Editing Lamb shift (section)

== 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|url-access=subscription}}</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 integral diverges as the wave number approaches zero or infinity. 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}}