Calculate scattering amplitude by delta function potential

1. Feb 7, 2013

liumylife

1. The problem statement, all variables and given/known data
I need to give scattering amplitude f(θ) in Born approximation to the first order in the case of delta function scattering potential δ(r). The problem is in spherical coordinate and I'll give major equation concerned.

2. Relevant equations
The equation for scattering amplitude is given by
f(θ) = C ∫_0^∞ dr rV(r)sinqr
where q is only a function of θ, so treat it as a constant in this formula, and C is just another constant, too.

The only issue left is to insert the delta function into the formula, and this is exactly the problem: no equation can be used in such case where the integral range begin just at zero.

3. The attempt at a solution

I look every textbook to find useful formulas, found nothing.
And I tried to create a new function such that the integral range is from -∞ to ∞,
then half the new integral to obtain a result, but it achieved nothing since it gives zero, so help me, please!

2. Feb 7, 2013

clamtrox

Your problem is that x δ(x) is zero everywhere, so the integral vanishes. Are you sure that delta function is one-dimensional? That seems strange. Perhaps it's 3-dimensional instead. Try massaging the identity $$\int dV \delta^3(r) = 1$$ to see the magic happen.

3. Feb 8, 2013

andrien

that is not the scattering amplitude with any potential,use the three dimensional delta function and calculate the fourier transform of it to obtain amplitude.

4. Feb 8, 2013