(* 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[ 26470, 721] NotebookOptionsPosition[ 25270, 673] NotebookOutlinePosition[ 25608, 688] CellTagsIndexPosition[ 25565, 685] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "\nParticle in a Box\n0 \[LessSlantEqual] x \[LessSlantEqual] 1\nperiod T = ", "\[HBar]/", "(", Cell[BoxData[ FormBox[ SubscriptBox["E", "m"], TraditionalForm]], "None"], "-", Cell[BoxData[ FormBox[ SubscriptBox["E", "n"], TraditionalForm]], "None"], Cell[BoxData[ FormBox[")", TraditionalForm]], "None"] }], "Subtitle", CellChangeTimes->{{3.463701601945812*^9, 3.463701612385776*^9}, { 3.4637025715588913`*^9, 3.4637026317539673`*^9}, {3.46370266404038*^9, 3.463702741787971*^9}, {3.4637549820352707`*^9, 3.463754987421644*^9}}, TextAlignment->Center, Background->GrayLevel[1]], Cell[CellGroupData[{ Cell[TextData[{ "Consider an arbitrary eigenstate, ", Cell[BoxData[ FormBox[ SubscriptBox["\[Psi]", "n"], TraditionalForm]], "None", FormatType->"TraditionalForm"], "(x)" }], "Section", CellChangeTimes->{{3.463701842329862*^9, 3.463701853083794*^9}, { 3.463702025758068*^9, 3.463702068201302*^9}, 3.4637029308176317`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"Psi", "[", RowBox[{"n_", ",", "x_"}], "]"}], ":=", " ", RowBox[{ RowBox[{"Sqrt", "[", "2", "]"}], "*", RowBox[{"Sin", "[", RowBox[{"n", "*", "Pi", "*", "x"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.4637009708143177`*^9, 3.4637009917811623`*^9}, 3.46375304167283*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Start with a 50/50 mix and time-evolve the \nprobability ", Cell[BoxData[ FormBox[ RowBox[{"|", RowBox[{"\[Psi]", RowBox[{"(", RowBox[{"x", ",", "t"}], ")"}]}], SuperscriptBox["|", "2"], " ", RowBox[{"=", RowBox[{ FractionBox["1", "2"], " ", "|", SubscriptBox["\[Psi]", "1"]}]}]}], TraditionalForm]], "None", FormatType->"TraditionalForm"], "(x) ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"i", " ", SubscriptBox["\[Omega]", "1"], "t"}]], TraditionalForm]], "None", FormatType->"TraditionalForm"], " + ", Cell[BoxData[ FormBox[ SubscriptBox["\[Psi]", "2"], TraditionalForm]], "None"], "(x) ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"i", " ", SubscriptBox["\[Omega]", "2"], "t"}]], TraditionalForm]], "None", FormatType->"TraditionalForm"], Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["|", "2"]}], TraditionalForm]], "None", FormatType->"TraditionalForm"], " " }], "Section", CellChangeTimes->{{3.463701842329862*^9, 3.463701899707283*^9}, { 3.4637019344191637`*^9, 3.4637020070733013`*^9}, {3.463702249399989*^9, 3.4637025178100567`*^9}, {3.463702835799185*^9, 3.463702841281068*^9}, 3.463702935171937*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Animate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"0.5", "*", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"Psi", "[", RowBox[{"1", ",", "x"}], "]"}], ")"}], "^", "2"}], "+", RowBox[{ RowBox[{"(", RowBox[{"Psi", "[", RowBox[{"2", ",", "x"}], "]"}], ")"}], "^", "2"}], " ", "+", "\[IndentingNewLine]", RowBox[{"2", "*", RowBox[{"Psi", "[", RowBox[{"1", ",", "x"}], "]"}], "*", RowBox[{"Psi", "[", RowBox[{"2", ",", "x"}], "]"}], "*", RowBox[{"Cos", "[", RowBox[{"2", "*", "Pi", "*", "t"}], "]"}]}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "4"}]}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "5"}], "}"}], ",", RowBox[{"AnimationRunning", "\[Rule]", "False"}]}], "]"}]], "Input", CellChangeTimes->{{3.463700467180892*^9, 