# Mathematically rigorous Quantum Mechanics

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!

• $L^2(V)$ is isomorphic to $l^2$, but it is not uniquely isomorphic to $l^2$. There are other basis functions one could use than the Fourier basis--hence the distinction. May 24, 2019 at 15:51
• @eyeballfrog I completely agree. By Fourier series and coefficient I do not mean the analytical expansion in trygonometric polynomials, but the expansion in any complete system spanning $L^2(V)$, as the many basis used by physicists are (plane waves, eigenstates of Hamiltonians and any complete system). May 24, 2019 at 15:57
• Right, but later on you're going to encounter problems where you have to use the expansions of the state in multiple bases. In that case, distinguishing the underlying physical state in $L^2(V)$ from its multiple possible expansions in $l^2$ is important. May 24, 2019 at 16:04
• @eyeballfrog Ok, thanks I did not know about this and you just cleared up a doubt. Anyway do you happen to know something which could help me even notes or something: I mean you seem skilled May 24, 2019 at 16:13
• You could look at the Reed and Simon series. Google Books has excerpts (vol. 2) including Contents of other volumes. May 24, 2019 at 16:47

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

• That's not the book you want to look at. In general it is a terrible idea to require mathematical rigor in physics. I would instead recommend Galindo and Pascual's Quantum Mechanics published by Springer. I believe this is what the OP is looking for, it provides the reference to math but focus on the physics. Mar 22 at 4:05
• @Simplyorange could you elaborate on "In general it is a terrible idea to require mathematical rigor in physics"? Mar 29 at 17:34
• Physics cares about different things than mathematics. If you insist on mathematical rigor you will get nowhere in physics. Consider your example of Hilbert space in QM, mathematicians focus on things such as rigged Hilbert spaces, distributions, Schwartz functions etc. But these are the wrong things to worry about. In QM you rarely know what the Hilbert space is precisely except for the most trivial case such as free particles or hydrogen atom. The Dirac notation works perfectly fine, no need to invent some other complicated unwieldy math to attempt to justify it that works only in rare cases Mar 29 at 22:55

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.

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.

• I mean, I am really interested in the physical interpretation, I chose this course for this very reason, but it seems to be buried under a load of calculations. And I am really interested in the foundations of the model, rather than its developments, as an example, I got very interested when the Heisenberg formalism was introduced, much less when we treated some.atomic model. May 24, 2019 at 16:27
• The "load of calculations" is pretty much the point of the course. You really can't grasp the physical interpretations until you are comfortable with the calculations. All the great minds of the 1920's to 40's that were debating physical interpretations were first really well steeped in the calculation aspects. Heisenberg was quite expert at the Schrodinger hydrogen model, and DIrac equation scattering theory -- even though he felt the more insightful interpretation was his viewpoint. May 25, 2019 at 6:33
• I was puzzled by the impression received that QM fields were not mathematical fields. I'm pretty sure that QM fields have addition and multiplication operations, necessary associative and commutative behavior (modulo Abelian and non-Abelian characteristics) and identity elements for each.
– DWin
Nov 25, 2019 at 1:46
• The QM concept of a field is more similar to the math concept of a map from physical space-time to the complex numbers or to members of a spin or Clifford algebra. The notion of multiplying two electron fields is not well-formed, and although one might be tempted to associate the inverse of an electron with a positron, the math does not work that way. BTW, the physics community "got there first" -- the Faraday/Maxwell notion of an electromagnetic field considerably pre-dates the use of "field" that mathematians use today. Nov 28, 2019 at 2:51
• If you want an association of physics "fields" with a math structure, the right one is that what physicists call a field is in many cases what mathematicians call a fiber bundle. Nov 28, 2019 at 2:52