|
|
| Line 1: |
Line 1: |
| {{Quantum field theory}}
| | == Mont Blanc Australia except for video == |
| [[Image:Hydrogen fine structure.svg|thumb|[[Fine structure]] of energy levels in hydrogen - relativistic corrections to the [[Bohr model]]]]
| |
|
| |
|
| In [[physics]], the '''Lamb shift''', named after [[Willis Lamb]] (1913–2008), is a small difference in [[energy]] between two [[energy level]]s <sup>2</sup>''S''<sub>1/2</sup> and <sup>2</sup>''P''<sub>1/2</sup> (in [[term symbol]] notation) of the [[hydrogen atom]] in [[quantum electrodynamics]] (QED). According to the [[Dirac equation]], the <sup>2</sup>''S''<sub>1/2</sup> and <sup>2</sup>''P''<sub>1/2</sup> orbitals should have the same energies. However, the interaction between the electron and the vacuum (which is not accounted for by the Dirac equation) causes a tiny energy shift which is different for states <sup>2</sup>''S''<sub>1/2</sup> and <sup>2</sup>''P''<sub>1/2</sup>. Lamb and [[Robert Retherford]] measured this shift in 1947,<ref name=Aruldhas>
| | Everytime I have moved I have lost them for a time but thankfully they have never gotton damaged. My favorite is still that gothy looking elf drawing. The difficulties they encountered were past the imagination of the decisionmakers in Paris.The climate was sweltering, particularly round the Nanxi River valley area, where summer temperatures could exceed 40; [http://www.nadya.com.au/graphics/popups/settings.asp Mont Blanc Australia] it had been humid and oppressive, and infections from tropical diseases and plague were always possible. Statistics reveal that during those six years, 12,000 souls lie buried along this 465kmlong track, among which 10,000 died within the Nanxi River valley, most of them Chinese laborers who gave up their lives in order to earn a living. <br><br>The report provides info on adopted children, adoptive families and birth mothers. It also provides data on the number of requests for information and the number of contact and information vetoes lodged by parties to the adoption.. Upload Limits: Free accounts can upload a limitless number of individual files of no more than 10 MB each. Pro accounts can upload [http://www.romavilla.com/custom/ordersummary.asp Mbt Shoes Australia] a limitless number of individual files of no more than 250 MB each (except for video, video can be as much as 1 GB per file). <br><br>Want to write? Write an article for one from the hobby magazines. Heck, post your build here on RCSB! The options are endless. Brothers to Living Easy with Dr. Joyce Brothers.[10]She also had a monthly column in Good Housekeeping magazine for almost four decades, and a syndicated newspaper column that they began writing in the 1970s and which at its height was printed in additional than 300 newspapers.[10][11] She's published several bestselling books, including the 1982 book, What Every Woman Should Know About Men, and also the 1992 book, Widowed, [http://www.matesinconstruction.com.au/flux/plugins/activation/inc_activation.asp Nike Free 3.0] inspired by the loss of her husband. <br><br>Is it feasible?The "embed a list of pages" menu allows me to "show pages tagged". I'd like an option to "hide pages tagged". [Update 20110924: The initial test deployment and stage 1 have gone well, with only minor glitches that we've mostly cleared up. Stage 2 and 3 are currently on schedule. <br><br>These [http://www.matesinconstruction.com.au/flux-system/plugins/resources.asp Pandora Australia] buses will drop you off at the bus stop right across from us. Taxis:In the city fares are approx. Actually a clinical trial with this homeopathic gave impressive results, treating the hyperthyroidism in many of the cats. Give 1 tab every 12 hours and assess the response after 30 days.. <br><br>The only candidate I've seen so far that will challenge the status quo is, Chowai Lumumba. I heard so many people say he is too ratical for that city, I decided to look or myself. Bare in mind though that you can find some dangers in on the net shopping that you will have to be cautious about. For one, look for a reputable online shop that sells only the highest quality products.<ul> |
| | | |
| {{cite book |title=Quantum Mechanics |chapter=§15.15 Lamb Shift |url=http://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=81-203-3635-6}}
| | <li>[http://220.194.55.213:8086/forum.php?mod=viewthread&tid=932148 http://220.194.55.213:8086/forum.php?mod=viewthread&tid=932148]</li> |
| | | |
| </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]] and [[Shinichiro Tomonaga]]. Lamb won the [[Nobel Prize in Physics]] in 1955 for his discoveries related to the Lamb shift.
