File:Quantum simple pendulum.webm
Summary
| Description |
English: A quantum simple pendulum.
The pendulum position is spread out, with opacity here being proportional to the probability that the pendulum is at that position at a given time. The average position of the quantum dynamics is the same as the classical pendulum dynamics (Ehrenfest theorem). Technicalities: I used the Crank-Nicholson method to evolve the system in time. This is a 1D problem, and the only variable I considered was the angle, with the initial state being a Gaussian. |
| Date | |
| Source | https://mathstodon.xyz/@j_bertolotti/112038394907883281 |
| Author | Berto |
| Permission (Reusing this file) |
https://mathstodon.xyz/@j_bertolotti/111363365323269417 |
Mathematica 14.0 code
L = 100; (*System size*) dx = 0.1; (*Space step*) dt = 0.0025; (*time step*) k0 = 0; (*Momentum of the incident packet*)
V1 = Table[50 (Cos[(2 \[Pi])/L x] + 1), {x, 0, L, dx}]; (*Potential*)
dim = Dimensions[V1][[1]]
(*For a undergraduate level explanation of why this works, see physik.uni-graz.at/~pep/Theses/BachelorThesis_Wachter_2017.pdf *)
\[Alpha] = (I dt)/(2 dx^2);
\[Xi]1 = Table[1 + (I dt)/2 (2/dx^2 + V1[[j]]), {j, 1, dim}];
\[Gamma]1 = Table[1 - (I dt)/2 (2/dx^2 + V1[[j]]), {j, 1, dim}];
U1 = SparseArray[{Band[{1, 1}] -> \[Xi]1, Band[{2, 1}] -> -\[Alpha], Band[{1, 2}] -> -\[Alpha]}, {dim, dim}];
U2 = SparseArray[{Band[{1, 1}] -> \[Gamma]1, Band[{2, 1}] -> \[Alpha], Band[{1, 2}] -> \[Alpha]}, {dim, dim}];
\[Psi]0 = Table[E^(-((x - L/2 + L/4)^2/(2 2^2))) E^(I k0 x), {x, 0, L, dx}]; (*Initial condiotion*)
\[Psi]0 = \[Psi]0/Sqrt[Total[Abs[\[Psi]0]^2]]; (*Normalization*)
nsteps = 20000; (*Number of time steps*)
U = Inverse[U1] . U2; (*Evolution operator*)
evo1 = NestList[Dot[U, #] &, \[Psi]0, nsteps]; (*Apply the evolution operator repeatedly to obtain the soltion at each time step*)
xTo\[Theta][x_] := x/dim 2 \[Pi];
timedim = Dimensions[evo1][[1]];
expect = Table[xTo\[Theta]@Range[dim] . Abs[evo1[[t]] ]^2, {t, 1, timedim, 1}];
frames = Table[
Graphics[{
Black, Thickness[0.02], Line[{{0, 0}, {Sin[expect[[timestep]]], Cos[expect[[timestep]]]}}], Disk[{Sin[expect[[timestep]]], Cos[expect[[timestep]]]}, 0.12],
White, Thickness[0.01], Line[{{0, 0}, {Sin[expect[[timestep]]], Cos[expect[[timestep]]]}}], Disk[{Sin[expect[[timestep]]], Cos[expect[[timestep]]]}, 0.11],
Thick, Black,
Table[{Opacity[5*Abs[evo1[[timestep, j]] ]^2], Line[{{0, 0}, {Sin[xTo\[Theta][j]], Cos[xTo\[Theta][j]]}}], Disk[{Sin[xTo\[Theta][j]], Cos[xTo\[Theta][j]]}, 0.1]}, {j, 1, dim}], Red, Point[{0, 0}]},
PlotRange -> {{-1.2, 1.2}, {-1.2, 1.2}}]
, {timestep, 1, timedim, 100}];
ListAnimate[frames]
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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