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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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I tried a book called "Quantum mechanics for Dummies" but they had a bid different definition on "dummies" than I have. –  Per Alexandersson Dec 16 '12 at 20:59

41 Answers 41

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.

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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.

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I concur. Missing out on classical mechanics before QM is sort of like learning algebraic topology before calculus. It's doable but you miss out on much of the context and motivation for the subject. I suppose there are people that pick up formalism quickly, but speaking only for myself, if I see formalism without a compelling historical context, without some sense of the ambient context that the creator of the idea had, more often than not the ideas just wash by me. –  Ryan Budney Nov 6 '09 at 8:38
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Math packaged as "physics" sounds alright to me! –  Harry Gindi Jul 10 '10 at 8:07
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I wholeheartedly agree with the idea that you should not begin with an overly mathematical course - basically all the essential features of QM can be learned in the context of quantum information and finite-dimensional systems, without delving into the subtleties of functional analysis. This way you will get a clearer picture of fundamental concepts, not obscured by technicalities. –  Michal Kotowski Dec 4 '10 at 16:54
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I firmly disagree. I am a physicist who has studied classical mechanics extensively, and researches quantum mechanics for a living. Classical mechanics is mostly irrelevant. There's a quick review of what I consider a good approach to classical mechanics in Terry Tao's blog: terrytao.wordpress.com/2009/11/26/… But nevermind classical mechanics. If you want to learn quantum mechanics, learn quantum mechanics. For that, I recommend Nielsen and Chuang's superb Quantum Computation and Quantum Information. –  Mateus Araújo May 12 '11 at 4:06

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.

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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).

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I think Noah refers to David Griffiths's "Introduction to quantum Mechanics" (Pearson/Prentice Hall), which seems indeed to be be one of the most widespread textbooks in American universities. –  Georges Elencwajg Oct 28 '09 at 8:55

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.

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I think the thickness of the book is a good thing. Shankar explains concepts well, and there is a large focus on examples. It is a self-contained, axiomatic treatment of quantum mechanics, and it contains a review of classical mechanics, so it's a great introduction to the subject. Plus, it's a good read. –  Dave Penneys Oct 28 '09 at 3:42

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".

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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.

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

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Seconded (and was going to be my answer). I like it better than the physics-y books (Sakurai, etc.). –  Theo Johnson-Freyd Oct 28 '09 at 0:08
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Skimming through this on Google Books... looks great! –  Reid Barton Oct 28 '09 at 1:09
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Yeah, it's very mathematical, and it gets to SUSY a bit. I worked through a good chunk of it before attempting to tackle Folland's QFT book. –  Charles Siegel Oct 28 '09 at 2:10

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.

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I really disagree here. I love Shor's algorithm as much as the next guy, but there's a ton of great quantum mechanics that has nothing whatsoever to do with quantum information. Go work out how to estimate the ground state energy of a beryllium atom first, I'd say. –  Scott Morrison Oct 29 '09 at 3:27

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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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.
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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.

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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.

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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.

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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.

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I realize this is unrelated to the question at hand, but does anyone know if there is a math equivalent to 't Hooft's page that Scott links to here? –  Justin Curry Jan 13 '10 at 1:20

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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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.

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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).

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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.

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The above links appear to be dead now. I found the following online, and I'm guessing it's by the same guy: sites.google.com/site/winitzki/sergei-winitzkis-files Scroll down on the page to find the 4 sets of QM notes mentioned above. –  user5768 Dec 19 '11 at 0:43

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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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.

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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.

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I haven't read this, but while the hydrogen atom is a wonderful place to start, it's not where most of today's mathematicians will want to! It's great for learning about spherical harmonics ... I initially learnt a lot of "quantum chemistry" (ie, learn everything there is to know about hydrogen, then helium, then berrylium, etc) even before I knew much maths. I thought it was fun at the time, but I think I wouldn't enjoy it now! –  Scott Morrison Nov 6 '09 at 1:26

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

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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!


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.


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

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I absolutely love the last sentence :-). –  Barbara Jul 10 '10 at 14:30

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.

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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.

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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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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'.

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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.

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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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