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Quantum Good books on QM with rigorous math

  1. Jun 11, 2015 #1
    I want to learn QM but in a way that i can learn both the deep mathematics, the physical intuition behind QM and the intuition behind the maths behind the QM(not many books have this).
    Note that this is my first attempt to learn QM and i will be watching some video lectures online,so i won't be on my own!
    I do not want something on graduate level,but i want something that is an introduction but on a more advanced undergrad level with rigorous math,emphasizes on intuition and-if possible- it also has some intuition on the mathematics that are or should be used.
    Thank you
     
  2. jcsd
  3. Jun 11, 2015 #2

    verty

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    For stuff like this, where there are many books and many opinions, I always think it is best to go with what most people like. And in my time at these forums, the one I've seen most liked is Shankar. Just putting it out there.
     
  4. Jun 11, 2015 #3
    Shankar is a great book as far as intuition goes,but it doesn't differ from other popular books like Griffith's as far as mathematics go.
     
  5. Jun 11, 2015 #4

    George Jones

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    Be careful with the term "rigorous mathematics", as my guess is that you don't really mean this. My guess is that you want suggestions for texts that would be suitable for third- and fourth-year physics majors.
     
  6. Jun 11, 2015 #5
    Yeah,something like that.Sorry..
     
  7. Jun 11, 2015 #6

    George Jones

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    Actually, if both Griffiths and Shankar are unsuitable, now I am a bit confused. You could try

    Taking the term "rigorous mathematics" very seriously, there is "Quantum Theory for Mathematicians" by Hall. This looks to be a good book, but it requires substantial background in pure mathematics (e.g., real analysis), and it omits some of the most interesting bits of quantum theory, e.g., entanglement.
     
    Last edited by a moderator: May 7, 2017
  8. Jun 11, 2015 #7
    I will check Townsend's book out for sure.Hall's book might not be what i am looking for though.I will check that out too.Thank you!
     
  9. Jun 20, 2015 #8
    You kind of get to pick and choose how rigorous you want to be. Obviously there are a slew of "modern physics" books that teach simplified quantum theory using algebra and calculus. To learn it "properly," though, you probably want to learn it in terms of abstract Hilbert spaces and linear operators. Unfortunately, there's a big difference between using those things and using them rigorously. At an introductory level, there are a bunch of books like Shankar, Miller, Zetilli, and Griffiths. These books use the Hilbert space formalism, but tend to sweep a lot of technicalities (like delta functions) under the rug or hand-wave them away. If you want to get a little more rigorous after reading that level of book, Ballentine (particularly chapter 1) seems to be a good way to go. Unfortunately (or fortunately), Ballentine just sort of points you in the right direction and doesn't really get into the details of spectral theory. I think the books by Galindo and Pascual are a little more rigorous, but probably similar to Ballentine. If you want to be really rigorous, you'll need to learn some functional analysis from proper math books (e.g., Kreyszig for introductory and the series by Reed and Simon for more topics you'll need), and you'll probably want to learn about Rigged Hilbert Spaces (the thesis of de la Madrid is pretty good; also, the original sequence of books on generalized functions by Gelfand is still one of the best references on the subject).

    As you can probably tell, I went down this rabbit hole myself a while ago. To make quantum mechanics very rigorous is actually quite difficult if you're not a math major. You end up having to learn a very long sequence of difficult concepts. (And that's not even getting into the group theory stuff or quantum field theory, neither of which I'm very familiar with). Part of the problem is that you start getting to a point where all the references are written by mathematicians for mathematicians, and they get very difficult to read if you don't have training as a mathematician. I've pretty much given up at this point, and I just trust that someone has proved that the things we do actually work.
     
  10. Jun 20, 2015 #9
    If this is your first time tackling QM I would definitely go with Griffiths. When you say "mathematically rigorous" I take it that you don't want to go digging around in the joyous land of Functional Calculus and infinite dimensional vector spaces since that kind of thing tends to be best handled AFTER you've already got a grip on how to do basic QM (wave mechanics and Schrodinger Equation).
    Griffiths is an excellent pedagogue and you will quickly learn the rudiments and techniques applicable at your level. You can then progress to something like Liboff, Eisberg and Resnick or Brandsen and Joachain - these will all give you a broader range of quantum mechanical systems to attack. You will also learn Dirac notation.

    Once you're pretty in depth hit J.J Sakurai's "Modern Quantum Mechanics". This will give you a proper algebraic and rigorous treatment. However if you've never done QM it will make no sense to you unless you've done a maths degree with a significant component of the relevant material.
     
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