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CellChangeTimes->{3.569523769814023*^9}, CellID->1940978731] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "ManipulateCaptionSection"], Cell[TextData[{ "The eigenfunctions in spherical coordinates for the hydrogen atom are ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[Phi]", RowBox[{"n", ",", RowBox[{"\[ScriptL]", ".", "m"}]}]], "(", RowBox[{"r", ",", "\[Theta]", ",", "\[CurlyPhi]"}], ")"}], "=", RowBox[{ RowBox[{ SubscriptBox["R", RowBox[{"n", ",", "\[ScriptL]"}]], "(", "r", ")"}], RowBox[{ SubsuperscriptBox["Y", "\[ScriptL]", "m"], "(", RowBox[{"\[Theta]", ",", "\[CurlyPhi]"}], ")"}]}]}], TraditionalForm]], "InlineMath"], ", where ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["R", RowBox[{"n", ",", "\[ScriptL]"}]], "(", "r", ")"}], TraditionalForm]], "InlineMath"], " and ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["Y", "\[ScriptL]", "m"], "(", RowBox[{"\[Theta]", ",", "\[CurlyPhi]"}], ")"}], TraditionalForm]], "InlineMath"], " are the solutions to the radial and angular parts of the \ Schr\[ODoubleDot]dinger equation, respectively, and ", Cell[BoxData[ FormBox["n", TraditionalForm]], "InlineMath"], ", ", Cell[BoxData[ FormBox["\[ScriptL]", TraditionalForm]], "InlineMath"], ", and ", Cell[BoxData[ FormBox["m", TraditionalForm]], "InlineMath"], " are the principal, orbital, and magnetic quantum numbers with allowed \ values ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"n", "=", "0"}], ",", "1", ",", "2", ",", "\[Ellipsis]", ",", RowBox[{"n", "-", "1"}], ","}], TraditionalForm]], "InlineMath"], " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[ScriptL]", "=", "0"}], ",", "1", ",", "2", ",", "\[Ellipsis]", ",", RowBox[{"n", "-", "1"}]}], TraditionalForm]], "InlineMath"], ", and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"m", "=", "0"}], ",", RowBox[{"\[PlusMinus]", "1"}], ",", RowBox[{"\[PlusMinus]", "2"}], ",", "\[Ellipsis]", ",", RowBox[{"\[PlusMinus]", "\[ScriptL]"}]}], TraditionalForm]], "InlineMath"], ". The ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["Y", "\[ScriptL]", "m"], "(", RowBox[{"\[Theta]", ",", "\[CurlyPhi]"}], ")"}], TraditionalForm]], "InlineMath"], " are the spherical harmonics and the radial functions are ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["R", RowBox[{"n", ",", "\[ScriptL]"}]], "(", "r", ")"}], "=", RowBox[{ SqrtBox[ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"n", "-", "1", "-", "\[ScriptL]"}], ")"}], "!"}], RowBox[{"2", " ", SuperscriptBox[ RowBox[{"n", " ", "[", RowBox[{ RowBox[{"(", RowBox[{"n", "+", "\[ScriptL]"}], ")"}], "!"}], "]"}], "3"]}]]], SuperscriptBox[ RowBox[{"(", FractionBox["2", RowBox[{"n", " ", SubscriptBox["a", "0"]}]], ")"}], RowBox[{"\[ScriptL]", "+", RowBox[{"3", "/", "2"}]}]], SuperscriptBox["r", "\[ScriptL]"], SuperscriptBox["e", RowBox[{ RowBox[{ RowBox[{"-", "r"}], "/", "n"}], " ", SubscriptBox["a", "0"]}]], RowBox[{ SubsuperscriptBox["L", RowBox[{"n", "+", "\[ScriptL]"}], RowBox[{ RowBox[{"2", " ", "\[ScriptL]"}], "+", "1"}]], "(", RowBox[{"2", " ", RowBox[{"r", "/", "n"}], " ", SubscriptBox["a", "0"]}], ")"}]}]}], TraditionalForm]], "InlineMath"], ", where ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["L", "p", "a"], "(", "x", ")"}], TraditionalForm]], "InlineMath"], " is the ", Cell[BoxData[ FormBox[ SuperscriptBox["p", "th"], TraditionalForm]], "InlineMath"], "-order associated Laguerre polynomial