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Why Eigenvectors/Eigenvalues?

  1. Jul 20, 2012 #1
    I am currently working on trying to understand some of the details in quantum physics.

    So far, Schrodinger's equation and wave mechanics seems to provide (at least when imposing semiclassical E-M interactions with an E-M field) a decent mechanism for why electrons end up in stationary states represented by the eigenfunctions of the wave equation. (These are the ones for which the wavefunction is not accelerating, thus not disturbing the E-M field.)

    I have looked through second quantization, but haven't seen any mechanism there for quantizing photons. It's sort of asserted in the axioms of setting it up.

    I've seen in it stated in several places that when particles interact, they always end up in an eigenfunction of some given operator. In general, I don't see the mechanism which would cause this. First, it seems the operators are arbitrarily chosen.

    In the case of electrons, if you ping a bound electron with another electron, the other electron will eventually end up in some sort of stationary state by radiation (and eventually, the ground state). But if you have some generic particle that doesn't have a mechanism for dissipating excess energy, it doesn't seem like there should be any reason that it ends up in a stationary state versus some arbitrary time-varying state.

    PS - for arbitrary time-varying states, it seems like the eigenfunctions are just some arbitrary basis to decompose them into. What is special about the eigenfunctions?

    I've got a model I'm playing with now, with imaginary 1d particle A that is bound in a harmonic oscillator, and imaginary 1d particle B, that starts at one end of the domain and travels as a wave packet. There is some potential between them, so they interact when they "collide", and you end up with a wave for B that either propagates through (associated with one relative state for A), or reflects, associated with another relative state for A. But A, it seems, has no preference as to where it ends up, whether eigenvalue or not.

    You have to come up with some specification for B's state anyway, before you get A's state. And you can specify whatever you want for B, for greater or lesser portions of the amplitude. Am I supposed to be seeing large amounts of the amplitude associated with eigenvectors?
     
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  3. Jul 20, 2012 #2

    atyy

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    A stationary state is an eigenstate of the Hamiltonian.

    In general, a system will not end up in a stationary state.

    When a measurement corresponding to an operator is made on system, the system will "collapse" from whatever state it is into an eigenstate of the operator corresponding to the measurement.

    If the measurement is of the system's energy, then the system will collapse into a stationary state.

    This "collapse" of the wavefunction occurs in the "shut up and calculate" interpretation of QM. It's obviously bizarre, and is one of the problems that say MWI tries to fix.
     
  4. Jul 20, 2012 #3

    bhobba

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    Hi Mad Rocket

    Its got to do with how to encode the results of observations. Imagine you have some kind of observational apparatus that has n outcomes and you associate n real numbers with those outcomes. You can list them out and you immediately recognize its a vector. Trouble is go to another basis and the values change - what you want is a way of expressing it that does not depend on a particular basis. To do that QM replaces the basis vectors |bi> by operators |bi><bi| to give the hermitian operator sigma yi|bi><bi| where the yi are the n real numbers. In this way it does not depend on any particular basis - the outcomes are the eigenvalues of the operator and the states are the eigenvectors.

    In fact you can develop all of QM this way by imposing the condition any Hermitian operator in principle represents some observation and applying Gleasons Theorem - but that is another story.

    Thanks
    Bill
     
  5. Jul 21, 2012 #4
    The main reason given is repeatability: the measurement gives the same result if repeated. That means the state remains in the subset of state space corresponding to the value of the first measurement.

    The "value" is not just a number on a gauge, it's also all the knowledge that went into the setup of the experiment, ie the entire known state.
     
  6. Jul 22, 2012 #5

    Rap

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    I know this is off-topic, and I don't want to sidetrack this thread, but I would be interested to know what you think of this statement: "The idea that the speed of light is constant irrespective of the speed of the observer is obviously bizarre, but the theory of relativity basically says "shut up and calculate".
     
  7. Jul 22, 2012 #6

    bhobba

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    It's not bizarre - it follows from the POR and Maxwell's equations. The thing with relativity is you are faced with an inconsistency - namely between locality with a finite propagation speed of effects and time as an absolute - both are heuristically very plausible but only one can be correct - it turns out to be locality that is correct.

    Decoherence has solved many of the issues with QM IMHO, but I don't want this to be a discussion about that - it has be thrashed out in many threads on this forum already.

    Thanks
    Bill
     
  8. Jul 22, 2012 #7

    Rap

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    Ok. I disagree, but I never posed the problem that particular way to one of your persuasion. Thanks for the answer.
     
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