Editing Lamb shift (section)

== Lamb–Retherford experiment ==
In 1947 Willis Lamb and [[Robert Retherford]] carried out an experiment using [[microwave]] techniques to stimulate radio-frequency transitions between
<sup>2</sup>''S''<sub>1/2</sub> and <sup>2</sup>''P''<sub>1/2</sub> levels of hydrogen.<ref>{{cite journal|title=Fine Structure of the Hydrogen Atom by a Microwave Method|first=Willis E.|last=Lamb|author2=Retherford, Robert C. |author-link=Willis Lamb|journal=[[Physical Review]]|volume=72|issue=3|pages=241–243|year=1947|doi=10.1103/PhysRev.72.241|bibcode = 1947PhRv...72..241L |doi-access=free}}</ref> 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 1000&nbsp;MHz (0.03&nbsp;cm<sup>−1</sup>) of the <sup>2</sup>''S''<sub>1/2</sub> level above the <sup>2</sup>''P''<sub>1/2</sub> level.

This particular difference is a [[one-loop effect]] of [[quantum electrodynamics]], and can be interpreted as the influence of virtual [[photon]]s that have been emitted and re-absorbed by the atom. In quantum electrodynamics the electromagnetic field is quantized and, like the [[harmonic oscillator]] in [[quantum mechanics]], its lowest state is not zero. Thus, there exist small [[zero-point energy|zero-point]] oscillations that cause the [[electron]] to execute rapid oscillatory motions. The electron is "smeared out" and each radius value is changed from ''r'' to ''r'' + ''δr'' (a small but finite perturbation).

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

:<math>\langle E_\mathrm{pot} \rangle=-\frac{Ze^2}{4\pi\epsilon_0}\left\langle\frac{1}{r+\delta r}\right\rangle.</math>

The Lamb shift itself is given by

:<math>\Delta E_\mathrm{Lamb}=\alpha^5 m_e c^2 \frac{k(n,0)}{4n^3}\ \mathrm{for}\ \ell=0\, </math>

with ''k''(''n'', 0) around 13 varying slightly with ''n'', and

:<math>\Delta E_\mathrm{Lamb}=\alpha^5 m_e c^2 \frac{1}{4n^3}\left[k(n,\ell)\pm \frac{1}{\pi(j+\frac{1}{2})(\ell+\frac{1}{2})}\right]\ \mathrm{for}\ \ell\ne 0\ \mathrm{and}\ j=\ell\pm\frac{1}{2},</math>

with log(''k''(''n'',{{ell}})) a small number (approx. −0.05) making ''k''(''n'',{{ell}}) close to unity.

For a derivation of Δ''E''<sub>Lamb</sub> see for example:<ref>{{cite book |last1=Bethe |first1=H.A. |last2=Salpeter |first2=E.E.| title=Quantum Mechanics of One- and Two-Electron Atoms| publisher=Springer |orig-year=1957 |page=103 |chapter-url={{GBurl|nxz2CAAAQBAJ|p=103}}
 |chapter=c) Radiative and other corrections §21. Fine structure and the Lamb shift |isbn=978-3-662-12869-5 |date=2013}}</ref>