and ", Cell[BoxData[ FormBox[ SubscriptBox["a", "0"], TraditionalForm]], "InlineMath"], " is the Bohr radius. The left graphic shows the radial probability density ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["r", "2"], "|", RowBox[{ SubscriptBox["R", RowBox[{"n", ",", "\[ScriptL]"}]], "(", "r", ")"}], SuperscriptBox["|", "2"]}], TraditionalForm]], "InlineMath"], " and the expectation value ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox[ RowBox[{"\[LeftAngleBracket]", "r", "\[RightAngleBracket]"}], RowBox[{"n", ",", "\[ScriptL]"}]], "\[Congruent]", RowBox[{"\[LeftAngleBracket]", RowBox[{ SubscriptBox["\[Phi]", RowBox[{"n", ",", RowBox[{"\[ScriptL]", ".", "m"}]}]], "|", "r", "|", SubscriptBox["\[Phi]", RowBox[{"n", ",", RowBox[{"\[ScriptL]", ".", "m"}]}]]}], "\[RightAngleBracket]"}]}], "=", RowBox[{ SuperscriptBox["n", "2"], RowBox[{ SubscriptBox["a", "0"], "[", RowBox[{"1", "+", RowBox[{ FractionBox["1", "2"], RowBox[{"(", RowBox[{"1", "-", FractionBox[ RowBox[{"\[ScriptL]", "(", RowBox[{"\[ScriptL]", "+", "1"}], ")"}], SuperscriptBox["n", "2"]]}], ")"}]}]}], "]"}]}]}], TraditionalForm]], "InlineMath"], ", and the right graphic shows the radial function." }], "ManipulateCaption", CellChangeTimes->{ 3.35696210375764*^9, {3.471529068929021*^9, 3.4715293165026093`*^9}, { 3.4715293831639*^9, 3.471529437061977*^9}, {3.4715295487840357`*^9, 3.47152981401045*^9}, {3.47152985069398*^9, 3.4715298662521133`*^9}, { 3.471789183048913*^9, 3.471789183048913*^9}, {3.471789525117981*^9, 3.4717895308993416`*^9}, {3.4717895892285867`*^9, 3.4717898109047174`*^9}, {3.471814078224716*^9, 3.4718140791921043`*^9}, { 3.4718889185129423`*^9, 3.471888972939502*^9}}, CellID->47933019] }, Open ]], Cell[CellGroupData[{ Cell["", "ThumbnailSection"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`L$$ = 2, $CellContext`n$$ = 3, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`n$$], 3, "quantum numbers n"}, {1, 2, 3, 4}}, {{ Hold[$CellContext`L$$], 2, Style["l", Italic]}, Dynamic[ Range[0, $CellContext`n$$ - 1]]}}, Typeset`size$$ = { 600., {147., 153.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`n$69255$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`L$$ = 2, $CellContext`n$$ = 3}, "ControllerVariables" :> { Hold[$CellContext`n$$, $CellContext`n$69255$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`L$$ >= $CellContext`n$$, $CellContext`L$$ = \ $CellContext`n$$ - 1]; $CellContext`label = Which[$CellContext`L$$ == 0, Style["s", Italic], $CellContext`L$$ == 1, Style["p", Italic], $CellContext`L$$ == 2, Style["d", Italic], $CellContext`L$$ == 3, Style["f", Italic]]; $CellContext`p1 = Quiet[ Plot[$CellContext`r^2 $CellContext`R[$CellContext`n$$, \ $CellContext`L$$, $CellContext`r]^2, {$CellContext`r, 0, Part[$CellContext`rmax, $CellContext`n$$]}, PlotRange -> {{0, Part[$CellContext`rmax, $CellContext`n$$]}, {0, Part[$CellContext`yrange, $CellContext`n$$]}}, Filling -> Axis, Ticks -> Part[$CellContext`ticksmat, $CellContext`n$$, 1], TicksStyle -> Directive[16, Gray], PlotStyle -> Thick, PlotLabel -> Style[ Row[{ NumberForm[$CellContext`n$$, 1], $CellContext`label}], 18, Darker[Blue]], AxesLabel -> $CellContext`label1]]; $CellContext`av = Graphics[{Thick, Dashed, Blue, Line[{{ $CellContext`rav[$CellContext`n$$, $CellContext`L$$], 0}, { $CellContext`rav[$CellContext`n$$, $CellContext`L$$], Part[$CellContext`yrange, $CellContext`n$$]}}], Text[ Style[ Row[{"\[LeftAngleBracket]", Style["r", Italic], "\!