Where does a math person go to learn quantum mechanics?

Question

My undergraduate advisor said something very interesting to me the other day; it was something like "not knowing quantum mechanics is like never having heard a symphony." I've been meaning to learn quantum for some time now, and after seeing it come up repeatedly in mathematical contexts like Scott Aaronson's blog or John Baez's TWF, I figure I might as well do it now.

Unfortunately, my physics background is a little lacking. I know some mechanics and some E&M, but I can't say I've mastered either (for example, I don't know either the Hamiltonian or the Lagrangian formulations of mechanics). I also have a relatively poor background in differential equations and multivariate calculus. However, I do know a little representation theory and a little functional analysis, and I like q-analogues! (This last comment is somewhat tongue-in-cheek.)

Given this state of affairs, what's my best option for learning quantum? Can you recommend me a good reference that downplays the historical progression and emphasizes the mathematics? Is it necessary that I understand what a Hamiltonian is first?

(I hope this is "of interest to mathematicians." Certainly the word "quantum" gets thrown around enough in mathematics papers that I would think it is.)

Answer 1

It could be just my own personal bias, but I think it is difficult to learn quantum mechanics without first learning classical mechanics. I recommend taking a 1 semester course, either graduate or advanced undergraduate, in classical mechanics and then taking a quantum mechanics course. I also think it would be a mistake to start with an overly mathematically-oriented QM course. You want to learn how physicists think and how they use this stuff to come up with real physical predictions. Otherwise, you're just learning math packaged as "physics". You shouldn't have much trouble later figuring out how to translate the physics back into math. But if you focus too much on the math at the beginning, you make it less likely you'll ever understand the physics.

Answer 2

Takhtajan's fairly recent "Quantum Mechanics for Mathematicians" should suit the bill.

Answer 3

I've just finished teaching the first semester of a year-long "Quantum Mechanics for Mathematicians" course. Some of the references I found most useful are

1. A good, clear, physics textbook. Shankar's "Principles of Quantum Mechanics" that many have mentioned fits the bill.

2. Faddeev and Yakubovskii, "Lectures on Quantum Mechanics for Mathematics Students" is short and to the point. Takhtajan's "Quantum Mechanics for Mathematicians" is at a higher level that I was aiming for, but quite good.

3. The first 60 or so pages of Folland's "Quantum Field Theory" are an excellent introduction to physics in general and QM in particular (and the rest of the book is a great QFT textbook).

Finally, I should point out that I've put up my course notes, which try to cover basic QM from a representation theory point of view, at the lowest level possible, they're here.

Answer 4

I second what Noah says. You don't have to take mechanics first, just linear algebra. I would recommend reading Shankar's Principles of Quantum Mechanics. It's better than Griffiths' book for a math person.

Answer 5

I think there are some excellent recommendations above. I learned quantum mechanics for real from Shankar, I think it's a great choice. Griffiths is also a great physics text. I would also recommend these following less famous books:

Physical chemistry and materials science textbooks. I would also highly recommend newer textbooks in physical chemistry as a perhaps less obvious place to look for excellent introductions to quantum mechanics - as Dirac famously said once, it's really the foundation for all of chemistry. An excellent physical chemistry is Physical Chemistry by Berry, Rice and Ross. Presumably there are also good introductions in materials science books, although I don't have any to recommend.

Not Feynman. In my opinion Feynman's Lectures in Physics is great for insight, but it's a terrible idea to learn anything from it the first time - remember that when Feynman actually lectured, most of the freshmen and sophomores (the intended audience) dropped the course, and were replaced by senior students!

Weyl (group theory). I'm surprised no one's mentioned Hermann Weyl's textbook "Theory of groups and quantum mechanics". It's an oldie but goodie, and perhaps best appreciated with someone with a good background in group theory.

Lieb (analysis). I recommend Elliott Lieb's Analysis GSM textbook - on the surface, it looks like it's about functional analysis, but it's secretly also a text on quantum mechanics!

