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Quantum Physics and mathematics advice wanted

  1. Apr 2, 2005 #1
    Hi

    A huge problem I'm finding with my study of QM is that its shrouded in maths and I'm becoming really bogged down in trying to understand things like hermite polynomials, assosiated legendre polymials, bessels and laguerres equations and so on. In the process, I feel that I'm loosing sight of the big p[hysical picture.

    I've got exams coming up soon and whilst I really want to understand all the mathermatics behind the physics, can I gloss over or it or accept such concepts as fact, if I want to get a holisitc understanding of the subject?

    Thanks in advance
     
  2. jcsd
  3. Apr 2, 2005 #2

    dextercioby

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    Most books treat the maths (special functions & ODE-s in QM) pretty well,while only a few really emphasize the physics behind equations...If you don't have such books,then it's up to you to do that,to extract physicsal information from a solution to an ODE...

    Of course,that depends solely on you,your will and the time you have & u wanna dedicate to studying QM properly.

    Daniel.
     
  4. Apr 2, 2005 #3
    Thanks Daniel (also for your help for that maths Q).

    Its not just special functions… I mean to give you an idea of what I’m having difficulty with: simple things like how quantum numbers are interdepent… fiddling with vector idenites in spherical polars to get equations which are a pain to normalise and so on, I often forget what we’re trying to do in the first place. And I’m not sure how much I’m gaining from trundling through all the maths. That’s just one example of many but isn’t it enough to know the magnetic moment is 2L+1 degenerate and take integer values only?
     
  5. Apr 2, 2005 #4

    Dr Transport

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    All great pohysicists are good or excellent mathemticians, it is part of the territory. There is a certain level of competance that is neccesary and it just happens that special functions is part of it. The more you work with them, the better you'll get, I still go back and mess with them occasionally to keep in practice.
     
  6. Apr 2, 2005 #5
    and some great physicsists do have occasional spelling glitches.

    :tongue:
     
  7. Apr 3, 2005 #6

    selfAdjoint

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    You want spelling, go to a lit-crit forum. Einstein struggled with English syntax, but that didn't diminish his value. :frown:
     
  8. Apr 3, 2005 #7
    Help with Math in Quantum

    Hi,

    I joined this forum recently so I'm a little late in posting. For Math used in QM
    refer the 2-volume Dover book "Mathematics of Classical and Quantum Physics" by Byron and Fuller. Although Shanker's book is considered a grad level text you could still go to it for Hilbert Spaces. Remember that the notion of "completeness" occurs in two different contexts: as it relates to spaces and to functions.

    For worked examples, look up "Quantum Mechanics: Concepts and Applications" N. Zettili. For a good conceptual grasp and problems done at the intro. level you may want to refer Griffiths' QM book. Bransden and Joachain also go into some detail about spectra. For a more advanced treatment of spectra esp. applications to Chemistry, refer Cohen-Tannoudji et. al. (Most QM books have the title <some variation on QM>, so it's more efficient to search by the author.) Whenever the topic comes up with friends who were in my undergrad program I suggest to them that our dearly unlamented prof who taught the subject should title his QM text "QM made difficult".
     
    Last edited: Apr 3, 2005
  9. Apr 4, 2005 #8

    dextercioby

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    A very rigurous text (though not by mathematicians) is Gallindo & Pascual "Quantum Mechanics" (2vol-s),Springer Verlag.


    Daniel.
     
  10. Apr 4, 2005 #9

    Dr Transport

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    Thanks selfAdjoint, my fingers do not work as well as my mind.....
     
  11. Apr 4, 2005 #10
    One of my profs told me that Niels Bohr was so dyslexic that Bohr got someone else to type his Ph.D. thesis.
     
  12. Apr 4, 2005 #11

    dextercioby

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    Maybe,but he was still a brilliant physicist.:wink:

    Daniel.
     
  13. Apr 4, 2005 #12

    robphy

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    One thing that bothered me when I was first learning these subjects is that "the physics" and "the math" were (to me) all jumbled together. More specifically, it was hard (for me) to distinguish the "physical theory and its mathematical structures" from the "specific applications with their own specialized mathematical techniques" (e.g., the various special functions).