\(\*SubscriptBox[\(\[RightAngleBracket]\), \(n, l\)]\)"}], 18], {1.48 $CellContext`rav[$CellContext`n$$, $CellContext`L$$], 0.8 Part[$CellContext`yrange, $CellContext`n$$]}]}]; Pane[ If[$CellContext`L$$ < $CellContext`n$$, GraphicsRow[{ Show[$CellContext`p1, $CellContext`av], Plot[ $CellContext`R[$CellContext`n$$, $CellContext`L$$, \ $CellContext`r], {$CellContext`r, 0, Part[$CellContext`rmax, $CellContext`n$$]}, PlotRange -> All, Ticks -> Part[$CellContext`ticksmat, $CellContext`n$$, 2], TicksStyle -> Directive[16, Gray], PlotStyle -> Thick, AxesLabel -> $CellContext`label2]}, ImageSize -> {600, 300}], Null], ImageSize -> {600, 300}]), "Specifications" :> {{{$CellContext`n$$, 3, "quantum numbers n"}, {1, 2, 3, 4}}, {{$CellContext`L$$, 2, Style["l", Italic]}, Dynamic[ Range[0, $CellContext`n$$ - 1]]}}, "Options" :> { ControlType -> SetterBar, TrackedSymbols :> {$CellContext`n$$, $CellContext`L$$}}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{643., {202., 207.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>(($CellContext`F[ Pattern[$CellContext`n, Blank[]], Pattern[$CellContext`L, Blank[]], Pattern[$CellContext`x, Blank[]]] := (($CellContext`x^$CellContext`L E^((-$CellContext`x)/2)) Factorial[$CellContext`n + $CellContext`L]) LaguerreL[$CellContext`n - $CellContext`L - 1, 2 $CellContext`L + 1, $CellContext`x]; $CellContext`R[ Pattern[$CellContext`n, Blank[]], Pattern[$CellContext`L, Blank[]], Pattern[$CellContext`r, Blank[]]] := (($CellContext`a^((-3)/2) (2/$CellContext`n^2)) ( Factorial[$CellContext`n - 1 - $CellContext`L]/ Factorial[$CellContext`n + $CellContext`L]^3)^ Rational[1, 2]) $CellContext`F[$CellContext`n, $CellContext`L, 2 ($CellContext`r/($CellContext`n $CellContext`a))]; \ $CellContext`rav[ Pattern[$CellContext`np, Blank[]], Pattern[$CellContext`Lp, Blank[]]] := $CellContext`np^2 (1. + 0.5 (1. - $CellContext`Lp (($CellContext`Lp + 1.)/$CellContext`np^2))); $CellContext`a = 1; $CellContext`rmax = {5, 15, 30, 50, 70}; $CellContext`yrange = { 0.55, 0.2, 0.12, 0.07}; $CellContext`ticksmat = ConstantArray[0., {5, 2}]; Part[$CellContext`ticksmat, 1, 1] = {{0, 2, 4}, {0.25, 0.5}}; Part[$CellContext`ticksmat, 1, 2] = {{0, 2, 4}, Automatic}; Part[$CellContext`ticksmat, 2, 1] = {{0, 6, 12}, {0.05, 0.15}}; Part[$CellContext`ticksmat, 2, 2] = {{0, 6, 12}, Automatic}; Part[$CellContext`ticksmat, 3, 1] = {{0, 10, 20}, {0.05, 0.1}}; Part[$CellContext`ticksmat, 3, 2] = {{0, 10, 20}, Automatic}; Part[$CellContext`ticksmat, 4, 1] = {{0, 20, 40}, {0.03, 0.06}}; Part[$CellContext`ticksmat, 4, 2] = {{0, 20, 40}, Automatic}; $CellContext`label1 = { Text[ Style[ Row[{ Style["r", Italic], " / ", Subscript[ Style["a", Italic], 0]}], 18]], Text[ Style[ Row[{Style["r", Italic]^2, " ", Subscript[ Style["R", Italic], Row[{ Style["n", Italic], ", ", Style["l", Italic]}]], "(", Style["r", Italic], ")"^2}], 18]]}; $CellContext`label2 = { Text[ Style[ Row[{ Style["r", Italic], " / ", Subscript[ Style["a", Italic], 0]}], 18]], Text[ Style[ Row[{ Subscript[ Style["R", Italic], Row[{ Style["n", Italic], ", ", Style["l", Italic]}]], "(", Style["r", Italic], ")"}], 18]]}); Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->1957055702] }, Open ]], Cell[CellGroupData[{ Cell["", "SnapshotsSection"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`L$$ = 1, $CellContext`n$$ = 3, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`n$$], 3, "quantum numbers n"}, {1, 2, 3, 4}}, {{ Hold[$CellContext`L$$], 1, Style["l", Italic]}, Dynamic[ Range[0, $CellContext`n$$ - 1]]}}, Typeset`size$$ = { 600., {147., 153.