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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lamb shift ： ウィキペディア英語版 Lamb shift In physics, the Lamb shift, named after Willis Lamb (1913–2008), is a small difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom in quantum electrodynamics (QED). According to the Dirac equation, the 2''S''1/2 and 2''P''1/2 orbitals should have the same energy. 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 2''S''1/2 and 2''P''1/2. Lamb and Robert Retherford measured this shift in 1947,〔 〕 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 Sin-Itiro Tomonaga. Lamb won the Nobel Prize in Physics in 1955 for his discoveries related to the Lamb shift. == Derivation == This heuristic derivation of the electrodynamic level shift following Welton is from ''Quantum Optics''. The fluctuation in the electric and magnetic fields associated with the QED vacuum perturbs the electric potential due to the atomic nucleus. This perturbation causes a fluctuation in the position of the electron, which explains the energy shift. The difference of potential energy is given by :$\backslash Delta\; V\; =\; V(\backslash vec+\backslash delta\; \backslash vec)-V(\backslash vec)=\backslash delta\; \backslash vec\; \backslash cdot\; \backslash nabla\; V\; +\; \backslash frac\; (\backslash delta\; \backslash vec\; \backslash cdot\; \backslash nabla)^2V(\backslash vec)+...$ Since the fluctuations are isotropic, :$\backslash langle\; \backslash delta\; \backslash vec\; \backslash rangle\; \_\; =0$ :$\backslash langle\; (\backslash delta\; \backslash vec\; \backslash cdot\; \backslash nabla\; )^2\; \backslash rangle\; \_\; =\; \backslash frac\; \backslash langle\; (\backslash delta\; \backslash vec)^2\backslash rangle\; \_\; \backslash nabla\; ^2$. So we can obtain :$\backslash langle\; \backslash Delta\; V\backslash rangle\; =\backslash frac\; \backslash langle\; (\backslash delta\; \backslash vec)^2\backslash rangle\; \_\backslash left\backslash langle\; \backslash nabla\; ^2\backslash left(\backslash frac\backslash right)\backslash right\backslash rangle\; \_$. The classical equation of motion for the electron displacement (''δr'') induced by a single mode of the field of wave vector and frequency ''ν'' is :$m\backslash frac\; (\backslash delta\; r)\_\}$, and this is valid only when the frequency ''ν'' is greater than ''ν''0 in the Bohr orbit, ''ν'' > ''πc''/''a''0. 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'', :$\backslash delta\; r(t)\backslash cong\; \backslash delta\; r(0)e^+c.c.$, therefore :$(\backslash delta\; r)\_\; E\_\; \backslash mathcal\; \_\}e^\}+h.c.)$. By the summation over all $\backslash vec$, :$\backslash langle\; (\backslash delta\; \backslash vec\; )^2\backslash rangle\; \_=\backslash sum\_\; \backslash right)^2\backslash langle\; 0|(E\_\}\; \backslash left(\backslash frac\; \backslash right)^2\backslash left(\backslash frac\; \backslash right)$, where $\backslash Omega$ is some large normalization volume (the volume of the hypothetical "box" containing the hydrogen atom) and :$\backslash mathcal\; \_$. The summation is changed into an integral because of the continuity of , $\backslash sum\_\; \backslash int\; d^3\; k$, so that :$\backslash langle\; (\backslash delta\; \backslash vec\; )^2\backslash rangle\; \_=2\backslash frac4\backslash pi\; \backslash int\; dkk^2\backslash left(\backslash frac\; \backslash right)^2\backslash left(\backslash frac\backslash right)=\backslash frac\backslash left(\backslash frac\backslash right)\backslash left(\backslash frac\backslash right)^2\backslash int\; \backslash frac$. 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 ''ν'' > ''πc''/''a''0, or equivalently ''k'' > ''π''/''a''0. 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. :$\backslash langle(\backslash delta\backslash vec)^2\backslash rangle\_\backslash cong\backslash frac\backslash left(\backslash frac\backslash right)\backslash left(\backslash frac\backslash right)^2\backslash ln\backslash frac$. For the atomic orbital and the Coulomb potential, :$\backslash left\backslash langle\backslash nabla^2\backslash left(\backslash frac\backslash right)\backslash right\backslash rangle\_=\backslash frac\backslash int\; d\backslash vec\backslash psi^$ *(\vec)\nabla^2\left(\frac\right)\psi(\vec)=\frac|\psi(0)|^2, since we know that :$\backslash nabla^2\backslash left(\backslash frac\backslash right)=-4\backslash pi\backslash delta(\backslash vec)$. 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, :$\backslash psi\_(0)=\backslash frac$. Therefore :$\backslash left\backslash langle\backslash nabla^2\backslash left(\backslash frac\backslash right)\backslash right\backslash rangle\_=\backslash frac|\backslash psi\_(0)|^2=\backslash frac$. Finally, the difference of the potential energy becomes :$\backslash langle\backslash Delta\; V\backslash rangle=\backslash frac\backslash frac\backslash frac\backslash left(\backslash frac\backslash right)^2\backslash frac\backslash ln\backslash frac\; =\; \backslash alpha^5\; mc^2\; \backslash frac\; \backslash ln\backslash frac$, where $\backslash alpha$ is the Fine-structure constant. This shift is about 1 GHz, very similar with the observed energy shift. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lamb shift」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.