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Lippmann-Schwinger equation

  1. Oct 7, 2009 #1
    This is probably a very stupid question as usual. I don't understand the Lippmann-Schwinger equation.

    First we have the Schrödinger equation [tex](H+V)\psi=E\psi[/tex], and we just rearrange it to [tex]\psi=\frac{1}{E-H}V\psi[/tex]. But now, somehow magically this becomes [tex]\psi=\phi+\frac{1}{E-H}V\psi[/tex] where [tex]\phi[/tex] is a solution to [tex]H\phi=E\phi[/tex]. Then we add a little imaginary quantity to the denominator just for the sake of being able to take an integral and let this imaginary quantity go to 0 in the end. This last step is not my problem. My question is: Where does the [tex]\phi[/tex] come from?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 7, 2009 #2

    gabbagabbahey

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    If you apply the operator [itex]E-H[/itex] to both sides of each equation, [itex]|\psi\rangle=\frac{1}{E-H}V|\psi\rangle[/itex] and [itex]|\psi\rangle=|\phi\rangle+\frac{1}{E-H}V|\psi\rangle[/itex], you get the same thing in each instance (since [itex](E-H)|\phi\rangle=0[/itex] ), so it appears that either could be the correct solution.

    However, you expect that [itex]|\psi\rangle\to|\phi\rangle[/itex] in the limit [itex]V\to0[/itex] and only the second solution satisfies that condition.
     
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