3.46370056928338*^9}, 3.4637006561995897`*^9, {3.4637011316312103`*^9, 3.463701143427863*^9}, { 3.463701347419118*^9, 3.46370134977527*^9}, {3.4637015529517307`*^9, 3.463701554130015*^9}, {3.46375305024275*^9, 3.463753089640717*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`t$$ = 2.9912796020507812`, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`t$$], 0, 5}}, Typeset`size$$ = {360., {113., 117.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`t$643$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`t$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`t$$, $CellContext`t$643$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Plot[0.5 ($CellContext`Psi[1, $CellContext`x]^2 + $CellContext`Psi[ 2, $CellContext`x]^2 + 2 $CellContext`Psi[1, $CellContext`x] $CellContext`Psi[ 2, $CellContext`x] Cos[2 Pi $CellContext`t$$]), {$CellContext`x, 0, 1}, PlotRange -> 4], "Specifications" :> {{$CellContext`t$$, 0, 5, AnimationRunning -> False, AppearanceElements -> { "ProgressSlider", "PlayPauseButton", "FasterSlowerButtons", "DirectionButton"}}}, "Options" :> { ControlType -> Animator, AppearanceElements -> None, SynchronousUpdating -> True, ShrinkingDelay -> 10.}, "DefaultOptions" :> {}], ImageSizeCache->{403., {150.34375, 155.65625}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{{3.4637011339544773`*^9, 3.463701159473605*^9}, { 3.463701351433354*^9, 3.463701365196908*^9}, {3.463701555424629*^9, 3.4637015652395973`*^9}, {3.463702858553855*^9, 3.463702863934799*^9}, { 3.463703261484024*^9, 3.463703266841008*^9}, {3.463752970261004*^9, 3.463752983069715*^9}, {3.463753100607477*^9, 3.463753110430161*^9}, 3.463754785257732*^9, {3.463754824205204*^9, 3.463754837269174*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Time-evolve a mixture \n ", Cell[BoxData[ FormBox[ RowBox[{"|", RowBox[{"\[Psi]", RowBox[{"(", RowBox[{"x", ",", "t"}], ")"}]}], SuperscriptBox["|", "2"], " ", RowBox[{"=", RowBox[{ FractionBox["1", "2"], " ", "|", SubscriptBox["\[Psi]", "1"]}]}]}], TraditionalForm]], "None", FormatType->"TraditionalForm"], "(x) ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"i", " ", SubscriptBox["\[Omega]", "1"], "t"}]], TraditionalForm]], "None", FormatType->"TraditionalForm"], " + ", Cell[BoxData[ FormBox[ SubscriptBox["\[Psi]", "3"], TraditionalForm]], "None"], "(x) ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"i", " ", SubscriptBox["\[Omega]", "3"], "t"}]], TraditionalForm]], "None", FormatType->"TraditionalForm"], Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["|", "2"]}], TraditionalForm]], "None", FormatType->"TraditionalForm"], " " }], "Section", CellChangeTimes->{{3.463701842329862*^9, 3.463701853083794*^9}, { 3.463702025758068*^9, 3.463702068201302*^9}, {3.463702099557661*^9, 3.463702232945278*^9}, {3.4637029069747267`*^9, 3.463702924796238*^9}, { 3.463703068687538*^9, 3.4637030732010107`*^9}}], Cell["\<\ Note that the period decreases by a factor of 3/8 but I \"scale this out.\".