大学物理 ›› 2022, Vol. 41 ›› Issue (9): 4-.doi: 10.16854/j.cnki.1000-0712.210619

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量子力学的数学基础

杨师杰   

  1. 北京师范大学 物理学系,北京100875
  • 收稿日期:2021-12-15 修回日期:2022-02-18 出版日期:2022-09-20 发布日期:2022-09-30
  • 作者简介:杨师杰(1966—),男,湖南资兴人,北京师范大学物理学系教授,博士,主要从事凝聚态物理理论研究工作.

On the mathematical fundations of quantum mechanics

YANG Shi-jie   

  1. Department of Physics, Beijing Normal University, Beijing 100875, China
  • Received:2021-12-15 Revised:2022-02-18 Online:2022-09-20 Published:2022-09-30

摘要: 本文系统地表述量子力学的数学基础,讨论了线性内积空间、希尔伯特空间完备性、函数空间以及连续基表示,以及自伴算符、施图姆-刘维尔系统的本征值问题和自然边界条件等,并以量子力学具体对象为例,讨论了非正规奇点情形下的自然边界条件.这些条件不是附加的,而是薛定谔方程本身所具备的.

关键词: 希尔伯特空间, 自伴算符, 本征值理论, 自然边界条件, 非正规奇点

Abstract: In this paper we present a comprehensive description of the mathematical foundations of quantum mechanics. The relevant topics include linear inner product, the completeness of Hilbert space, the functional space and continuous bases representations. We also discuss the self-adjoint operator, the eigen-value problem of the Sturm-Liouville system and the natural boundary conditions. We employ quantum problems as examples to discuss natural boundary condition under the non-regular singularity cases. These conditions are not externally applied but contain in the Schrdinger equation implicitly.

Key words: Hilbert space, self-adjoint operator, eigen-value theory, natural boundary condition, non-regular singularity