---
title: The Lamb Shift and QED Radiative Corrections
module: QED Corrections and Hyperfine Structure
moduleNumber: 4
lessonNumber: 1
order: 401
summary: >
  The Dirac equation makes the 2S₁/₂ and 2P₁/₂ levels of hydrogen exactly
  degenerate. Lamb and Retherford measured a splitting of about 1058 MHz that
  the Dirac theory cannot produce. The gap comes from the electron's coupling to
  the quantized electromagnetic field: self-energy, vacuum polarization, and the
  anomalous magnetic moment. Welton's vacuum-fluctuation estimate reproduces the
  size and shows why the effect lands almost entirely on s-states, and the same
  radiative corrections make hydrogen the most stringent test of QED.
topics: [QED Corrections and Hyperfine Structure]
sources:
  - book: Foot
    ref: "Ch. 2 — The Hydrogen Atom; §2.3 The Lamb Shift; Ch. 5"
  - book: Bransden & Joachain
    ref: "Ch. 5 — One-Electron Atoms; §5.4–5.5 Radiative Corrections and the Lamb Shift"
  - book: Griffiths & Schroeter
    ref: "Ch. 7 — Time-Independent Perturbation Theory; afterword on QED"
draft: false
---

The [fine-structure formula](/atomic-physics/fine-structure-and-the-dirac-atom/darwin-term-fine-structure-formula)
and the exact
[Dirac spectrum](/atomic-physics/fine-structure-and-the-dirac-atom/dirac-equation-hydrogen)
agree on a sharp prediction: the energy of a hydrogen level depends only on $n$
and the total angular momentum $j$, never on the orbital label $\ell$
separately. The two $n=2$, $j=\tfrac12$ states — $2S_{1/2}$ (with $\ell=0$) and
$2P_{1/2}$ (with $\ell=1$) — are therefore predicted to sit at exactly the same
energy. This is not an approximation of the Dirac theory; it is exact to all
orders in the Coulomb interaction.

In 1947 Willis Lamb and Robert Retherford drove microwave transitions between
these two levels and found them split by about $1058\ \mathrm{MHz}$, with
$2S_{1/2}$ lying **above** $2P_{1/2}$.[^lr] No amount of care with the Dirac
Coulomb problem produces this gap, because the gap is not a property of an
electron in a fixed external potential at all. It is a property of the electron
coupled to the quantized electromagnetic field — the vacuum that the Dirac
equation, treating the field as a classical background, leaves out. Explaining
the number launched quantum electrodynamics as a predictive theory.

## The degeneracy the Dirac theory protects

The Dirac energy levels for a point Coulomb potential are

$$
E_{n j} = m c^2\left[1 + \left(\frac{Z\alpha}{\,n - \delta_j\,}\right)^2\right]^{-1/2},
\qquad
\delta_j = \left(j+\tfrac12\right) - \sqrt{\left(j+\tfrac12\right)^2 - (Z\alpha)^2},
$$

a function of $n$ and $j$ only. Expanding to order $(Z\alpha)^4$ recovers the
gross structure, the fine structure, and the statement that $2S_{1/2}$ and
$2P_{1/2}$ coincide. The coincidence is a genuine degeneracy of the Dirac
Coulomb Hamiltonian, tied to a hidden symmetry of the $1/r$ potential, and it is
stable against every correction that stays inside the one-particle theory:
relativistic kinematics, spin-orbit coupling, and the Darwin term are already
included in $E_{nj}$.

