 - \frac{4\cdot2}{2}\right) - \left(3 - \frac{4\cdot2}{1}\right)\right]
> = \frac{(E_2)^2}{2mc^2}\cdot 4 = 4.5\times10^{-5}\ \text{eV}.
> $$
> In frequency this is $\Delta\nu = \Delta E/h \approx 10.9\ \text{GHz}$, matching the
> measured $2p$ fine-structure interval. It sits five orders of magnitude below the
> $10.2\ \text{eV}$ Lyman gap between $n = 1$ and $n = 2$, the factor of order
> $\alpha^2$ that names the fine structure.

## The Darwin term

The spin–orbit formula carries $\ell$ in its denominator and fails for $s$
states ($\ell = 0$), which nonetheless must obey the $n,j$ formula since a
$j = \tfrac12$ $s$ state is degenerate with the corresponding $p$ state. The gap
is filled by the **Darwin term**, a correction with no classical analogue that
arises from the electron's inability to be localized more finely than its
Compton wavelength (the _Zitterbewegung_ smearing of the point charge over a
region of size $\hbar/mc$). It smears the potential, and the leading effect is
proportional to its Laplacian,

$$
\hat H_D' = \frac{\hbar^2}{8m^2c^2}\nabla^2 V
= \frac{\hbar^2}{8m^2c^2}\frac{e^2}{\epsilon_0}\,\delta^3(\vec r),
$$

using $\nabla^2(1/r) = -4\pi\delta^3(\vec r)$. Because it is a contact term it
acts only where the wavefunction is nonzero at the origin, which is only for
$\ell = 0$. With $\lvert\psi_{n00}(0)\rvert^2 = 1/\pi n^3 a_0^3$,

$$
E_D^1 = \frac{\hbar^2 e^2}{8m^2c^2\epsilon_0}\,\lvert\psi_{n00}(0)\rvert^2,
$$

and this exactly equals the value the fine-structure formula assigns to an
$\ell = 0$, $j = \tfrac12$ state. The three corrections therefore combine into
one formula valid for all $\ell$: relativistic and Darwin for $s$ states,
relativistic and spin–orbit for $\ell \ge 1$.

$$
% caption: For an s state the spin–orbit term vanishes and the Darwin contact
% term supplies the shift; for higher orbital angular momentum the Darwin term
% vanishes and spin–orbit supplies it. In every case the relativistic term adds
% and the total matches the n, j formula.
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  bar/.style={draw, minimum width=13mm, align=center}]
\definecolor{acc}{HTML}{4A6FA5}
% s-state stack
\node[anchor=south, black!70] at (1.0,3.2) {$s$ state};
\fill[acc!14] (0.4,0) rectangle (1.6,1.4);
\draw[acc, thick] (0.4,0) rectangle (1.6,1.4);
\node at (1.0,0.7) {rel.};
\fill[black] (0.4,1.4) rectangle (1.6,2.6);
\draw[black, thick, dashed] (0.4,1.4) rectangle (1.6,2.6);
\node at (1.0,2.0) {Darwin};
% p-state stack
\node[anchor=south, black!70] at (4.0,3.2) {$p$ state};
\fill[acc!14] (3.4,0) rectangle (4.6,1.4);
\draw[acc, thick] (3.4,0) rectangle (4.6,1.4);
\node at (4.0,0.7) {rel.};
\fill[black] (3.4,1.4) rectangle (4.6,2.6);
\draw[black, thick, dashed] (3.4,1.4) rectangle (4.6,2.6);
\node[align=center] at (4.0,2.0) {spin\\orbit};
\end{tikzpicture}
$$

## The Lamb shift

The degeneracy of $2S_{\frac12}$ and $2P_{\frac12}$ survives the entire
fine-structure calculation and the exact Dirac equation, but not experiment.
Lamb and Retherford (1947) measured a splitting of about $1057\ \text{MHz}$
($4.4\times10^{-6}\ \text{eV}$), with $2S_{\frac12}$ lying above $2P_{\frac12}$.
The **Lamb shift** is a quantum-electrodynamic effect: the electron interacts
with the fluctuating vacuum electromagnetic field, which slightly smears its
position and shifts $s$ states (nonzero at the origin) more than $p$ states. It
sits at order $\alpha^5 mc^2$, one power of $\alpha$ below fine structure, and it
is beyond the scope of a Schrödinger-equation perturbation, requiring the
quantized radiation field. It stands here as the boundary where the
single-particle theory ends.

