Mathematically rigorous Quantum Mechanics

Question

I am a student of mathematics attending a course in Quantum Mechanics. This course is held by a physicist, and it is really confusing for me to follow his reasonments. With this, I do not mean to be arrogant or disrespectful, it is only a matter of backgrounds. I was wondering if anyone knows a textbook which is mathematically rigorous, in the sense that the functional analytical structure of the subject (distributions, measures, Hilbert spaces, duality) is properly handled. Instead of doing inconceivable (for me) things such as considering Dirac's delta a function or distinguishing between a state in $L^2(V)$ and its Fourier coefficients in $l^2$, when we have a nice isomorphism between these two.

I have background in Functional Analysis, Measure Theory and Topology, but a reference which explains also some mathematics would be good, you never know. The course covers all the basis of quantum mechanics, hence: Schroedinger and Dirac's picture with examples (hclassical problems, harmonic oscillator, hydrogen atom), spins and reaches the helium atom. Mostly it follows Schwabl Quantum Mechanics https://www.springer.com/it/book/9783540719328, which I consulted but does not provide what I search for.

I really want to make it clear: I really appreciate physicists and their stunning mind elasticity: after all, most of distribution theory was developed to underpin what Dirac had grasped with intuition. But I am just a student trying to do my best at college, and I need to cope with my background!

Thanks in advance.

Answer 1

You might try Hall, "Quantum Theory for Mathematicians".

Answer 2

To just place most of the course on a firm math footing, you could read Von Neumann's " Mathematical Foundations of Quantum Mechanics ISBN-13: 978-0691028934, particularly chapter 2. You would still be missing the Clifford algebra that underlies the Dirac equation, but there are several good math books on that.

But you are clearly missing the point of the class. Form a purely intellectual point of view, the professor is trying to convey applied math techniques that are relevant to the physics of elementary particles/fields. (BTW, "field" in this sense is not the same word as "field" in the sense of a set with two operations obeying certain properties.)

So for example, the course should cover some aspects of perturbation theory, some aspects of Fourier theory, more orthogonal function theory (Laguerre polynomials and spherical harmonics, for example), Green's functions, and so forth. If you get stuck on the foundational aspects of each of these, you will never make it to an understanding of the calculational techniques, which should be the fun part of the course for a strong mathematician. And you also will never make it to an understanding of the underlying physics ideas (which you might not care about, and that's OK).

You need to treat this course as an excursion into mathematics which is different than the math you have already studied. The fact that the Delta functions and Fourier analysis used are closely related to concepts you are already comfortable with should not be dwelt on, or you will miss the real content of the course.

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As a test of whether you grasp what the professor is trying to teach, try finding the energy levels and orbital structures oa 4-dimensional Hydrogen atom (still assuming a $\frac1{r^2}$ central attractive force). That should lead you to Gegenbaur polynomials instead of Legendre polynomials.

Then redo that using a $\frac1{r^3}$ force, which would be necessary to have the same Gauss's Law relation between charge and flux in 4 space dimensions. Based on the different nature of that problem, you could speculate why our usual physics only works in three dimensions.

Answer 3

Not a textbook, but Frederic Schuller's Lectures on Quantum Theory might be what you're looking for. Simon Rae and Richie Dadhley have created a set of notes from the lectures.