Integrate[
      q (E^(I q r - (q^2) 
                (d^2))) ((E^((2  (d^2) - I t) k q)) - 
            (E^(-(2  (d^2) - I t) k q))), {q, 
        0, ∞}]

Simplify[%]

Simplify[
      Expand[(2 (-(((E^(((((2 (d^2) k + I (r - k t)))^2)/(4 (d^2))))) π (((-2 (d^2) k + I (-r + k t)))^3))/(8 ((((d^2)))^(3/2)))) - 
                    (1/16) (E^(((((2 (d^2) k + I (r - k t)))^2)/(4 (d^2))))) (Sqrt[π]) (((((((2 (d^2) k + I (r - k t)))^2)/(d^2))))^(3/2)) (-2 (Sqrt[π]) - ((4 (E^(-((((2 (d^2) k + I (r - k t)))^2)/(4 (d^2))))))/(Sqrt[((((2 (d^2) k + I (r - k t)))^2)/(d^2))])) + 2 (Sqrt[π]) (1 - 
                                Erf[(1/2) (Sqrt[((((2 (d^2) k + I (r - k t)))^2)/(d^2))])]))))/((Sqrt[π]) (((-2 (d^2) k - I r + I k t))^2)) - (2 (-(((E^(((((2 (d^2) k - I (r + k t)))^2)/(4 (d^2))))) π (((2 (d^2) k - I (r + k t)))^3))/(8 ((((d^2)))^(3/2)))) - 
                    (1/16) (E^(((((2 (d^2) k - I (r + k t)))^2)/(4 (d^2))))) (Sqrt[π]) (((((((2 (d^2) k - I (r + k t)))^2)/(d^2))))^(3/2)) (-2 (Sqrt[π]) - ((4 (E^(-((((2 (d^2) k - I (r + k t)))^2)/(4 (d^2))))))/(Sqrt[((((2 (d^2) k - I (r + k t)))^2)/(d^2))])) + 2 (Sqrt[π]) (1 - 
                                Erf[(1/2) (Sqrt[((((2 (d^2) k - I (r + k t)))^2)/(d^2))])]))))/((Sqrt[π]) (((2 (d^2) k - I r - I k t))^2))]]

out = % ((2  π (E^(-(k^2) (d^2))))/(2  (d^2) k));

Plot[(((((d^2)/(2  (π^3)))))^(3/2)) (k^2) 
          4  π (((Abs[out]))^2) /. {k -> 1, t -> -30, 
          d -> 10}, {r, 0, 100}, 
      PlotRange -> {0, 0.001}]

\(Do[Plot[\(\((d\^2\/\(2  π\^3\))\)\^\(3/2\)\) \(k\^2\) 
            4  π \((Abs[out])\)\^2 /. {k -> 1, t -> i, 
            d -> 10}, {r, 0, 100}, PlotRange -> {0, 0.001}, 
        AxesLabel -> {r, \*""4 π (r^2)|ψ\!\(\( | \^2\)\)""}, PlotLabel -> {"t=", i}], {i, \(-70\), 70}]\)