diff --git a/23_NLTE_ionization.pdf b/23_NLTE_ionization.pdf new file mode 100644 index 0000000..6439465 Binary files /dev/null and b/23_NLTE_ionization.pdf differ diff --git a/Ulrich.ps b/Ulrich.ps new file mode 100644 index 0000000..e69de29 diff --git a/hw7.motes b/hw7.motes new file mode 100644 index 0000000..527fbd6 --- /dev/null +++ b/hw7.motes @@ -0,0 +1,78 @@ +Problem 1 +━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ +a) The wave function ψ(ξ) = ξᵖ exp(-ξ²/2) L(ξ²), should satisfy the boundary condition for either even or odd fuctions of ξ with p=0 or p=1, respectively, for any function L(u). + + Recall ξ = √(mω/ħ) x. + + dξ = √(mω/ħ) dx = α dx. + + ψ(ξ) = ξᵖ exp(-ξ²/2) L(ξ²). + + ψ'(ξ) = d/dξ (ξᵖ exp(-ξ²/2) L(ξ²)) + = (d/dξ ξᵖ) exp(-ξ²/2) L(ξ²) + + ξᵖ (d/dξ exp(-ξ²/2)) L(ξ²) + + ξᵖ exp(-ξ²/2) (d/dξ L(ξ²)). + + For an even state, ψ'(0) = 0 and p=0, so + + ψ'(ξ) = (d/dξ 1) exp(-ξ²/2) L(ξ²) + + 1 d/dξ exp(-ξ²/2) L(ξ²) + + 1 exp(-ξ²/2) d/dξ L(ξ²). + + ψ'(ξ) = - ξ exp(-ξ²/2) L(ξ²) + + exp(-ξ²/2) L'(ξ²). + + ψ'(0) = L'(0). + + This must be wrong, somehow... in this case the boundary condition only applies if L'(0) = 0. + + ∎ + + For an odd state, ψ(0) = 0 and p=1, so + + ψ(0) = 0ᵖ * ... = 0. This is trivial. + + ∎ + + +b) Equation 2.72 from Griffiths: + + dHₙ/dξ = 2nHₙ₋₁(ξ). + +Substituting the state ψ(ξ), I can obtain the associated Laguerre differential equation + + uL'' + (p - ½ + 1 - u)L' + kL = 0, with u = ξ². + +A better and equivalent substitution uses + ψ(x,t) = A exp(ι(kx - (ħk²/2m)t)). + + h = ξᵖ L(ξ²). + + u = ξ². + + ξᵖ = ξ² + + h = ξᵖ L(ξ²) + + ψ(x,t) = A exp( ι(kx - k²/4πm ξᵖ L(ξ²) t) ) + = A exp( ι(kx - k²/4πm ξᵖ L(u) t) ). + + This... is probably not the "h" you meant. + + h = ξᵖ L(ξ²). + + dHₙ/dξ = 2nHₙ₋₁(ξ). + + h'(ξ) = dh/dξ = pξᵖ⁻¹ L(ξ²) + ξᵖ L'(ξ²). + + h''(ξ) = pξᵖ⁻¹ L(ξ²) + ξᵖ L'(ξ²) = p(p-1)ξᵖ⁻² L(ξ²) + pξᵖ⁻¹ L'(ξ²) + pξᵖ⁻¹ L'(ξ²) + ξᵖ L''(ξ²). + + p(p-1)ξᵖ⁻² L(ξ²) + pξᵖ⁻¹ L'(ξ²) + pξᵖ⁻¹ L'(ξ²) + ξᵖ L''(ξ²). + + Hₙ in Griffiths maps to hₚ in the problem sheet, I'll assume. + + Hₙ = ξⁿ. + + + + diff --git a/hw7.motes.ps b/hw7.motes.ps new file mode 100644 index 0000000..0c3108f --- /dev/null +++ b/hw7.motes.ps @@ -0,0 +1,3279 @@ +%!PS-Adobe-3.0 +%%Title: Otho Ulrich - HW 7 +%%Creator: paps version 0.6.7 by Dov Grobgeld +%%Pages: (atend) +%%BoundingBox: 0 0 595 841 +%%BeginProlog +%%Orientation: Portrait +/papsdict 1 dict def +papsdict begin + +/inch {72 mul} bind def +/mm {1 inch 25.4 div mul} bind def + +% override setpagedevice if it is not defined +/setpagedevice where { + pop % get rid of its dictionary + /setpagesize { + 3 dict begin + /pageheight exch def + /pagewidth exch def + /orientation 0 def + % Exchange pagewidth and pageheight so that pagewidth is bigger + pagewidth pageheight gt { + pagewidth + /pagewidth pageheight def + /pageheight exch def + /orientation 3 def + } if + 2 dict + dup /PageSize [pagewidth pageheight] put + dup /Orientation orientation put + setpagedevice + end + } def +} +{ + /setpagesize { pop pop } def +} ifelse +/duplex { + statusdict /setduplexmode known + { statusdict begin setduplexmode end } {pop} ifelse +} def +/tumble { + statusdict /settumble known + { statusdict begin settumble end } {pop} ifelse +} def +% Turn the page around +/turnpage { + 90 rotate + 0 pageheight neg translate +} def +% User settings +/pagewidth 595 def +/pageheight 841 def +pagewidth pageheight setpagesize +/column_width 523 def +/bodyheight 755 def +/lmarg 36 def +/ytop 791 def +/do_separation_line true def +/do_landscape false def +/do_tumble true def +/do_duplex true def +% Procedures to translate position to first and second column +/lw 20 def % whatever +/setnumcolumns { + /numcolumns exch def + /firstcolumn { /xpos lmarg def /ypos ytop def} def + /nextcolumn { + do_separation_line { + xpos column_width add gutter_width 2 div add % x start + ytop lw add moveto % y start + 0 bodyheight lw add neg rlineto 0 setlinewidth stroke + } if + /xpos xpos column_width add gutter_width add def + /ypos ytop def + } def +} def + +1 setnumcolumns +/showline { + /y exch def + /s exch def + xpos y moveto + column_width 0 rlineto stroke + xpos y moveto /Helvetica findfont 20 scalefont setfont s show +} def +/paps_bop { % Beginning of page definitions + papsdict begin + gsave + do_landscape {turnpage} if + % ps2pdf gets wrong orientation without this! 