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A Bloch wave (also called Bloch state or Bloch function or Bloch wave function), named after Swiss physicist Felix Bloch, is a type of wavefunction for a particle in a periodically-repeating environment, most commonly an electron in a crystal. A wavefunction ψ is a Bloch wave if it has the form: : where r is position, ψ is the Bloch wave, ''u'' is a periodic function with the same periodicity as the crystal, k is a vector of real numbers called the crystal wave vector, ''e'' is Euler's number, and ''i'' is the imaginary unit. In other words, if you multiply a plane wave by a periodic function, you get a Bloch wave. Bloch waves are important because of Bloch's theorem, which states that the energy eigenstates for an electron in a crystal can be written as Bloch waves. (More precisely, it states that the electron wave functions in a crystal have a basis consisting entirely of Bloch wave energy eigenstates.) This fact underlies the concept of electronic band structures. These Bloch wave energy eigenstates are written with subscripts as ψ''n'' k, where ''n'' is a discrete index, called the ''band index'', which is present because there are many different Bloch waves with the same k (each has a different periodic component ''u''). Within a band (i.e., for fixed ''n''), ψ''n'' k varies continuously with k, as does its energy. Also, for any reciprocal lattice vector K, ψ''n'' k = ψ''n'',(k+K). Therefore, all distinct Bloch waves occur for k-values within the first Brillouin zone of the reciprocal lattice. ==Applications and consequences== 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「Bloch wave」の詳細全文を読む スポンサード リンク
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