It may not have the specific topics you want, but I like Leon Takhtajan's book entitled, coincidentally, Quantum Mechanics for Mathematicians.
Books like Dirac tended to frustrate me with drawn-out developments of "this is a ket, this is a bra, they have such-and-such properties and we combine them in the following ways." Takhtajan is kind enough to come out and say "take a Hilbert space and a self-adjoint operator."
This answer is quite late, so I'll make it general for those wondering about how to jump into QM with an undergraduate or higher background in math. A word of caution to mathematicians entering the physics realm: though there is a great overlap in material, the emphasis, pedagogy, and approach of a physicist can be quite different than that of a mathematician. You may (or may not) be frustrated by the lack of rigor, and amount of "guesswork and validation" to find the solutions you are required.
The tools widely used in QM include:
- differential equations / partial differential equations
- linear algebra
- vector operations / vector spaces
- basic complex analysis
A great conceptual introduction to the physics of QM can be found here:
- Wikipedia pages (Schrodinger Equation, Quantum Mechancs, and associated wiki links)
- Introduction to Quantum Mechanics (David J. Griffiths).
The latter, in particular, I found quite exceptional. Though other texts may be more complete references, Griffiths makes the physical ideas behind QM very plain and intuitive - something often obscured elsewhere.
As for whether to take the undergraduate or graduate course: My experience is that the undergraduate course will focus more on the conceptual aspects of QM, while the graduate course will assume some of that and focus on more difficult problems; due to this, I personally would recommend beginning with the undergraduate course.
Best Answer
Going in line with what Methemagician1234 has told you I will put on the table other books on functional analysis that make connections to quantum mechanics. However, being a quantum chemist by formation I recognize that a lot of research does not deal directly with the infinite dimensional Hilbert space, but with its simulation on a finite dimensional space. So, a deep understanding of linear algebra is not only useful for understanding the foundation of functional analysis but also for the computer modeling of quantum phenomena. These are the resources:
As for the linear algebra books: