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 ____________________________________________________________________