物理与工程

精确对角化是求解量子力学问题最直观的数值方法,广泛应用于少体与多体系统,是计算物理课程的核心内容之一。本文基于“由浅入深”的教学理念,系统介绍了如何在本科计算物理课程中讲授精确对角化方法:从单粒子系统出发,逐步拓展至多体量子系统。通过一系列精心设计的教学示例,学生不仅掌握了基矢选取、对称性使用、矩阵构建与对角化等关键步骤,也深入理解了该方法的优势与局限。这一教学过程有效衔接了量子力学形式体系与前沿多体数值方法,为学生后续学习密度矩阵重整化群、张量网络等高级多体计算方法奠定了坚实基础。

Exact diagonalization serves as the most intuitive numerical approach for solving quantum problems and is widely applied in both few-body and many-body systems, making it a core component of computational physics curricula. Guided by a scaffolded teaching philosophy that progresses from fundamental to advanced concepts, this paper systematically outlines the instruction of exact diagonalization in undergraduate computational physics courses—beginning with single-particle systems and gradually advancing to quantum many-body systems. Through a series of carefully designed pedagogical examples, students not only master key procedural steps such as basis selection, symmetry utilization, matrix construction, and diagonalization, but also develop a deeper comprehension of the method's strengths and limitations. This instructional framework effectively bridges the formalism of quantum mechanics with cutting-edge many-body numerical techniques, laying a solid groundwork for students' future exploration of advanced many-body computational methods such as the density matrix renormalization group and tensor networks.