Re: DiracDelta Integrals
- To: mathgroup at smc.vnet.net
- Subject: [mg9326] Re: DiracDelta Integrals
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Sat, 1 Nov 1997 03:33:16 -0500
- Organization: University of Western Australia
- Sender: owner-wri-mathgroup at wolfram.com
Peter Jay Salzman wrote:
> just out of curiosity, i tried
>
> Integrate[ Exp[I k x], {x, -Infinity, Infinity}]
>
> and Mathematica said "the integral is 0 if Im[k]==0, otherwise, i give
> up". so, thinking i was being slick, i loaded the
> Calculus`DiracDelta` package.
>
> i tried the integral again, but Mathematica said the same thing. then,
> knowing that this comes up all the time in fourier transforms, i loaded
> Calculus`FourierTransform`. this time, when i entered the integral,
> Mathematica told me: "hey, bubba. that integral doesn't converge on
> the domain of integration".
>
> surely, there MUST be a package that allows Mathematica to do this
> integral and give me 2 Pi DiracDelta[k]. what do i need to load to
> make Mathematica recognize this?
You are computing a Fourier Transform. Hence
In[1]:= <<Calculus`FourierTransform`
In[2]:= FourierTransform[1, x, k]
Out[2]= 2 Pi DiracDelta[k]
>
> the reason why i'm asking is because i'm trying to do the integral
>
> Exp[-Abs[k]/c] Exp[I k x] Exp[i p x/hbar], {x,-Infinity,Infinity},
>
> {k, -Infinity, Infinity}
In[3]:= FourierTransform[Exp[I p x/hbar], x, k] Out[3]=
p
2 Pi DiracDelta[k + ----]
hbar
In[4]:= Integrate[Exp[-(Abs[k]/c)] 2 Pi DiracDelta[k + p/hbar],
{k, -Infinity, Infinity}]
Out[4]=
p
Abs[----]
hbar
2 Exp[-(---------)] Pi
c
> by the way, are there any packages that allow bra-ket notation on
> Mathematica?
Have a look at Notation Maker (the Notebook is NotationMaker.nb and is
available from the on-line help under Demos/Notebook Demos. There is a
section there on notations for Bras, Kets and Operators in Quantum
Mechanics.
Cheers,
Paul
____________________________________________________________________
Paul Abbott Phone: +61-8-9380-2734
Department of Physics Fax: +61-8-9380-1014
The University of Western Australia Nedlands WA 6907
mailto:paul at physics.uwa.edu.au AUSTRALIA
http://www.pd.uwa.edu.au/~paul
God IS a weakly left-handed dice player
____________________________________________________________________