This fall I’m teaching on quantum mechanics for mathematicians, at the undergraduate level. There’s a web-page with more information here. I’ll be writing up lecture notes, which should appear on that web-page as the course goes on, starting Wednesday.

We’ll see how this works, but the plan is to teach many of the standard topics, although starting from a different point. Most quantum mechanics classes start out with classical mechanics, then somehow try and motivate quantum mechanics from there, following the historical logic of the subject. I’ll instead start with the simplest purely quantum systems, especially the two-state, spin-1/2 system, now famous as the “qubit” of quantum computation. This is also a central example for the theory of Lie groups, Lie algebras and representations, so something that every mathematician should become familiar with. Another advantage of starting here is that there’s no analysis, just linear algebra, and one can easily do everything rigorously.

Later on in the course I’ll get to the standard material about wave-functions and quantum particles in potentials. The emphasis will be though not on the analytical machinery needed as a rigorous foundation for this subject in general, but on specific problems and their symmetries, and the use of these symmetries to do real calculations, ending up with the spectrum of the hydrogen atom.

We’ll see how this goes, and what the students think of it. As lecture notes appear, corrections and suggestions of how to improve them would be appreciated.

Peter,

You might want to check out anything you can find about the course Herb Bernstein has been teaching at Hampshire since the early 70’s – I think Lee Smolin took it when he was there. It assumes no math at all that I remember, and no physics either, but he does QM from the operator perspective, rather than the way I (and I assume most people) first learned it. He was of course aiming at a different audience (many if not most of the students were first or second year), but it might be interesting for you to check out. Not sure what if anything has been written about it, pretty sure Herb is still at Hampshire though he must be close to retirement at this point, plus he has his institute.

There is a good reason standard QM courses and textbooks start off as they do. The intended audience (physics undergraduates) have 2-3 years of experience with classical dynamics — Newton, then Lagrange and maybe Hamilton — also electrodynamics and relativity, also optics and statistical mechanics. It makes perfectly good sense to talk about blackbody radiation and the photoelectric effect. It also makes perfectly good sense (and is very good physics) to teach students that the equations of classical physics were derived on the basis of macroscopic objects, and now they must learn that there is no reason for thos equations to be true at the atomic level, and the fact is that new ideas (not just new equations) are needed. It is a cautionary tale that we should not assume that the equations we know apply all the way down to the smallest, and also the largest, length scales. It is the business of a physicist to recognize that there may be new physics in domains of energy, momentum, length, time, whatever, which have not been explored yet. QM at the atomic level is an example of this. Feynman has a good piece about this in the Feynman Lectures on Physics, although I cannot cite the exact page reference from memory.

It makes sense to begin with a “free particle”, then a “particle in a box”, then a “particle hitting a barrier” (includes tunneling through a barrier) then a harmonic oscillator and then the hydrogen atom. These are all concepts the students can relate to, at some level, with material in their prerequisite courses. They can solve some partial differential equations. They have already solved Laplace’s equation and the diffusion equation in many problems. But Lie groups? How many physics undergrads know about Lie groups? SU(2)?

This seems like a good perspective for mathematicians, but I really enjoyed the comparison of classical physics with QM when I took the intro to QM class. I am guessing most math majors have some exposure to basic physics problems so it’s probably OK to discuss some of these great examples (potential wells, harmonic oscillators) for a good intuitive basis of comparison between NM (Newtonian) and QM.

Peter,

Feynman did a similar experiment in Feynman lectures vol III treating spin/rotation as central before moving onto Schrodinger’s equation. Interested in your comments on his approach.

Good luck!

Paul

qm for physicists,

Undergrad math students rarely know anything about Lie groups or SU(2) either. Part of the idea of the course is to teach the theory of SU(2) and its representations, assuming no knowledge of the subject. No intention to get into the general theory of Lie groups.

Paul Wells,

I’ve gotten some inspiration from Feynman’s lectures and his starting this way. He has all sorts of wonderful material about the physics of spin, and I guess was trying to get students familiar with that as a way of getting into the subject. I’ve heard this wasn’t very successful, perhaps partially because it’s very challenging stuff to absorb. Unlike Feynman, I’ll be concentrating more on the mathematics (which Feynman was kind of avoiding) hoping that if students get comfortable with the mathematical ideas in this simple context, it will help them in further study of math, as well as in understanding these physical systems.

