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Rigorous Quantum Mechanics text or online

  1. Apr 27, 2012 #1
    Hi everybody, I'm taking Quantum Mechanics 1 this semester. Thing is many of my classmates and I feel the professor is horrible and not a good teacher. He is often disorganized and unprepared for class. Problem is there's no way to change him so we're stuck with him.

    He apparently wants to teach the course focusing on the mathematical formalism and I've been having a difficult time finding texts which resemble what he is doing just so I could follow his class. I really want to understand this subject as it is important for a Physics Major.

    Can anyone recommend any rigorous quantum mechanics textbooks? Any online courses or video lectures would be awesome as well.

    By the way, he's been talking about Hilbert space operators, functions on operators like exp(A) where A is an operator and continous and discrete basis (which I did not understand at all). Material which covers those topics would be greatly appreciated.
  2. jcsd
  3. Apr 27, 2012 #2
  4. Apr 27, 2012 #3

    Jano L.

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    Hello danjordan,
    welcome to PF.

    There is no single perfect book, even the best suffer from omissions/inaccuracies so it is important to read more books simultaneously, from authors with different perspective, for the sake of comparison and equilibrated view (what one author mystificates other one explains and vice versa).

    Very highly praised recently published book is

    Sakurai, J.J., Modern Quantum Mechanics.

    You will find bra kets, "coordinate representations" and delta-functions there. I quite like it but it is rather too brief on some important things outside particle physics.

    Complementary parts closer to original development of the theory (spectroscopy, atoms and molecules) are covered with distinct clarity and in the book

    Slater, J. C. (1968). Quantum Theory of Matter (2nd ed.). New York: McGraw-Hill.

    I really recommend this one. It focuses on physics instead of abstract notions and symbols and also gives informed comments and references to original papers. I wish I knew this book sooner.

    Older books and papers by the contributors to the subject are also very interesting and valuable. Search

    van der Waerden, Sources of Quantum Mechanics, (discontinuous view of the theory)

    E. Schroedinger, Four Lectures on Wave Mechanics (continuous wave view), and also Papers on Wave Mechanics (for more details).

    Very good textbook (explains discontinuous view) is also

    D. Bohm, Quantum Theory.

    This is very carefully written book full of original and interesting comments.

    Also very famous book is
    P.A.M. Dirac, Principles of Quantum Mechanics, (formal analogies with classical mechanics, quite formal approach based on noncommutativity of variables, introduced bra-kets).

    Hope that helps,

  5. Apr 27, 2012 #4
    The "Mathematical Tools..." chapter in Zettili's Quantum Mechanics: Concepts and Applications should serve you well. And in my opinion, Dirac is too outdated. A better reference would be Ballentine's book which is often highly recommended in this forums.
  6. Apr 27, 2012 #5


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    Which textbook are you using now? Has the professor required or recommended any textbooks, or is he relying completely on his own materials?
  7. Apr 27, 2012 #6


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    There are no rigorous QM textbooks. (I think I've read in this forum that the textbook that's the closest to rigorous is the one by Galindo and Pascual). The mathematics of QM is far too difficult for a typical physics student. Typically, you would have to supplement what you know with a course on real analysis, a course on topology, a course on measure and integration theory, and then probably two courses on functional analysis. A typical physics student would also need to study linear algebra again.

    If I understand your needs correctly, then I think what you should do is to buy a copy of Axler's "Linear algebra done right" and study that on your own. It can certainly help you understand inner product spaces and linear operators much better.

    Most of the stuff that the previous posters recommended seem completely wrong for you (sorry guys), assuming that I have correctly understood what you're looking for. (It seems to me that you want to understand the mathematical foundations, not learn how to solve problems that aren't covered by the course you're taking now).

    I also recommend "Lectures on quantum theory: Mathematical and structural foundations" by Chris Isham. It doesn't go into the details of the rigorous mathematics, but it explains what the theory says very nicely. It's short, cheap, and easy to read. It can't replace a standard textbook, but it's an excellent supplement.
  8. Apr 27, 2012 #7
    I think people have misinterpreted the OP, because they took the word "rigorous" a bit too seriously. I don't think the OP really wants to understand the mathematical foundations of QM in the sense of functional analysis and the like. Rather, he seems to just want to understand QM with the level of rigor typical for an advanced undergraduate or graduate quantum mechanics course. That means working with bra's and kets, linear operators, continuous and discrete eigenstates, exponentials, etc. So I think an appropriate quantum mechanics book with things like that would be either the graduate textbook "Modern Quantum Mechanics" or, if that's too hard, the undergraduate textbook "A Modern Approach to Quantum Mechanics" by Townsend.
  9. Apr 27, 2012 #8


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    I agree that Sakurai's "Modern quantum mechanics" presents the basics (eigenstates and stuff) pretty well. It's very far from rigorous though.

