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Elementary questions about inner product interpretation

  1. Jun 11, 2013 #1
    When one says that <[itex]\varphi[/itex]|[itex]\psi[/itex]> is the probability that [itex]\psi[/itex] collapses to [itex]\varphi[/itex], does this "collapse" necessarily involve a measurement (so that one would have to find the implicit Hamiltonian)? Or does this just exist as part of the evolution of the wave function, perhaps the vacuum energy playing a role in the Schrödinger equation?
     
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  3. Jun 11, 2013 #2

    kith

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    Yes. (Also it's not the probability but the probability amplitude.)

    In order to do what?

    The vacuum energy is a concept from quantum field theory. It doesn't play a role in ordinary QM. However, there are some speculative extensions to QM which use an explicit physical mechanism to achieve collapse. See http://en.wikipedia.org/wiki/Interpretation_of_quantum_mechanics#Objective_collapse_theories.
     
  4. Jun 11, 2013 #3
    Thanks, kith.
    Ah, oops, right. There's many a slip 'twixt cup and lip.
    Sorry, I mean the implicit operator. I was considering that the collapse would be brought about by some operator in the measuring process, so that one would be looking at <[itex]\varphi[/itex]|[itex]\psi[/itex]> = <M[itex]\psi[/itex]|[itex]\psi[/itex]> for some operator M.

    Thanks for the link.
     
  5. Jun 11, 2013 #4

    kith

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    This is a good question. In the textbook description of measurements, there is no such operator M. We have an initial state |ψ> and an observable O and are able to calculate the probabilities for the occurances of the eigenstates of O as final states. The measurement itself has no representation in the mathematical framework.

    Now we can try to describe the measurement dynamically. A measuring device and a system interact, so in principle, we should be able to write down an interaction Hamiltonian for them. If we look at the time evolution of the system only, this interaction leads to decoherence in one basis. From this basis, we can construct the observable O. The emergence of the preferred basis is called "environmentally induced superselection". Decoherence means that an initial superposition loses its ability to interfere. This leads to several possible interpretations which get rid of the collapse. But note that your operator M is not a possible description of what happens. The dynamical description doesn't yield a single outcome but contains a probabilistic element.
     
    Last edited: Jun 11, 2013
  6. Jun 11, 2013 #5
    Just a clarification, the probability is [itex]|<\varphi[/itex]|[itex]\psi>|^2[/itex], not <[itex]\varphi[/itex]|[itex]\psi[/itex]>. The latter is the probability amplitude.
     
  7. Jun 11, 2013 #6
    Thanks, kith. Very interesting. Now I shall work on the gritty details ....

    thanks for the clarification, LastOneStanding.
     
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