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`n$69306$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`L$$ = 1, $CellContext`n$$ = 3}, "ControllerVariables" :> { Hold[$CellContext`n$$, $CellContext`n$69306$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`L$$ >= $CellContext`n$$, $CellContext`L$$ = \ $CellContext`n$$ - 1]; $CellContext`label = Which[$CellContext`L$$ == 0, Style["s", Italic], $CellContext`L$$ == 1, Style["p", Italic], $CellContext`L$$ == 2, Style["d", Italic], $CellContext`L$$ == 3, Style["f", Italic]]; $CellContext`p1 = Quiet[ Plot[$CellContext`r^2 $CellContext`R[$CellContext`n$$, \ $CellContext`L$$, $CellContext`r]^2, {$CellContext`r, 0, Part[$CellContext`rmax, $CellContext`n$$]}, PlotRange -> {{0, Part[$CellContext`rmax, $CellContext`n$$]}, {0, Part[$CellContext`yrange, $CellContext`n$$]}}, Filling -> Axis, Ticks -> Part[$CellContext`ticksmat, $CellContext`n$$, 1], TicksStyle -> Directive[16, Gray], PlotStyle -> Thick, PlotLabel -> Style[ Row[{ NumberForm[$CellContext`n$$, 1], $CellContext`label}], 18, Darker[Blue]], AxesLabel -> $CellContext`label1]]; $CellContext`av = Graphics[{Thick, Dashed, Blue, Line[{{ $CellContext`rav[$CellContext`n$$, $CellContext`L$$], 0}, { $CellContext`rav[$CellContext`n$$, $CellContext`L$$], Part[$CellContext`yrange, $CellContext`n$$]}}], Text[ Style[ Row[{"\[LeftAngleBracket]", Style["r", Italic], "\!\(\*SubscriptBox[\(\[RightAngleBracket]\), \(n, l\)]\)"}], 18], {1.48 $CellContext`rav[$CellContext`n$$, $CellContext`L$$], 0.8 Part[$CellContext`yrange, $CellContext`n$$]}]}]; Pane[ If[$CellContext`L$$ < $CellContext`n$$, GraphicsRow[{ Show[$CellContext`p1, $CellContext`av], Plot[ $CellContext`R[$CellContext`n$$, $CellContext`L$$, \ $CellContext`r], {$CellContext`r, 0, Part[$CellContext`rmax, $CellContext`n$$]}, PlotRange -> All, Ticks -> Part[$CellContext`ticksmat, $CellContext`n$$, 2], TicksStyle -> Directive[16, Gray], PlotStyle -> Thick, AxesLabel -> $CellContext`label2]}, ImageSize -> {600, 300}], Null], ImageSize -> {600, 300}]), "Specifications" :> {{{$CellContext`n$$, 3, "quantum numbers n"}, {1, 2, 3, 4}}, {{$CellContext`L$$, 1, Style["l", Italic]}, Dynamic[ Range[0, $CellContext`n$$ - 1]]}}, "Options" :> { ControlType -> SetterBar, TrackedSymbols :> {$CellContext`n$$, $CellContext`L$$}}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{643., {202., 207.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>(($CellContext`F[ Pattern[$CellContext`n, Blank[]], Pattern[$CellContext`L, Blank[]], Pattern[$CellContext`x, Blank[]]] := (($CellContext`x^$CellContext`L E^((-$CellContext`x)/2)) Factorial[$CellContext`n + $CellContext`L]) LaguerreL[$CellContext`n - $CellContext`L - 1, 2 $CellContext`L + 1, $CellContext`x]; $CellContext`R[ Pattern[$CellContext`n, Blank[]], Pattern[$CellContext`L, Blank[]], Pattern[$CellContext`r, Blank[]]] := (($CellContext`a^((-3)/2) (2/$CellContext`n^2)) ( Factorial[$CellContext`n - 1 - $CellContext`L]/ Factorial[$CellContext`n + $CellContext`L]^3)^ Rational[1, 2]) $CellContext`F[$CellContext`n, $CellContext`L, 2 ($CellContext`r/($CellContext`n $CellContext`a))]; \ $CellContext`rav[ Pattern[$CellContext`np, Blank[]], Pattern[$CellContext`Lp, Blank[]]] := $CellContext`np^2 (1. + 0.5 (1. - $CellContext`Lp (($CellContext`Lp + 1.)