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Homework Help: Free particle -> bound particle

  1. Mar 12, 2012 #1
    Free particle --> bound particle

    1. The problem statement, all variables and given/known data

    A free neutron meets a finite square well of depth [itex]V_{0}[/itex], and width 2a centered around origo.

    However, the probability that the neutron emits a photon when it meets the potential well, and thus decreasing its energy is proportional to the integral [itex]\int^{t_{1}}_{t_{0}}\int^{a}_{-a} |\Psi(x,t)|^{2} dx dt[/itex]. Where [itex]t_{1}-t_{0}[/itex] is the time it takes the neutron to cross the well.

    The question then is: "What energy is the most advantageous for the neutron to have, in order to be trapped by the potential well?"

    3. The attempt at a solution
    The initial energy is [itex]E_{0}[/itex], the energy of the photon is [itex]E_{p}[/itex]

    I'm guessing I have to find a value for [itex]E_{0}[/itex], so as to make the integral a large as possible.
    Last edited: Mar 12, 2012
  2. jcsd
  3. Mar 12, 2012 #2


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    Homework Helper

    Re: Free particle --> bound particle

    Is there some form of ψ here that you are neglecting to tell us (like it is the wavefunction of the neutron)? If ψ is independent of E (or E0), then I don't see how it would make a difference (unless it's as silly as to realize to treat the neutron classically, so that t1-t0 depends inversely on the square root of E, which it may be, since it talks about "the time it takes the neutron to cross the well").
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