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Is this the correct way to solve this QM integral problem?

  1. Apr 3, 2016 #1
    < Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown >

    Problem is:

    If the behavior of ψ( r,t ) as r->inf is dominated by r-n, what values can n assume if the integral
    A(ψ*∇ψ-ψ∇ψ*)⋅nda
    taken over the surface at infinity is to vanish.

    I considered ψ as ar-n calculate like below
    ψ*∇ψ≈ar-n⋅a*(-nr-n-1)=-naa*r-2n-1
    ψ∇ψ*≈a*r-n⋅a(-nr-n-1)=-naa*r-2n-1
    So... ψ*∇ψ-ψ∇ψ*=0 at anywhere. Thus, n does not affect to integration.

    Well, this result is so ridiculous :/
     
  2. jcsd
  3. Apr 3, 2016 #2

    TSny

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    Welcome to PF!

    Suppose ##\psi(x) = \Large{\frac{e^{ikr}}{r^n}}##. Would this be considered a function that is dominated by r-n? (I think so, but I don't know the precise definition of "dominated by".)
     
  4. Apr 3, 2016 #3
    Thanks for reply!
    I just tried and got the following result
    ##\psi^{*}\nabla\psi - \psi\nabla\psi^{*} = \Large{\frac{2ki}{r^{2n}}}##
    So, now ##n## must be larger than 0. Quite acceptable result :)
     
  5. Apr 3, 2016 #4

    TSny

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    Are there any factors of r in the area element da?
     
  6. Apr 3, 2016 #5
    Oh. r^2 dependency... So n>1

    OMG I've submitted my homework lol
     
    Last edited: Apr 3, 2016
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