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File:Many Interacting Worlds-Ground state of harmonic oscilator.gif

Many_Interacting_Worlds-Ground_state_of_harmonic_oscilator.gif ‎(576 × 189 pixels, file size: 8.15 MB, MIME type: image/gif, looped, 1,000 frames)

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Captions

Captions

Zero point energy in the Many Interacting World interpretation for a harmonic oscillator.

Summary edit

Description
English: In the "Many Interacting Worlds" interpretation of QM, each particle interact classically with the rest of the world, but it also interacts (weakly) with all the copies of itself that exist in different worlds, resulting in a "quantum" behavior.

In the animation: copies of the same mass in a harmonic potential with (viscous) friction. In the classical case all copies set on the lowest energy level, but in the quantum case they repel each other, and thus spread out in energy (blue lines are the energy of each copy).

[Refs]

Date
Source https://twitter.com/j_bertolotti/status/1118076348659007488
Author Jacopo Bertolotti
Permission
(Reusing this file)		 https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 11.0 code edit 

Clear[pos, posold, posoldold, acc];
np = 20;
m = 1;
k = 10;
dt = 0.01;
b = 0.1;
pos = Table[RandomReal[{-1, 1}], np];
v0 = Table[0, np];
nstep = 1000;
posold = pos;
posoldold = pos;
evo = Reap[Do[
      acc = Table[-k pos[[j]] - 1 (pos[[j]] - posold[[j]])/dt, {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}];
      , nstep];][[2, 1]];
pos = Table[RandomReal[{-1, 1}], np];
v0 = Table[0, np];
posold = pos;
posoldold = pos;
evoq = Reap[Do[
      P = (Exp[-Sqrt[(x - #)^2]/b] &) /@ pos;
      acc = Table[-k pos[[j]] - 1 (pos[[j]] - posold[[j]])/dt - (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}];
      , nstep];][[2, 1]];
ptot = Table[
   GraphicsRow[{
     Show[
      Plot[x^2, {x, -1, 1}, PlotStyle -> {Red, Thick, Dashed}, Ticks -> None, AxesLabel -> {"x", "E"}, LabelStyle -> {Black, Bold}, PlotLabel -> "Classic case:\n copies do not interact"],
      Graphics[{Table[{Opacity[0.3], Disk[{evo[[j, 1, n]], 0}, 0.05], Blue, Opacity[1], Line[{{-1, evo[[j, 2, n]]/3}, {1, evo[[j, 2, n]]/3} }]
          }, {n, 1, np}]}, PlotRange -> {{-1., 1.}, {-1, 1}}]
      ],
     Show[
      Plot[x^2, {x, -1, 1}, PlotStyle -> {Red, Thick, Dashed}, Ticks -> None, AxesLabel -> {"x", "E"}, LabelStyle -> {Black, Bold}, PlotLabel -> "Quantum case:\n copies do interact (U=|\[Del]\[Rho]/\[Rho]\!\
\(\*SuperscriptBox[\(|\), \(2\)]\))"], Graphics[{Table[{Opacity[0.3], Disk[{evoq[[j, 1, n]], 0}, 0.05]
          , Blue, Opacity[1], Line[{{-1, evoq[[j, 2, n]]/3}, {1, evoq[[j, 2, n]]/3} }]
          }, {n, 1, np}]}, PlotRange -> {{-1., 1.}, {-1, 1}}]
      ]
     }, ImageSize -> Large]
   , {j, 1, nstep}];
ListAnimate[ptot]

Licensing edit

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero 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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Date/TimeThumbnailDimensionsUserComment
current08:02, 17 April 2019Thumbnail for version as of 08:02, 17 April 2019576 × 189 (8.15 MB)Berto (talk | contribs)User created page with UploadWizard

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