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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 Landau quantization ： ウィキペディア英語版 Landau quantization Landau quantization in quantum mechanics is the quantization of the cyclotron orbits of charged particles in magnetic fields. As a result, the charged particles can only occupy orbits with discrete energy values, called Landau levels. The Landau levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. Landau quantization is directly responsible for oscillations in electronic properties of materials as a function of the applied magnetic field. It is named after the Soviet physicist Lev Landau. ==Derivation== Consider a two-dimensional system of non-interacting particles with charge and spin confined to an area in the plane. Apply a uniform magnetic field $\backslash mathbf\; =\; \backslash begin0\backslash \backslash 0\backslash \backslash B\backslash end$ along the -axis. In CGS units, the Hamiltonian of this system is :$\backslash hat=\backslash frac(\backslash hat\}/c)^2.$ Here, p̂ is the canonical momentum operator and  is the electromagnetic vector potential, which is related to the magnetic field by :$\backslash mathbf=\backslash mathbf\backslash times\; \backslash hat\}=$ \begin0\\Bx \\0 \end. where =|B| and ''x̂'' is the component of the position operator. In this gauge, the Hamiltonian is :$\backslash hat\; =\; \backslash frac\; +\; \backslash frac\; \backslash left(\backslash hat\_y\; -\; \backslash frac\backslash right)^2.$ The operator $\backslash hat\_y$ commutes with this Hamiltonian, since the operator ''ŷ'' is absent by the choice of gauge. Thus the operator $\backslash hat\_y$ can be replaced by its eigenvalue . The Hamiltonian can also be written more simply by noting that the cyclotron frequency is , giving :$\backslash hat\; =\; \backslash frac\; +\; \backslash frac\; m\; \backslash omega\_c^2\; \backslash left(\; \backslash hat\; -\; \backslash frac\; \backslash right)^2.$ This is exactly the Hamiltonian for the quantum harmonic oscillator, except with the minimum of the potential shifted in coordinate space by . To find the energies, note that translating the harmonic oscillator potential does not affect the energies. The energies of this system are thus identical to those of the standard quantum harmonic oscillator, :$E\_n=\backslash hbar\backslash omega\_c\backslash left(n+\backslash frac\backslash right),\backslash quad\; n\backslash geq\; 0~.$ The energy does not depend on the quantum number , so there will be degeneracies. For the wave functions, recall that $\backslash hat\_y$ commutes with the Hamiltonian. Then the wave function factors into a product of momentum eigenstates in the direction and harmonic oscillator eigenstates $|\backslash phi\_n\backslash rangle$ shifted by an amount 0 in the direction: :$\backslash Psi(x,y)=e^\; \backslash phi\_n(x-x\_0)~.$ In sum, the state of the electron is characterized by two quantum numbers, and . 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「Landau quantization」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.