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Wave functions

  1. Aug 31, 2007 #1
    Why is it that when observing an electron or photon it causes the wave function to collapse, surely the photons that cause this collapse were still "colliding" with it when we wern't looking. Or does it only collapse the wave function from the observers viewpoint?

    (I'm abit of a noob so if there are formulae could you explain them, but I might be talking crud so tel me if I am)
     
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  3. Aug 31, 2007 #2

    dextercioby

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    There are fully credible formulations of QM in which the "collapse" of the wavefunctions doesn't exist. I'll let those believing in the existence of such "collapse" answer your question.
     
  4. Aug 31, 2007 #3

    ZapperZ

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    It is a postulate of quantum mechanics and is a result of the way it is described mathematically. In other words, your question is currently analogous to asking "Why is the speed of light always c in all inertial reference frame?"

    Zz.
     
  5. Aug 31, 2007 #4
    What you propose appears to me quite similar to the theory of decoherence.
     
  6. Aug 31, 2007 #5
    No, the universe splits into many copies.
     
  7. Aug 31, 2007 #6

    dextercioby

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    Incidentally at the link you mentioned it's not present among the given postulates. What and why is it a result of the way it(probably the quantum system) is described mathematically ?
     
    Last edited: Aug 31, 2007
  8. Aug 31, 2007 #7

    jtbell

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    "Observing" does not necessarily mean the same thing as "a human looking at something with the eye, using photons."
     
  9. Aug 31, 2007 #8

    ZapperZ

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    While those postulates are described in words, they are based on strict mathematical formalism. Terms such as "hermitian", etc. are solidly grounded in it. So to me, these postulates are nothing more than a bunch of mathematical statements.

    I think this is unlike SR, which truly states a "physical" situation. While you can translate SR's postulates into mathematical operations, I believe Einstein made them starting with physical intuition, which is why they are easier to comprehend. The QM postulates, on the other hand, makes no sense if one doesn't understand the mathematics, which ties in to my other assertion on why QM is so difficult to understand for most people.

    Zz.
     
  10. Aug 31, 2007 #9
    I think there are two issues with the wavefunctions collapse :
    - the mathematical formalism which I have no problem to accept
    - the physical interpretation which is subject to many interpretations

    Why such assertion ? Is it physics or metaphysics ?
     
  11. Aug 31, 2007 #10
    I hope this is not off topic to this thread. But is their anything in the algebra of wave functions that require them to be complex numbers? IIRC the usual development starts with the assumption that the wave function is complex, and then goes on to develop inner products and such. But I wonder if the algebra and the inner product of wavefunctions can be justified by other reasons, then does this mathematically require the wave function to be complex? Thanks.
     
  12. Aug 31, 2007 #11
    I think if you describe the wave by a cos(px - Et + phi) then you violate the relativity principle (the classical one from Galilei).
    If for one observer the wave cancels every L = pi/p
    For a second observer at speed V wrt the first, the wave cancels every L = pi/p' with p' = p + mV.
    I guess a similar argument can be found in relastivistic quantum mecanics.
     
  13. Aug 31, 2007 #12

    ZapperZ

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    Er.. solve the differential equation! That is what dictates the nature of your solution, i.e. the wave function. It just doesn't appear out of thin air.

    Zz.
     
  14. Aug 31, 2007 #13
    As I mentioned earlier, QM does not necessarily require complex wavefunctions. Example (Shroedinger, Nature (1952), v.169, p.538): for any solution of the equations of the Klein-Gordon-Maxwell electrodynamics (a scalar charged field \psi interacting with electromagnetic field) there exists a physically equivalent solution with a real (not complex) field, which can be obtained from the original solution by a gauge transform. Thus, the entire range of physical phenomena described by the Klein-Gordon-Maxwell electrodynamics may be described using real fields only. Shroedinger's comment: "That the wave function ... can be made real by a change of gauge is but a truism, though it contradicts the widespread belief about 'charged' fields requiring complex representation."
     
  15. Aug 31, 2007 #14
    This is a good question. One interesting way to formulate quantum mechanics (which I actually prefer) is to start from axioms of measurements and logic. This approach is called "quantum logic"

    G. Birkhoff, J. von Neumann, "The logic of quantum mechanics", Ann. Math., 37 (1936), 823.

    G. W. Mackey, "The mathematical foundations of quantum mechanics", (W. A. Benjamin, New York, 1963), see esp. Section 2-2

    It has been proven by C. Piron that these axioms imply a theory constructed in a Hilbert space over R (real numbers), or C (complex numbers) or Q (quaternions).

    C. Piron, "Foundations of Quantum Physics", (W. A. Benjamin, Reading, 1976)

    I haven't seen many works about R-quantum mechanics. Apparently it leads to some contradictions with experiment, though I am not sure exactly which. One old reference is

    E. C. G. Stueckelberg, "Quantum theory in real Hilbert space", Helv. Phys. Acta, 33 (1960), 727.

    There were many attempts to build a quaternionic quantum mechanics. It is known that in the 1-particle sector it is equivalent to the usual QM in C-Hilbert spaces. It is also known that the definition of the tensor product of quaternionic Hilbert spaces is rather tricky, so it is not clear (to me) how compound systems can be described there. A good reference about Q-quantum mechanics is

    J. M. Jauch, "Projective representation of the Poncare group in a quaternionic Hilbert space", in Group theory and its applications, edited by E.M. Loebl, (Academic Press, New York, 1971)

    It appears that our usual C-quantum mechanics satisifes all theoretical and experimental needs. That's the best theory we have at the moment.

