Summary edit

Mathematica 11.0 code edit

Clear[pos, posold, posoldold, acc];
np = 30;
m = 1;
k = 10;
dt = 0.002;
b = 0.1;
pos = Table[RandomReal[{-1, -0.2}], np];
v0 = Table[0, np];
nstep = 2000;
V = x^2 + 1. E^(-(x^2/(2 0.01^2)));
dV = D[V, x];
posold = pos;
posoldold = pos;
evoq = Reap[Do[
      P = (Exp[-Sqrt[(x - #)^2]/b] &) /@ pos;
      acc = Table[
        -k (dV /. {x -> pos[[j]]}) - (Evaluate[D[0.01 (Total@Drop[D[P, x], {j}]/Total@Drop[P, {j}])^2, x]] /. {x -> pos[[j]]})
        , {j, 1, np}];
      posoldold = posold;
      posold = pos;
      pos = 2 posold - posoldold + acc dt^2;
      Sow[{pos, m/2 ((pos - posold)/dt)^2 + k/2 pos^2}];
      (*Print[posoldold,posold,pos,acc];*)
      , nstep];][[2, 1]];
p1q = Table[
   Show[
    Plot[V, {x, -1, 1}, PlotStyle -> {Red, Thick, Dashed}, Ticks -> None, AxesLabel -> {"x", "E"}, LabelStyle -> {Black, Bold}, PlotLabel -> 
      "Quantum case:\n copies do interact (U=|\!\(\*FractionBox[\(\\[Del]\[Rho]\), \(\[Rho]\)]\)\!\(\*SuperscriptBox[\(|\), \(2\)]\))"],    Graphics[{Table[{Opacity[0.3], Disk[{evoq[[j, 1, n]], 0}, 0.05]}, {n, 1, np}]}, PlotRange -> {{-1., 1.}, {-1, 1}}]
    ]
   , {j, 1, nstep, 10}];
ListAnimate[p1q]

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