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Rigged Hilbert Space's And Quantum Mechanics

  1. Oct 24, 2011 #1

    bhobba

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    Ages ago when I first learned about QM the textbooks all said QM was formulated in a Hilbert Space. Yet you had this Dirac Delta function and infinite norms. Got a hold of Von Neumans book and yea it resolved it but by using the Stieltjes Integral and resolutions of the identity however that is not what physics books used. Looks like I was stuck with either something that looked like a mathematical slight of hand or a rigorous treatment no one seemed to use.

    Then I heard about Rigged Hilbert spaces and the Generalised Spectral Theorem that solved the issue. Got a hold of some literature on it and it looked the goods.

    Only trouble is why don't the physics books do it that way? I have Ballentine - QM - A Modern Approach but its the only one I know of.

    Any ideas guys?

    Thanks
    Bill
     
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  3. Oct 24, 2011 #2

    dextercioby

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    There's no textbook written with full degree of rigorosity. I find Galindo & Pascual's 2 volume set the most satisfactory source. The literature on RHS is not too extended but I find it enough to fill the need of mathematical rigor. The only model which I haven't seen addressed by people who published on RHSs is a 3D one, namely the H-atom. I remember only some article, but very sketchy, by some guy Komy in the proceedings of the Trieste Intl. School of Theoretical Physics.

    Anyways, search for papers published by Arno Boehm and his disciples M. Gadella and especially Rafael de la Madrid. See also the sources quoted here: https://www.physicsforums.com/showthread.php?t=435123
     
    Last edited: Oct 24, 2011
  4. Oct 24, 2011 #3

    bhobba

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    I thought Von Neumann's book was pretty rigorous.

    Thanks for the shinny though. Do you mean Arno Bohm? - I know he wrote a textbook about the Rigged Hilbert Space formulation but it seems old and out of print.

    I am looking for a text to complement Ballantine. I did a bit of a search on Amazon and cant find anything about it. Any idea where you can get it from - I may fork out for it if its not too expensive.

    Thanks
    Bill
     
  5. Oct 24, 2011 #4

    dextercioby

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    Von Neumann's book IS rigorous, but he doesn't consider spectral equations and their solutions for arbitrary self-adjoint operators in Hilbert spaces. His solution to the spectral problems proposes POVMs and direct integrals but these 2 concepts are really hiding distributions as measures. Von Neumann's book was partly re-written by Reed and Simon's 4 volumes and by <Quantum Mechanics in Hilbert Space> (2 editions) by Eduard Prugovecky.

    It's Boehm, for I can't put the Umlaut on the 'o' from the laptop's keyboard. :biggrin:

    Anyways, if you don't afford/don't find Galindo & Pascual, then I reccomend A. Capri's book :<Nonrelativistic Quantum Mechanics> or even A. Boehm's own textbook. I hope it's cheaper and/or easier to find.
     
  6. Oct 24, 2011 #5

    bhobba

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    Again - thanks for the skinny.

    Thanks
    Bill
     
  7. Oct 24, 2011 #6

    strangerep

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    They usually just think of it as "Hilbert space with distributions and a delta-valued inner product". That's easier to think about, and faster to reach useful calculational techniques.

    It depends on what precisely you mean by "complement Ballentine"?
    Every paper I've seen by Arno Bohm and his students/collaborators stops short of an actual proof of the nuclear spectral theorem which underpins the whole approach. For that, you need a math book like Gelfand & Vilenkin vol 4 -- which is a challenging read.

    Maybe you should say what questions/issues you find unanswered in Ballentine's introductory treatment?
     
  8. Oct 24, 2011 #7

    bhobba

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    Well the proof of the generalised spectral theorem for one. However I have dug up some papers from the Internet that does prove it a while ago and have gone through the proof eg:
    http://www.math.neu.edu/~king_chris/GenEf.pdf [Broken]

    Its just I was hoping to have it all in one place.

    Ballentine is by far my favourite QM textbook but having a background in math I am always hoping for bit more mathematical detail.

    Thanks
    Bill
     
    Last edited by a moderator: May 5, 2017
  9. Oct 25, 2011 #8

    dextercioby

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    The paper you found is a copy-paste from one of the appendices of Berezin & Shubin's <The Schrödinger Equation>. It's based on Hilbert-Schmidt riggings, a concept developed by Berezanskii at the end of the 50's. Another proof you can find is in Maurin's book on Generalized Eigenfunctions (K. Maurin, Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups, Polish Scientific Publishers, Warsaw, 1968).
     
  10. Oct 25, 2011 #9

    bhobba

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    Man you know a lot of stuff off top of your head. Didn't know any of that, but in the case of a Hilbert-Schmidt rigging of a Rigged Hilbert space it does seem to justify the key idea.

    Thanks
    Bill
     
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