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arXiv 0806.3198cond-mat.mtrl-sci
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周期固体中电子波函数的正确形式

Correct form of the electron wavefunction in periodic solids

T. R. S. Prasanna

摘要:

布洛赫波函数要么导致数学上不可能的结果,要么表明当详细考虑大晶体的几何形状时,基态能量是尺寸和形状的函数。这与Born-von Karman周期边界条件的假设是不相容的。困难的来源是布洛赫波函数对波数指数k的不正确依赖。如果把电子波函数的周期部分表示为un(r),则可以克服数学上不可能的结果,un(r)只取决于能带指数n,而与波数指数k无关。波函数的这种正确形式与布洛赫定理和布洛赫波函数的所有其他性质是一致的。正确的形式也与Born-von Karman周期边界条件一致。在周期性固体中,电子波函数的正确形式具有深远的意义。它简化了电子结构的计算,因为每个带只需要计算一个波函数un(r)。它使物理学家喜欢的能带图和化学家喜欢的键图在概念上统一起来。电子波函数的正确形式将简化对涉及价电子的许多现象的理解。

英文摘要:

The Bloch wavefunction leads either to mathematically impossible consequences or suggests that the ground state energy is a function of size and shape when the geometry of large crystals is considered in detail. It is incompatible with the assumption underlying the Born-von Karman periodic boundary condition. The source of the difficulty is the incorrect dependence of the Bloch wavefunction on the wavenumber index k. The mathematically impossible consequences can be overcome if the periodic part of the electron wavefunction is represented as un(r), which is dependent only on the band index, n, and is independent of the wavenumber index k. This correct form of the wavefunction is consistent with the Bloch theorem and with all other properties of Bloch wavefunctions. The correct form is also consistent with the Born-von Karman periodic boundary condition. The correct form of the electronic wavefunction in a periodic solid has profound consequences. It simplifies the calculation of electronic structure as only one wavefunction per band, un(r), needs to be evaluated. It brings about a conceptual unification between the band picture favored by physicists and the bond picture favored by chemists. The correct form of the electron wavefunction will simplify the understanding of many phenomena involving valence electrons.

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