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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.)

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Hi Qiaochu, Try Roger Penrose's great volume Road to Reality: a Complete Guide to the Laws of the Universe. It is meant to be popular science, but written in a completely rigorous way and take you far and deep into the cutting edge mathematical physics. Also it's a great way to build physics intuition without bogged down by the formalism.

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  • $\begingroup$ I've tried to read that book. I got distracted by the mathematics (in particular, the calculus on manifolds) and never actually got to the physics. $\endgroup$
    – Zhen Lin
    Commented Feb 24, 2011 at 23:34
  • $\begingroup$ Glad someone actually tried to read it. My favorite part is that he explains GR quite well, with all the physics motivations that can be found in say the book GR for mathematicians of Loring and Tu. $\endgroup$
    – John Jiang
    Commented Feb 28, 2011 at 18:52
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I would highly recommend Shankar's "Principles of Quantum Mechanics", a book from which I learnt quantum mechanics in 12th grade (no, this says nothing about my abilities, but instead goes to show how fantastically good Shankar is in explaining the principles of quantum mechanics to a high school student interested in self-studying it before college). This, followed by Feynman's Volume III, were my first taste of quantum mechanics before I even started undergraduate studies in physics.

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As far as I can tell, this has not been suggested yet. I very highly recommend Quantum Theory, Groups and Representations: An Introduction by Peter Woit. It's available for free here. It is extremely well written and takes a fully mathematical (or as close as it can get) approach to QM and QFT. It is quite comprehensive and emphasizes symmetry groups and their representations.

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Charles already posted Takhtajan's book, which is my first choice --- it's geared towards early graduate students. A more elementary book, geared towards math undergrads, is by Fadeev (who is, incidentally, Takhtajan's adviser, and Takhtajan's book can be understood as some sort of sequel).

The book Mirror Symmetry by Hori et al. includes a whirlwind overview of quantum mechanics and quantum field theory. It's not mathematical in the sense that it contains no proofs and even fewer definitions, but it doesn't require any particular background.

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For a short, well-written and fairly basic introduction to QM you can check Gillespie's classic "A Quantum Mechanics Primer". You won't need any previous background.

If you want to get a grasp / read a summary about what the book has to offer, you may visit my post:

Gillespie’s A Quantum Mechanics Primer (I)

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I agree with the comment saying take a course if you can.

That said, for self study, I found Bohm's quantum theory (dover ; cheap) very usable, given an engineering undergrad education. It is old fashioned, and very verbose (perhaps too much), but also doesn't skimp on the math.

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Matt Leifer once told me people should start (for a "conceptual" background) with John Polkinghorne's Quantum Theory: A Very Short Introduction. Then, to delve into the concepts in depth, I would recommend getting (when it is released in a month or two) Q-PSI: Quantum Processes, Systems, and Information by Ben Schumacher and Michael Westmoreland (Cambridge U. Press). I've been using it to teach undergrad. QM for a bit over a year. I would supplement it with J.J. Sakurai's Modern Quantum Mechanics (not Advanced Quantum Mechanics - they're completely different books) and, to truly appreciate a different viewpoint, Quantum Paradoxes: Quantum Theory for the Perplexed by Yakir Aharonov and Daniel Rohrlich.

If you get interested in the quantum information side of things (which is highly mathematical), pick up Protecting Information: From Classical Error Correction to Quantum Cryptography by Susan Loepp and Bill Wootters and the "bible" of QI, Quantum Computation and Quantum Information (known popularly as "Mike and Ike") by Michael Nielsen and Isaac Chuang.

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I happen to know about what appears to be a most appropriate book to teach quata to mathematicians : DANIEL T. GILLESPIE : A QUANTUM MECHANICS PRIMER, An Elementary Introduction to the Formal Theory of Non-relativistic Quantum Mechanics. Open University, England, 1973, ISBN 0 7002 2290 1. I taught quanta from it to several generations of mathematicians.

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David Mermin wrote a quantum textbook with a similar audience in mind (no diff. eqs. required): http://www.amazon.com/Quantum-Computer-Science-David-Mermin/dp/0521876583

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Quantum Mechanics and Quantum Field Theory: A Mathematical Primer.

I haven't read it (yet), though, but, given the Preface, it was written specifically for math students seeking a mathematically-rigorous introduction to QM and QFT.

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There are two recent texbooks that are very beautiful, full of pictures, if very unorthodox. They use category as foundation, and only treat finite-dimensional quantum mechanics. They are most suited for quantum information theory and philosophy and foundations of quantum mechanics.

  1. Bob Coecke and Aleks Kissinger, Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning (Cambridge, United Kingdom ; New York, NY, USA: Cambridge University Press, 2017).
  2. Christiaan Johan Marie Heunen and Jamie Vicary, Categories for Quantum Theory: An Introduction, Oxford Graduate Texts in Mathematics, 28, First edition (Oxford ; New York, NY: Oxford University Press, 2019).

First book is very slow and easy both in its physics and math. Second book is brisker in math, and more suited for mathematicians.

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There is a wonderful book by David Morin called "Introduction to Classical Mechanics". This book treats mechanics in a problems-oriented format; worked exercises dominate the book, making it very hands on. This was written for the honors freshman mechanics class at Harvard, and the math is simple but rich. It is basically a mix of differential equations and Euclidean geometry (and physics) and yet are problems there to challenge everybody. The book moves efficiently and he covers Lagrangians by Chapter 6. He explains the least action principle in a simple and natural way using only basic calculus.

In my opinion, you can learn quantum mechanics straight from a variety of sources, but it will seem unmotivated. Then again, Fenynman says "I think I can safely say that nobody understands quantum mechanics."

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Quantum Mechanics, Enrico Fermi's work is helpful.I think mathematicians study quantum mechanics,to learn how physics is processed with math,the way and the insight physicists do work with math,and the physical interpretation as well.It is an excellent book although the content is somehow oldfashioned.

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