## Hyperfine structure and the 21 cm line

The proton is not a static point charge but a spin-$\tfrac12$ particle with its
own magnetic moment $\vec\mu_p = g_p(e/2m_p)\vec S_p$, with $g_p \approx 5.59$.
The electron's magnetic moment couples to the field of the proton's dipole. For
the $\ell = 0$ ground state the dominant piece is the **Fermi contact
interaction**, proportional to $\vec S_p\cdot\vec S_e$ evaluated at the electron
density at the origin,

$$
E_{hf}^1 = \frac{\mu_0 g_p e^2}{3\pi m_p m_e}\,\frac{1}{a_0^3}\,\frac{\braket{\vec S_p\cdot\vec S_e}}{\hbar^2}.
$$

The two spins add to a total $\vec F = \vec S_p + \vec S_e$ with
$\vec S_p\cdot\vec S_e = \tfrac12(F^2 - S_p^2 - S_e^2)$, giving
$+\tfrac14\hbar^2$ for the triplet ($F = 1$) and $-\tfrac34\hbar^2$ for the
singlet ($F = 0$). The ground state splits in two, with the singlet lower.

> **Worked example.** The hydrogen ground-state hyperfine splitting between the
> $F = 1$ and $F = 0$ levels is
> $$
> \Delta E_{hf} = \frac{4 g_p \hbar^4}{3 m_p m_e^2 c^2 a_0^4}
> \approx 5.88\times10^{-6}\ \text{eV}.
> $$
> The photon emitted in the triplet-to-singlet transition has frequency
> $$
> \nu = \frac{\Delta E_{hf}}{h} \approx 1420\ \text{MHz},
> \qquad \lambda = \frac{c}{\nu} \approx 21\ \text{cm}.
> $$
> This is the **21 cm line**, the most important spectral line in radio
> astronomy: the transition is highly forbidden (magnetic dipole, mean lifetime
> about $11\ \text{million years}$), but the vast column of neutral hydrogen in
> the galaxy makes it observable and maps the structure and rotation of the
> Milky Way.

$$
% caption: The hydrogen 1s ground state splits by the electron–proton spin–spin
% interaction into an upper F = 1 triplet and a lower F = 0 singlet; the
% transition between them radiates the 21 cm line at 1420 MHz.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% unperturbed ground state
\draw[black, line width=2pt] (0,1.4) -- (1.8,1.4);
\node[anchor=east, black!70] at (0,1.4) {$1S_{\frac{1}{2}}$};
% split
\draw[acc, very thick] (3.6,2.2) -- (5.4,2.2);
\draw[acc, very thick] (3.6,0.6) -- (5.4,0.6);
\node[anchor=west, acc] at (5.5,2.2) {$F = 1$ (triplet)};
\node[anchor=west, acc] at (5.5,0.6) {$F = 0$ (singlet)};
\draw[black, dashed] (1.8,1.4) -- (3.6,2.2);
\draw[black, dashed] (1.8,1.4) -- (3.6,0.6);
% transition
\draw[black, ->, very thick] (4.5,2.2) -- (4.5,0.6);
\node[anchor=west, black] at (4.55,1.4) {$21$ cm};
\end{tikzpicture}
$$

The corrections continue below the hyperfine scale (the proton's finite size,
higher QED orders), but each is another power of $\alpha$ or the mass ratio
smaller, and hydrogen spectroscopy is now the most precisely tested prediction
in physics. The
[next lesson](/quantum-mechanics/approximation-methods/the-zeeman-and-stark-effects)
places the atom in external fields, where the same coupled basis competes with
laboratory-scale perturbations and the good states depend on which dominates.

[^griffiths-fs]: **Griffiths & Schroeter**, _Introduction to Quantum Mechanics_ 3rd ed., §7.3 — the relativistic and spin–orbit corrections derived as first-order shifts, their combination into the $n,j$ fine-structure formula (Eqs. 7.51, 7.67, 7.69); §7.4 the Darwin term; §7.5 hyperfine splitting and the 21 cm line. Cambridge, 2018.
[^sakurai-fs]: **Sakurai & Napolitano**, _Modern Quantum Mechanics_ 3rd ed., §5.3 — fine structure of hydrogenlike atoms, the coupled $\ket{n,\ell,j,m_j}$ basis, and the relation to the Dirac-equation result. Cambridge, 2021.
[^codata]: **CODATA 2018 recommended values**, National Institute of Standards and Technology — fine-structure constant $\alpha^{-1} = 137.035999$, Rydberg energy $13.605693\ \text{eV}$, Bohr radius $a_0 = 5.29177\times10^{-11}\ \text{m}$. [physics.nist.gov/cuu/Constants](https://physics.nist.gov/cuu/Constants/).