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{{Short description|Difference in energy of hydrogenic atom electron states not predicted by the Dirac equation
}}
{{Use American English|date=January 2019}}{{Quantum field theory}}
[[File:Hydrogen_fine_structure2.svg|thumb|[[Fine structure]] of energy levels in hydrogen – relativistic corrections to the [[Bohr model]]]]

In [[physics]], the '''Lamb shift''', named after [[Willis Lamb]], is an anomalous difference in energy between two electron orbitals in a [[hydrogen atom]]. The difference was not predicted by theory and it cannot be derived from the [[Dirac equation]], which predicts identical energies. Hence the Lamb ''shift'' is a deviation from theory seen in the differing [[energy|energies]] contained by the  <sup>2</sup>''S''<sub>1/2</sub> and <sup>2</sup>''P''<sub>1/2</sub> [[energy levels|orbitals]] of the hydrogen atom.

The Lamb shift is caused by interactions between the [[virtual photon|virtual photons]] created through [[Quantum fluctuation|vacuum energy fluctuations]] and the electron as it moves around the hydrogen nucleus in each of these two orbitals. The Lamb shift has since played a significant role through vacuum energy fluctuations in theoretical prediction of [[Hawking radiation]] from [[black hole]]s.

This effect was first measured in 1947 in the '''Lamb–Retherford experiment'''<!--boldface per WP:R#PLA--> on the hydrogen microwave spectrum<ref name=Aruldhas>
{{cite book |title=Quantum Mechanics |chapter=§15.15 Lamb Shift |chapter-url=https://books.google.com/books?id=4HLB6884s9IC&pg=PA404 |page=404 |edition= 2nd |publisher=Prentice-Hall of India Pvt. Ltd. |author=G Aruldhas |year=2009 |isbn=978-81-203-3635-3}}
</ref> and this measurement provided the stimulus for [[renormalization]] theory to handle the divergences. It was the harbinger of modern [[quantum electrodynamics]] developed by [[Julian Schwinger]], [[Richard Feynman]],  [[Ernst Stueckelberg]], [[Sin'ichirō Tomonaga|Sin-Itiro Tomonaga]] and [[Freeman Dyson]]. Lamb won the [[Nobel Prize in Physics]] in 1955 for his discoveries related to the Lamb shift. [[Victor Weisskopf]] regretted that his insecurity about his mathematical abilities may have cost him a Nobel Prize when he did not publish results (which turned out to be correct) about what is now known as the [[Lamb shift]].<ref>{{cite web |url=https://www.nasonline.org/wp-content/uploads/2024/06/weisskopf-victor.pdf|first1=Kurt |last1=Gottfried |first2=J. David |last2=Jackson |title=Victor Frederick Weisskopf, 1908–2002, A Biographical Memoir |page=16 |quote=I might even have shared the Nobel Prize with Lamb}}</ref>

== Importance ==
In 1978, on Lamb's 65th birthday, [[Freeman Dyson]] addressed him as follows: "Those years, when the Lamb shift was the central theme of physics, were golden years for all the physicists of my generation. You were the first to see that this tiny shift, so elusive and hard to measure, would clarify our thinking about particles and fields."<ref>{{cite journal|title=Willis E. Lamb, Jr. 1913—2008|journal=Biographical Memoirs of the National Academy of Sciences|year=2009|pages= 6|url=https://www.nasonline.org/wp-content/uploads/2024/06/lamb-jr-willis.pdf}}</ref>

== Derivation ==
This heuristic derivation of the electrodynamic level shift follows [[Theodore A. Welton]]'s approach.<ref>{{cite book|author1=Marlan Orvil Scully |author2=Muhammad Suhail Zubairy |title=Quantum Optics|year=1997|publisher=Cambridge University Press|location=Cambridge UK|isbn=0-521-43595-1|url=https://books.google.com/books?id=20ISsQCKKmQC&pg=PA430|pages=13–16}}</ref><ref>{{Cite journal|last=Welton|first=Theodore A.|date=1948-11-01|title=Some Observable Effects of the Quantum-Mechanical Fluctuations of the Electromagnetic Field|url=https://link.aps.org/doi/10.1103/PhysRev.74.1157|journal=Physical Review|language=en|volume=74|issue=9|pages=1157–1167|doi=10.1103/PhysRev.74.1157|bibcode=1948PhRv...74.1157W |issn=0031-899X}}</ref>

The fluctuations in the electric and magnetic fields associated with the [[QED vacuum]] perturbs the [[electric potential]] due to the [[atomic nucleus]]. This [[Perturbation theory (quantum mechanics)|perturbation]] causes a fluctuation in the position of the [[electron]], which explains the energy shift. The difference of [[potential energy]] is given by

