0
$\begingroup$

While evaluating the final wavefunction of a free spin-0 particle in one dimension in terms of Green's functions (propagators) and considering the initial conditions as Gaussian wavepackets, I arrived at an expression that consists of two definite integrals each of which has to be evaluated separately. The integration is with respect to the variable $z$ (the relative position). The two integrals $I_{1}$ and $I_{2}$ are:

I1 = Integrate[(7.2288*
     BesselJ[1, Sqrt[-2 + 2*Sqrt[2]]*Sqrt[200*Sqrt[2] - z^2]]*
          Sin[x + z])/(E^(1.25*(x + z)^2)*Sqrt[200*Sqrt[2] - z^2]), 
     {z, -16.8179, 16.8179}]

and

I2 = Integrate[
     -((0.0181*
       BesselJ[0, Sqrt[-2 + 2*Sqrt[2]]*Sqrt[200*Sqrt[2] - z^2]]*
       BesselK[1, Sqrt[0.04 + (x + z)^2]]*Sin[x + z])/
         Sqrt[0.04 + (x + z)^2]), 
     {z, -16.8179, 16.8179}]

A guidance on how to go about evaluating this further would be appreciated.

$\endgroup$
  • $\begingroup$ this will need to be done numerically. Specify x and use NIntegrate $\endgroup$ – george2079 Apr 15 '17 at 18:05
2
$\begingroup$

Maybe so:

I1[x_?NumericQ] := NIntegrate[(7.2288 Exp[-1.25 (x + z)^2] BesselJ[1, 
Sqrt[-2 + 2 Sqrt[2]] Sqrt[200 Sqrt[2] - z^2]] Sin[x + z])/Sqrt[
200 Sqrt[2] - z^2], {z, -16.8179, 16.8179}]

ListLinePlot@Table[{x, I1[x]}, {x, -5, 5, 1/5}]

enter image description here

I2[x_?NumericQ] := NIntegrate[-((0.0181 BesselJ[0, 
Sqrt[-2 + 2 Sqrt[2]] Sqrt[200 Sqrt[2] - z^2]] BesselK[1, Sqrt[
0.04 + (x + z)^2]] Sin[x + z])/Sqrt[0.04 + (x + z)^2]), {z, -16.8179, 16.8179}]

ListLinePlot@Table[{x, I2[x]}, {x, -5, 5, 1/5}]

enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.