    Eugene.
     
  16. Sep 1, 2007 #15
    Relativity theory is as guilty as Quantum Theory in reducing a physical situation into a mathematics - rubber sheet geometry (manifolds) vs Hilbert space state vectors.
    Time dilation vs collapsing the wave-particle duality.
    The common folks out there especially those who play with car engines have a hard time swallowing all that.
     
  17. Sep 1, 2007 #16
    The “collapse” is the experimental demonstration of the validity of the Spectral Decomposition Theorem, proven by D.Hilbert. Therefore, it tells us that the functional analysis is the adequate mathematical framework to describe quantum physics.

    However, the “postulates” presented at the suggested link only states that. If so, then one may claim that the use of classical analysis is the postulate of the Newtonian mechanics or the use of vector analysis is the postulate of Maxwell ED. I consider such statements absurd.

    I did not find any reason to believe in anything but my eyes (and others). Fortunately or unfortunately I look the results of the hydrogen atom spectroscopy and see the spectral lines. So far I am not familiar with any fully credible formulation of QM that demonstrates that they do not exist.

    Regards, Dany.
     
  18. Sep 1, 2007 #17

    Fra

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    I've also scanned some of those logic deductions and while they are interesting I've not yet seen one to my full satisfaction (but perhaps I'm too stupid). IMO, representation, it's parametrisation during changes and it's evolution is entangled up with each other. So the question IMO is as much, where you start. The complex phase is as far as I see it related to the notion of change.

    My own limited personal thinking along these lines, has lead me to conclude that this "logic alone" (at least what I have seen so far) doesn't seem to reach complete satisfaction. But I think there maybe on consistent way forward, if one consider an evolutionary perspective, which I'd like to call a "drifting logic", in the spirit of learning logic, if you add some constraints of the observers learning power (memory/mass/computing power/stability). The whole process is supposedly driven by real life feedback. So far at least I haven't found any fool proof papers in this spirit. But then I'm a crappy librarian, that's part of why I like this place... to see if other people has their attention of something essential.

    If anyone knows any state of the art papers in this field I would very much like to read it.

    /Fredrik
     
  19. Sep 1, 2007 #18
    I afraid my answers make situation with “the common folks out there especially those who play with car engines” even worse.

    The non-relativistic QM as well as the Newtonian Mechanics may be roughly considered as the empty stage, platform. In order to give it the physical content one should introduce the interactions. If you consider that on the fundamental level, so far we know four and only four fundamental interactions and you use the presence of the phases in the description of the QM system (local gauge invariance). The simplest case is QED (minimal coupling) which require one phase: U(1). That already show you why you need the complex numbers numerical system to comply with the experimental evidence. The detailed theoretical and experimental demonstration is given in the AB and A.Tonomura et al. papers. However, to describe weak (actually electroweak) interactions you need more phases: U(2). Even before electroweak, W.Pauli and P.A.M. Dirac “spoiled” everything through introduction of spin. They demonstrated that the complex numbers are not enough. Later we saw that the electron carry not only the electric charge but also the weak charges. And now came E.Cartan and “spoiled” everything again. To describe QM system we need to use the unitary groups. But the unitary groups form the infinite chain. Why U(2)? Which one out of Glashow prostitutes fit? Here E.Cartan gives us a hint: there is one finite chain - exceptional Lie groups. Why it is finite? You find that it is the underlined numerical system of the Cayley numbers responsible for that. And now you meet the remarkable mathematical result (the Hurwitz algebras and Hurwitz theorem).

    I prefer to use a physical jargon instead of rigorous mathematical definitions. In contrast to infinite diversity of all possible algebras, the Hurwitz algebras are the number systems. In the literature they are called quadratic normal division algebras or, shortly, composition algebras. Below, I will describe conditions that they should satisfy, but now I present the result of the famous Hurwitz theorem: The dimensionalities of these algebras are respectively 1, 2, 4 and 8. That’s all. There are no others. Most popular examples are:

    1)1-dim algebra of real numbers;
    2)2-dim algebra of complex numbers with real coefficients;
    3)4-dim algebra of real quaternions (Hamilton numbers);
    4)8-dim algebra of real octonions (Cayley numbers);

    Roughly, the conditions they should satisfy are the following:

    1) division - the equations a*x=b and x*a=b have unique
    solution.
    Translation to physics say: It is required repeatability for
    the results of the physical measurements and their independence
    from the experimentalist or interpreter subjective choice.
    2) composition – N(x*y)=N(x)*N(y).
    Translation to physics say: It is possible to construct
    many-particles physical system without interaction (ensemble
    of completely independent physical objects first and to
    introduce the interaction at later stage).

    The post #14 above contains small part of the relevant refs and wrong completely in each of its statements. QQM with the complex scalar product is U(2) gauge field theory and the composition property assures that the consistent solution for the Fock second quantization procedure exists. OQM with the complex scalar product is U(4) gauge field theory. CQM with the real scalar product is dispersion free field theory and thus allow the wave mechanical generalization of Newtonian Mechanics and the partial solution of the measurement problem (for the non-relativistic limit).

    I worked on the problem of the tensor products in QQM with the complex scalar product during four years. I was convinced that the problem as well as the solution is trivial. Therefore I was convinced that I am simply stupid. Only S.L. Adler papers on the algebraic chromodynamics indicated to me where I was wrong and allows finding the solution. It turns out not so trivial. The solution allows defining the mathematical architecture of all of the physics according to W.R. Hamilton and E. Schrödinger conjecture.

    Regards, Dany.
     
    Last edited: Sep 1, 2007
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