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Finite Spherical Potential Well

  1. Mar 4, 2009 #1
    This is more a qualitative question than a specific homework question, but a homework problem got me wondering about it.

    I was solving the finite potential well.

    [tex]V(r) = 0 \hspace{1cm} r \geq a[/tex]
    [tex]V(r) = -V_0\hspace{1cm} r < a[/tex]

    I am trying to solve for the ground state energy. When I find the forms of the solution in the interior of the well, I find that I get

    [tex] \frac{c_1 \sin{(kr)} + c_2 \cos{(kr)}}{r} [/tex]

    I know from doing other reading that I should end up throwing away the cosine term, but I do not understand why.

    I can see that it blows up at [tex] r=0 [/tex], but it still looks like it will be normalizable to me since a volume integral in spherical coordinates provides an extra factor of [tex]r^2[/tex]
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 4, 2009 #2

    Avodyne

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    Science Advisor

    It doesn't satisfy the Schrodinger equation, because [itex]\nabla^2(1/r) \propto \delta^3(\vec x)[/itex].
     
  4. Mar 4, 2009 #3
    I don't understand that reply. I got it by solving the schrodinger equation, so it must satisfy it, no?
     
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