边界条件如何进入哈密顿量：基于SSH链的紧束缚模型教学设计

How Boundary Conditions Enter the Hamiltonian: A Design for Teaching Tight-Binding Models with the SSH Chain

摘要: 紧束缚近似是固体物理中连接微观晶格结构与电子能带理论的重要方法。然而在实际教学中，课程内容往往直接从布洛赫波和动量空间哈密顿量展开。学生虽能完成傅里叶变换和能带求解，却难以阐明矩阵元的来源，也难以准确理解开放边界条件与周期边界条件对哈密顿量结构、系统对称性及物理性质的具体影响。本文以一维Su-Schrieffer-Heeger (SSH)链为教学载体，构建了由“晶格连接–实空间哈密顿量–边界项–平移对称性–体能带–边缘态”逐步展开的教学路径。首先设计基于晶格连接关系构造矩阵元的任务，强化对紧束缚哈密顿量的结构化理解，继而比较两种边界条件下矩阵角元的差异，说明周期边界条件并非抽象附加规则，而是一个明确的跨边界跃迁项。随后在周期边界下利用平移对称性完成动量空间分块，并在开放边界下通过有限链能谱和本征态概率分布展示边缘态的能谱特征及空间局域性。该设计旨在帮助学生从会写公式走向能够依据研究对象选择模型，并建立对紧束缚近似、边界条件和体–边关系之间联系的整体认识。

Abstract: The tight-binding approximation provides an important framework for connecting microscopic lattice structures with electronic band theory in solid-state physics. However, in many teaching practices, the related concepts are often introduced directly from Bloch waves and momentum-space Hamiltonians. Although students can perform Fourier transformations and calculate band structures, they may still have difficulty identifying the physical origin of Hamiltonian matrix elements and understanding how open and periodic boundary conditions modify the Hamiltonian structure, system symmetries, and corresponding physical properties. In this work, a one-dimensional Su-Schrieffer-Heeger (SSH) chain is employed as a teaching model, and a conceptual learning pathway is constructed from “lattice connectivity-real-space Hamiltonian-boundary terms-translational symmetry-bulk bands-edge states.” First, a lattice-to-matrix construction task is designed to strengthen students’ understanding of the structure of tight-binding Hamiltonians. The differences between Hamiltonian matrix elements under open and periodic boundary conditions are then compared, demonstrating that periodic boundary conditions are not merely abstract computational assumptions but correspond to explicit hopping terms across the boundary. Furthermore, translational symmetry under periodic boundary conditions is used to introduce momentum-space block diagonalization, while finite-chain spectra and eigenstate probability distributions under open boundary conditions are employed to reveal the spectral characteristics and spatial localization of edge states. This teaching design aims to guide students beyond formula manipulation toward selecting appropriate models according to physical questions, while establishing an integrated understanding of tight-binding approximation, boundary conditions, and bulk-boundary correspondence.