    What was helpful (to me) was to find alternative textbooks with more emphasis on the underlying mathematical structure of the physical theory. One book I like is "Mathematical and Conceptual Foundations of 20th Century Physics" by G.G. Emch (ISBN 0444875859) [check your local library: http://www.worldcatlibraries.org/wcpa/ow/0247483c16f88d04a19afeb4da09e526.html ]. In my opinion, a book like this gives one way to look at the big picture. Note: this does not work out the details of, e.g., the hydrogen atom. For that, you should look in the standard textbooks.
     
  14. Apr 5, 2005 #13
    QM is really difficult because it uses a great amount of mathematics. This includes real analysis, differential equations, linear algebra, complex analysis + special functions, calculus of variations, fourier analysis (fourier series, fourier transforms and abstract harmonic analysis), functional analysis (hilbert spaces, linear operators and spectral theory), probability + statistics, set theory + logic, abstract algebra (with a fundamental emphasis on group theory) and topology (which is also abstract).

    I think the rest comes from practice and intuition...
     
    Last edited: Apr 5, 2005
  15. Apr 5, 2005 #14

    dextercioby

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    Yet most people really hate the interrpretation and not the formalism...

    Daniel.
     
  16. Apr 5, 2005 #15
    i gettit......the darned s :tongue:
     
  17. Apr 5, 2005 #16

    ZapperZ

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    What you have pointed out is exactly the kind of issues that I tried to address in one of my "So You Want To Be A Physicist" essay[1]. I devoted an entire discussion on mathematical preparations, and why it isn't the most efficient setting to have the student learn the mathematics at the same time as he/she is learning the physics. As has been proven here, one CAN lose sight of the physics because one is being bogged down in the mathematics. It is difficult enough to learn QM, but to learn the math at the same time makes it a daunting task. A QM class should not be the first time one hears the word "orthornormal".

    Note that the same can be said about E&M. The first 3 or 4 chapters of Jackson, for example, would have you burried in partial differential equations, Bessel functions, Legendre polynomials, etc. More often than not, E&M is difficult due to the math.

    I have mentioned Mary Boas text Mathematical Methods in the Physical Sciences more than once on here, but I'll mention it again. If you are an undergrad in physics, engineering, or any other physical sciences, GET THIS TEXT. I say that with ZERO hesitation. If you have had the full sequences of intro Calculus, and a bit of differential calculus, you are ready for this text. It is meant for students in their late sophormore into early Junior year, and can be used as a self-study aid. If you include the Students Solution Manual, you have a self-contained mathematical physics text that will guide you through practically all the mathematics you would need as an undergrad, and beyond (I still use this text even today, which is why my copy is in a sad condition).

    Zz.

    [1] ZapperZ "So You Want To Be A Physicist - Part 3" PF Journal [08-26-2004 06:50 AM].
     
  18. Apr 5, 2005 #17

    Haelfix

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    I always felt special functions were decidedly the easy part about learning QMs.

    I have some equation that I understand well, how it was derived and where it came from. I don't know how to solve it. Oh look, some mathematician has figured it out already in some completely unrelated context and just gives me the answer. Thats just memorization, at that stage of the game, and pretty easy =)

    Of course when you take some grad class in analysis or somesuch, or a good physics class taught by a more mathematically oriented physcisit you then learn where it comes from and appreciate the beauty a bit more (they're usually examples of a more general class of orthonormal functions called Sturm-Liouville functions)
     
  19. Apr 6, 2005 #18
    Haelfix, I heartily agree with your remarks about Sturm-Liouville functions.
    :cool: I wish that profs who teach Math methods (and even EM, QM and CM classes) really emphasize the connection between different types of functions, etc. Students often see these individual Math solution methods in terms of "tricks" or recipes to be used in isolation w/o seeing these approaches as part of a more unified scheme.
     
  20. Apr 6, 2005 #19

    dextercioby

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    Any Math.Phys. course not dealing with generalized hypergeometric functions is incomplete...


    Daniel.
     
  21. Apr 12, 2005 #20
    Mine came in the mail yesterday. Best $60 (or so, don't remember) I've spent in a long time.
     
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