    I don't know why people say it's a "graduate" text. It's appropriate for a second course on QM, but it doesn't assume that you know anything from the first course.

    For the OP: A book or article at "the level of rigor typical for an advanced undergraduate or graduate quantum mechanics course" typically does everything as if the methods we use to deal with finite-dimensional Hilbert spaces work without modification with infinite-dimensional vector spaces too. A vast majority of physicists don't know the mathematics of QM any better than this. Note that you need to understand (rigorous) linear algebra pretty well just to understand the mathematics of QM at this (non-rigorous) level, so I still think that's what you will need to study.
  10. Apr 27, 2012 #9
    Well, it is intended as a text for a second course in quantum mechanics, but it demands a level of mathematical maturity generally expected of graduate students. In contrast, a text like Townsend's will hold your hand while introducing the mathematical formalism of QM, giving concrete examples before diving into abstract notation.
  11. Apr 27, 2012 #10


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    Hm, I don't really agree about the mathematical maturity. It requires more maturity than you have when you start at the university, that's for sure, but I'd say that the maturity of someone who has studied calculus (with at least a few hard proofs) and linear algebra should be sufficient. To me "graduate" means fifth year, but I'd say that a third year student should definitely be able to handle it.
  12. May 23, 2012 #11
    Hi everybody, thanks for the many responses and sorry for the lateness in mine (midterms have taken much of my time). I'll be checking out the suggestions you all made.
    To clarify, the teacher has recommended many books but doesn't follow the progression of anyone in particular (which means we have to check out many books for any one topic :( ). Something that's a little annoying is he keeps referencing Griffiths book saying it's all there but there none of the math he is using.

    right now we're looking at representation theory and it's kicking me in the head right now :)
  13. May 23, 2012 #12
    danjordan, is this an undergraduate class or a graduate class?
  14. May 23, 2012 #13


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    You might get more constructive replies if you posted some of the specific points (and the math) you're having trouble with. So far, your complaints seem rather general and vague.
  15. May 24, 2012 #14
  16. May 24, 2012 #15


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    Get Ballentine's - QM - A Modern Development. The first 3 chapters contain all you need to lay the foundations including a much better development of Schroedinger's equation etc than I have seen elsewhere (it based on its real foundation - Galilean invariance - like much of physics symmetry is what really underlies it). Quite rigorous too.

    But if you want real rigour see Von Neumann - Mathematical Foundations of QM - be warned though - rigorous it is, and the math is not too difficult, but his approach excludes the usual formalism used by physicists which uses Dirac Delta functions etc. The forward is an interesting read to see what a genuine mathematician thinks of the approach usually used by physicists.

  17. May 24, 2012 #16
    I think the OP stated things in a misleading way. He doesn't want to master rigorous QM at the level of Ballentine. He just wants to understand his professor, who is just introducing a little more rigor than Griffiths, the standard undergraduate intro to QM. So the topics being covered are things like discrete and continuous eigenstates, the exponential of an operator, the connection between symmetry and conservation laws, all presumably at a level that is relatively elementary compared to Ballentine and his discussions of central extensions, rigged Hilbert spaces, and the like.
  18. May 24, 2012 #17


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    In that case I recommend the Structure and Interpretation of QM by Hughs:

    Excellent companion to a first course in QM IMHO. Good companion to Griffiths. While Griffiths is an excellent introductory textbook it is the lets try and learn how to solve some problems type text and leaves many important questions unanswered - if you are the type that worries about that sort of thing then it can confuse. Hugh's fills the gap.

    In fact I think Griffiths and Hugh's is a good preparation for the real deal in a book like Ballentine.

    I spent at least 10 years pulling my hair out about mathematical issues in QM - this maddening Dirac Delta Function and saying its based on a Hilbert space when obviously it wasn't - then there is all these axioms - yet Dirac pulled them out from analogies with Poisson Brackets - something else was obviously going on - the issues went on and on. I read countless books in the local university library about it and gradually, very gradually, what was really going on emerged. If I had read those three books in that order it would have made things a lot easier.

    Aren't that the truth :rolleyes::rolleyes::rolleyes::rolleyes:

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