(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 3208351, 54878] NotebookOptionsPosition[ 3196627, 54526] NotebookOutlinePosition[ 3197040, 54544] CellTagsIndexPosition[ 3196997, 54541] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.4276382031460752`*^9, 3.427638203515709*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.427638206525607*^9, 3.427749010441477*^9, 3.427846911746355*^9, 3.5687528514525347`*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.427638270622589*^9}], Cell[CellGroupData[{ Cell["Quantum Motion of a Particle in a Box", "Title", CellChangeTimes->{{3.427638289012788*^9, 3.4276383037985353`*^9}}], Cell["\<\ Nicholas Wheeler ÌÇÐÄÊÓÆµ Physics Department 13 August 2008\ \>", "Text", CellChangeTimes->{{3.427638310552442*^9, 3.427638326702342*^9}}, FontSize->10], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.427638339742312*^9}], Cell[CellGroupData[{ Cell["Introduction", "Subsection", CellChangeTimes->{{3.427638350096182*^9, 3.427638354253036*^9}}], Cell["\<\ It is by way of preparation for discussion of \"quantum Liossajous figures\" \ and \"quantum vorticity\" that I look here to aspects\[LongDash]specifically, \ dynamics aspects\[LongDash]of the simplest and most familiar of all quantum \ mechanical systems: the \"particle-in-a-box\" problem. 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Some of what I will have to say parallels work \ reported in Cimarron Wortham's ÌÇÐÄÊÓÆµ thesis: \"Motion of launched wave \ packets in the infinite square well\" (2004)." }], "Text", CellChangeTimes->{{3.427639862352906*^9, 3.427639899386907*^9}, { 3.4276399445096397`*^9, 3.427640039203033*^9}, {3.427640122741301*^9, 3.427640288540469*^9}}], Cell["We are interested in properties of the solutions of", "Text", CellChangeTimes->{{3.427640310669792*^9, 3.427640328690339*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", FractionBox[ SuperscriptBox["\[HBar]", "2"], RowBox[{"2", "m"}]]}], RowBox[{ SubscriptBox["\[PartialD]", RowBox[{"x", ",", "x"}]], RowBox[{"\[Psi]", "[", RowBox[{"x", ",", "t"}], "]"}]}]}], "=", RowBox[{"\[ImaginaryI]", " ", "\[HBar]", " ", RowBox[{ SubscriptBox["\[PartialD]", "t"], " ", RowBox[{"\[Psi]", "[", RowBox[{"x", ",", "t"}], "]"}]}]}]}]], "DisplayFormula", CellChangeTimes->{{3.427640334827813*^9, 3.427640452472559*^9}, 3.4276405128154716`*^9, {3.4276768886001167`*^9, 3.427676891868166*^9}}], Cell["that conform to the boundary conditions", "Text", CellChangeTimes->{{3.4276404786269217`*^9, 3.427640487698247*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"\[Psi]", "[", RowBox[{"0", ",", "t"}], "]"}], "=", RowBox[{ RowBox[{"\[Psi]", "[", RowBox[{"a", ",", "t"}], "]"}], " ", ":", " ", RowBox[{"all", " ", "t"}]}]}]], "DisplayFormula", CellChangeTimes->{{3.427640500919813*^9, 3.427640549028561*^9}}], Cell["If we adopt units in which", "Text", CellChangeTimes->{{3.427640818301907*^9, 3.4276408231779003`*^9}}], Cell[BoxData[ RowBox[{"\[HBar]", "=", RowBox[{ RowBox[{"2", "m"}], "=", RowBox[{"a", "=", "1"}]}]}]], "DisplayFormula", CellChangeTimes->{{3.427640827206102*^9, 3.427640841733309*^9}}], Cell["we have", "Text", CellChangeTimes->{{3.4276408692819843`*^9, 3.4276408699777822`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"-", RowBox[{ SubscriptBox["\[PartialD]", RowBox[{"x", ",", "x"}]], RowBox[{"\[Psi]", "[", RowBox[{"x", ",", "t"}], "]"}]}]}], "=", RowBox[{"\[ImaginaryI]", " ", RowBox[{ SubscriptBox["\[PartialD]", "t"], " ", RowBox[{"\[Psi]", "[", RowBox[{"x", ",", "t"}], "]"}]}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Psi]", "[", RowBox[{"0", ",", "t"}], "]"}], "=", RowBox[{ RowBox[{"\[Psi]", "[", RowBox[{"1", ",", "t"}], "]"}], " ", ":", " ", RowBox[{"all", " ", "t"}]}]}]}], "DisplayFormula", CellChangeTimes->{{3.4276408890139523`*^9, 3.427640912845538*^9}}], Cell["\<\ which is easier to work with. 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Observable quantum motion arises \ when such functions are superimposed, as an interference effect. 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Which amounts to shrinking the time scale " }], "Text", CellChangeTimes->{{3.427822299155775*^9, 3.427822365186771*^9}, { 3.427822398161037*^9, 3.4278225208172197`*^9}, 3.4279891343461723`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"t", "\[RightArrow]", "\[Theta]"}], "=", RowBox[{"\[Alpha]", " ", RowBox[{"t", " ", ":", " ", RowBox[{"\[Alpha]", "\[LessLess]", " ", "1"}]}]}]}]], "DisplayFormula", CellChangeTimes->{{3.4278225261425743`*^9, 3.427822581874565*^9}}], Cell["\<\ We have encountered the familiar fact which lies at the base of the Feynman \ formalism\[LongDash]namely, that \"quantum mechanics is briefly classical.\" \ I propose therefore to look to the short-time motion of a launched wavepacket \ (in sufficiently long times it will execute\[LongDash]exactly\[LongDash]all \ the complicated gyrations described above, and will, in particular, display \ recurrence).\ \>", "Text", CellChangeTimes->{{3.4278225989962473`*^9, 3.42782280849666*^9}}], Cell["\<\ Notice in this connection that in the most recent of the preceding figures \ the value of \[LeftAngleBracket]x\[RightAngleBracket] oscillates back and \ forth during the brief interval\ \>", "Text", CellChangeTimes->{{3.4280212249927197`*^9, 3.428021344077156*^9}, 3.428021393640823*^9}], Cell[BoxData[ RowBox[{"0", "<", "t", "<", RowBox[{ FractionBox["20", "600"], SubscriptBox["T", "recurrence"]}]}]], "DisplayFormula", CellChangeTimes->{{3.428021349121915*^9, 3.428021370135173*^9}}], Cell["\<\ with seemingly constant frequency and slowly diminishing amplitude. 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