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There are some subjects that none of the introductory quantum mechanics texts I've read ever do a satisfactory job of explaining, and I think are really worth following up after Shankar or another such book. The most important ones I think are:

• Many-body phenomena. This is really where some of the strangest predictions of quantum mechanics come from, like the EPR paradox and spin statistics. Levine's Physical Chemistry is an excellent place to start. Another great book is Blaizot and Ripka's Quantum Theory of Finite Systems, which does a superb job with boson and fermion statistics.

• Dynamics (time-dependent quantum mechanics. I cannot recommend Tannor's Introduction to Quantum Mechanics: A Time-dependent Perspective enough as a really fantastic resource for learning how practicing physicists and chemists actually do these calculations, beyond the really simplistic calculations presented in most introductory texts. That could also work as a first textbook.

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You know, I'm in the building next to you. Maybe you should come by and talk sometime. :)

Answer 6

A great starting point are the lecture notes for physics 219 at Caltech:

http://www.theory.caltech.edu/people/preskill/ph229/#lecture

If you pick up any QM textbook that doesn't contain "no cloning theorem", "entanglement", "EPR (Einstein-Poldolsky-Rosen)", "threshold theorem", "quantum cryptography", "Shor's algorithm", or the word "Channel" in it then it is out-of-date. (Similarly, don't take a QM course that doesn't have most of these words on the sylibus.)

The quantum information community has really made abstract quantum theory interesting in the last 20 years or so. 20 years ago it would seem heretical that one could study quantum theory from an abstract point of view without specific interest in a particular physical system. Now there are at multiple papers on the quant-ph arXiv every day that do just that. The surprising fact: there are lots of interesting research problems available even in the finite-dimensional Hilbert space case. ("Interesting" here means that solutions of such problems appear quite frequently in Physical Review Letters, Journal of Mathematical Physics, Communications of Mathematical Physics, Nature, ect.)

Note: QM can get technical much faster than classical mechanics does. If you want to actually analyze atoms besides hydrogen then it's going to be a quite a while before you can do it rigorously. It's analysis. Reed and Simon's "Methods of Mathematical Physics" books are a must if you want to go in that direction.

BTW, if you want to understand what a Hamiltonian is, go get a hold of Arnold's GTM book on classical mechanics. It is clearly the best.

Answer 7

I wonder why no one mentioned von Neumann's Mathematical Foundations of Quantum Mechanics. It is a little outdated and contains some minor errors but besides that it is one of the best books i have ever read. I would especially recommend it for people who study mathematics since it's written from this point of view and not in the sometimes strange physicists language (aka no bra-ket notation, no inexact use of mathematical facts etc). Since he is using (and for the first time formulating(!)) the theory of separable Hilbert spaces this is already accessible to anyone who knows linear algebra and even more so if you know some functional analysis. Most importantly von Neumann was a genius who could structure his thoughts in an extremely clear, highly logical fashion which makes it very enjoyable to read this book.

Answer 8

To me the answer to your question is clear:
Mackey's Mathematical Foundations of Quantum Mechanics either in conjunction with, or followed by Dirac's Principles of Quantum Mechanics.

After those perhaps Weyl's book `The theory of groups and quantum mechanics'.

Answer 9

Assuming that you know enough representation theory, I suggest Folland's QFT book right from the start. It covers what you need to know about classical mechanics in the first (or second) chapter, and then there is a chapter about non-relativistic quantum mechanics—this is the best introduction to the subject you will find around, since it explains in a rational way why the state space is a Hilbert space, why observables are represented as self-adjoin operators, etc.

In case your knowledge of representation theory is lacking, I suggest you take some time to learn more about it; it will pay off to see how spin comes out of rotational symmetries of the system, and later on, how to derive relativistic wave equations from relativistic symmetry (which is something you can find in the Weinberg book too, but Folland's explanation is better for a mathematician, modulo learning the Mackey machine).

If you are willing to study some C*-algebras, another suggestion is Strocchi's An Introduction to the Mathematical Structure of Quantum Mechanics. It does not assume any background in physics, but it is quite heavy on the math side (it is used in a third year course in QM for mathematicians at the SNS of Pisa.) It does present the subject from what is usually referred to as the algebraic point of view, that is, you assume, after some considerations, that you can get a C*-algebra of observables out of the set of all measurement instruments. This algebra is commutative for classical mechanics, while for quantum mechanics it is not (this underlines the fundamental role of the uncertainty principle in quantum mechanics).