:<math>\Delta V = V(\vec{r}+\delta \vec{r})-V(\vec{r})=\delta \vec{r} \cdot \nabla V (\vec{r}) + \frac{1}{2} (\delta \vec{r} \cdot \nabla)^2V(\vec{r})+\cdots</math>

Since the fluctuations are [[isotropic]],

:<math>\langle \delta \vec{r} \rangle _{\rm vac} =0,</math>
:<math>\langle (\delta \vec{r} \cdot \nabla )^2 \rangle _{\rm vac} = \frac{1}{3} \langle (\delta \vec{r})^2\rangle _{\rm vac} \nabla ^2.</math>

So one can obtain

:<math>\langle \Delta V\rangle =\frac{1}{6} \langle (\delta \vec{r})^2\rangle _{\rm vac}\left\langle \nabla ^2\left(\frac{-e^2}{4\pi \epsilon _0r}\right)\right\rangle _{\rm at}.</math>

The classical [[equation of motion]] for the electron displacement (''δr'')<sub>{{vec|''k''}}</sub> induced by a single mode of the field of [[wave vector]] {{vec|''k''}} and [[frequency]] ''ν'' is

:<math>m\frac{d^2}{dt^2} (\delta r)_{\vec{k}}=-eE_{\vec{k}},</math>

and this is valid only when the [[frequency]] ''ν'' is greater than ''ν''<sub>0</sub> in the Bohr orbit, <math>\nu > \pi c/a_0</math>. The electron is unable to respond to the fluctuating field if the fluctuations are smaller than the natural orbital frequency in the atom.

For the field oscillating at ''ν'',

:<math>\delta r(t)\cong \delta r(0)(e^{-i\nu t}+e^{i\nu t}),</math>

therefore

:<math>(\delta r)_{\vec{k}} \cong \frac{e}{mc^2k^2} E_{\vec{k}}=\frac{e}{mc^2k^2} \mathcal{E} _{\vec{k}} \left (a_{\vec{k}}e^{-i\nu t+i\vec{k}\cdot \vec{r}}+h.c. \right) \qquad \text{with} \qquad  \mathcal{E} _{\vec{k}}=\left(\frac{\hbar ck/2}{\epsilon _0 \Omega}\right)^{1/2},</math>

where <math>\Omega</math> is some large normalization volume (the volume of the hypothetical "box" containing the hydrogen atom), and <math>h.c.</math> denotes the hermitian conjugate of the preceding term. By the summation over all <math>\vec{k},</math>

:<math>\begin{align}
\langle (\delta \vec{r} )^2\rangle _{\rm vac} &=\sum_{\vec{k}} \left(\frac{e}{mc^2k^2} \right)^2 \left\langle 0\left |(E_{\vec{k}})^2 \right |0 \right \rangle \\
&=\sum_{\vec{k}} \left(\frac{e}{mc^2k^2} \right)^2\left(\frac{\hbar ck}{2\epsilon _0 \Omega} \right) \\
&=2\frac{\Omega}{(2\pi )^3}4\pi \int dkk^2\left(\frac{e}{mc^2k^2} \right)^2\left(\frac{\hbar ck}{2\epsilon_0 \Omega}\right) && \text{since continuity of } \vec{k} \text{ implies } \sum_{\vec{k}} \to 2 \frac{\Omega}{(2\pi)^3} \int d^3 k \\
&=\frac{1}{2\epsilon_0\pi^2}\left(\frac{e^2}{\hbar c}\right)\left(\frac{\hbar}{mc}\right)^2\int \frac{dk}{k}
\end{align}</math>

This result diverges when no limits about the integral (at both large and small frequencies). As mentioned above, this method is expected to be valid only when <math>\nu > \pi c/a_0</math>, or equivalently <math>k > \pi/a_0</math>. It is also valid only for wavelengths longer than the [[Compton wavelength]], or equivalently <math>k < mc/\hbar</math>. Therefore, one can choose the upper and lower limit of the integral and these limits make the result converge.

:<math>\langle(\delta\vec{r})^2\rangle_{\rm vac}\cong\frac{1}{2\epsilon_0\pi^2}\left(\frac{e^2}{\hbar c}\right)\left(\frac{\hbar}{mc}\right)^2\ln\frac{4\epsilon_0\hbar c}{e^2}</math>.

