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A spherically symmetrc attractive potential

  1. Jan 24, 2007 #1
    Given a spherically symmetric attractive potential of the form V(r) = -V * e^(-r/a)
    To solve the schroedinger equation and obtain the quantization condition for the energy eigenvalues.
    Hint : - Use the Substitution x = e ^ (-r/2a)





    I did the usual separation of variables psi = R * spherical harmonics
    Also i substituted R = U/r and got the radial differential equation in U.
    But now, I am stuck.. I know the asymptotic behavior of U. U tends to 0 as r goes to 0. As r tends to infinity, for bound state U will go as e ^ (-kr)
    But i dont know what to do next... pls help.
    thanks.
     
  2. jcsd
  3. Jan 24, 2007 #2
    After you do the substitiution, you should get a second order differential equation of U with respect to r.
    i.e

    [tex] \frac{d^2u}{dr^2} = f(r,V(r)) [/tex]

    Integrate the equation twice and apply the boundary condition, you will get U... becareful about the boundary condition...
    Hint: What is the value of dU/dr as r goes to infinity? How about r=0?
     
    Last edited: Jan 24, 2007
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