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.

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.