| | <li>[http://www.vvmeishi.com/forum.php?mod=viewthread&tid=59020&fromuid=23179 http://www.vvmeishi.com/forum.php?mod=viewthread&tid=59020&fromuid=23179]</li> |
| | | |
| == Derivation ==
| | <li>[http://bbs.qieos.com/forum.php?mod=viewthread&tid=34880&fromuid=45722 http://bbs.qieos.com/forum.php?mod=viewthread&tid=34880&fromuid=45722]</li> |
| This heuristic derivation of the electrodynamic level shift following Welton is from ''Quantum Optics''.<ref>{{cite book|author=Marlan Orvil Scully & Muhammad Suhail Zubairy|title=Quantum optics|year=1997|publisher=Cambridge University Press|location=Cambridge UK|isbn=0-521-43595-1|url=http://books.google.com/?id=20ISsQCKKmQC&pg=PA430|pages=13–16}}</ref>
| | |
| | | <li>[http://bbs.yuanmengjihua.com/forum.php?mod=viewthread&tid=137974 http://bbs.yuanmengjihua.com/forum.php?mod=viewthread&tid=137974]</li> |
| The fluctuation in the electric and magnetic fields associated with the [[QED vacuum]] perturbs the [[Coulomb 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 + \frac{1}{2} (\delta \vec{r} \cdot \nabla)^2V(\vec{r})+...</math>
| | <li>[http://0738go.com/forum.php?mod=viewthread&tid=3216275 http://0738go.com/forum.php?mod=viewthread&tid=3216275]</li> |
| Since the fluctuations are [[isotropic]],
| | |
| :<math>\langle \delta \vec{r} \rangle _{vac} =0</math>
| | </ul> |
| :<math>\langle (\delta \vec{r} \cdot \nabla )^2 \rangle _{vac} = \frac{1}{3} \langle (\delta \vec{r})^2\rangle _{vac} \nabla ^2</math>.
| |
| So we can obtain
| |
| :<math>\langle \Delta V\rangle =\frac{1}{6} \langle (\delta \vec{r})^2\rangle _{vac}\left\langle \nabla ^2\left(\frac{-e^2}{4\pi \epsilon _0r}\right)\right\rangle _{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, ''ν'' > ''πc''/''a''<sub>0</sub>. | |
| | |
| For the field oscillating at ''ν'',
| |
| :<math>\delta r(t)\cong \delta r(0)e^{-i\nu t}+c.c.</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}}(a_{\vec{k}}e^{-i\nu t+i\vec{k}\cdot \vec{r}}+h.c.)</math>.
| |
| | |
| By the summation over all <math>\vec{k}</math>,
| |
| :<math>\langle (\delta \vec{r} )^2\rangle _{vac}=\sum_{\vec{k}} \left(\frac{e}{mc^2k^2} \right)^2\langle 0|(E_{\vec{k}})^2|0\rangle =\sum_{\vec{k}} \left(\frac{e}{mc^2k^2} \right)^2\left(\frac{\hbar ck}{2\epsilon _0 \Omega} \right)</math>,
| |
| where <math>\Omega</math> is some large normalization volume (the volume of the hypothetical "box" containing the hydrogen atom) and
| |
| :<math>\mathcal{E} _{\vec{k}}=(\hbar ck/2\epsilon _0 \Omega)^{1/2}</math>.
| |
| | |
| The summation is changed into an integral because of the continuity of {{vec|''k''}}, <math>\sum_{\vec{k}} \rightarrow 2 \frac{\Omega}{(2\pi)^3} \int d^3 k </math>, so that
| |
| :<math>\langle (\delta \vec{r} )^2\rangle _{vac}=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)=\frac{1}{2\epsilon_0\pi^2}\left(\frac{e^2}{\hbar c}\right)\left(\frac{\hbar}{mc}\right)^2\int \frac{dk}{k}</math>.
| |
| | |
| This result diverges when there is no limit about the integral. But this method is valid only when ''ν'' > ''πc''/''a''<sub>0</sub>, or equivalently ''k'' > ''π''/''a''<sub>0</sub>. It is also valid only for wavelengths longer than the [[Compton wavelength]], or equivalently ''k'' < ''mc''/''ħ''. Therefore we can choose the upper and lower limit of the integral and these limits make the result converge.
| |
| :<math>\langle(\delta\vec{r})^2\rangle_{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_{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 we know 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, 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_{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}</math>.
| |
| This shift is about 1 GHz, very similar with the observed energy shift.
| |
| | |
| == Experimental work ==
| |
| 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</sup> and <sup>2</sup>''P''<sub>1/2</sup> levels of hydrogen.<ref>{{cite journal|title=Fine Structure of the Hydrogen Atom by a Microwave Method|first=Willis E.|last=Lamb|coauthors= Retherford, Robert C.|authorlink=Willis Lamb|journal=[[Physical Review]]|volume=72|issue=3|pages=241–243|year=1947|doi=10.1103/PhysRev.72.241|bibcode = 1947PhRv...72..241L }}</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 MHz of the <sup>2</sup>''S''<sub>1/2</sup> level above the <sup>2</sup>''P''<sub>1/2</sup> 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]] oscillations that cause the [[electron]] to execute rapid oscillatory motions. The electron is "smeared out" and the radius is changed
| |
| from ''r'' to ''r'' + ''δr''.
| |
| | |
| 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 ''k''(''n'',{{ell}}) a small number (< 0.05).
| |
| | |
| For a derivation of Δ''E''<sub>Lamb</sub> see for example:<ref>{{cite book | author=Bethe, H.A.& Salpeter, E.E.| title=Quantum Mechanics of One- and Two-Electron
| |
| Atoms| publisher=Springer |year=1957 |page=103}}</ref>
| |
| | |
| ==Lamb shift in the hydrogen spectrum==
| |
| 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]]. 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]].