/$CellContext`np^2))); $CellContext`a = 1; $CellContext`rmax = {5, 15, 30, 50, 70}; $CellContext`yrange = { 0.55, 0.2, 0.12, 0.07}; $CellContext`ticksmat = ConstantArray[0., {5, 2}]; Part[$CellContext`ticksmat, 1, 1] = {{0, 2, 4}, {0.25, 0.5}}; Part[$CellContext`ticksmat, 1, 2] = {{0, 2, 4}, Automatic}; Part[$CellContext`ticksmat, 2, 1] = {{0, 6, 12}, {0.05, 0.15}}; Part[$CellContext`ticksmat, 2, 2] = {{0, 6, 12}, Automatic}; Part[$CellContext`ticksmat, 3, 1] = {{0, 10, 20}, {0.05, 0.1}}; Part[$CellContext`ticksmat, 3, 2] = {{0, 10, 20}, Automatic}; Part[$CellContext`ticksmat, 4, 1] = {{0, 20, 40}, {0.03, 0.06}}; Part[$CellContext`ticksmat, 4, 2] = {{0, 20, 40}, Automatic}; $CellContext`label1 = { Text[ Style[ Row[{ Style["r", Italic], " / ", Subscript[ Style["a", Italic], 0]}], 18]], Text[ Style[ Row[{Style["r", Italic]^2, " ", Subscript[ Style["R", Italic], Row[{ Style["n", Italic], ", ", Style["l", Italic]}]], "(", Style["r", Italic], ")"^2}], 18]]}; $CellContext`label2 = { Text[ Style[ Row[{ Style["r", Italic], " / ", Subscript[ Style["a", Italic], 0]}], 18]], Text[ Style[ Row[{ Subscript[ Style["R", Italic], Row[{ Style["n", Italic], ", ", Style["l", Italic]}]], "(", Style["r", Italic], ")"}], 18]]}); Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->1375646062], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`L$$ = 3, $CellContext`n$$ = 4, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`n$$], 4, "quantum numbers n"}, {1, 2, 3, 4}}, {{ Hold[$CellContext`L$$], 3, Style["l", Italic]}, Dynamic[ Range[0, $CellContext`n$$ - 1]]}}, Typeset`size$$ = { 600., {147., 153.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`n$69357$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`L$$ = 3, $CellContext`n$$ = 4}, "ControllerVariables" :> { Hold[$CellContext`n$$, $CellContext`n$69357$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`L$$ >= $CellContext`n$$, $CellContext`L$$ = \ $CellContext`n$$ - 1]; $CellContext`label = Which[$CellContext`L$$ == 0, Style["s", Italic], $CellContext`L$$ == 1, Style["p", Italic], $CellContext`L$$ == 2, Style["d", Italic], $CellContext`L$$ == 3, Style["f", Italic]]; $CellContext`p1 = Quiet[ Plot[$CellContext`r^2 $CellContext`R[$CellContext`n$$, \ $CellContext`L$$, $CellContext`r]^2, {$CellContext`r, 0, Part[$CellContext`rmax, $CellContext`n$$]}, PlotRange -> {{0, Part[$CellContext`rmax, $CellContext`n$$]}, {0, Part[$CellContext`yrange, $CellContext`n$$]}}, Filling -> Axis, Ticks -> Part[$CellContext`ticksmat, $CellContext`n$$, 1], TicksStyle -> Directive[16, Gray], PlotStyle -> Thick, PlotLabel -> Style[ Row[{ NumberForm[$CellContext`n$$, 1], $CellContext`label}], 18, Darker[Blue]], AxesLabel -> $CellContext`label1]]; $CellContext`av = Graphics[{Thick, Dashed, Blue, Line[{{ $CellContext`rav[$CellContext`n$$, $CellContext`L$$], 0}, { $CellContext`rav[$CellContext`n$$, $CellContext`L$$], Part[$CellContext`yrange, $CellContext`n$$]}}], Text[ Style[ Row[{"\[LeftAngleBracket]", Style["r", Italic], "\!\(\*SubscriptBox[\(\[RightAngleBracket]\), \(n, l\)]\)"}], 18], {1.48 $CellContext`rav[$CellContext`n$$, $CellContext`L$$], 0.8 Part[$CellContext`yrange, $CellContext`n$$]}]}]; Pane[ If[$CellContext`L$$ < $CellContext`n$$, GraphicsRow[{ Show[$CellContext`p1, $CellContext`av], Plot[ $CellContext`R[$CellContext`n$$, $CellContext`L$$, \ $CellContext`r], {$CellContext`r, 0, Part[$CellContext`rmax, $CellContext`n$$]}, PlotRange -> All, Ticks -> Part[$CellContext`ticksmat, $CellContext`n$$, 2], TicksStyle -> Directive[16, Gray], PlotStyle -> Thick, AxesLabel -> $CellContext`label2]}, ImageSize -> {600, 300}], Null], ImageSize -> {600, 300}]), "Specifications" :> {{{$CellContext`n$$, 4, "quantum numbers n"}, {1, 2, 3, 4}}, {{$CellContext`L$$, 3, Style["l", Italic]}, Dynamic[ Range[0, $CellContext`n$$ - 1]]}}, "Options" :> { ControlType -> SetterBar, TrackedSymbols :> {$CellContext`n$$, $CellContext`L$$}}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{643., {202., 207.