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Partial wave analysis - incoming/outgoing?

  1. Apr 13, 2013 #1
    In the chapter on partial wave analysis in Griffiths's Introduction to Quantum Mechanics, he considers a spherically symmetric potential and says that for large r, the radial part of Schrodinger's equation becomes,

    [itex]\frac{d^{2}u}{dr^{2}}≈-k^{2}u[/itex]

    with a general solution of

    [itex]u(r)=C\exp{ikr}+D\exp{-ikr}[/itex].

    He then says the first term represents the outgoing wave and the second term is the incoming wave. Why is that the case? These two differ by [itex]\pi[/itex] in the complex plane, but I don't see how that enables one to make the "incoming"/"outgoing" distinction.
     
  2. jcsd
  3. Apr 13, 2013 #2

    fzero

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    Act on the solution with the radial momentum operator ##\hat{p}_r = -i\hbar \partial_r## and compare the signs of the two terms.
     
  4. Apr 13, 2013 #3
    Thanks! The operator yields the momentum for the first term and (-1)*momentum for the second. A negative magnitude for momentum does not make sense, so therefore it is to be evaluated for times t<0 making it the incoming wave. Is that the argument?
     
  5. Apr 13, 2013 #4

    fzero

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    We're computing the momentum, not just the magnitude. If ##k## is real, then the magnitude of momentum is always positive. We're also not discussing time dependence here. The argument is simply that, if ##k>0##, then one solution carries momentum in the ##+r## direction, while the other carries it in the ##-r## direction. If we did add in the time dependence, we could see this more explicitly, but it isn't necessary.
     
  6. Apr 15, 2013 #5

    Meir Achuz

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    A factor [itex]e^{-i\omega t}[/itex] is understood. This makes one form outgoing and the other ingoing.
     
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