\ \>", "Text", CellChangeTimes->{{3.4637030850687304`*^9, 3.463703094280373*^9}, { 3.4637031275381813`*^9, 3.463703200270104*^9}, {3.4637550498808107`*^9, 3.463755060335939*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Animate", "[", " ", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"0.5", "*", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"Psi", "[", RowBox[{"1", ",", "x"}], "]"}], ")"}], "^", "2"}], "+", RowBox[{ RowBox[{"(", RowBox[{"Psi", "[", RowBox[{"3", ",", "x"}], "]"}], ")"}], "^", "2"}], " ", "+", "\[IndentingNewLine]", RowBox[{"2", "*", RowBox[{"Psi", "[", RowBox[{"1", ",", "x"}], "]"}], "*", RowBox[{"Psi", "[", RowBox[{"3", ",", "x"}], "]"}], "*", RowBox[{"Cos", "[", RowBox[{"2", "*", "Pi", "*", "t"}], "]"}]}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "4"}]}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "5"}], "}"}], ",", RowBox[{"AnimationRunning", "\[Rule]", "False"}]}], "]"}]], "Input", CellChangeTimes->{{3.463700467180892*^9, 3.46370056928338*^9}, 3.4637006561995897`*^9, {3.463701020353881*^9, 3.463701041252326*^9}, { 3.463701201120122*^9, 3.463701201678919*^9}, {3.463701522457654*^9, 3.4637015318906107`*^9}, {3.4637031489632683`*^9, 3.4637031533939323`*^9}, {3.4637531248097754`*^9, 3.463753137068675*^9}, { 3.463753311599677*^9, 3.463753320599177*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`t$$ = 1.0360560417175293`, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`t$$], 0, 5}}, Typeset`size$$ = {360., {113., 117.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`t$689$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`t$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`t$$, $CellContext`t$689$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Plot[0.5 ($CellContext`Psi[1, $CellContext`x]^2 + $CellContext`Psi[ 3, $CellContext`x]^2 + 2 $CellContext`Psi[1, $CellContext`x] $CellContext`Psi[ 3, $CellContext`x] Cos[2 Pi $CellContext`t$$]), {$CellContext`x, 0, 1}, PlotRange -> 4], "Specifications" :> {{$CellContext`t$$, 0, 5, AnimationRunning -> False, AppearanceElements -> { "ProgressSlider", "PlayPauseButton", "FasterSlowerButtons", "DirectionButton"}}}, "Options" :> { ControlType -> Animator, AppearanceElements -> None, SynchronousUpdating -> True, ShrinkingDelay -> 10.}, "DefaultOptions" :> {}], ImageSizeCache->{403., {150.34375, 155.65625}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{{3.4637010478315153`*^9, 3.463701073938692*^9}, { 3.463701203417741*^9, 3.463701206256494*^9}, {3.463701519204517*^9, 3.46370154432629*^9}, {3.463753141358794*^9, 3.4637531562796993`*^9}, { 3.46375478560172*^9, 3.463754818495574*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Time-evolve a mixture \n ", Cell[BoxData[ FormBox[ RowBox[{"|", RowBox[{"\[Psi]", RowBox[{"(", RowBox[{"x", ",", "t"}], ")"}]}], SuperscriptBox["|", "2"], " ", RowBox[{"=", RowBox[{ FractionBox["1", "2"], " ", "|", SubscriptBox["\[Psi]", "2"]}]}]}], TraditionalForm]], "None", FormatType->"TraditionalForm"], "(x) ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"i", " ", SubscriptBox["\[Omega]", "2"], "t"}]], TraditionalForm]], "None", FormatType->"TraditionalForm"], " + ", Cell[BoxData[ FormBox[ SubscriptBox["\[Psi]", "3"], TraditionalForm]], "None"], "(x) ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"i", " ", SubscriptBox["\[Omega]", "3"], "t"}]], TraditionalForm]], "None", FormatType->"TraditionalForm"], Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["|", "2"]}], TraditionalForm]], "None", FormatType->"TraditionalForm"], " " }], "Section", CellChangeTimes->{{3.463701842329862*^9, 