The Lamb Shift and QED Radiative Corrections

The fine-structure formula and the exact Dirac spectrum 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 $n=2$, $j=\frac{1}{2}$ 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 $2P_{1/2}$.¹ 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

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 $\frac{1}{2}\hslash \omega$ 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.

Vacuum fluctuations and Welton's estimate

The cleanest physical picture is due to Theodore Welton.² 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

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

The upper cutoff is the electron's Compton frequency ${\omega }_{\max }\sim m{c}^2/\hslash$ (above it the non-relativistic treatment fails); the lower cutoff is the atomic orbital frequency ${\omega }_{\min }\sim (Z\alpha {)}^{2}m{c}^2/\hslash$ (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,

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,

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

Only $s$-states have $|{\psi }_{n\ell }(0){|}^2\ne 0$; 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.

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 $\alpha (Z\alpha {)}^{4}m{c}^2$ 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 $Ze$. 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.

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.³ The self-energy of a bound electron is the second-order shift from emitting and reabsorbing a photon,

where the sum runs over intermediate atomic states $|m⟩$ and the momentum integral is cut off at $K=mc/\hslash$. 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,

with $⟨E{⟩}_n$ the Bethe logarithm, an average excitation energy of the atom. For $2S$ the Bethe logarithm is $\ln (⟨E⟩/\mathrm{Ry})\approx 2.81$, 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.

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 $\alpha /\pi$,⁴

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

known to twelve significant figures.⁵ 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.

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.

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.