Breaking it requires new physics. The electron in a hydrogen atom is not alone
with the proton; it is immersed in the electromagnetic field, whose modes have
a zero-point energy $\tfrac12\hbar\omega$ per mode even in the vacuum. The
electron continuously emits and reabsorbs virtual photons, and the proton's
Coulomb field continuously creates and annihilates virtual electron-positron
pairs. These processes shift the bound levels, and — decisively — they shift
$2S_{1/2}$ and $2P_{1/2}$ by different amounts, because the two states have
different probability densities at the nucleus.

$$
% caption: The n = 2 levels of hydrogen. The Dirac theory (left) makes 2S₁/₂ and
% 2P₁/₂ exactly degenerate; the measured spectrum (right) lifts 2S₁/₂ above 2P₁/₂
% by the Lamb shift, about 1058 MHz.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% Dirac column
\node[anchor=south] at (2.1,3.3) {Dirac theory};
\draw[black, very thick] (0.3,2.7) -- (2.1,2.7) node[right, black] {$2P_{\frac{3}{2}}$};
\draw[acc, very thick] (0.3,0.6) -- (1.3,0.6);
\draw[acc, very thick] (1.5,0.6) -- (2.5,0.6);
\node[black] at (0.8,0.25) {$2S_{\frac{1}{2}}$};
\node[black] at (2.0,0.25) {$2P_{\frac{1}{2}}$};
\node[black, anchor=east] at (0.25,0.6) {degenerate};
% observed column
\node[anchor=south] at (7.1,3.3) {observed};
\draw[black, very thick] (5.3,2.7) -- (7.1,2.7) node[right, black] {$2P_{\frac{3}{2}}$};
\draw[acc, very thick] (5.3,0.95) -- (6.3,0.95) node[left=8mm, black] {$2S_{\frac{1}{2}}$};
\draw[acc, very thick] (6.5,0.55) -- (7.5,0.55) node[right, black] {$2P_{\frac{1}{2}}$};
% fine-structure gap bracket
\draw[<->, black] (8.0,0.55) -- (8.0,2.7);
\node[black, anchor=west, align=left] at (8.1,1.62) {f\/ine\\structure\\11 GHz};
% Lamb-shift gap bracket
\draw[<->, black] (4.9,0.55) -- (4.9,0.95);
\node[black, anchor=east, align=right] at (4.75,0.75) {Lamb shift\\1058 MHz};
\end{tikzpicture}
$$

## Vacuum fluctuations and Welton's estimate

The cleanest physical picture is due to Theodore Welton.[^welton] The quantized
electromagnetic field has fluctuating electric and magnetic fields even in its
ground state. A bound electron responds to the fluctuating field $\vec E_{\!f}$
by jittering about its mean position. Write the extra displacement $\delta\vec
r$ driven by the fluctuations and treat the electron classically for its
response: the equation of motion for a Fourier component of angular frequency
$\omega$ is

$$
m\,\frac{\d^2 (\delta\vec r)_\omega}{\d t^2} = -e\,\vec E_{\!f,\omega}
\quad\Longrightarrow\quad
(\delta\vec r)_\omega = \frac{e}{m\omega^2}\,\vec E_{\!f,\omega}.
$$

The mean-square displacement sums the contributions of all field modes. The
zero-point field has $\langle E_{\!f,\omega}^2\rangle$ spread over a spectrum,
and carrying out the mode sum gives a logarithmically divergent integral cut off
at both ends,

$$
\langle (\delta r)^2\rangle
= \frac{2\alpha}{\pi}\left(\frac{\hbar}{m c}\right)^{2}
\int \frac{\d\omega}{\omega}
\approx \frac{2\alpha}{\pi}\left(\frac{\hbar}{m c}\right)^{2}
\ln\!\frac{\omega_{\max}}{\omega_{\min}}.
$$

The upper cutoff is the electron's Compton frequency $\omega_{\max}\sim mc^2/\hbar$
(above it the non-relativistic treatment fails); the lower cutoff is the atomic
orbital frequency $\omega_{\min}\sim (Z\alpha)^2 mc^2/\hbar$ (below it the
electron is not free to jitter — the binding responds). The ratio of cutoffs is
$1/(Z\alpha)^2$, so the logarithm is $\ln[1/(Z\alpha)^2]$, a number of order
$10$ for hydrogen.

