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Summary
| Description |
English: Trajectories of a particle in a box (also called an infinite square well) in classical mechanics (A) and quantum mechanics (B-F). In (A), the particle moves at constant velocity, bouncing back and forth. In (B-F), wavefunction solutions to the Time-Dependent Schrodinger Equation are shown for the same geometry and potential. The horizontal axis is position, the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. (B,C,D) are stationary states (energy eigenstates), which come from solutions to the Time-Independent Schrodinger Equation. (E,F) are non-stationary states, solutions to the Time-Dependent but not Time-Independent Schrodinger Equation. Both (E) and (F) are randomly-generated superpositions of the four lowest-energy eigenstates, (B-D) plus a fourth not shown. |
| Date | |
| Source | Own work |
| Author | Sbyrnes321 |
(*Source code written in Mathematica 6.0 by Steve Byrnes, Apr. 2011.
This source code is public domain.*)
(*Shows classical and quantum trajectory animations for an infinite-square-well potential.
Assumes L=hbar=1, m=2*pi^(-2), so that the nth energy eigenstate has energy n^2.*)
ClearAll["Global`*"]
(***Wavefunctions of the energy eigenstates***)
psi[n_, x_] := Sin[n*Pi*x]*2^(1/2);
energy[n_] := n^2;
psit[n_, x_, t_] := psi[n, x] Exp[-I*energy[n]*t];
(***A random time-dependent state***)
SeedRandom[1];
CoefList = Table[Random[]*Exp[2*Pi*I*Random[]], {n, 1, 4}];
CoefList = CoefList/Norm[CoefList];
Randpsi[x_, t_] := Sum[CoefList[[n]]*psit[n, x, t], {n, 1, 4}];
(***Another random time-dependent state***)
SeedRandom[2];
CoefList2 = Table[Random[]*Exp[2*Pi*I*Random[]], {n, 1, 3}];
CoefList2 = CoefList2/Norm[CoefList2];
Randpsi2[x_, t_] := Sum[CoefList2[[n]]*psit[n, x, t], {n, 1, 3}];
(***Set default style for plots***)
SetOptions[Plot,
{PlotRange -> {{-.05, 1.05}, {-2.5, 2.5}}, Ticks -> None,
PlotStyle -> {Directive[Thick, Blue], Directive[Thick, Pink]},
Axes -> {True, False}}];
SetOptions[ListPlot, {PlotRange -> {{-.05, 1.05}, {-2.5, 2.5}}, Axes -> False}];
(***Draw walls***)
walls = ListPlot[{{{0, -2.5}, {0, 2.5}}, {{1, -2.5}, {1, 2.5}}},
Joined -> True, PlotStyle -> {{Thick, Black}, {Thick, Black}}];
(***Make the classical plot...a red ball bounces back and forth.***)
classicaltrajectory[t_, left_, right_] := 2*(right - left)*Abs[t - Round[t]] + left;
classicalball[t_, left_, right_] := ListPlot[{{classicaltrajectory[t, left, right], 0}},
PlotStyle -> Directive[Red, AbsolutePointSize[15]]];
classical[t_, label_] := Show[walls, classicalball[t, .1, .9], PlotLabel -> label];
(***Make the quantum plots***)
plotpsi[n_, t_, label_] := Show[walls,
Plot[{Re[psit[n, x, t]], Im[psit[n, x, t]]}, {x, 0, 1}],
PlotLabel -> label, Axes -> {True, False}, Ticks -> None];
plotrand[t_, label_] := Show[walls,
Plot[{Re[Randpsi[x, t]], Im[Randpsi[x, t]]}, {x, 0, 1}],
PlotLabel -> label, Axes -> {True, False}, Ticks -> None];
plotrand2[t_, label_] := Show[walls,
Plot[{Re[Randpsi2[x, t]], Im[Randpsi2[x, t]]}, {x, 0, 1}],
PlotLabel -> label, Axes -> {True, False}, Ticks -> None];
(***Put all the plots together***)
MakeFrame[t_] := GraphicsGrid[
{{classical[3 t/(4 Pi), "A"], plotpsi[1, t, "B"]},
{plotpsi[2, t, "C"], plotpsi[3, t, "D"]},
{plotrand[t, "E"], plotrand2[t, "F"]}},
Frame -> All, ImageSize -> 300];
output = Table[MakeFrame[t], {t, 0, 4 Pi*138/139, 4 Pi/139}];
SetDirectory["C:\\Users\\Steve\\Desktop"]
Export["test.gif", output, "DisplayDurations" -> 10]
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
 
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
| The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 06:39, 27 April 2011 | 300 × 280 (1,006 KB) | Sbyrnes321 | {{Information |Description ={{en|1=Trajectories of a particle in a box (also called an infinite square well) in classical mechanics (A) and quantum mechanics (B-F). In (A), the particle moves at constant velocity, bouncing back and forth. In (B-F), wav |
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