       Re: Different Integration Results

• To: mathgroup at smc.vnet.net
• Subject: [mg30366] Re: [mg30344] Different Integration Results
• From: BobHanlon at aol.com
• Date: Sun, 12 Aug 2001 02:29:54 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```In a message dated 2001/8/11 4:38:06 AM, Harald.Grossauer at uibk.ac.at writes:

>I have got a problem with the attached notebook. In the last two lines,
>if I use Integrate[ ] the result is 99/35, NIntegrate[ ] says it is
>"1.". Due to the nature of the problem (quantum theory, fourier
>transform) I would expect the result to be 1 exactly. What could cause
>this difference?
>

Use FullSimplify when evaluating psi and psiconj.  The form of the rho is
then very simple and integrating rho works fine.

phi[p_]:= Evaluate[Simplify[
18/Sqrt*UnitStep[p]*p*(Exp[-p]-(1/6)*Exp[-p/2])]];

Integrate[phi[p]^2,{p,-Infinity,Infinity}]

1

psi[x_]:=Evaluate[FullSimplify[
(1/Sqrt[2*Pi])*Integrate[phi[p]*Exp[I*p*x],
{p,-Infinity,Infinity}],Element[x,Reals]]];

psiconj[x_]:= Evaluate[FullSimplify[
(1/Sqrt[2*Pi])*Integrate[phi[p]*Exp[-I*p*x],
{p,-Infinity,Infinity}],Element[x,Reals]]];

rho[x_]:= Evaluate[Simplify[psi[x]*psiconj[x]]];

Plot[rho[x],{x,-10,10}];

Integrate[rho[x],{x,-Infinity,Infinity}]

1

NIntegrate[rho[x],{x,-Infinity,Infinity}]

0.9999999999999989

Bob Hanlon
Chantilly, VA  USA

```

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