A spherically symmetrc attractive potential

[anjor]

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 don't know what to do next... pls help.
thanks.

 

[chanvincent]

After you do the substitiution, you should get a second order differential equation of U with respect to r.
i.e

$$\frac{d^2u}{dr^2} = f(r,V(r))$$

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?