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>(($CellContext`F[ Pattern[$CellContext`n, Blank[]], Pattern[$CellContext`L, Blank[]], Pattern[$CellContext`x, Blank[]]] := (($CellContext`x^$CellContext`L E^((-$CellContext`x)/2)) Factorial[$CellContext`n + $CellContext`L]) LaguerreL[$CellContext`n - $CellContext`L - 1, 2 $CellContext`L + 1, $CellContext`x]; $CellContext`R[ Pattern[$CellContext`n, Blank[]], Pattern[$CellContext`L, Blank[]], Pattern[$CellContext`r, Blank[]]] := (($CellContext`a^((-3)/2) (2/$CellContext`n^2)) ( Factorial[$CellContext`n - 1 - $CellContext`L]/ Factorial[$CellContext`n + $CellContext`L]^3)^ Rational[1, 2]) $CellContext`F[$CellContext`n, $CellContext`L, 2 ($CellContext`r/($CellContext`n $CellContext`a))]; \ $CellContext`rav[ Pattern[$CellContext`np, Blank[]], Pattern[$CellContext`Lp, Blank[]]] := $CellContext`np^2 (1. + 0.5 (1. - $CellContext`Lp (($CellContext`Lp + 1.)/$CellContext`np^2))); $CellContext`a = 1; $CellContext`rmax = {5, 15, 30, 50, 70}; $CellContext`yrange = { 0.55, 0.2, 0.12, 0.07}; $CellContext`ticksmat = ConstantArray[0., {5, 2}]; Part[$CellContext`ticksmat, 1, 1] = {{0, 2, 4}, {0.25, 0.5}}; Part[$CellContext`ticksmat, 1, 2] = {{0, 2, 4}, Automatic}; Part[$CellContext`ticksmat, 2, 1] = {{0, 6, 12}, {0.05, 0.15}}; Part[$CellContext`ticksmat, 2, 2] = {{0, 6, 12}, Automatic}; Part[$CellContext`ticksmat, 3, 1] = {{0, 10, 20}, {0.05, 0.1}}; Part[$CellContext`ticksmat, 3, 2] = {{0, 10, 20}, Automatic}; Part[$CellContext`ticksmat, 4, 1] = {{0, 20, 40}, {0.03, 0.06}}; Part[$CellContext`ticksmat, 4, 2] = {{0, 20, 40}, Automatic}; $CellContext`label1 = { Text[ Style[ Row[{ Style["r", Italic], " / ", Subscript[ Style["a", Italic], 0]}], 18]], Text[ Style[ Row[{Style["r", Italic]^2, " ", Subscript[ Style["R", Italic], Row[{ Style["n", Italic], ", ", Style["l", Italic]}]], "(", Style["r", Italic], ")"^2}], 18]]}; $CellContext`label2 = { Text[ Style[ Row[{ Style["r", Italic], " / ", Subscript[ Style["a", Italic], 0]}], 18]], Text[ Style[ Row[{ Subscript[ Style["R", Italic], Row[{ Style["n", Italic], ", ", Style["l", Italic]}]], "(", Style["r", Italic], ")"}], 18]]}); Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->674420432], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`L$$ = 2, $CellContext`n$$ = 3, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`n$$], 3, "quantum numbers n"}, {1, 2, 3, 4}}, {{ Hold[$CellContext`L$$], 2, Style["l", Italic]}, Dynamic[ Range[0, $CellContext`n$$ - 1]]}}, Typeset`size$$ = { 600., {147., 153.