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Thread: When Schrödinger's wavefunction reacts on rapid change of potential?

  1. #1 When Schrödinger's wavefunction reacts on rapid change of potential? 
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    We are used to stationary Schrödinger equation. Slowly varying potential makes it more complicated. Physics should smoothen rapidly varying potential ... but let us discuss what's happening while theoretical rapid change of potential (no adiabatic approximation).
    For example imagine that potential has one minimum before the switch moment and a different one after (e.g. capacitor charged in one way then in opposite one) - like in this picture:



    In minus infinity electron should be in the ground state of one potential and in plus infinity in ground state of the other - the question is how the transition of wavefunction would look like?
    The main problem is that quantum mechanics is time symmetric - such transition shouldn't be instant, so this symmetry suggests that the middle of this transition is the switch moment ... but it means that the wavefunction has started evolving before the switch???

    I have to admit I don't understand the situation from perspective of quantum mechanics.
    The above picture used Maximal Entropy Random Walk instead (page 48 of http://arxiv.org/abs/1111.2253 ) - corrected Brownian motion to finally become thermodynamical model - not only approximate maximum uncertainty principle, but really maximizing entropy. Thanks of it, it doesn't longer disagree with thermodynamical predictions of quantum mechanics - the equilibrium dynamical state has probability density being exactly the squares of the lowest energy eigenfunction of Schrödinger's Hamiltonian.
    So this model agrees with the ground state in plus/minus infinities, but also naturally explains the transition - it indeed starts before the switch, but this time there is nothing strange about it: this model is thermodynamical - not fundamental but effective: we already know the history of potential and it allows us to estimate the best probability distribution of the particle. For example knowing that later it will be in another potential well allows us to tell that earlier it should be nearby.
    There is another strange thing about above picture - its Ehrenfest Newton's equation has opposite sign - the particle accelerates uphill then decelerates downhill ... but it's just required e.g. to transport density between these minimums ...

    But what's going on here in standard quantum mechanics?
    Would the wavefunction start transforming in the moment of change or before?


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  3. #2  
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    In what context are we used to a stationary schrodinger equation?


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    Ok, maybe not all of us, but while quantum mechanics courses there were mainly stationary wells, stationary atoms (and wavefunction collapses...) and time dependence of potential is considered mainly in perturbative approximation ... and what I have seen I would rather classify as adiabatic approximations - I don't think I have seen considered rapid changes ?

    ----------------------------------

    Here is some basic feeling - the situation should evolve to the final ground state having the same energy as initial, so the first thought would be there is no place for wavefunction collapse ... but in fact there is - the trick is that while changing the potential, we gave the particle energy ...
    But there is something wrong with this thinking what bothers me:
    - it's perfectly time symmetric situation, so shouldn't we expect time symmetric solution?
    - the Ehrenfest equation stubbornly wants to have opposite sign here - how to see it in QM?

    The paper is preliminary version of my current PhD I'm planning to defend soon and so I would really gladly discuss to better understand also e.g. looking standard issues it brought - here are briefly presented a bit less standard, but I would gladly expand and discuss them if someone would be interested:
    - the form of Bose-Hubbard Hamiltonian - MERW mathematically can be seen as nearly Feynman path integrals in imaginary time and so it goes to the ground state probability distribution ... the problem is that while deriving Hamiltonian in B-H case in two situations it perfectly agrees: for single particle without interaction and in continuous limit (lattice constant -> 0) ... but generally B-H is only approximation of what these "discrete path integrals" leads to. So there appears a natural question: which one is approximation and which one is the proper one? Maybe in discrete case we should be more careful about just guessing Hamiltonian ...
    - proper understanding of Pauli exclusion principle, anti symmetrization - where exactly it is required? It is said that for two electrons, but if we don't make standard approximation - two particle wavefunction as tensor product of single particles, the Coulomb repulsion itself says that 6 dimensional wavefunction is practically nonzero only when they are in nearly opposite positions - there is no additional exclusion principle required to make them "excluded". So it seems to be needed for the third electron ... but Coulomb repulsion and that minimal energy configuration of magnetic moments is only for two of them - it itself shouldn't allow more than two electrons to fit in a single state ... do we really need additional exclusion principle?
    Last edited by Jarek Duda; February 23rd, 2012 at 10:59 PM.
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  5. #4  
    Moderator Moderator Markus Hanke's Avatar
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    Have you considered the fact that the switch itself is not actually instantaneous ? In the example of the capacitor, the charge/decharge cycle takes a finite amount of time, meaning the potential is a smooth, continuous function everywhere, and the moment of the switch is nothing but an extremum of the function.
    To be honest, I fail to see the issue ? Can you explain further ?
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