0
$\begingroup$

Consider a Gaussian wave packet coming from the left (for one-dimensional problem) with a positive momentum and encountering a step potential. I would like to evaluate the transmitted scattered wave function.

The expression for the transmitted wave function is evaluating using the following integral(taken from this paper https://arxiv.org/pdf/1304.5179.pdf: 

$Assumptions = k ∈ Reals && ℏ ∈ Reals && 
  k0 ∈ Reals && m > 0 && l0 > 0 && t > 0;

A0[l0_, k0_, k_] = ((2 l0^2)/π)^(1/4) (Exp[-l0^2 (k - k0)^2]);

T0[k_, κ_, d_] = (4 k^2 Sqrt[1 - Sign[v0] (k0/k)^2])/(
 (k + k Sqrt[1 - Sign[v0] (k0/k)^2])^2;

ψtr[l0_, k0_, v0_,m_, ℏ_, x_, t_] = 
 1/Sqrt[2 π]
   Integrate[
   A0[l0, k0, k]*Sqrt[T0[k, v0, k0]]*
    Exp[I (k x + Sign[v0] ( 
2 ArcTan[Sqrt[((k Sqrt[1 - Sign[v0] (k0/k)^2])/k)^
   Sign[v0]]] - \[Pi]/2) - (ℏ k^2 t)/(
        2 m))], {k, -∞, ∞}]

Is there any way to get a closed form of the integral?

Improve this question
$\endgroup$
8
  • $\begingroup$ Are you even sure your integral has a closed form expression? $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Oct 25, 2017 at 9:32
  • $\begingroup$ @J.M. I expect to have some expression as this is a well known problem in quantum mechanics. $\endgroup$
    – user0322
    Commented Oct 25, 2017 at 9:39
  • $\begingroup$ Sure, but not all the well-known problems have closed form solutions. In any event, why not mention the "well-known problem" in your question for context? $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Oct 25, 2017 at 9:42
  • $\begingroup$ @J.M. there is exactly the same expression (Eq. 7) in the following paper: arxiv.org/abs/1303.6181 $\endgroup$
    – user0322
    Commented Oct 25, 2017 at 9:53
  • 1
    $\begingroup$ I think @J.M. is right to wonder whether you can expect a closed-form expression for this. I would hazard that most problems in QM don't have analytic solutions. You can give Mathematica a fighting chance by adding Assuming[l0 > 0 && k0 > 0 && k > 0 && v0 > 0 && m > 0 && \[HBar] > 0, Integrate[ ... ]] but I think it's unlikely you'll get a closed-form expression. $\endgroup$
    – b3m2a1
    Commented Dec 11, 2017 at 7:19

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.