3.463701853083794*^9}, { 3.463702025758068*^9, 3.463702068201302*^9}, {3.463702099557661*^9, 3.463702232945278*^9}, {3.4637029069747267`*^9, 3.463702924796238*^9}, { 3.463703068687538*^9, 3.4637030732010107`*^9}, {3.463753383035862*^9, 3.463753389556665*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Animate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"0.5", "*", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"Psi", "[", RowBox[{"2", ",", "x"}], "]"}], ")"}], "^", "2"}], "+", RowBox[{ RowBox[{"(", RowBox[{"Psi", "[", RowBox[{"3", ",", "x"}], "]"}], ")"}], "^", "2"}], " ", "+", "\[IndentingNewLine]", RowBox[{"2", "*", RowBox[{"Psi", "[", RowBox[{"2", ",", "x"}], "]"}], "*", RowBox[{"Psi", "[", RowBox[{"3", ",", "x"}], "]"}], "*", RowBox[{"Cos", "[", RowBox[{"2", "*", "Pi", "*", "t"}], "]"}]}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "4"}]}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "5"}], "}"}], ",", RowBox[{"AnimationRunning", "\[Rule]", "False"}]}], "]"}]], "Input", CellChangeTimes->{{3.463700467180892*^9, 3.46370056928338*^9}, 3.4637006561995897`*^9, {3.463701020353881*^9, 3.463701041252326*^9}, { 3.463701084501889*^9, 3.463701088392593*^9}, {3.4637029575666933`*^9, 3.46370297509114*^9}, {3.4637032139312773`*^9, 3.463703227822137*^9}, { 3.463703277346394*^9, 3.463703283637162*^9}, {3.463753165822855*^9, 3.4637532088413267`*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`t$$ = 0.749213695526123, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`t$$], 0, 5}}, Typeset`size$$ = {360., {113., 117.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`t$728$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`t$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`t$$, $CellContext`t$728$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Plot[0.5 ($CellContext`Psi[2, $CellContext`x]^2 + $CellContext`Psi[ 3, $CellContext`x]^2 + 2 $CellContext`Psi[2, $CellContext`x] $CellContext`Psi[ 3, $CellContext`x] Cos[2 Pi $CellContext`t$$]), {$CellContext`x, 0, 1}, PlotRange -> 4], "Specifications" :> {{$CellContext`t$$, 0, 5, AnimationRunning -> False, AppearanceElements -> { "ProgressSlider", "PlayPauseButton", "FasterSlowerButtons", "DirectionButton"}}}, "Options" :> { ControlType -> Animator, AppearanceElements -> None, SynchronousUpdating -> True, ShrinkingDelay -> 10.}, "DefaultOptions" :> {}], ImageSizeCache->{403., {150.34375, 155.65625}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.46370108981936*^9, {3.463702976060535*^9, 3.463703003558474*^9}, { 3.463703264053486*^9, 3.4637032875535727`*^9}, {3.463753216576277*^9, 3.4637532272973957`*^9}, {3.463753407113701*^9, 3.463753418415181*^9}, { 3.4637547857836723`*^9, 3.463754808816654*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Time-evolve a mixture \n ", Cell[BoxData[ FormBox[ RowBox[{"|", RowBox[{"\[Psi]", RowBox[{"(", RowBox[{"x", ",", "t"}], ")"}]}], SuperscriptBox["|", "2"], " ", RowBox[{"=", RowBox[{ FractionBox["1", "3"], " ", "|", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Psi]", RowBox[{"1", " "}]], "(", "x", ")"}], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"i", " ", SubscriptBox["\[Omega]", "1"], "t"}]]}], " ", "+", " ", SubscriptBox["\[Psi]", "2"]}]}]}]}], TraditionalForm]], "None"], "(x) ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"i", " ", SubscriptBox["\[Omega]", "2"], "t"}]], TraditionalForm]], "None"], " + \n ", Cell[BoxData[ FormBox[ SubscriptBox["\[Psi]", "3"], TraditionalForm]], "None"], "(x) ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"i", " ", SubscriptBox["\[Omega]", "3"], "t"}]], TraditionalForm]], "None"], Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["|", "2"]}], TraditionalForm]], "None"], " \n Note ", Cell[BoxData[ FormBox[ SubscriptBox["\[Omega]", "3"], TraditionalForm]], "None"], " -", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ SubscriptBox["\[Omega]", RowBox[{"2", " "}]], "\[NotEqual]", " ", SubscriptBox["\[Omega]", "2"], " "}]}], TraditionalForm]], "None"], "- ", Cell[BoxData[ RowBox[{" ", SubscriptBox["\[Omega]", "1"], " "}]]], ", affects cosines" }], "Section", CellChangeTimes->{{3.463701842329862*^9, 3.463701853083794*^9}, { 3.463702025758068*^9, 3.463702068201302*^9}, {3.463702099557661*^9, 3.463702232945278*^9}, {3.4637029069747267`*^9, 3.463702924796238*^9}, { 3.463703068687538*^9, 3.4637030732010107`*^9}, {3.463753383035862*^9, 3.463753389556665*^9}, {3.463753464890751*^9, 3.4637535080994797`*^9}, { 3.463753546199374*^9, 3.463753626695706*^9}, {3.463753803686619*^9, 3.463753937439124*^9}, {3.4637551026762877`*^9, 3.463755110604026*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Animate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"1.", "/", "3."}], "*", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"Psi", "[", RowBox[{"1", ",", "x"}], "]"}], ")"}], "^", "2"}], "+", RowBox[{ RowBox[{"(", RowBox[{"Psi", "[", RowBox[{"2", ",", "x"}], "]"}], ")"}], "^", "2"}], "+", RowBox[{ RowBox[{"(", RowBox[{"Psi", "[", RowBox[{"3", ",", "x"}], "]"}], ")"}], "^", "2"}], " ", "+", "\[IndentingNewLine]", RowBox[{"2", "*", RowBox[{"Psi", "[", RowBox[{"1", ",", "x"}], "]"}], "*", RowBox[{"Psi", "[", RowBox[{"2", ",", "x"}], "]"}], "*", RowBox[{"Cos", "[", RowBox[{"2", "*", "Pi", "*", "t"}], "]"}]}], "+", "\[IndentingNewLine]", RowBox[{"2", "*", RowBox[{"Psi", "[", RowBox[{"2", ",", "x"}], "]"}], "*", RowBox[{"Psi", "[", RowBox[{"3", ",", "x"}], "]"}], "*", RowBox[{"Cos", "[", RowBox[{ RowBox[{"10", "/", "3"}], "*", "Pi", "*", "t"}], "]"}]}], "+", "\[IndentingNewLine]", RowBox[{"2", "*", RowBox[{"Psi", "[", RowBox[{"1", ",", "x"}], "]"}], "*", RowBox[{"Psi", "[", RowBox[{"3", ",", "x"}], "]"}], "*", RowBox[{"Cos", "[", RowBox[{ RowBox[{"16", "/", "3"}], "*", "Pi", "*", "t"}], "]"}]}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "4"}]}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "5"}], "}"}], ",", RowBox[{"AnimationRunning", "\[Rule]", "False"}]}], "]"}]], "Input", CellChangeTimes->{{3.463700467180892*^9, 3.46370056928338*^9}, 3.4637006561995897`*^9, {3.463701020353881*^9, 3.463701041252326*^9}, { 3.463701084501889*^9, 3.463701088392593*^9}, {3.4637029575666933`*^9, 3.46370297509114*^9}, {3.4637032139312773`*^9, 3.463703227822137*^9}, { 3.463703277346394*^9, 3.463703283637162*^9}, {3.463753165822855*^9, 3.4637532088413267`*^9}, {3.46375360286228*^9, 3.463753697243444*^9}, { 3.463753728580593*^9, 3.4637537911374493`*^9}, 