Answer 10

I really enjoyed "Road to Reality" by Roger Penrose. He wrote it for "general audience", and for a while it was selling in the big bookstores like Barnes and Noble, but I suspect that his definition of "general audience" is what most mathematicians mean: graduate level sophistication without research experience in the subject.

Answer 11

I took undergraduate quantum mechanics as the only Physics class I took in college. You're in better shape knowing linear algebra well and not knowing any physics than you would be the other way around. So one option is just to take a normal quantum mechanics course or read a normal undergrad text (we used the one with Schrodinger's cat on the cover).

Answer 12

I disagree (though not particularly strongly) with the comments claiming that you should learn classical mechanics first. You don't need much physics background to learn to do basic calculations with wave functions, and to pick up ideas like observables-as-differential-operators. If you want a list of good web resources and textbooks together with good organizing commentary, I recommend looking at Gerardus 't Hooft's page: Theoretical physics as a challenge. You can safely ignore the paragraphs in the beginning about his page being only for people who want to win the Nobel prize in physics.

Answer 13

I think learning QM makes only sense if one knows the basics of Lagrange- and Hamilton-Jacobi-mechanics, where I found the old but very beautifull and readable "Classical Mechanics" by Goldstein very good. It needs no preknowledge and covers the development of early QM from that too. The formalism in classical mechanics described there in a very nice way is THE fundamental piece on which field theory and quantum field theory is built on. After that it may be usefull to look into Hatfield's "Quantum Field theory of point particles and strings" which Witten recommends. Then there is a tourist guide and lectures from an IAS summer school on that.

Answer 14

It doesn't stress the physical intuition very much, but I learned a lot of basic quantum from the first chapter of Nielsen and Chuang. If you're planning on doing quantum computing or QIT stuff, you'll likely need the book anyway, although there's probably better resources if you're not.

Answer 15

I agree with Scott firmly here. The Lagrangian and Hamiltonian formulation of mechanics is a beautiful subject with immediate doorways to symplectic geometry, but unnecessary for the appreciation of the basic postulates of QM ("The Universe is a vector space..."). Honestly, if Qiaochu wants to appreciate this clean formulation and not worry about "perturbative expansions of blah blah blah," this is sufficient.

I am also a huge fan of Griffith's "Introduction to Quantum Mechanics" as a first book. It can be read without serious knowledge of electromagnetism and classical mechanics. I know that the MIT physics undergraduate curriculum does not require Lagrangian/Hamiltonian Classical Mechanics before their 3 term QM sequence. This means for Qiaochu that 8.01 and 8.02 is sufficient, 18.03 is more important, and I have heard of many a MIT student taking 8.05 without 8.04, but don't expect a physics professor to say that to you. (My apologies for the MIT-speak)

The problem with learning "grown-up" mechanics (at MIT 8.07 or Sussman's 6.946J, which I recommend highly) before QM is that this path leads more naturally to understanding more differential-geometric concepts and consequently takes serious time. This is fine for a physicist and is probably a wise move before taking GR, but for a growing mathematician, I would advise understanding smooth manifold theory before trying to learn the more sophisticated and elegant approach to mechanics. This inevitably entails a much more firm grounding in ODE theory and I recommend V.I. Arnold's ODEs book for that.

Once you have these two perspectives in hand you can then sit back and wonder how the QM Universe-as-a-Hilbert-space viewpoint and the GR Universe-as-a-C^2-manifold description can ever be reconciled.

Answer 16

I recommend against Shankar, he is not conceptually very careful.

One can, in fact, learn the foundations of QM without knowing Classical Mechanics at all, and I recommend reading the first six chapters of Dirac, The principles of quantum mechanics, to get a feeling for the physical concepts. although he is not mathematically rigorous, he never even clearly defines what space his operators are operating on, this is an advantage for once.

Then, in order to get the same conceptual clarity from a mathematician, and better mathematical rigour, I always recommend Anthony Sudbery Quantum Mechanics and the Particles of Nature.

