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What branch of mathematics to study to understand Quantum Mechanics?

  1. Dec 13, 2011 #1
    I just got a book on Quantum Mechanics, but at a loss with the maths. I had Algebra, Trigo, Calculus, differential equations, etc, but that was a long time ago, now as a programmer, I hardly ever think of those and forgotten them as a result.

    I believe I still have strong enough concept of algebra and trigonometry but what branch of mathematics would be most suited to understanding and applying quantum mechanics? Thank you!
  2. jcsd
  3. Dec 13, 2011 #2
    It really depends on the level of understanding you want.

    If you don't mind handwaving, then Linear Algebra and Calculus is almost enough to be able to start with books like that of Ballentine for example.

    The one from Galindo and Pascual is quite in the middle (in rigour).

    If you want full mathematical rigour, then you will need a lot of Functional Analysis (among several other branches of Mathematics) to be able to start with books like those of Reed and Simon, or Kadison and Ringrose. Gerald Teschl has a quite small and lovely book as well, and Strochi has another lovely one too.

    All this speaking about Non-Relativistic Quantum Mechanics.

    By the way, Arnold Neumaier wrote an excellent (in my opinion) book trying to highlight the Algebraic Structure of the subject, a way of better grasping the differences and similarities with Classical Mechanics.

    Quantum Field Theory is quite a different "monster".
  4. Dec 13, 2011 #3


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    Complex numbers, linear algebra (linear operators, eigenvectors and eigenvalues, matrices, the relationship between linear operators and matrices, orthonormal bases) and some calculus (in particular partial derivatives). People always mention differential equations in these threads (yes, there are lots of them), but I have never considered that to be particularly important since the QM book will tell you how to solve the equations that you need to be able to solve.

    This is to understand the standard non-rigorous presentation of QM. To understand a rigorous presentation of QM and the associated mathematics, you need to study topology, functional analysis and a little abstract algebra. This is hard as **** (it could take years), and most physicists don't do it.
  5. Dec 13, 2011 #4


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    You should learn Hilbert-space theory and some elementary representation theory for Lie groups and algebras.
  6. Dec 13, 2011 #5

    Vanadium 50

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    It probably also depends on the book you're trying to understand.
  7. Dec 13, 2011 #6
    Linear algebra is probably the most important thing. Group theory can help. Lie algebras and Lie groups might also be relevant.

    As far as differential equations, it's nice to have a good understanding of the classical harmonic oscillator (and coupled harmonic oscillators) before you study the quantum one. Partial diff eq may help.

    Also, it's not just a matter of math. It also helps to know classical mechanics (Lagrangians and Hamiltonians), and maybe the physics of waves.

    But, you have to tell us what book you are trying to study because there are many different approaches to QM. Also, it might not be the best idea to rely on one book, especially if you are somewhat unprepared.
  8. Dec 14, 2011 #7
    Calculus and linear algebra at the very least. If your interested in applied stuff, you may want to bone up on differential equations as well. Although nowadays it's easier to solve DEs with Mathematica than with pencil and paper.
  9. May 24, 2012 #8
    Sorry for the very late reply, but thanks a lot guys!! Linear Algebra, calculus, and DEs amazing!!

    I'm trying to learn QM for practical applications, mostly for energy and propulsion applications. Atm, I'm working out solutions for energy use and I keep getting defeated by classical assumptions.

    My enthusiasm for more dramatic applications is due to my engineering background, intense interest in aviation/space flight, and inventing, mostly on RC aircraft, sustainability and improvising in adverse situations. I might team up with physics Phd buddy but still had to learn these.
  10. May 24, 2012 #9
    If you want an easier introduction where just a decent understanding of calculus will probably be enough to get you through, try David Griffiths Intro to QM text.

    You'll still get pretty good coverage of the field and then probably be ready for a more advanced QM text.
  11. May 24, 2012 #10


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    Which book? Some people here may be familiar with it and can suggest what you need to study in order to understand it. Different books take different approaches and have different mathematical prerequisites.
  12. May 25, 2012 #11


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    I already had a degree in math where I did two subjects on advanced analysis with Hilbert spaces, Lebesque Integration etc. But even that was just the behginning - it took me 10 years of part time study to come to grips with Rigged Hilbert Spaces, Lie Algebras, Group Theory, Noethers Theroem and other stuff to really get a proper mathematical ubderstanding of what was going on.

    Based on that my reccomendation would be a standard QM book like Griffiths, The Structure And Interpretion Of QM by Hughs and QM A Modern Development by Ballentine. A good source for the math involved is Mathematical Physics by Geroch.

    Last edited: May 25, 2012
  13. May 25, 2012 #12
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