You should take a look at Schumacher and Westmoreland’s book if you have not done so already http://www.amazon.com/Quantum-Processes-Information-Benjamin-Schumacher/dp/052187534X

Hi Professor Woit, I have little physics knowledge but very high level math skills–would I survive this course?

Peter,

If you want a good undergrad resource for Lie Groups, check out http://www.amazon.com/Lie-Groups-Introduction-Mathematical-Association/dp/0883857596/ref=sr_1_4?ie=UTF8&qid=1346719859&sr=8-4&keywords=pollatsek

I took algebra with Harriet as an undergrad, and she did Lie Groups the second semester, I think the book is basically taken from her teaching notes.

The starting point seems very nice. It’s for mathematicians rather than physicists, so a historical “why we needed this” view seems pointless. As a coder without any real education in QM but yet an interest in what’s happening in physics, I’ve been thinking about reading some basics of quantum computing to get an idea of what “quantum” really means.

You should look at

Julian Schwinger, “Quantum Mechanics” Springer

My introductory QM course at McGill started off the with the Stern Gerlach experiment. We used the textbook by townsend which didn’t get to the wave function until chapter 6.

Hello Dr. Woit,

Any way to audit this course through the internet?

Warm regards.

Postscript:

A very useful reference:

Intermediate Spectral Theory and Quantum Dynamics (Progress in Mathematical Physics)

Link: amazon.com/Intermediate-Spectral-Dynamics-Progress-Mathematical/dp/3764387947

@ JollyJoker : having an historical point of view is never pointless especially in Quantum Mechanics. Even if the course is geared towards students of mathematics, giving them an idea of why the things are the way they are in QM is a good way to teach the subject. Then you can go on exploring exotic topics, or even formulate QM is a purely group theoretic way. Remember QM is not mathematics, it is physics so in ultimate analysis physics provides the justification for mathematics not the other way around.

Thanks to all for the book suggestions, most of which I wasn’t aware of. The Townsend book that David mentions (A Modern Approach to Quantum Mechanics, John S. Townsend) looks quite good, starting as I’d like to try with spin. The book that Matt Leifer mentions also (like most books oriented towards quantum information theory) starts with spin, looks very interesting, although headed more in a different direction than the course I’m planning to teach. JeffM, that looks like a very nice intro book on groups and Lie algebras. I do though want to emphasize much more the representations, which that approach doesn’t cover.

Emma Woodhouse, Sadiq Ahmed,

The only prerequisite for the class is calculus/linear algebra. In particular no physics will be assumed (although for some later topics, some physics background would be helpful). I’ll be writing up notes as I go along, anyone who wants to is invited to try following along as those appear. We’ll see how far they get before I run out of steam.

One thing I’m also thinking of doing is providing a much shorter, summary version of the notes, one aimed at mathematicians with a lot of background. This would contain no explanations of the math, just explanations of the physics in terms of the math, much of which would be a translation effort.

I really think this is the best way of teaching quantum mechanics. The approach that starts with wavefunctions and infinite-dimensional Hilbert spaces is just too mathematically complicated for a student seeing the subject for the first time. The only problem with this simpler approach is that there aren’t many interesting physical systems. Besides spin-1/2 particles (and particles with higher spin), what systems can you study using only finite-dimensional Hilbert spaces?

When I was first learning quantum mechanics, one of the things that bothered me was the arbitrariness of the postulates. I think you can minimize the number of unmotivated postulates if you take a more mathematical approach like the one described here. For example, in the C*-algebraic formalism, you start by postulating that the observable quantities in a quantum mechanical system correspond to the self-adjoint elements of a unital C*-algebra. From this one assumption, you can derive the most of the theory. For example, it follows from the operational meaning of the word “state” that the states of a quantum system correspond to functionals on the C*-algebra. You can then recover the Hilbert space using the GNS construction, and you can derive Born’s rule from the Riesz representation theorem. You can also get Schrodinger’s equation from Stone’s theorem on one-parameter unitary groups.

If I were teaching a course on quantum mechanics for mathematicians, that’s how I would do it.

Bob Jones,

Well, you can take tensor products…. You could also do the SU(3) classification of states using flavor in particle theory, but that’s way beyond this kind of course. The theory of spherical harmonics and thus the classification of atomic orbitals by angular momentum quantum numbers is also essentially finite-dimensional.

At least for a course at this level, I don’t want to try and set up general theory, just work out the parts of the theory and examples where most everything comes down to representation theory (i.e. not do general potentials, just quadratic and 1/r, and maybe set up perturbation theory).