If you are reading QM for "culture", as I gather you are, the books which work well as intro texts for professionals are really irrelevant, so forget Sakurai, Gottfried, Landau, et hoc genus omne. I except Feynman, but only after reading the above two, you could benefit from the chapters in his undergrad Lectures on Physics and benefit from his book on Path Integrals and QM (joint with Hibbs).

I recommend against Mackey or von Neumann, they have no physical insight, Varadarajan is even worse.

Answer 17

Dan Dugger has wonderful mostly-finished notes on this here:

http://pages.uoregon.edu/ddugger/qftbook.pdf

It's written for topologists, very clearly, and he does a great job of giving both physical and "mathematical" explanations/intuitions.

Really well done.

Answer 18

For a good introduction requiring minimal background you could try D.J. Griffiths, Introduction to Quantum Mechanics, but keep in mind that it is aimed at physics or engineering students.

Also a good book is Landau and Lifshitz, Quantum Mechanics, but you also need to read a at least a good part of Mechanics, by the same authors in order to understand it.

Answer 19

A concise introduction to quantum probability, quantum mechanics, and quantum computation (36 pages) by Greg Kuperberg: http://www.math.ucdavis.edu/~greg/intro-2005.pdf

Answer 20

Take a class! I took the core graduate Quantum Mechanics class offered by the Physics department at my school (using Sakurai's book), and it was one of the best classes I ever took.

Answer 21

To sum up and structure what others have said (and add my grain of salt):

a. there are a few core "quantum" concepts to learn. The normal route follows the historical one up to the 1960s (skipping too physical considerations for you), and this is indeed done well in Takhtajan's book (chapters 1 to 5). It does go through classical mechanical aspects, and requires some knowledge of undergrad math (linear algebra, Hilbert spaces, multivariate calculus, ODEs & PDEs, and a little differential geometry).

Of course be aware that this reference still does skip a lot of issues that physicists are aware of, a good physics textbook for beginners is Physics of Atoms and Molecules by Bransden and Joachain. To which one must add more recent aspects, not covered neither there or in Takhtajan, from the 1980s-1990s as in Preskill's notes. Beyond that there is field theory and ever more physics, but the basic stuff is in those three references.

b. you can then take any one mathematical aspect and push it very far. It could be the geometry, or functional analysis, or representation theory, or semiclassical limit, or complexity theory... Then it's not quantum mechanics per se anymore, but explains why some objects are labelled quantum, or studied in a certain way.

c. don't be blinded by the math if you talk to physics or engineering friends: it can look all very neat but yet a key thing to know is that most quantum systems just cannot be solved explicitely , which is why physicists introduce all sorts of approximations and asymptotics, and use computers. It's not about lack of rigor, it's dealing with non-solvable systems (e.g. only the H atom and H2+ ion can can be solved explicitely, already the He atom has a too little symmetry group to do that).

Answer 22

1) I !ike Winitzki's style a lot. He is a mathematical physicist of Russian origin and seems to write in a way that will appeal to you. His homepage is

http://homepages.physik.uni-muenchen.de/~winitzki/

Here are the first four relevant handouts on Quantum Mechanics

http://homepages.physik.uni-muenchen.de/~winitzki/QM_notes_1.pdf

http://homepages.physik.uni-muenchen.de/~winitzki/QM_notes_2.pdf

http://homepages.physik.uni-muenchen.de/~winitzki/QM_notes_3.pdf

http://homepages.physik.uni-muenchen.de/~winitzki/QM_notes_4.pdf

2) An element of the tensor product of two vector spaces is not necessarily a tensor product of two vectors, but sometimes a sum of such. This might be considered a mathematical shenanigan but if you start with the state vectors of two quantum systems it exactly corresponds to the notorious notion of entanglement which so displeased Einstein (not to mention Podolski and Rosen). This is one of the most beautiful facts I have ever learned. So you are absolutely right in wanting to study Quantum Mechanics: it is incredibly exciting, beautiful and intellectually rewarding.

Answer 23

This is a question, not an answer, but: Has anyone here read Singer's "Linearity, symmetry and prediction in the hydrogen atom"? I'm a sucker for a good example, and it seems like this could be a good grounding for the daunting task of understanding quantum mechanics.

