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A spherically symmetrc attractive potential

  • Thread starter anjor
  • Start date
23
0
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.
 

Answers and Replies

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:

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