Fourier transform with Mathematica (Dirac mean position eigenfunction)

[Jacob]

Mathematica can't calculate Fourier transform (Dirac mean position eigenfunction)

Hi, I'm attempting to use Mathematica to calculate a mean-position eigenfunction of the Dirac equation. To do so I need to evaluate Fourier transforms (from p-space to r-space) of wavefunctions dependent on:

$$\frac1{{\sqrt{{({1+{k^2}+{\sqrt{1+{k^2}}}})}}}}$$

where k is in units of the Compton wavevector.

Mathematica is unable to evaluate the FT of the above (either Fourier sine transform or normal FT). Can anyone give any suggestions as to how I could evaluate it?

More specifically, I am making a reverse Foldy-Wouthuysen transformation of a mean-position eigenfunction in p-space, then transforming the result into r-space assuming spherical symmetry. The first component of the r-space eigenfunction is given by the Fourier sine transform of:

$$k\,{\sqrt{1+{\frac1{\sqrt{1+{k^2}}}}}}$$

Thanks for any help.

 

[Jacob]

Hi again, have I posted this in the right forum? If not please suggest where I'm most likely to get an answer!

Otherwise I would be very grateful if anyone could tell me either how to Fourier transform:

$$k\,{\sqrt{1+{\frac1{\sqrt{1+{k^2}}}}}}$$

or that it is not possible to do so.

 

[Galileo]

I sure can't, but the function isn't square integrable over k-space in any case.

Also, it is in very good approximation equal to k.

 

[Jacob]

Thanks :).

Unfortunately it's the difference from k that's important as it is the exact spatial extent of the wavefunction which is of interest (it is a Dirac delta in the untransformed Foldy-Wouthuysen representation since it's a mean position eigenfunction and mean position = r in that representation).