3.463753968206603*^9, { 3.463754004149967*^9, 3.463754035483061*^9}, {3.463754077659938*^9, 3.46375407987393*^9}, 3.4637541325191708`*^9, {3.463754198309054*^9, 3.4637542138816643`*^9}, {3.463754244121582*^9, 3.463754251760161*^9}, { 3.463754360684902*^9, 3.4637543739351883`*^9}, 3.463754487334032*^9, { 3.463754525170158*^9, 3.4637545880695143`*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`t$$ = 0.22290468215942383`, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`t$$], 0, 5}}, Typeset`size$$ = {360., {113., 117.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`t$767$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`t$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`t$$, $CellContext`t$767$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Plot[(1./3.) ($CellContext`Psi[1, $CellContext`x]^2 + $CellContext`Psi[ 2, $CellContext`x]^2 + $CellContext`Psi[3, $CellContext`x]^2 + 2 $CellContext`Psi[1, $CellContext`x] $CellContext`Psi[ 2, $CellContext`x] Cos[2 Pi $CellContext`t$$] + 2 $CellContext`Psi[2, $CellContext`x] $CellContext`Psi[ 3, $CellContext`x] Cos[(10/3) Pi $CellContext`t$$] + 2 $CellContext`Psi[1, $CellContext`x] $CellContext`Psi[ 3, $CellContext`x] Cos[(16/3) Pi $CellContext`t$$]), {$CellContext`x, 0, 1}, PlotRange -> 4], "Specifications" :> {{$CellContext`t$$, 0, 5, AnimationRunning -> False, AppearanceElements -> { "ProgressSlider", "PlayPauseButton", "FasterSlowerButtons", "DirectionButton"}}}, "Options" :> { ControlType -> Animator, AppearanceElements -> None, SynchronousUpdating -> True, ShrinkingDelay -> 10.}, "DefaultOptions" :> {}], ImageSizeCache->{403., {150.34375, 155.65625}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.463753971232367*^9, {3.463754005902774*^9, 3.4637540487121687`*^9}, 3.463754084796802*^9, 3.46375413452985*^9, {3.463754215225874*^9, 3.4637542182765207`*^9}, {3.463754253043355*^9, 3.4637542671615047`*^9}, { 3.4637543096953793`*^9, 3.463754341233758*^9}, 3.4637543763801928`*^9, 3.4637544891563253`*^9, {3.4637545988861322`*^9, 3.463754610271884*^9}, { 3.463754786075276*^9, 3.463754799747613*^9}}] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{640, 652}, WindowMargins->{{4, Automatic}, {Automatic, 4}}, FrontEndVersion->"7.0 for Mac OS X x86 (32-bit) (February 18, 2009)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[567, 22, 621, 18, 121, "Subtitle"], Cell[CellGroupData[{ Cell[1213, 44, 335, 9, 67, "Section"], Cell[1551, 55, 328, 9, 27, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[1916, 69, 1310, 42, 103, "Section"], Cell[CellGroupData[{ Cell[3251, 115, 1310, 33, 58, "Input"], Cell[4564, 150, 2448, 47, 323, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[7061, 203, 1303, 42, 103, "Section"], Cell[8367, 247, 270, 5, 26, "Text"], Cell[CellGroupData[{ Cell[8662, 256, 1421, 35, 58, "Input"], Cell[10086, 293, 2275, 45, 323, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[12410, 344, 1352, 43, 103, "Section"], Cell[CellGroupData[{ Cell[13787, 391, 1413, 35, 58, "Input"], Cell[15203, 428, 2304, 46, 323, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[17556, 480, 2151, 65, 157, "Section"], Cell[CellGroupData[{ Cell[19732, 549, 2731, 65, 88, "Input"], Cell[22466, 616, 2764, 52, 323, "Output"] }, Open ]] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)