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def +/eBA { start_ol +504 6510 m +504 7023 l +504 7184 515 7267 x +527 7350 627 7437 x +727 7524 927 7524 x +1247 7524 1319 7344 x +5247 7344 l +5393 7344 5468 7331 x +5544 7320 5634 7231 x +5724 7143 5724 6973 x +5724 6855 5682 6792 x +5641 6730 5461 6559 x +5187 6266 4912 5949 x +4639 5632 4251 5037 x +3864 4443 3575 3795 x +3287 3148 3074 2228 x +2862 1308 2844 335 x +2844 95 2725 -23 x +2608 -144 2439 -144 x +2016 -144 2016 423 x +2016 999 2118 1654 x +2221 2310 2473 3160 x +2727 4010 3249 4919 x +3772 5828 4505 6610 x +1354 6610 l +1354 6378 1343 6284 x +1332 6190 1235 6099 x +1138 6009 927 6009 x +643 6009 573 6138 x +504 6266 504 6510 x +6336 fwd_x +end_ol + } def +/fBA { start_ol +432 3676 m +432 5290 1222 6388 x +2014 7488 3111 7488 x +3891 7488 4512 6948 x +4658 7282 4739 7384 x +4821 7488 5002 7488 x +5358 7488 5364 7005 x +5364 5365 l +5364 5209 5349 5128 x +5334 5049 5235 4965 x +5136 4882 4944 4882 x +4576 4882 4536 5224 x +4401 6063 3988 6409 x +3575 6754 3159 6754 x 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@@ -11,7 +11,7 @@ covered: Correspondence Principle -This seems to be primarily historical motivation. +This seems to be primarily historical motivation. The correspondence principle, I'm not sure I've ever attempted to compute for even a simple system. Hmm... I see the note here "What is the Correspondence principle? Derive the quantum condition from it." I don't know what is meant by the "quantum condition". I guess just that the states of a system are quantized. I have no clue how I would show that using the correspondence principle. Should ask about this. diff --git a/journal/R11 b/journal/R11 index ab1e3d8..12f28c5 100644 --- a/journal/R11 +++ b/journal/R11 @@ -1,9 +1,4 @@ Ehrenfest Theorems -

= m d/dt and <-∂/∂x V> = d/dt

- - -───────────── - - +

= m d/dt and <-∂/∂x V> = d/dt

\ No newline at end of file diff --git a/journal/chap2.motes b/journal/chap2.motes index d2cf8d2..9729406 100644 --- a/journal/chap2.motes +++ b/journal/chap2.motes @@ -226,12 +226,6 @@ Regarding group and phase velocities, Here, ω represents the phase velocity, and ϕ is narrowly peaked about k₀. -<<<<<<< HEAD -*** dispersion notes: derive group velocity from 2 frequencies - -*** dispersion comes from curvature of ω(k) - -======= A Taylor expansion helps elucidate the situation ω(k) = ω₀ + ω′₀(k-k₀) @@ -247,4 +241,3 @@ At t=0, ψ(x,0) = 1/√(π) ∫[-∞,∞]ds ϕ(k₀ + s) exp(ι(kₒ+s)x), ψ(x,t) = 1/√(π) exp(ι(-ω₀ + kₒ ω′₀)t∫[-∞,∞]ds ϕ(k₀ + s) exp(ι(kₒ+s)x) ->>>>>>> 9d60318cfe5d81c018be9b1fdbf72e03e8733ea0 diff --git a/project/Group3_Documentation.pdf b/project/Group3_Documentation.pdf new file mode 100644 index 0000000..9699747 Binary files /dev/null and b/project/Group3_Documentation.pdf differ diff --git a/project/HydrogenAtomRadialFunctions.nb b/project/HydrogenAtomRadialFunctions.nb new file 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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. Delete a locator by \ +clicking it\\nwhile holding down the \"", + FrameBox[ + "Alt", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> + GrayLevel[0.9]], "\" key.