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.
I'll just make my comments into an answer.
- Reed and Simon Volume 1: Functional Analysis (Methods of Modern Mathematical Physics) here on amazon has a table of contents.
- Reed and Simon Volume 2: Fourier Analysis, Self-Adjointness (Methods of Modern Mathematical Physics) here on amazon, again you can see toc.
- Cohen-Tannoudji - Quantum Mechanics (2 vol. set). Reasonably rigorous and may fit Schuller to some extent - lots of end of chapter appendices. amazon link
- Messiah Quantum Mechanics (2 Volumes in 1) - two volumes has a lot in it, might not be as rigorous as you want it. amazon link
- Quantum Mechanics in Hilbert Space: Second Edition, suggested by user254665 - The preface to the first edition starts "This book was developed from a fourth-year undergraduate course given at the University of Toronto to advanced undergraduate and first-year graduate students in physics and mathematics. It is intended to provide the inquisitive student with a critical presentation of the basic mathematics of nonrelativistic quantum mechanics at a level which meets the present standards of mathematical rigor." Seems to fit the course reasonably well judging by toc - amazon, toc, preface etc.
R&S Volume one introduces Hilbert spaces, Banach spaces, spectral theorem etc. and leads from bounded to unbounded operators and the fourier transform.
R&S Volume 2 is very physics orientated, with topics on fourier transforms, hamiltonians in non-rel QM, and talks about self adjoint operators, and a bit about time dependent Hamiltonians.
As a note on quantum states, there's various definitions. They can be
- vectors in a Hilbert space $\mathcal{H}$, but specifically one that satisfies a Schrodinger equation you're interested in (Assuming you're in Schrodinger picture in non-rel QM). State space would be a subspace of a Hilbert space.
- elements of the projective hilbert space $\Bbb P\mathcal{H}$
- traceclass positive operators of trace $1$, normally called $\textit{density matrices}$.
- A rank one projection operator.
It depends on what you want to do with them.
The first I think is the most common, as when teaching quantum mechanics, the wave functions usually belong to an $L^2$ space, and are found by solving the Schrodinger PDE.
The second last one is useful for statistical mixtures and open quantum systems, you can have pure and mixed states. The pure states can be identified with the last item on the list.
As a sidenote this was asked by a different user on physics stack, same question on schullers course, and it was closed, even though it's physics, and a pretty reasonable request. It might be useful to check there for the one answer that was able to be posted before it was closed.
https://physics.stackexchange.com/questions/259583/good-texts-on-quantum-mechanics-to-accompany-this-online-course#comment579079_259583
Best Answer
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."