A jittering electron samples the Coulomb potential over a small smeared region
rather than at a point. Averaging $V(\vec r + \delta\vec r)$ over the isotropic
fluctuation and Taylor-expanding,

$$
\langle V(\vec r + \delta\vec r)\rangle - V(\vec r)
= \tfrac16\,\langle (\delta r)^2\rangle\,\nabla^2 V(\vec r) + \cdots,
$$

because the linear term averages to zero and the second-order term contracts to
one-sixth of the mean-square displacement times the Laplacian. The Coulomb
Laplacian is a contact term,

$$
\nabla^2 V = \nabla^2\!\left(-\frac{Z e^2}{4\pi\varepsilon_0 r}\right)
= \frac{Z e^2}{\varepsilon_0}\,\delta^3(\vec r),
$$

so the level shift is proportional to the electron density at the nucleus:

$$
\Delta E_{n\ell}
= \tfrac16\,\langle (\delta r)^2\rangle\,\frac{Z e^2}{\varepsilon_0}\,
|\psi_{n\ell}(0)|^2 .
$$

Only $s$-states have $|\psi_{n\ell}(0)|^2\neq 0$; every $\ell\ge 1$ state
vanishes at the origin and receives no contact shift. This is the mechanism that
splits the degeneracy: $2S_{1/2}$ is raised, $2P_{1/2}$ is (to this order)
untouched.

> **Theorem (Welton estimate of the Lamb shift).** Averaging the Coulomb
> potential over the zero-point fluctuation displacement gives, for a hydrogenic
> $ns$ level with $|\psi_{ns}(0)|^2 = Z^3/(\pi n^3 a_0^3)$,
> $$
> \Delta E_{ns} = \frac{4}{3}\,\frac{\alpha^5 m c^2}{\pi\, n^3}\,Z^4\,
> \ln\!\frac{1}{(Z\alpha)^2}.
> $$
> For the $2S_{1/2}$ state of hydrogen this evaluates to about $1000\ \mathrm{MHz}$,
> within ten percent of the measured $1058\ \mathrm{MHz}$.

$$
% caption: A point electron sees the full 1/r Coulomb singularity; vacuum
% fluctuations smear its position over a Compton-scale region, softening the
% potential it samples at the origin and shifting only s-states.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% axes
\draw[->] (-3.2,0) -- (3.2,0) node[right, black] {$r$};
\draw[->] (0,-2.7) -- (0,0.6) node[above, black] {$V(r)$};
% bare Coulomb -1/r (both sides), steep
\draw[acc, very thick, domain=0.32:3.0, samples=60] plot (\x, {-0.85/\x});
\draw[acc, very thick, domain=-3.0:-0.32, samples=60] plot (\x, {-0.85/(-\x)});
\node[acc, anchor=west] at (1.4,-0.75) {bare Coulomb};
% smeared potential: bounded near origin
\draw[black, very thick, dashed, domain=-3.0:3.0, samples=120]
  plot (\x, {-0.85/sqrt(\x*\x + 0.36)});
\node[black] at (-2.2,-1.6) {smeared};
% jitter cloud at origin
\foreach \p in {(-0.18,-1.28),(0.16,-1.42),(-0.05,-1.55),(0.22,-1.18),(-0.28,-1.5),(0.08,-1.3)}
  \fill[black] \p circle (0.9pt);
\end{tikzpicture}
$$

## The radiative corrections separately

The Welton picture captures the dominant piece — the electron self-energy — but
the full shift is a sum of distinct QED processes, each with a definite sign and
magnitude. To order $\alpha(Z\alpha)^4 mc^2$ the $n=2$ Lamb shift breaks into
three contributions.

- **Electron self-energy.** The electron emits and reabsorbs a virtual photon.
  This dresses its interaction with the Coulomb field and is the process
  Welton's estimate models. It is the largest term and it raises $s$-states.
  Numerically it contributes roughly $+1010\ \mathrm{MHz}$ to the
  $2S_{1/2}$–$2P_{1/2}$ splitting.
- **Vacuum polarization.** A virtual electron-positron pair briefly screens the
  proton's charge, so an electron that penetrates to small $r$ sees slightly
  more charge than $Z e$. This **Uehling** effect deepens the potential for
  $s$-states and shifts $2S_{1/2}$ **down**, contributing about
  $-27\ \mathrm{MHz}$. It is the one term with the opposite sign, and in muonic
  atoms — where the heavier lepton orbits far closer to the nucleus — it
  dominates.
- **Anomalous magnetic moment.** The electron's $g$-factor is not exactly $2$;
  the same virtual-photon cloud gives it an extra magnetic moment
  $a_e = (g-2)/2$. This modifies the spin-orbit coupling and adds about
  $+68\ \mathrm{MHz}$, felt through the $p$-states.

The three sum to close to the measured $1058\ \mathrm{MHz}$; the small remainder
is reduced-mass and higher-order corrections.

$$
% caption: The three radiative contributions to the 2S₁/₂–2P₁/₂ splitting. Self-
% energy (about +1010 MHz) and the anomalous moment (about +68 MHz) raise the gap;
% vacuum polarization (about −27 MHz) lowers it. The signed sum is near 1058 MHz.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% baseline
\draw[black] (-0.5,0) -- (7.5,0);
\node[anchor=east] at (-0.55,0) {0};
% self-energy: tall up bar
\fill[acc!18, draw=acc, very thick] (0.2,0) rectangle (1.4,3.03);
\node[anchor=north, align=center, text=acc] at (0.8,2.92) {self-\\energy};
% vacuum polarization: short down bar
\fill[black, draw=black, very thick] (2.2,0) rectangle (3.4,-0.55);
\node[anchor=north, align=center] at (2.8,-0.6) {vac.\\pol.};
% anomalous moment: medium up bar
\fill[black, draw=black, very thick, dashed] (4.2,0) rectangle (5.4,1.36);
\node[anchor=south, align=center] at (4.8,1.4) {$g$ factor};
% total marker
\draw[black, very thick, dashed] (-0.5,3.17) -- (7.5,3.17);
\node[anchor=south] at (6.0,3.22) {total 1058 MHz};
\end{tikzpicture}
$$

## The self-energy shift from Bethe's calculation

Hans Bethe produced the first quantitative Lamb-shift number within weeks of the
measurement, using a non-relativistic treatment of the self-energy with a
relativistic cutoff.[^bethe] The self-energy of a bound electron is the
second-order shift from emitting and reabsorbing a photon,

$$
\Delta E_n^{\text{self}}
= -\frac{2\alpha}{3\pi m^2 c^2}\sum_{m}
\frac{|\langle m|\,\vec p\,|n\rangle|^2\,(E_m - E_n)}
{\ }\,
\int_0^{K}\frac{(E_m - E_n)\,\d k}{E_m - E_n + \hbar c k},
$$

where the sum runs over intermediate atomic states $|m\rangle$ and the momentum
integral is cut off at $K = mc/\hbar$. The free-electron self-energy — the same
process for an unbound electron — is already absorbed into the electron's
physical mass, a step called **mass renormalization**. Subtracting it removes
the leading (linear) divergence and leaves a logarithm,

$$
\Delta E_{ns}^{\text{self}}
= \frac{4\alpha^5 m c^2}{3\pi\,n^3}\,Z^4
\left[\ln\frac{m c^2}{2\langle E\rangle_n} + \frac{11}{24} - \ln 2 + \cdots\right],
$$

with $\langle E\rangle_n$ the **Bethe logarithm**, an average excitation energy
of the atom. For $2S$ the Bethe logarithm is $\ln(\langle E\rangle/\mathrm{Ry})
\approx 2.81$, and the bracket evaluates the self-energy term to about
$1010\ \mathrm{MHz}$, matching the breakdown above. The renormalization step is
the conceptual heart of the calculation: the divergent free-electron self-energy
is unobservable and is folded into the measured mass, and only the
**difference** between the bound and free self-energies — finite, and
$\ell$-dependent through $|\psi(0)|^2$ — is a physical level shift.

