Problem with Fourier bessel transform of Yukawa potential

You could try doing a numerical integration instead.In summary, the speaker is having trouble finding the Fourier Bessel Transform for the Yukuwa potential and is comparing the results from the analytical and numerical transforms. They are also looking for tips on how to improve the numerical transform for the (screened) coulomb potential. The use of a sine transform is not ideal and a numerical integration may be a better solution.
  • #1
praban
13
0
Hello,

I am trying to find Fourier Bessel Transform (i.e. Hankel transform of order zero) for Yukuwa potential of the form
f(r) = - e1*e2*exp(-kappa*r)/(r) (e1, e2 and kappa are constants). I am using the discrete sine transform routine from FFTW ( with dst routine). For this potential there is analytical result - f^hat (k) = - 4*pi *e1*e2/(kappa^2 + k^2).

I was comparing the results from numerical and analytical transform. However, I see that there is a significant difference (delR =0.3, 1st point is at 0.1, 4096 points are used for the numerical transform but the error remains even if I increase it 16384). Is there any trick to get better numerical transform for (screened) coulomb potential?

analytical numerical r
-711437635.18197799 -748996019.05573177 0.1
-275116156.66050136 -261467385.57794687 0.4
-136050696.75080600 -143418942.44284841 0.7
-79670334.886979684 -75837581.595151573 1.0
-51976687.575621709 -54872935.399959348 1.3

thanks,

Pradipta
 
  • #3
A sine transform is a poor choice since your function is not zero at r=0. You might have better luck with a cosine transform. In any case you will have a problem with a numerical transform since the function is infinite at the origin.
 

1. What is the Fourier Bessel transform of the Yukawa potential?

The Fourier Bessel transform of the Yukawa potential is a mathematical tool used in quantum mechanics to describe the interaction between particles. It involves transforming the potential function from position space to momentum space.

2. What is the problem with the Fourier Bessel transform of the Yukawa potential?

The problem with this transform is that it does not converge for all values of the potential function, specifically when the potential function has a long-range component. This leads to inaccuracies in the calculations and makes it difficult to accurately model certain physical systems.

3. How is this problem addressed in quantum mechanics?

In quantum mechanics, this problem is addressed by using a modified version of the Fourier Bessel transform known as the screened Fourier Bessel transform. This takes into account the long-range component of the potential and provides more accurate results.

4. Can the Fourier Bessel transform of the Yukawa potential be used in all physical systems?

No, the Fourier Bessel transform of the Yukawa potential is not suitable for all physical systems due to its convergence issues. It is most commonly used in simple models of atomic and molecular interactions.

5. Are there any other alternatives to the Fourier Bessel transform for the Yukawa potential?

Yes, there are other mathematical techniques such as the Hartree-Fock method and the coupled-cluster method that can also be used to describe the interaction between particles in the presence of a Yukawa potential. These methods may provide more accurate results in certain cases.

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