发表机构
Izmir Institute of Technology（伊兹密尔理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于算子的Bloch定理推导，通过构造哈密顿量并证明其与平移算子对易，重现Kittel方程并推导新公式，应用于Kronig-Penney模型展示能带与电子离域。
AI 中文摘要
我们提出了一种基于算子的Bloch定理推导方法。在该方法中，我们首先通过将势能部分表示为动量平移算子的无穷级数，构造出哈密顿量的显式算子形式。接着，利用指数重排恒等式，我们证明了该哈密顿量与对应于晶格平移的位置平移算子对易。然后，我们基于Merzbacher对可对易算子同时对角化的处理，重现了Kittel的中心方程，并推导出对应于特定能量本征值的联合本征矢的新公式。此外，通过Born规则，我们精确描述了Bloch态的位置和动量概率分布，并探讨了晶体动量的概念。最后，我们将所得结果应用于Kronig-Penney模型，数值展示了能带结构，并阐明了单电子的离域特性。
英文摘要
We present an operator based derivation of Bloch's theorem. In this approach, we first construct an explicit operator form of the Hamiltonian by writing the potential part in terms of an infinite sum of the momentum translation operators. Next, we prove that it commutes with the position translation operator corresponding to a lattice translation, using the exponential reordering identity. We then build on Merzbacher's treatment of simultaneous diagonalization of commuting operators to reproduce Kittel's central equation and also to derive a new formula for the simultaneous eigenket corresponding to a specific energy eigenvalue. Moreover, through the Born rule, we provide a precise account of the position and momentum probability distributions for Bloch states, and we examine the concept of crystal momentum. Finally, we apply our findings to the Kronig Penney model, where we numerically depict the band structure and illustrate the delocalized nature of the single electron.
Comments7 pages, 2 figures, published in Am. J. Phys
