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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 Klein paradox ： ウィキペディア英語版 Klein paradox In 1929, physicist Oskar Klein obtained a surprising result by applying the Dirac equation to the familiar problem of electron scattering from a potential barrier. In nonrelativistic quantum mechanics, electron tunneling into a barrier is observed, with exponential damping. However, Klein’s result showed that if the potential is of the order of the electron mass, $V\backslash sim\; mc^2$, the barrier is nearly transparent. Moreover, as the potential approaches infinity, the reflection diminishes and the electron is always transmitted. The immediate application of the paradox was to Rutherford's proton–electron model for neutral particles within the nucleus, before the discovery of the neutron. The paradox presented a quantum mechanical objection to the notion of an electron confined within a nucleus.〔 〕 This clear and precise paradox suggested that an electron could not be confined within a nucleus by any potential well. The meaning of this paradox was intensely debated at the time.〔 == Massless particles == Consider a massless relativistic particle approaching a potential step of height $V\_0$ with energy  The particle's wave function, $\backslash psi$, follows the time-independent Dirac equation: :: And $\backslash sigma\_x$ is the Pauli matrix: :: Assuming the particle is propagating from the left, we obtain two solutions — one before the step, in region (1) and one under the potential, in region (2): ::$\backslash psi\_1=Ae^\backslash left(\; \backslash begin\; 1\; \backslash \backslash \; 1\backslash end\; \backslash right)+A\text{'}e^\backslash left(\; \backslash begin\; -1\; \backslash \backslash \; 1\backslash end\; \backslash right)\; ,\backslash quad\; p=E\_0\; \backslash ,$ ::$\backslash psi\_2=Be^\backslash left(\; \backslash begin\; 1\; \backslash \backslash \; 1\backslash end\; \backslash right)\; ,\backslash quad\; \backslash left|k\backslash right|=V\_0-E\_0\; \backslash ,$ Where the coefficients , and are complex numbers. Both the incoming and transmitted wave functions are associated with positive group velocity (Blue lines in Fig.1), whereas the reflected wave function is associated with negative group velocity. (Green lines in Fig.1) We now want to calculate the transmission and reflection coefficients, $T,\; R.$ They are derived from the probability amplitude currents. The definition of the probability current associated with the Dirac equation is: ::$J\_i=\backslash psi\_i^\backslash dagger\; \backslash sigma\_x\; \backslash psi\_i,\backslash \; i=1,2\; \backslash ,$ In this case: ::$J\_1=2\backslash left(A\; \backslash right|^2-\backslash left|\; A\text{'}\; \backslash right|^2\backslash right),\; \backslash quad\; J\_2=2\backslash left|\; B\; \backslash right|^2\; \backslash ,$ The transmission and reflection coefficients are: ::$R=\backslash frac\; ,\; \backslash quad\; T=\backslash frac\; \backslash ,$ Continuity of the wave function at $x=0$, yields: ::$\backslash left|A\backslash right|^2=\backslash left|B\backslash right|^2\; \backslash ,$ ::$\backslash left|A\text{'}\backslash right|^2=0\; \backslash ,$ And so the transmission coefficient is 1 and there is no reflection. One interpretation of the paradox is that a potential step cannot reverse the direction of the group velocity of a massless relativistic particle. This explanation best suits the single particle solution cited above. Other, more complex interpretations are suggested in literature, in the context of quantum field theory where the unrestrained tunnelling is shown to occur due to the existence of particle–antiparticle pairs at the potential. 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「Klein paradox」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.