# Inhomogenous Schrödinger equation

Please help me out in solving this inhomogeneous Schrödinger equation in cylindrical co-ordinates [You may suggest if I have to go for mathematics]:

$$\ddot R + \frac1r\dot R+\left(b^2-d^2-\frac{a^2}{r^2}\right) R - \frac{c^2}r \delta(r-r_0) R ~=~ F e^{ikr}.$$ This is only the radial part.

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If you provide more information, people will be more willing to help you:-). What have you tried? Where have you looked it up? Where did you find it? – Ferdinando Randisi Oct 29 '12 at 14:51
Ok. This is the radial part of schrodinger equation in Cylindrical co-ordinates. I am solving the potential scattering problem, the potential is potential barrier. I tried it with mathematica, but could get the solution. It was tried to solve in mathematics also without source term [RHS]. But now there is the source as the plane wave. – nagendra Oct 29 '12 at 15:09
Yes, but what potential barrier? This is not a simple free particle confined in a sphere, is it? If it were, it should not have the $\delta$ term, if i remember correctly. Also, what is the $Fe^{ikr}$ term? If you tell us were the equation comes from, we might remember something about it. – Ferdinando Randisi Oct 29 '12 at 15:15
Actually, the equation has the derivative of the potential and the potential is v(r) = vo in 0<r<ro and 0 in r> ro. At r = ro the derivative turns out to be d_function. The term in RHS is the source term, the plane wave. Actually, this equation is around r = ro. Derivative is spatial derivative. – nagendra Oct 29 '12 at 15:28
Right Hand Side is Plane wave. Does any one have any idea to solve this please? – nagendra Nov 21 '12 at 18:14

If Mathematica refuses to solve this, it is likely that no exact solution exists. You might look for approximate solutions then, maybe expanding in power series (even though that wouldn't allow you to keep track of the delta term). Or you could generalize whatyou do in simple one-dimension potential scattering.

WKB method could help, too.

What have you tried? Do you have an expert you can ask at hand?

If you are a student, you can ask a professor at your university, otherwise you can try sending an email to someone who is expert in these sort of differential equations (maybe a professor of mathematical methods for physics, or of quantum mechanics).

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Yes I am a student and trying to solve the scattering problem in Quantum Mechanics. So far I have not found who could solve this. I am trying trying with the Green function technique as the rest of the terms look like the source. I will share my updates. Thanks. – nagendra Nov 22 '12 at 14:45