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F'x
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It is well known from theory that the Coulomb potential can be obtained as a Fourier transform in the following way:

$$ \int \frac{\mathrm{d}^3p}{\left( 2 \pi \right)^3} \frac{e^{\mathrm{i} \mathbf{p} \cdot \mathbf{r}}}{\mathbf{p}^2+m^2} = \frac{e^{-rm}}{4 \pi r} \hspace{20pt} r \equiv \sqrt{x^2+y^2+z^2}$$

(The mass term is here for mathematical convergence reasons, one can set the mass to zero after the transform. The Fourier transform also has a precise physical meaning, but that's out of the scope of this question.) I am trying to reproduce this result with Wolfram Mathematica; skipping some unnecessary constants I use the following code:

FourierTransform[1/(k*k + q*q + r*r + m*m), {k, q, r}, {x, y, z}]

which does not yield the expected result... I'm not an expert of Mathematica, by any means, but I cannot see why it gets this Fourier transform wrong.

Thank you in advance for any reply!

It is well known from theory that the Coulomb potential can be obtained as a Fourier transform in the following way:

$$ \int \frac{\mathrm{d}^3p}{\left( 2 \pi \right)^3} \frac{e^{\mathrm{i} \mathbf{p} \cdot \mathbf{r}}}{\mathbf{p}^2+m^2} = \frac{e^{-rm}}{4 \pi r} \hspace{20pt} r \equiv \sqrt{x^2+y^2+z^2}$$

(The mass term is here for mathematical convergence reasons, one can set the mass to zero after the transform. The Fourier transform also has a precise physical meaning, but that's out of the scope of this question.) I am trying to reproduce this result with Wolfram Mathematica; skipping some unnecessary constants I use the following code:

FourierTransform[1/(k*k + q*q + r*r + m*m), {k, q, r}, {x, y, z}]

which does not yield the expected result... I'm not an expert of Mathematica, by any means, but I cannot see why it gets this Fourier transform wrong.

Thank you in advance for any reply!

It is well known from theory that the Coulomb potential can be obtained as a Fourier transform in the following way:

$$ \int \frac{\mathrm{d}^3p}{\left( 2 \pi \right)^3} \frac{e^{\mathrm{i} \mathbf{p} \cdot \mathbf{r}}}{\mathbf{p}^2+m^2} = \frac{e^{-rm}}{4 \pi r} \hspace{20pt} r \equiv \sqrt{x^2+y^2+z^2}$$

(The mass term is here for mathematical convergence reasons, one can set the mass to zero after the transform. The Fourier transform also has a precise physical meaning, but that's out of the scope of this question.) I am trying to reproduce this result with Wolfram Mathematica; skipping some unnecessary constants I use the following code:

FourierTransform[1/(k*k + q*q + r*r + m*m), {k, q, r}, {x, y, z}]

which does not yield the expected result... I'm not an expert of Mathematica, by any means, but I cannot see why it gets this Fourier transform wrong.

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zakk
zakk
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It is well known from theory that the Coulomb potential can be obtained as a Fourier transform in the following way:

$$ \int \frac{\mathrm{d}^3p}{\left( 2 \pi \right)^3} \frac{e^{\mathrm{i} \mathbf{p} \cdot \mathbf{r}}}{\mathbf{p}^2+m^2} = \frac{e^{-rm}}{4 \pi r} $$$$ \int \frac{\mathrm{d}^3p}{\left( 2 \pi \right)^3} \frac{e^{\mathrm{i} \mathbf{p} \cdot \mathbf{r}}}{\mathbf{p}^2+m^2} = \frac{e^{-rm}}{4 \pi r} \hspace{20pt} r \equiv \sqrt{x^2+y^2+z^2}$$

(The mass term is here for mathematical convergence reasons, one can set the mass to zero after the transform. The Fourier transform also has a precise physical meaning, but that's out of the scope of this question.) I am trying to reproduce this result with Wolfram Mathematica; skipping some unnecessary constants I use the following code:

FourierTransform[1/(k*k + q*q + r*r + m*m), {k, q, r}, {x, y, z}]

which does not yield the expected result... I'm not an expert of Mathematica, by any means, but I cannot see why it gets this Fourier transform wrong.

Thank you in advance for any reply!

It is well known from theory that the Coulomb potential can be obtained as a Fourier transform in the following way:

$$ \int \frac{\mathrm{d}^3p}{\left( 2 \pi \right)^3} \frac{e^{\mathrm{i} \mathbf{p} \cdot \mathbf{r}}}{\mathbf{p}^2+m^2} = \frac{e^{-rm}}{4 \pi r} $$

(The mass term is here for mathematical convergence reasons, one can set the mass to zero after the transform. The Fourier transform also has a precise physical meaning, but that's out of the scope of this question.) I am trying to reproduce this result with Wolfram Mathematica; skipping some unnecessary constants I use the following code:

FourierTransform[1/(k*k + q*q + r*r + m*m), {k, q, r}, {x, y, z}]

which does not yield the expected result... I'm not an expert of Mathematica, by any means, but I cannot see why it gets this Fourier transform wrong.

Thank you in advance for any reply!

It is well known from theory that the Coulomb potential can be obtained as a Fourier transform in the following way:

$$ \int \frac{\mathrm{d}^3p}{\left( 2 \pi \right)^3} \frac{e^{\mathrm{i} \mathbf{p} \cdot \mathbf{r}}}{\mathbf{p}^2+m^2} = \frac{e^{-rm}}{4 \pi r} \hspace{20pt} r \equiv \sqrt{x^2+y^2+z^2}$$

(The mass term is here for mathematical convergence reasons, one can set the mass to zero after the transform. The Fourier transform also has a precise physical meaning, but that's out of the scope of this question.) I am trying to reproduce this result with Wolfram Mathematica; skipping some unnecessary constants I use the following code:

FourierTransform[1/(k*k + q*q + r*r + m*m), {k, q, r}, {x, y, z}]

which does not yield the expected result... I'm not an expert of Mathematica, by any means, but I cannot see why it gets this Fourier transform wrong.

Thank you in advance for any reply!

Source Link
zakk
zakk
  • 978
  • 1
  • 7
  • 17

Coulomb potential as a Fourier transform

It is well known from theory that the Coulomb potential can be obtained as a Fourier transform in the following way:

$$ \int \frac{\mathrm{d}^3p}{\left( 2 \pi \right)^3} \frac{e^{\mathrm{i} \mathbf{p} \cdot \mathbf{r}}}{\mathbf{p}^2+m^2} = \frac{e^{-rm}}{4 \pi r} $$

(The mass term is here for mathematical convergence reasons, one can set the mass to zero after the transform. The Fourier transform also has a precise physical meaning, but that's out of the scope of this question.) I am trying to reproduce this result with Wolfram Mathematica; skipping some unnecessary constants I use the following code:

FourierTransform[1/(k*k + q*q + r*r + m*m), {k, q, r}, {x, y, z}]

which does not yield the expected result... I'm not an expert of Mathematica, by any means, but I cannot see why it gets this Fourier transform wrong.

Thank you in advance for any reply!