Time Evolution of the
Free Particle

Definition

Needs["Graphics`MultipleListPlot`"]
\(b[m_, x0_, σ_, k0_] := 
      Block[{n = 2\^m, dx = 1/n}, 
		x\ \ \ \  = \ 
          Range[0, n - 1]\ *\ 
            dx; 
		fr\ \  = \ \(E\^\(\(-\((x - x0)\)\^2\)\/\(2\ σ\^2\)\)\) E\^\(\(-\ I\)\ 2\ π\ k0\ x\); 
		fk\  = \ Fourier[fr]; 
		Do[
          Block[{}, 
		fkt\  = \ 
              fk\ *\ Table[E\^\(\(-I\) \(\((2  π\ j)\)\^2\) t\), {j, 0, n - 1}]; 
		frt\  = \ 
              InverseFourier[fkt]; 
		MultipleListPlot[
              Re[frt], Abs[frt]\^2, 
              PlotJoined -> 
                True, 
		SymbolShape -> 
                None, 
		PlotStyle -> {{Thickness[ .005], Hue[0.6]}, {Thickness[ .005], 
                    Hue[1.0]}}, 
		PlotRange -> {{0, 
                    n}, {0, 1}}];
		], {t, 
            0,  .001,  .00005}]; 
		SelectionMove[
          EvaluationNotebook[], After, 
          EvaluationCell]; 
		SelectionMove[
          EvaluationNotebook[], Next, 
          CellGroup]; 
		FrontEndTokenExecute["OpenCloseGroup"]
		]\)

Time Evolution

b[m = 8, x0 =  .6, σ =  .01, k0 = 80];