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Continuous eigenvalues

  1. Sep 12, 2010 #1
    Dear all,
    in basic QM books the position and momentum operators (continuous eigenvectors) are introduce by means of the dirac delta and some analogies are made with the infinite dimensional, but discrete case in order to provide some intuition for the integral formulas presented. My knowledge is limited, but it seems to me that thee formulas apply in the distribution sense. Can anybody explain to me how the position and momentum operators are treated in a rigorous way? An outline will do.

    Thanks for any help!

    Goldbeetle
     
  2. jcsd
  3. Sep 12, 2010 #2
    This is not an easy subject. The keywords to search for are "http://en.wikipedia.org/wiki/Rigged_Hilbert_space" [Broken]" and "Gelfand triples"
     
    Last edited by a moderator: May 4, 2017
  4. Sep 12, 2010 #3

    Fredrik

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    You need the RHS if you insist that these operators should have "eigenvectors", but you can also just accept that they don't, and study a spectral theorem for unbounded operators in a book on functional analysis.
     
  5. Sep 12, 2010 #4
    Remember that infinities in physics are simply shorthands for limit procedures. Add to this a physicist's lack of need for mathematical rigour, and loose reasoning with infinities, and you should conclude that whenever such things arise, you should start looking for what that limit procedure actually is...
     
  6. Sep 13, 2010 #5
    Fredrik: the second option you mention is what Geroch does in chapters 55-56 of his book "Mathematical Analysis"?

    To everybody: thanks!
     
  7. Sep 13, 2010 #6

    Fredrik

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    Yes, that looks like the sort of stuff I had in mind. I still don't know much of it myself, so I don't have an opinion about Geroch's presentation. My own plan for learning the rigorous treatment of both bounded and "not necessarily bounded" operators, is to study "Functional analysis: Spectral theory" by V.S. Sunder, but I've been planning to do that for a long time now.

    (A comment mainly for other people reading this: Geroch's book is titled: "Mathematical physics", not "Mathematical analysis").
     
  8. Sep 14, 2010 #7
    That's for the correction, Fredrik! By the way, the book you mention is available online for free on the author's webpage.
     
    Last edited: Sep 14, 2010
  9. Sep 14, 2010 #8

    George Jones

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    Another very readable exposition of functional analysis and spectral theory is given in chapters 1 - 3, 7 - 11 of Introductory Functional Analysis by Erwin Kreyzig.

    For a rigourous overview of rigged Hilbert spaces (Gelfand triples) and Dirac notation, I recommend highly sections 11.2, 11.3, and 12.2 from Quantum Field Theory I: Basics in Mathematics and Physics (A Bridge Between Mathematicians and Physicists) and subsection 7.6.4 from Quantum Field Theory II: Quantum Electrodynamics (A Bridge Between Mathematicians and Physicists) by Eberhard Zeidler.
     
  10. Sep 14, 2010 #9
    George,
    thanks. I have the book Kreyzig and it's extremely readable. I'm also reading in parallel two introductions that make use also of Lebesgue integration.

    I'll check out Zeidler's material because it looks very promising.
     
  11. Sep 16, 2010 #10
    Fredrik, George: the material you suggested is wonderful! Thanks again!

    George: have you read also the two books on functional analysis by Zeidler? How are they?
     
  12. Sep 16, 2010 #11

    George Jones

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    No, I haven't looked at Zeidler's functional analysis books.
     
  13. Sep 18, 2010 #12
    The first place you should always check for questions of this nature, in my opinion, is Landau. If you are not aware of it, Landau wrote the books on every subject in theoretical physics and his proofs are often quite rigorous and beautiful. In this case, the reference you are seeking appears only on page 15 of vol3, 3rd edition, section 5, "the continuous spectrum". In this section, all of the relevant details are discussed exhaustively.
     
  14. Sep 18, 2010 #13
    I would dare to say that Landau does not warn the reader strong enough that formal manipulation with "eigenstates" for the continuum spectrum can lead to apparent paradoxes. In my opinion he should. Many other textbooks and course of QM are similarly guilty.
     
  15. Sep 19, 2010 #14
    calhoun137 and arkajad: thanks!
     
  16. Sep 25, 2010 #15
    George,
    sorry to bother you again about this, but I'm lost....
    Where exactly in Zeidler's QFT I 12.2 is the proof of the completeness relation for continuous eigenvalues?
    Thanks a lot!
    Ciao.
    Goldbeetle
     
  17. Sep 25, 2010 #16

    George Jones

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    I don't have Zeidler home with me, but, if I remember correctly, Zeidler doesn't prove this. I think Zeidler gives a reference to Gelfand's nuclear spectral theorem. This is why I wrote "overview"; sorry.
     
  18. Sep 25, 2010 #17
    Thanks. If you have good references for Geldfand's spectral theorem, please post them.
     
  19. Sep 25, 2010 #18

    George Jones

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  20. Sep 26, 2010 #19
  21. Sep 26, 2010 #20
    There is a nice, short overview in Chapter 8.4 of "Nonrelativistic Quantum Mechanics" by Anton Z. Capri, World Scientific (2002). Chapter 8 is about distributions and Fourier transforms.
     
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