\""}], + TooltipDelay->0.35]], "ControlSuggestions", + FontFamily->"Verdana", + CellTags->"CreateAndDeleteLocators"], + +Cell[BoxData[ + TooltipBox[ + RowBox[{ + CheckboxBox[False], Cell[" Slider Zoom"]}], + RowBox[{"\"Hold down the \"", + FrameBox[ + "Alt", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> + GrayLevel[0.9]], + "\" key while moving a slider to make fine adjustments in the slider \ +value.\\nHold \"", + FrameBox[ + "Ctrl", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> + GrayLevel[0.9]], "\" and/or \"", + FrameBox[ + "Shift", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> + GrayLevel[0.9]], "\" at the same time as \"", + FrameBox[ + "Alt", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> + GrayLevel[0.9]], "\" to make ever finer adjustments.\""}], + TooltipDelay->0.35]], "ControlSuggestions", + FontFamily->"Verdana", + CellTags->"SliderZoom"], + +Cell[BoxData[ + TooltipBox[ + RowBox[{ + CheckboxBox[False], Cell[" Gamepad Controls"]}], + "\"Control this Demonstration with a gamepad or other\\nhuman interface \ +device connected to your computer.\"", + TooltipDelay->0.35]], "ControlSuggestions", + CellChangeTimes->{3.35696210375764*^9, 3.3895522232313623`*^9}, + FontFamily->"Verdana", + CellTags->"GamepadControls"], + +Cell[BoxData[ + TooltipBox[ + RowBox[{ + CheckboxBox[False], Cell[" Automatic Animation"]}], + RowBox[{"\"Animate a slider in this Demonstration by clicking the\"", + AdjustmentBox[ + Cell[ + GraphicsData[ + "CompressedBitmap", + "eJzzTSzJSM1NLMlMTlRwL0osyMhMLlZwyy8CCjEzMjAwcIKwAgOI/R/IhBKc\n\ +/4EAyGAG0f+nTZsGwgysIJIRKsWKLAXGIHFmEpUgLADxWUAkI24jZs+eTaEt\n\ +IG+wQKRmzJgBlYf5lhEA30OqWA=="], "Graphics", ImageSize -> {9, 9}, ImageMargins -> + 0, CellBaseline -> Baseline], BoxBaselineShift -> 0.1839080459770115, + BoxMargins -> {{0., 0.}, {-0.1839080459770115, 0.1839080459770115}}], + "\"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 -> "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->"BookmarkAnimation"] +}, Open ]], + +Cell["", "SearchTermsSection"], + +Cell[CellGroupData[{ + +Cell["", "RelatedLinksSection"], + +Cell[TextData[ButtonBox["Hydrogen Orbitals", + BaseStyle->"Hyperlink", + ButtonData->{ + 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double R(double n, double l, double r) { - double rho = r/a*n; + double rho = r*a/n; double result = 1/r * pow(rho,l+1) * exp(-rho) * v(n,l,rho); } -normalize() - - double Y(double l, double m, double theta, double phi); double psi(double n, double l, double m, double r, double theta, double phi) - = R(n,l,r) * Y(l,m,theta,phi); - - +{ + norm(n,l) * R(n,l,r) * Y(l,m,theta,phi); +} +double norm(double n, double l) +{ + double result_squared = pow((2*a/n),3) * factorial(n-l-1) / (2*n) / pow(factorial(n+1),3); + return sqrt(result_squared); +} int main(int argc, char const *argv[]) diff --git a/project/writeup.mth b/project/writeup.mth index fed9705..40483b1 100644 --- a/project/writeup.mth +++ b/project/writeup.mth @@ -1,3 +1,20 @@ +PHY 520: Group Project – Written Documentation +Steven Wallbrown, Henry Colburn, Bruce He, Otho Ulrich + +We present descriptions of our models in three parts. In this document, the infinite square well and 1/r potential models are presented. The simple harmonic oscillator is described in word document, accompanying this submission. In each case, a model was built in mathematica that parallels the developments in "Introduction to Quantum Mechanics" [Griffiths] and Dr. Chris Crawford's course notes for the PHY 520 course at UK, 2017. + +The states were easy to model without normalization. We found that normalizing the states was a difficult consideration. We were pleased to see the qualitative behaviour we expected, at least, and look forward to developing similar models more fully. The simple harmonic oscillator is plotted in 1d and 2d, and the infinite square well and 1/r potential solutions are plotted in 1d. Mathematica programs are included