(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 3797603, 63289] NotebookOptionsPosition[ 3789532, 63042] NotebookOutlinePosition[ 3790507, 63075] CellTagsIndexPosition[ 3790328, 63068] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.531078729260962*^9, 3.531078732195829*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.531078733580008*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.531078737783594*^9}], Cell[CellGroupData[{ Cell["Iteration Techniques", "Title", CellChangeTimes->{{3.531078746356318*^9, 3.531078755056684*^9}}], Cell["\<\ Nicholas Wheeler 23 November 2011\ \>", "Text", CellChangeTimes->{{3.5310787607385197`*^9, 3.5310787724633293`*^9}}, FontSize->10], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.5310787985027227`*^9}], Cell[CellGroupData[{ Cell["Introduction", "Subsection", CellChangeTimes->{{3.531088834855207*^9, 3.531088841129381*^9}}], Cell["\<\ In my recent \"Experimental Mathematical Physics\" seminar (9 November 2011) \ I had occasion to remark (Slide 30: \[Section]5. 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One could introduce a \ \"more physical\" mechanism for introducing randomness\[Ellipsis]as is \ presumably contemplated in serious stochastic DE work\[Ellipsis]but again my \ motivation fails me. Note the clear violation of energy conservation, which I suppose is to be \ expected if randomness is considered to be a result of uncontrolled \ perturbative interaction with an \"environment.\" Energy non-conservation \ would be expected even if that interaction were \"more physical.\" If my undergraduate self were to sketch his idea to me today I would advise \ him to forget it, for it seems incapable of capturing any of the essentials \ of quantum mechanics. A worthy objective would be to demonstrate a mechanism \ whereby the classical distribution\ \>", "Text", CellChangeTimes->{{3.531237347106667*^9, 3.53123767103226*^9}, { 3.531262009045027*^9, 3.531262251187769*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"P", "[", RowBox[{"x", ",", "p", ",", "t"}], "]"}], "=", RowBox[{ RowBox[{"\[Delta]", "[", RowBox[{"x", "-", RowBox[{"x", "[", "t", "]"}]}], "]"}], RowBox[{"\[Delta]", "[", RowBox[{"p", "-", RowBox[{"p", "[", "t", "]"}]}], "]"}]}]}]], "Output", CellChangeTimes->{{3.531262257491983*^9, 3.5312623050947733`*^9}}], Cell["\<\ acquires features of the Wigner distribution, of which the most essential is\ \>", "Text", CellChangeTimes->{{3.531262311672757*^9, 3.5312623312118053`*^9}, { 3.531262374991515*^9, 3.5312623807288017`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"\[Integral]", RowBox[{ RowBox[{"P", "[", RowBox[{"x", ",", "p"}], "]"}], RowBox[{"\[DifferentialD]", "x"}], RowBox[{"\[DifferentialD]", "p"}]}]}], "=", "1"}]], "Output", CellChangeTimes->{{3.531262400871323*^9, 3.531262410356513*^9}, { 3.5312624408674097`*^9, 3.531262454210249*^9}}], Cell[TextData[{ "But that looks pretty hopeless to me at the moment (maybe in another 56 \ years?).\n\nPossibly more workable/interesting is this idea: ", StyleBox["Introduce a random jiggle into", FontColor->RGBColor[1, 0, 0]] }], "Text", CellChangeTimes->{{3.531262523244095*^9, 3.531262630968461*^9}}], Cell[BoxData[ StyleBox[ RowBox[{ SubscriptBox["\[Psi]", "t"], "=", RowBox[{ RowBox[{"\[DoubleStruckCapitalU]", "[", "t", "]"}], SubscriptBox["\[Psi]", "0"]}]}], "Output", FontColor->RGBColor[1, 0, 0]]], "Input", CellChangeTimes->{3.531262711110682*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Motion in a \"periodic Sina Zeytinoglu potential\"\ \>", "Subsection", CellChangeTimes->{{3.531163279020609*^9, 3.5311633033164988`*^9}}], Cell["\<\ We are led by my understanding of the picture Zeytinoglu has in mind to \ contemplate a dynamic random walk on the 2D plane (4D phase space), in the \ presence of a spatially periodic potential. I construct a couple of such \ potentials:\ \>", "Text", CellChangeTimes->{{3.5312377185908546`*^9, 3.53123783464988*^9}}], Cell[CellGroupData[{ Cell["Illustrative spatially periodic potentials in 2D", "Subsubsection", CellChangeTimes->{{3.53123784087386*^9, 3.531237855756629*^9}}], Cell[BoxData[""], "Input", CellChangeTimes->{{3.531237890657295*^9, 3.531237892560655*^9}}], Cell[TextData[StyleBox["A. 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