Notebook to show the time dependence of complex wave functions. Written by Steve Goldhaber.

Clear["Global`*"]
tscale=100;

tscale is the time scale for manipulate

First, define the real and imaginary parts of an ISW wave function

PnRe[n_Integer,x_Real,t_Real]:=Sin[n*Pi*x]*Cos[n*n*Pi*t*2/tscale];

PnIm[n_Integer,x_Real,t_Real]:=Sin[n*Pi*x]*Sin[-n*n*Pi*t*2/tscale]

Now, let's see what the ground state looks like as a function of time

Manipulate[ParametricPlot3D[{q,PnIm[1,q,t],PnRe[1,q,t]},{q,0,1},AxesLabel->{"x","Im","Re"},PlotStyle->{Thick,Red},PlotRange->{{0.0,1.0},{-1.0,1.0},{-1.0,1.0}},BoxRatios->{1,1,1}],{t,0,tscale,0.1}]

Next, let's plot the ground state and first excited state on the same graph

Manipulate[ParametricPlot3D[{{q,PnIm[1,q,t],PnRe[1,q,t]},{q,PnIm[2,q,t],PnRe[2,q,t]}},{q,0,1},AxesLabel->{"x","Im","Re"},PlotStyle->{{Thick,Red},{Thick,Blue}},PlotRange->{{0.0,1.0},{-1.0,1.0},{-1.0,1.0}},BoxRatios->{1,1,1}],{t,0,tscale,0.1}]

Finally, let's plot the sum as a single graph

Manipulate[ParametricPlot3D[{q,PnIm[1,q,t]+PnIm[2,q,t],PnRe[1,q,t]+PnRe[2,q,t]},{q,0,1},AxesLabel->{"x","Im","Re"},PlotStyle->{Thick,Purple},PlotRange->{{0.0,1.0},{-1.8,1.8},{-1.8,1.8}},BoxRatios->{1,1,1}],{t,0,tscale,0.1}]

What does it look like if you just see the projection in real space (you can grab it and turn it to see the whole thing)?

Manipulate[ParametricPlot3D[{q,PnIm[1,q,t]+PnIm[2,q,t],PnRe[1,q,t]+PnRe[2,q,t]},{q,0,1},AxesLabel->{"x","Im","Re"},PlotStyle->Thick,PlotRange->{{0.0,1.0},{-2.0,2.0},{-2.0,2.0}},BoxRatios->{1,1,1},ViewPoint->{0,-Infinity,0}],{t,0,tscale,0.1}]

Here is the simple probability density plot as a function of time for ((Subscript[u, 1])(x,t)+(Subscript[u, 2])(x,t))/(Sqrt[2])

Manipulate[Plot[(Sin[Pi*x]^2+Sin[2*Pi*x]^2+2*Sin[Pi*x]*Sin[2*Pi*x]*Cos[3*Pi*t*2/tscale])/2,{x,0,1},AxesLabel->{"x","|ψ((|)^2)"},PlotStyle->Thick,PlotRange->{{0.0,1.0},{-2.0,2.0}}],{t,0,tscale,0.1}]