$$
% caption: The two leading radiative processes as schematic diagrams. Left: the
% electron emits and reabsorbs a virtual photon (self-energy). Right: the exchanged
% photon briefly becomes an electron-positron pair (vacuum polarization).
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
% self-energy
\draw[acc, very thick, ->] (-3.4,0) -- (-2.2,0);
\draw[acc, very thick] (-2.2,0) -- (-0.6,0);
\draw[acc, very thick, ->] (-0.6,0) -- (0.6,0) node[right, black] {electron};
\draw[black, thick, dotted] (-2.2,0) .. controls (-1.9,1.2) and (-0.9,1.2) .. (-0.6,0);
\node[black, anchor=south] at (-1.4,1.0) {photon};
\node[black, anchor=north] at (-1.4,-0.3) {self-energy};
% vacuum polarization
\begin{scope}[xshift=6.2cm]
\draw[black, thick, dotted] (-2.6,0) -- (-1.1,0);
\draw[black, thick, dotted] (1.1,0) -- (2.6,0) node[right, black] {photon};
\draw[acc, very thick] (0,0) circle (1.1);
\draw[acc, very thick, ->] (-1.1,0.02) arc (180:90:1.1);
\draw[acc, very thick, ->] (1.1,-0.02) arc (0:-90:1.1);
\node[acc] at (0,0) {pair};
\node[black, anchor=north] at (0,-1.35) {vacuum polarization};
\end{scope}
\end{tikzpicture}
$$

## The anomalous magnetic moment

The Dirac equation predicts a gyromagnetic ratio $g=2$ for the electron exactly.
The virtual-photon cloud corrects it. Julian Schwinger's one-loop calculation
gives the first term of a series in $\alpha/\pi$,[^schwinger]

$$
a_e \equiv \frac{g-2}{2}
= \frac{\alpha}{2\pi} - 0.32848\left(\frac{\alpha}{\pi}\right)^2
+ 1.181\left(\frac{\alpha}{\pi}\right)^3 - \cdots,
$$

so $a_e = 0.0011614\ldots$ at leading order, against a measured value

$$
a_e^{\text{exp}} = 1.159\,652\,180\,59(13)\times 10^{-3},
$$

known to twelve significant figures.[^gm2] The QED prediction, carried to five
loops and including small hadronic and weak contributions, agrees to this
precision; the comparison is one of the most stringent tests of any physical
theory, and it is what fixes the best value of $\alpha$ outside of atom
interferometry. The same $a_e$ enters the Lamb-shift breakdown through the
spin-orbit interaction: the electron's magnetic moment is slightly larger than
Dirac's value, so its coupling to the internal magnetic field of the orbit is
correspondingly larger.

## Hydrogen as a QED laboratory

The Lamb shift is a small fraction of the fine structure, which is itself
$\alpha^2$ of the gross structure. The three scales stack in a fixed hierarchy.

| Structure | Scale | $2$-level example | Order of magnitude |
| --- | --- | --- | --- |
| Gross (Bohr) | $\alpha^2 m c^2$ | $-3.4\ \mathrm{eV}$ binding | $\mathrm{eV}$ |
| Fine | $\alpha^4 m c^2$ | $2P_{3/2}$–$2P_{1/2}$, $\sim 11\ \mathrm{GHz}$ | $10^{-4}\ \mathrm{eV}$ |
| Lamb (radiative) | $\alpha^5 m c^2 \ln(1/\alpha)$ | $2S_{1/2}$–$2P_{1/2}$, $1058\ \mathrm{MHz}$ | $10^{-6}\ \mathrm{eV}$ |

$$
% caption: The energy hierarchy in hydrogen, on a logarithmic scale. Each layer is
% roughly α² below the one above: gross structure near an eV, fine structure near
% 0.1 meV, the Lamb shift near a few μeV.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->] (-0.4,0) -- (10.6,0) node[right, black] {energy (log scale)};
% ticks
\foreach \x/\lab in {1/{a few eV}, 5/{0.1 meV}, 9/{a few micro-eV}}
  {\draw[black] (\x,0.12) -- (\x,-0.12); \node[anchor=north] at (\x,-0.2) {\lab};}
\node[black, anchor=south] at (1,0.2) {gross structure};
\node[black, anchor=south] at (5,0.2) {f\/ine structure};
\node[acc, anchor=south] at (9,0.2) {Lamb shift};
\end{tikzpicture}
$$