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`n$69408$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`L$$ = 2, $CellContext`n$$ = 3}, "ControllerVariables" :> { Hold[$CellContext`n$$, $CellContext`n$69408$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`L$$ >= $CellContext`n$$, $CellContext`L$$ = \ $CellContext`n$$ - 1]; $CellContext`label = Which[$CellContext`L$$ == 0, Style["s", Italic], $CellContext`L$$ == 1, Style["p", Italic], $CellContext`L$$ == 2, Style["d", Italic], $CellContext`L$$ == 3, Style["f", Italic]]; $CellContext`p1 = Quiet[ Plot[$CellContext`r^2 $CellContext`R[$CellContext`n$$, \ $CellContext`L$$, $CellContext`r]^2, {$CellContext`r, 0, Part[$CellContext`rmax, $CellContext`n$$]}, PlotRange -> {{0, Part[$CellContext`rmax, $CellContext`n$$]}, {0, Part[$CellContext`yrange, $CellContext`n$$]}}, Filling -> Axis, Ticks -> Part[$CellContext`ticksmat, $CellContext`n$$, 1], TicksStyle -> Directive[16, Gray], PlotStyle -> Thick, PlotLabel -> Style[ Row[{ NumberForm[$CellContext`n$$, 1], $CellContext`label}], 18, Darker[Blue]], AxesLabel -> $CellContext`label1]]; $CellContext`av = Graphics[{Thick, Dashed, Blue, Line[{{ $CellContext`rav[$CellContext`n$$, $CellContext`L$$], 0}, { $CellContext`rav[$CellContext`n$$, $CellContext`L$$], Part[$CellContext`yrange, $CellContext`n$$]}}], Text[ Style[ Row[{"\[LeftAngleBracket]", Style["r", Italic], "\!\(\*SubscriptBox[\(\[RightAngleBracket]\), \(n, l\)]\)"}], 18], {1.48 $CellContext`rav[$CellContext`n$$, $CellContext`L$$], 0.8 Part[$CellContext`yrange, $CellContext`n$$]}]}]; Pane[ If[$CellContext`L$$ < $CellContext`n$$, GraphicsRow[{ Show[$CellContext`p1, $CellContext`av], Plot[ $CellContext`R[$CellContext`n$$, $CellContext`L$$, \ $CellContext`r], {$CellContext`r, 0, Part[$CellContext`rmax, $CellContext`n$$]}, PlotRange -> All, Ticks -> Part[$CellContext`ticksmat, $CellContext`n$$, 2], TicksStyle -> Directive[16, Gray], PlotStyle -> Thick, AxesLabel -> $CellContext`label2]}, ImageSize -> {600, 300}], Null], ImageSize -> {600, 300}]), "Specifications" :> {{{$CellContext`n$$, 3, "quantum numbers n"}, {1, 2, 3, 4}}, {{$CellContext`L$$, 2, Style["l", Italic]}, Dynamic[ Range[0, $CellContext`n$$ - 1]]}}, "Options" :> { ControlType -> SetterBar, TrackedSymbols :> {$CellContext`n$$, $CellContext`L$$}}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{643., {202., 207.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>(($CellContext`F[ Pattern[$CellContext`n, Blank[]], Pattern[$CellContext`L, Blank[]], Pattern[$CellContext`x, Blank[]]] := (($CellContext`x^$CellContext`L E^((-$CellContext`x)/2)) Factorial[$CellContext`n + $CellContext`L]) LaguerreL[$CellContext`n - $CellContext`L - 1, 2 $CellContext`L + 1, $CellContext`x]; $CellContext`R[ Pattern[$CellContext`n, Blank[]], Pattern[$CellContext`L, Blank[]], Pattern[$CellContext`r, Blank[]]] := (($CellContext`a^((-3)/2) (2/$CellContext`n^2)) ( Factorial[$CellContext`n - 1 - $CellContext`L]/ Factorial[$CellContext`n + $CellContext`L]^3)^ Rational[1, 2]) $CellContext`F[$CellContext`n, $CellContext`L, 2 ($CellContext`r/($CellContext`n $CellContext`a))]; \ $CellContext`rav[ Pattern[$CellContext`np, Blank[]], Pattern[$CellContext`Lp, Blank[]]] := $CellContext`np^2 (1. + 0.5 (1. - $CellContext`Lp (($CellContext`Lp + 1.)/$CellContext`np^2))); $CellContext`a = 1; $CellContext`rmax = {5, 15, 30, 50, 70}; $CellContext`yrange = { 0.55, 0.2, 0.12, 0.07}; $CellContext`ticksmat = ConstantArray[0., {5, 2}]; Part[$CellContext`ticksmat, 1, 1] = {{0, 2, 4}, {0.25, 0.5}}; Part[$CellContext`ticksmat, 1, 2] = {{0, 2, 4}, Automatic}; Part[$CellContext`ticksmat, 2, 1] = {{0, 6, 12}, {0.05, 0.15}}; Part[$CellContext`ticksmat, 2, 2] = {{0, 6, 12}, Automatic}; Part[$CellContext`ticksmat, 3, 1] = {{0, 10, 20}, {0.05, 0.1}}; Part[$CellContext`ticksmat, 3, 2] = {{0, 10, 20}, Automatic}; Part[$CellContext`ticksmat, 4, 1] = {{0, 20, 40}, {0.03, 0.06}}; Part[$CellContext`ticksmat, 