# Unable to integrate function with several variables using Integrate

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?

• Are you even sure your integral has a closed form expression? Commented Oct 25, 2017 at 9:32
• @J.M. I expect to have some expression as this is a well known problem in quantum mechanics. Commented Oct 25, 2017 at 9:39
• 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? Commented Oct 25, 2017 at 9:42
• @J.M. there is exactly the same expression (Eq. 7) in the following paper: arxiv.org/abs/1303.6181 Commented Oct 25, 2017 at 9:53
• 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. Commented Dec 11, 2017 at 7:19