| |
| | |
| A different perspective relates [[Zitterbewegung]] to the Lamb shift.<ref name=Genz>
| |
| {{cite book
| |
| |author=Henning Genz
| |
| |title=Nothingness: the science of empty space
| |
| |publisher= Oxford: Perseus
| |
| |location=Reading MA
| |
| |page=245 ff.
| |
| |year=2002
| |
| |isbn=0-7382-0610-5
| |
| |url=http://books.google.com/?id=Cn_Q9wbDOM0C&printsec=frontcover&dq=%22empty+space%22}}
| |
| </ref> | |
| | |
| == See also ==
| |
| * [[Shelter Island Conference]]
| |
| * [[Zeeman effect]] used to measure the Lamb shift.
| |
| {{portal|Physics}}
| |
| | |
| ==References==
| |
| <references/> | |
| | |
| ==Further reading==
| |
| *{{cite book
| |
| |author=Boris M Smirnov
| |
| |title=Physics of atoms and ions
| |
| |year= 2003
| |
| |publisher=Springer
| |
| |location=New York
| |
| |isbn=0-387-95550-X
| |
| |url=http://books.google.com/?id=I1O8WYOcUscC&pg=PA166
| |
| |pages=39–41}}
| |
| *{{cite book
| |
| |author=Marlan Orvil Scully & [[Muhammad Suhail Zubairy]]
| |
| |title=Quantum optics
| |
| |year= 1997
| |
| |publisher=Cambridge University Press
| |
| |location=Cambridge UK
| |
| |isbn=0-521-43595-1
| |
| |url=http://books.google.com/?id=20ISsQCKKmQC&pg=PA430
| |
| |pages=13–16}}
| |
| | |
| ==External links==
| |
| *[http://webofstories.com/play/4569 Hans Bethe talking about Lamb-shift calculations] on [[Web of Stories]]
| |
| *[http://www.nobelprize.org/nobel_prizes/physics/laureates/1955/lamb.html Nobel Prize biography of Willis Lamb]
| |
| *[http://www.nobelprize.org/nobel_prizes/physics/laureates/1955/lamb-lecture.html Nobel lecture of Willis Lamb: ''Fine Structure of the Hydrogen Atom'']
| |
| {{QED}}
| |
| {{Quantum field theories}}
| |
| {{Particles}}
| |
| | |
| {{DEFAULTSORT:Lamb Shift}}
| |
| [[Category:Quantum field theory]]
| |
| [[Category:Concepts in physics]]
| |
Mont Blanc Australia except for video
Everytime I have moved I have lost them for a time but thankfully they have never gotton damaged. My favorite is still that gothy looking elf drawing. The difficulties they encountered were past the imagination of the decisionmakers in Paris.The climate was sweltering, particularly round the Nanxi River valley area, where summer temperatures could exceed 40; Mont Blanc Australia it had been humid and oppressive, and infections from tropical diseases and plague were always possible. Statistics reveal that during those six years, 12,000 souls lie buried along this 465kmlong track, among which 10,000 died within the Nanxi River valley, most of them Chinese laborers who gave up their lives in order to earn a living.
The report provides info on adopted children, adoptive families and birth mothers. It also provides data on the number of requests for information and the number of contact and information vetoes lodged by parties to the adoption.. Upload Limits: Free accounts can upload a limitless number of individual files of no more than 10 MB each. Pro accounts can upload Mbt Shoes Australia a limitless number of individual files of no more than 250 MB each (except for video, video can be as much as 1 GB per file).
Want to write? Write an article for one from the hobby magazines. Heck, post your build here on RCSB! The options are endless. Brothers to Living Easy with Dr. Joyce Brothers.[10]She also had a monthly column in Good Housekeeping magazine for almost four decades, and a syndicated newspaper column that they began writing in the 1970s and which at its height was printed in additional than 300 newspapers.[10][11] She's published several bestselling books, including the 1982 book, What Every Woman Should Know About Men, and also the 1992 book, Widowed, Nike Free 3.0 inspired by the loss of her husband.
Is it feasible?The "embed a list of pages" menu allows me to "show pages tagged". I'd like an option to "hide pages tagged". [Update 20110924: The initial test deployment and stage 1 have gone well, with only minor glitches that we've mostly cleared up. Stage 2 and 3 are currently on schedule.
These Pandora Australia buses will drop you off at the bus stop right across from us. Taxis:In the city fares are approx. Actually a clinical trial with this homeopathic gave impressive results, treating the hyperthyroidism in many of the cats. Give 1 tab every 12 hours and assess the response after 30 days..
The only candidate I've seen so far that will challenge the status quo is, Chowai Lumumba. I heard so many people say he is too ratical for that city, I decided to look or myself. Bare in mind though that you can find some dangers in on the net shopping that you will have to be cautious about. For one, look for a reputable online shop that sells only the highest quality products.