Re: Mathematica can't calculate Fourier transform (Dirac mean position eigenfunction)
- To: mathgroup at smc.vnet.net
- Subject: [mg54988] Re: Mathematica can't calculate Fourier transform (Dirac mean position eigenfunction)
- From: dh <dh at metrohm.ch>
- Date: Wed, 9 Mar 2005 06:34:15 -0500 (EST)
- References: <d0juvv$n6i$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Hallo Jacob,
in order that a FourierTransform exists, the function must be absolute
integrable. Now try to integrate ( (1 + k^2 + (1 + k^2)^(1/2) )^(-1/2)
from zero to infinity and you will see that it is diverging.
Sincerely, Daniel
Jacob wrote:
> 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 k-space to r-space) of wavefunctions dependent
> on:
>
> ( (1 + k^2 + (1 + k^2)^(1/2) )^(-1/2)
>
> where k is in units of the Compton wavevector.
>
> Cell expression:
>
> Cell[BoxData[
> FractionBox["1",
> SqrtBox[
> RowBox[{"1", "+",
> SuperscriptBox["k", "2"], "+",
> SqrtBox[
> RowBox[{"1", "+",
> SuperscriptBox["k", "2"]}]]}]]]], "Output"]
>
>
> 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 ( 1 + (1 + k^2)^(-1/2) )^(1/2)
>
> Cell[BoxData[
> RowBox[{"k", " ",
> SqrtBox[
> RowBox[{"1", "+",
> FractionBox["1",
> SqrtBox[
> RowBox[{"1", "+",
> SuperscriptBox["k", "2"]}]]]}]]}]], "Output"]
>
>
> Thanks for any help.
>