Learning quantum mechanics well requires two things:
1. Master the mathematical tools used to describe quantum mechanics.
2. Understand the thinking method of describing physical system with quantum mechanics.
The mathematical tools needed to learn quantum mechanics well are as follows:
1. Some basic knowledge of mathematical analysis, including basic real variable functions, complex variable functions, ordinary differential and partial differential equations, etc. Any of these scientific advanced mathematics or mathematical analysis courses will be involved.
2. Understand some basic special functions, such as spherical harmonic function and Bessel function. These will be introduced in the course of mathematical physics methods offered by the physics department, and of course you don't have to look them up yourself.
3. Have a good understanding of the basic concepts of linear algebra, including linear space, subspace, orthogonality, basis, matrix and linear transformation, eigenvalue and eigenvector. In particular, it is necessary to establish the concept that matrix is transformation and eigenvector is transformation into basis, because this is the basis of describing quantum mechanics. These concepts should also be clearly established in the undergraduate course of linear algebra.