File:Spin rotation.gif
Spin_rotation.gif (360 × 384 pixels, file size: 1.83 MB, MIME type: image/gif, looped, 126 frames)
Captions
Captions
Summary
editDescriptionSpin rotation.gif |
English: A (normalized) 2-components spinor can be described by 3 angles. Two are the spin orientation, and the third a global phase. When you make a 2π rotation this phase is not the same as before, and you need two full turn to get back to the original state. |
Date | |
Source | https://twitter.com/j_bertolotti/status/1142503314287943680 |
Author | Jacopo Bertolotti |
Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 11.0 code
edits ={Cos[\[Pi]/8] E^(-I \[Pi]/8), Sin[\[Pi]/8] E^(I \[Pi]/8)}; s = s/Norm[s]; evox = Reap[ For[\[Theta] = 0, \[Theta] <= 4 \[Pi], \[Theta] = \[Theta] + 0.1, tmp = MatrixExp[I/2 PauliMatrix[1]*\[Theta]].s; Sow[Chop[tmp]]; ];][[2, 1]]; (*The "circle3D" function definition was taken from https://mathematica.stackexchange.com/questions/79524/how-to-draw-a-circle-in-3d-on-a-sphere*) circle3D[centre_: {0, 0, 0}, radius_: 1, normal_: {0, 0, 1}, angle_: {0, 2 Pi}] := Composition[Line, Map[RotationTransform[{{0, 0, 1}, normal}, centre], #] &, Map[Append[#, Last@centre] &, #] &, Append[DeleteDuplicates[Most@#], Last@#] &, Level[#, {-2}] &, MeshPrimitives[#, 1] &, DiscretizeRegion, If][First@Differences@angle >= 2 Pi, Circle[Most@centre, radius], Circle[Most@centre, radius, angle]] (*The "Unwrap" function definition was taken from https://community.wolfram.com/groups/-/m/t/1340126*) ClearAll[Unwrap] Unwrap[lst_List] := Unwrap[lst, 2 Pi] (*phase jumps of 2Pi is the default because of trigonometric funtions*) Unwrap[lst_List, \[CapitalDelta]_] := Unwrap[lst, \[CapitalDelta], Scaled[0.5]] (*default tolerance is half the phase jump \[CapitalDelta]*) Unwrap[lst_List, \[CapitalDelta]_, tolerance_] := Module[{tol, jumps}, tol = If[Head[tolerance] === Scaled, \[CapitalDelta] tolerance[[1]], tolerance]; jumps = Differences[lst]; jumps = -Sign[jumps] Unitize[Chop[Abs[jumps], tol]]; jumps = \[CapitalDelta] Prepend[Accumulate[jumps], 0]; jumps + lst] (**) pole[s_] := {Conjugate[s].PauliMatrix[1].s, Conjugate[s].PauliMatrix[2].s, Conjugate[s].PauliMatrix[3].s}; flag[s_] := (Arg[s[[2]]] + Arg[s[[1]]])/2; \[Alpha] = Unwrap[flag /@ evox, \[Pi]]; \[Alpha] = \[Alpha] - Min[\[Alpha]]; \[Alpha] = \[Alpha]/Max[\[Alpha]]; psx = Table[ Graphics3D[{Text[Style["x", Bold, Large], {1.4, 0, 0}], Text[Style["y", Bold, Large], {0, 1.4, 0}], Text[Style["z", Bold, Large], {0, 0, 1.4}], Thick, Arrow[{{-1.2, 0, 0}, {1.3, 0, 0}}], Arrow[{{0, -1.2, 0}, {0, 1.3, 0}}], Arrow[{{0, 0, -1.2}, {0, 0, 1.3}}], Hue[\[Alpha][[j]]], PointSize[0.05], Point[pole[evox[[j]]]], Line[{{0, 0, 0}, pole[evox[[j]]]}], Opacity[0.8], White, Sphere[{0, 0, 0}, 1], Opacity[1], Gray, circle3D[{0, 0, 0}, 1, {0, 0, 1}], circle3D[{0, 0, 0}, 1, {1, 0, 0}], circle3D[{0, 0, 0}, 1, {0, 1, 0}]}, Boxed -> False, Lighting -> "Neutral", ViewPoint -> {2, 2, 2}, Epilog -> Inset[BarLegend[{Hue[#/(2 \[Pi])] &, {0, 2 \[Pi]}}, Ticks -> Table[j, {j, 0, 2 \[Pi], \[Pi]/2}], LegendLabel -> "Spinor's\nglobal phase"], {0.9, 0.5}] ] , {j, 1, dim}]; ListAnimate[psx]
Licensing
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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.
http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse |
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current | 18:49, 23 June 2019 | ![]() | 360 × 384 (1.83 MB) | Berto (talk | contribs) | User created page with UploadWizard |
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