For the [[atomic orbital]] and the [[Coulomb potential]],

:<math>\left\langle\nabla^2\left(\frac{-e^2}{4\pi\epsilon_0r}\right)\right\rangle_{\rm at}=\frac{-e^2}{4\pi\epsilon_0}\int d\vec{r}\psi^*(\vec{r})\nabla^2\left(\frac{1}{r}\right)\psi(\vec{r})=\frac{e^2}{\epsilon_0}|\psi(0)|^2,</math>

since it is known that

:<math>\nabla^2\left(\frac{1}{r}\right)=-4\pi\delta(\vec{r}).</math>

For ''p'' orbitals, the nonrelativistic [[wave function]] vanishes at the origin (at the nucleus), so there is no energy shift. But for ''s'' orbitals there is some finite value at the origin,

:<math>\psi_{2S}(0)=\frac{1}{(8\pi a_0^3)^{1/2}},</math>

where the [[Bohr radius]] is

:<math>a_0=\frac{4\pi\epsilon_0\hbar^2}{me^2}.</math>

Therefore,

:<math>\left\langle\nabla^2\left(\frac{-e^2}{4\pi\epsilon_0r}\right)\right\rangle_{\rm at}=\frac{e^2}{\epsilon_0}|\psi_{2S}(0)|^2=\frac{e^2}{8\pi\epsilon_0a_0^3}</math>.

Finally, the difference of the potential energy becomes:

:<math>\langle\Delta V\rangle=\frac{4}{3}\frac{e^2}{4\pi\epsilon_0}\frac{e^2}{4\pi\epsilon_0\hbar c}\left(\frac{\hbar}{mc}\right)^2\frac{1}{8\pi a_0^3}\ln\frac{4\epsilon_0\hbar c}{e^2} = \alpha^5 mc^2 \frac{1}{6\pi} \ln\frac{1}{\pi\alpha},</math>

where <math>\alpha</math> is the [[fine-structure constant]]. This shift is about 500&nbsp;MHz, within an order of magnitude of the observed shift of 1057&nbsp;MHz. This is equal to an energy of only 7.00 x 10^-25 J., or 4.37 x 10^-6 eV. 

Welton's heuristic derivation of the Lamb shift is similar to, but distinct from, the calculation of the [[Darwin term]] using [[Zitterbewegung]], a contribution to the [[fine structure]] that is of lower order in <math>\alpha</math> than the Lamb shift.<ref>{{cite book|last1=Itzykson |first1=Claude |author-link1=Claude Itzykson |last2=Zuber |first2=Jean-Bernard |author-link2=Jean-Bernard Zuber |title=Quantum Field Theory |publisher=Dover Publications |year=2012 |isbn=9780486134697 |oclc=868270376}}</ref>{{rp|80–81}}

== 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>

==In the hydrogen spectrum==
{{Main|Lyman series}}
In 1947, [[Hans Bethe]] was the first to explain the Lamb shift in the [[hydrogen spectrum]], and he thus laid the foundation for the modern development of [[quantum electrodynamics]]. Bethe was able to derive the Lamb shift by implementing the idea of mass renormalization, which allowed him to calculate the observed energy shift as the difference between the shift of a bound electron and the shift of a free electron.<ref name=BetheEmagShift>
{{cite journal |last=Bethe|first=H. A.|date=1947|title=The Electromagnetic Shift of Energy Levels|url=http://link.aps.org/doi/10.1103/PhysRev.72.339|journal=Phys. Rev.|volume=72|issue=4|pages=339–341|bibcode=1947PhRv...72..339B|doi=10.1103/PhysRev.72.339|s2cid=120434909 }}
</ref> The Lamb shift currently provides a measurement of the [[fine-structure constant]] α to better than one part in a million, allowing a [[Precision tests of QED|precision test of quantum electrodynamics]].

== See also ==
{{portal|Physics}}
* [[Uehling potential]], first approximation to the Lamb shift
* [[Shelter Island Conference]]
* [[Zeeman effect]] used to measure the Lamb shift

==References==
{{Reflist}}

==Further reading==
* {{cite book
|first=Boris M. |last=Smirnov
|title=Physics of atoms and ions
|year= 2003
|publisher=Springer
|isbn=0-387-95550-X
|url=https://books.google.com/books?id=I1O8WYOcUscC&pg=PA166
|pages=39–41}}
* {{cite book
|author-link=Marlan Scully |first1=M.O. |last1=Scully |author2-link=Muhammad Suhail Zubairy |first2=M.S. |last2=Zubairy
|title=Quantum optics
|year= 1997
|publisher=Cambridge University Press
|isbn=0-521-43595-1
|chapter=§1.3 Lamb shift
|chapter-url={{GBurl|20ISsQCKKmQC|p=13}}
|pages=13–16}}

==External links==
* [http://webofstories.com/play/4569 Hans Bethe talking about Lamb-shift calculations] on [[Web of Stories]]
* [https://www.nobelprize.org/nobel_prizes/physics/laureates/1955/lamb.html Nobel Prize biography of Willis Lamb]
* [https://www.nobelprize.org/nobel_prizes/physics/laureates/1955/lamb-lecture.html Nobel lecture of Willis Lamb: ''Fine Structure of the Hydrogen Atom'']
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{{Quantum field theories}}
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[[Category:Quantum electrodynamics]]
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