4, 2] = {{0, 20, 40}, Automatic}; $CellContext`label1 = { Text[ Style[ Row[{ Style["r", Italic], " / ", Subscript[ Style["a", Italic], 0]}], 18]], Text[ Style[ Row[{Style["r", Italic]^2, " ", Subscript[ Style["R", Italic], Row[{ Style["n", Italic], ", ", Style["l", Italic]}]], "(", Style["r", Italic], ")"^2}], 18]]}; $CellContext`label2 = { Text[ Style[ Row[{ Style["r", Italic], " / ", Subscript[ Style["a", Italic], 0]}], 18]], Text[ Style[ Row[{ Subscript[ Style["R", Italic], Row[{ Style["n", Italic], ", ", Style["l", Italic]}]], "(", Style["r", Italic], ")"}], 18]]}); Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->565511758] }, Open ]], Cell["", "DetailsSection"], Cell[CellGroupData[{ Cell["", "ControlSuggestionsSection"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[True], Cell[" Resize Images"]}], "\"Click inside an image to reveal its orange resize frame.\\nDrag any of \ the orange resize handles to resize the image.\"", TooltipDelay->0.35]], "ControlSuggestions", CellChangeTimes->{3.35696210375764*^9, 3.471789840374034*^9}, FontFamily->"Verdana", CellTags->"ResizeImages"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[False], Cell[" Rotate and Zoom in 3D"]}], RowBox[{ "\"Drag a 3D graphic to rotate it. Starting the drag near the center \ tumbles\\nthe graphic; starting near a corner turns it parallel to the plane \ of the screen.\\nHold down \"", FrameBox[ "Ctrl", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> GrayLevel[0.9]], "\" (or \"", FrameBox[ "Cmd", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> GrayLevel[0.9]], "\" on Mac) and drag up and down to zoom.\""}], TooltipDelay->0.35]], "ControlSuggestions", FontFamily->"Verdana", CellTags->"RotateAndZoomIn3D"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[False], Cell[" Drag Locators"]}], RowBox[{"\"Drag any locator (\"", GraphicsBox[ LocatorBox[ Scaled[{0.5, 0.5}]], ImageSize -> 20], "\", etc.) to move it around.\""}], TooltipDelay->0.35]], "ControlSuggestions", FontFamily->"Verdana", CellTags->"DragLocators"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[False], Cell[" Create and Delete Locators"]}], RowBox[{"\"Insert a new locator in the graphic by holding down the \"", FrameBox[ "Alt", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> GrayLevel[0.9]], "\" key\\nand clicking where you want it to be. 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"\"button\\nnext to the slider, and then clicking the play button that \ appears.\\nAnimate all controls by selecting \"", StyleBox["Autorun", FontWeight -> "Bold"], "\" from the\"", AdjustmentBox[ Cell[ GraphicsData[ "CompressedBitmap", "eJyNULENwyAQfEySIlMwTVJlCGRFsosokeNtqBmDBagoaZjAI1C8/8GUUUC6\n\ 57h7cQ8PvU7Pl17nUav7oj/TPH7V7b2QJAUAXBkKmCPRowxICy64bRvGGNF7\n\ X8CctGoDSN4xhIDGGDhzFXwUh3/ClBKrDQPmnGXtI6u0OOd+tZBVUqy1xSaH\n\ UqiK6pPe4XdEdAz6563tx/gejuORGMxJaz8mdpJn7hc="], "Graphics", ImageSize -> {10, 10}, ImageMargins -> 0, CellBaseline -> Baseline], BoxBaselineShift -> 0.1839080459770115, BoxMargins -> {{0., 0.}, {-0.1839080459770115, 0.1839080459770115}}], "\"menu.\""}], TooltipDelay->0.35]], "ControlSuggestions", FontFamily->"Verdana", CellTags->"AutomaticAnimation"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[False], Cell[" Bookmark Animation"]}], RowBox[{ "\"See a prepared animation of this Demonstration by selecting\\n\"", StyleBox["Animate Bookmarks", FontWeight -> 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