Bravo, Peter! If there’s

anythingI’m confident about in life, it’s that starting with qubits is the right way to teach quantum mechanics. I tried to set out the reasons in my Quantum Computing Since Democritus Lecture 9. Basically, yes, it can be great fun to keep students in suspense for an hour or so (piling on one strange phenomenon after another) before finally letting them in on a great secret (in this case, that Nature prefers the 2-norm over the 1-norm when doing probability theory). But it seems excessive to drag out such a shaggy-dog story for an entire semester!Two other sets of course lecture notes people might find helpful, if they’re interested in this approach to quantum mechanics, are Umesh Vazirani’s (from his undergrad course “Qubits, Quantum Mechanics, and Computers”) and John Preskill’s (from his graduate quantum computing course).

Peter,

Thanks for your response. Those are nice examples.

A pretty good book which formulates quantum mechanics in the way I described is

http://www.amazon.com/Introduction-Mathematical-Structure-Quantum-Mechanics/dp/9812835229/ref=la_B001JO5VSM_1_2?ie=UTF8&qid=1346783227&sr=1-2

Scott,

Thanks. I’ve already had a link to the Preskill notes, will look at the Vazirani ones.

I liked your lecture trying to motivate quantum, especially the QM as OS line. Still though, I think the answer to the “why” questions is something like “because unitary representations of groups are a fundamental unifying structure that is all over mathematics, so why shouldn’t it be fundamental for physics?

Excellent Peter, am looking forward to your notes!

May I suggest that you may want to introduce your students to the profoundly illuminating work of the late Itamar Pitowsky? I would in particular suggest the immediate relevance to your purposes of the fascinating and highly readable article

“George Boole’s ‘Conditions of Possible Experience’ and the Quantum Puzzle”

Brit. J. Phil. Sci. 45 (1994), 95-125

Here one finds what is probably the most penetrating and revealing account ever written of the reasons for the existence of (and general methods for constructing) “Bell-type inequalities” — which are, of course, badly violated in the real world of elementary particle interactions.

In its essentials, Pitowsky’s exposition is readily accessible to students with the background you assume, and I find it hard to imagine how any other approach could ever hope to equal his in clarifying the precise sense in which quantum phenomena are so extraordinarily bizarre.

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If you’re going to start with spin then be aware of the non-relativistic origin of spin famously argued by Levy LeBlond in this 1967 paper , (free download from Project euclid) and discussed by Walter Greiner in chapter 13 of his introductory Quantum Mechanics book: Quantum Mechanics: An Introduction

(more pages viewable with ‘Look Inside’ feature at amazon)

(Greiner’s text follows the more traditional introductory course)

Hi Peter,

Sakurai’s book “Modern quantum mechanics” also starts with a description of spin-1/2 particles and Stern-Gerlach apparatus, and is a pretty good book on the whole (as you probably are aware).

Peter again slightly OT, but would be curious to know your talk on the latest HEPAP

meeting

http://science.energy.gov/hep/hepap/meetings/20120827/

(apologies if I missed it)

Sakurai’s book is probably much too difficult for an introductory course.

Shantanu,

Thanks for the link, I didn’t know about that. But, seems to be not much news. For one thing, US federal budget prospects are likely to be up in the air until after the election.

The tell on Feynman Volume III is that the first year students (knew some) were lost, but since then many senior (was one) and graduate level quantum courses start with it to introduce the subject. I have a friend who used it as his only text for studying for and passing qualifiers. There is no doubt that its approach provides a much more sophisticated appreciation than the usual grind it out of PDEs one.

That being said, there is a strong argument that linear algebra is a much more important course for physicists than diffeq (which mostly looks like a bag of tricks to physicists) and PDE (a bag of even more tedious tricks).

For what its worth, I took quantum as a senior using Feynman III and loved it. In grad school the quantum course was taught by a Schwinger student and I found it tedious, but understandable given the foundation I had.

What is a good free online reference for the class? When does the class start?

A LaTeX suggestion: when writing your brackets don’t use > and } % bra

\newcommand{\bra}[1]{\left< #1 \right|} % ket

\newcommand{\braket}[2]{\left} % braket

Some examples:

\ket{\Psi}

\bra{\Phi}

\braket{\Phi}{\Psi}

Zen,

Thanks a lot for the suggestion, will try it out.

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