Answer 24

Igor Dolgachev's course notes "Introduction to Physics" starts with an introduction to classical mechanics and develops quantum mechanics from a mathematical point of view. It's a good place to start if you're strong mathematically but are having a difficulty understanding physicists' notations and perspectives.

Answer 25

There's quite a variety of resources available for learning quantum mechanics. Since you're interested in the mathematical perspective, my suggestion would be to start with one of the mathematically-oriented quantum field theory books.

Physics courses about quantum mechanics contain many "real-world" applications which may be of less interest to someone without the previous background.

Quantum field theory is really what is used in all mathematics constructions you keep hearing about, and it's possible to learn the necessary quantum mechanics on the way.

In a major research university, you also have quite a number of options of finding interesting people and asking them directly. Though the courses in the physics departments would be structured for physicists, a lecturer in one of those courses could be a good place to get loaded with references.

Let me note that the physics and mathematics departments in Cambridge, Massachusetts, have a reputation for constructing effective bridges between physics and mathematics. Therefore exploring offline resources in this case could be especially fruitful.

Answer 26

When I was in college, the books I was taught from were Sakurai's "Modern quantum mechanics" and Townsend's "A modern approach to quantum mechanics". Though, they are clearly physics books, they both take a non-historical more mathematical approach starting with the two-dimensional vector space of a spin 1/2 particle. They don't assume any classical mechanics, neither Newtonian, Lagrangian or Hamiltonian. The first chapter of Sakurai for example gives a nice intro to the basic ideas of quantum mechanics. If what you're interested in is quantum field theory from a mathematical point of view, I'd suggest Folland's recent book "Quantum field theory". Also, if you decide to go for classical mechanics from an advanced mathematical perspective, you could try Abraham and Marsden's "Foundations of Mechanics".

Answer 27

Try this book:

• Geometry of Quantum Theory. Varadarajan, V.S. This is a book about the mathematical foundations of quantum theory. Its aim is to develop the conceptual basis of modern quantum theory from general principles using the resources and techniques of modern mathematics.

Answer 28

I got a lot out of Anthony Sudbery's "Quantum Mechanics and the Particles of Nature: An Outline for Mathematicians". My favourite part was solving for the energy levels of the hydrogen atom using a bit of linear algebra and not a single partial differential equation. Beautiful! I remember the discussion of spin being particularly lucid too (and which you may find easy depending on how good your representation theory is). You may be able to find it in a library but it's hard to buy. It's not a hard-core pure math axiomatic approach to QM. It assumes undergrad stuff like linear algebra, really elementary differential equations and maybe a tiny bit of of group theory.

Answer 29

It does not seem that anyone has mentioned Hannabuss's An introduction to quantum theory . Although I have personally not lectured from it, it seemed good to us when we considered a quantum theory course for maths undergraduates in Edinburgh. (Alas, there was opposition from the physicists and the course ended up not being given.) At this level quantum mechanics is an application of linear algebra, so well suited for mathematics students. Having said that, I echo others' answers where they suggest trying to develop a "physical" intuition as well.

Answer 30

I had taught myself Quantum Theory from the excellent book by J.J. Sakurai called "Modern Quantum Mechanics". Its a great book to build the conceptual foundations strongly but lacks examples.

Then I had read the 2-volume book on Quantum Mechanics by Cohen-Tanoudji et. al. This book will give enormous number of examples of using Quantum Theory especially about using two-state system formalism.

It is extremely detailed to say the least and very clear, though at times lacks the elegance of the book by Sakurai. In terms of contents it subsumes the books by Shanker or Sakurai many times over.

The trickiest part of Quantum Theory is to get grips on the theory of angular momentum and spherical harmonics and Wigner's D-matrices. This is masterfully covered in the classic book by Edmonds. The pinnacle point of this journey is to understand and use Wigner-Eckart Theorem which Sakurai also does pretty well.

One can also look at the book by Giamarchi and the books by Frank Wilczek for some sophisticated issues. Especially about understanding the subtleties that happen when one tries to do Quantum Theory in 1 and 2 spatial dimensions.