double check the integral done in diracPotential.tex, and perform the first Fourier transformation.

ClearAll[ kappa, k, m, a, t, ee]
$Assumptions = kappa ∈ Reals  && kappa > 0 && ℏ ∈ Reals && ℏ > 0&& k ∈ Reals  && k > 0 && m ∈ Reals  && m > 0 && t ∈ Reals  && t > 0;

Integrate[ ( Sqrt[kappa] E^(-kappa Abs[x] ))^2, {x, -Infinity, Infinity}]

a[k_] = Sqrt[kappa/2/Pi] Integrate[ E^(-I k x - kappa Abs[x]), {x, -Infinity, Infinity}]
ee[kappa_] = -(kappa ℏ)^2/(2m)

a[k] /. kappa -> (*m (Subscript[V, 0])/ℏ^2*) κ // Simplify // TraditionalForm
a[k] /. kappa -> m (Subscript[V, 0])/ℏ^2 // Simplify // TraditionalForm

(((Sqrt[(2/π)])  (((m (Subscript[V, 0])))^(3/2))/ℏ)/((k^2) (ℏ^2)+(m^2) (Subscript[V, 0]^2)/(ℏ^2)))

psi[x_, t_] =(1/Sqrt[2 Pi])Integrate[ Exp[ I k x - I ℏ k^2 t/(2m)] a[k], {k, -Infinity, Infinity}] // Simplify

psi[x_, t_] =FourierTransform[ Exp[ - I ℏ k^2 t/(2m)] a[k], k,x]