The regularity of the ladder is what makes hydrogen a precision instrument. Each
successive layer is a smaller correction that a more careful theory must
reproduce, and each has been measured. Modern two-photon spectroscopy of the
$1S$–$2S$ transition reaches a fractional precision near $10^{-15}$, so the $1S$
Lamb shift — about $8172\ \mathrm{MHz}$, scaled up from the $2S$ value by the
$1/n^3$ density factor — is itself resolved to many digits. Comparing that
measurement against the QED calculation determines the Rydberg constant and the
proton charge radius, and a disagreement in the extracted radius (the "proton
radius puzzle" from muonic hydrogen) turned the Lamb shift into a probe of the
proton itself, treated in the
[isotope-shift lesson](/atomic-physics/qed-corrections-and-hyperfine-structure/nuclear-effects-isotope-shift).

> **Definition (Radiative correction).** A shift in an atomic energy level caused
> by the electron's coupling to the quantized electromagnetic field — emission
> and reabsorption of virtual photons (self-energy), creation of virtual pairs
> (vacuum polarization), and the resulting anomalous magnetic moment. These
> corrections are absent from the Dirac equation, which treats the field as a
> fixed classical background, and they lift the $\ell$-degeneracy the Dirac
> theory protects.

The lesson of the Lamb shift is that the vacuum is not empty. The zero-point
electromagnetic field, the constant traffic of virtual particles, produces
measurable shifts in the most carefully studied atom, and the theory that
computes those shifts agrees with experiment to a part in $10^{12}$. Everything
that follows in this module — hyperfine structure, isotope shifts, the nuclear
corrections — sits on top of the QED-corrected level scheme established here.

[^lr]: Lamb, W. E. & Retherford, R. C. (1947), "Fine Structure of the Hydrogen Atom by a Microwave Method," _Phys. Rev._ **72**, 241. The originally reported splitting was about $1000\ \mathrm{MHz}$; the currently accepted $2S_{1/2}$–$2P_{1/2}$ interval is $1057.8\ \mathrm{MHz}$. See also **Foot**, §2.3.
[^welton]: Welton, T. A. (1948), "Some Observable Effects of the Quantum-Mechanical Fluctuations of the Electromagnetic Field," _Phys. Rev._ **74**, 1157. The heuristic is reproduced in **Bransden & Joachain**, §5.4, and **Foot**, §2.3.
[^bethe]: Bethe, H. A. (1947), "The Electromagnetic Shift of Energy Levels," _Phys. Rev._ **72**, 339. The non-relativistic self-energy with mass renormalization and the Bethe logarithm; see **Bransden & Joachain**, §5.4.
[^schwinger]: Schwinger, J. (1948), "On Quantum-Electrodynamics and the Magnetic Moment of the Electron," _Phys. Rev._ **73**, 416 — the $\alpha/2\pi$ result. Higher coefficients: Aoyama, Hayakawa, Kinoshita, Nio (2012), _Phys. Rev. Lett._ **109**, 111807.
[^gm2]: Fan, X., Myers, T. G., Sukra, B. A. D., Gabrielse, G. (2023), "Measurement of the Electron Magnetic Moment," _Phys. Rev. Lett._ **130**, 071801. CODATA 2018 electron $g$-factor: $g_e = 2.002\,319\,304\,362\,56(35)$, [physics.nist.gov/cgi-bin/cuu/Value?gem](https://physics.nist.gov/cgi-bin/cuu/Value?gem).