In physics, the Lamb shift, named after Willis Lamb, is a small difference in energy between two energy levels $2s\_\{1/2\}$ and $2p\_\{1/2\}$ of the hydrogen atom in quantum mechanics. According to Dirac and Schrödinger theory, hydrogen states with the same $n$ and $j$ quantum numbers but different $l$ quantum numbers ought to be degenerate.

Experimental work

In 1947 Lamb and Robert Retherford carried out an experiment using microwave techniques to stimulate radio-frequency transitions between $2s\_\{1/2\}$ and $2p\_\{1/2\}$ levels. By using lower frequencies than for optical transitions the Doppler broadening could be neglected (Doppler broadening is proportional to the frequency). The energy difference Lamb and Retherford found was a rise of about 1060MHz of the $2s\_\{1/2\}$ level above the $2p\_\{1/2\}$ level.

This particular difference is a one-loop effect of quantum electrodynamics, and can be interpreted as the influence of virtual photons that have been emitted and re-absorbed by the atom. In quantum electrodynamics (QED) the electromagnetic field is quantized and as for the harmonic oscillator in quantum mechanics its lowest state is not zero. So there exist little zero-point oscillations that cause the electron to execute rapid oscillatory motions. The electron is kind of "smeared out" and the radius is changed by $r+\backslash delta\; r$.

The Coulomb potential is therefore perturbed by a small amount and the degeneration of the two energy levels is removed. The new potential can be approximated (using Atomic units) as follows:

$\backslash langle\; E\_\backslash mathrm\{pot\}\; \backslash rangle=-\backslash frac\{Ze^2\}\{4\backslash pi\backslash epsilon\_0\}\backslash left\backslash langle\backslash frac\{1\}\{r+\backslash delta\; r\}\backslash right\backslash rangle.$

The Lamb shift itself is given by

$\backslash Delta\; E\_\backslash mathrm\{Lamb\}=\backslash alpha^5\; m\_e\; c^2\; \backslash frac\{k(n,0)\}\{4n^3\}\backslash \; \backslash mathrm\{for\}\backslash \; \backslash ell=0\backslash ,$

and

$\backslash Delta\; E\_\backslash mathrm\{Lamb\}=\backslash alpha^5\; m\_e\; c^2\; \backslash frac\{1\}\{4n^3\}\backslash left\backslash frac\{1\}\{\backslash pi(j+\backslash frac\{1\}\{2\})(\backslash ell+\backslash frac\{1\}\{2\})\}\backslash right\backslash \; \backslash mathrm\{for\}\backslash \; \backslash ell\backslash ne\; 0\backslash \; \backslash mathrm\{and\}\backslash \; j=\backslash ell\backslash pm\backslash frac\{1\}\{2\},$

with $k(n,\backslash ell)$ a small number (< 0.05).

Quantum field theory

Lambverschiebung | Décalage de Lamb | Przesunięcie Lamba | Лэмбовский сдвиг | Lambskift