in this submission. + + +Modeling the infinite square well potential +───────────── + +The concept behind the simulation created is that of the infinite square well potential of a one-dimensional wave. The basics behind the idea is that a particle trapped in an area bordered by infinite potential must be composed of waves whose nodes correspond to the distance from one infinite potential barrier to another. The waves must be zero at the boundaries but are not required to have their derivatives be zero since there is no exponential decay of the wave, due to the infinite potential. Thus, the classical equation, sin(nπx/a) (where a is the well width and n is an integer) works for this situation where as the full quantum equation is not neccesary. This gives rise to the quantization of the energy levels so only discrete values are allowed. The program in Mathematica is designed so that up to three different waves can be plotted of different energy levels. The Y axis is the energy levels of the waves and the x axis is the width of the well. The program demonstrates the relationship between the number of nodes of a wave (entered in as n1, n2, and n3) and the energy levels associated with those waves. The simulation also draws attention to the spreading out of each successive wave from its predecessor in terms of separation energy, e.g. to go from n=5 to n=6 requires less energy input than n=100 to n=101. Within each wave function being plotted there is a horizontal line that represents the energy level of the wave that is on top of it. This is for a clearer representation of the energy level of the wave. The plotting range was also modified within each function so as to guarantee the plot would be large enough to fit up to the highest energy wave. With continued playing, the relationship between energy increases and node variation would become apparent to a user. + + +Modeling the 1/r potential - Hydrogen Atom +───────────── + To model a quantum particle in a radial potential, we addressed the derivation of the wave function parallel with the development in Griffiths, pg. 145, and in Dr. Crawford's course notes, and some various online sources. The potential V(r) = 1/r in spherical coordinates describes the system as having constants of motion in the angular coordinates theta and phi, thus conserving angular momentum. Still, it admits an effective potential that includes a centrifugal factor, @@ -46,9 +63,9 @@ The complete solution is then constructed assuming a power series v(ρ), The recursion relationship for the cⱼ coefficients is given by - ⎧ 2(j + l + 1) - ρ₀ ⎫ - cⱼ₊₁ = ⎨─────────────────⎬ cⱼ, - ⎩(j + 1)(j + 2l + 2)⎭ + ⎧ 2(j + l + 1) - ρ₀ ⎫ + cⱼ₊₁ = ⎨───────────────────⎬ cⱼ, + ⎩(j + 1)(j + 2l + 2)⎭ where c₀ is normalized by total probability equal to one. @@ -92,9 +109,9 @@ The model we've developed is nearly capable of handling the associated LaGuerre Substituting our new information back into the recursion relationship gives us the recursion in terms of the quantum numbers, - ⎧ 2(j + l + 1 - n) ⎫ - cⱼ₊₁ = ⎨────────────────⎬ cⱼ. - ⎩(j + 1)(j + 2l + 2)⎭ + ⎧ 2(j + l + 1 - n) ⎫ + cⱼ₊₁ = ⎨───────────────────⎬ cⱼ. + ⎩(j + 1)(j + 2l + 2)⎭ This is sufficient information to model the radial component of the wave function, except for normalizing the coefficient c₀. This is done using the expression of probability conservation @@ -102,19 +119,31 @@ This is sufficient information to model the radial component of the wave functio ∫ │Rₙₗ│² r² dr = 1. o -This is difficult to model numerically in the general case, and while interesting, the more practical approach is probably to normalize any states we're interested in by hand. The normalization of c₀ is different for each set of values {n,l}. +We want to simulate the hydrogen atom, where V(r) ≠ -1/r, but instead V(r) = -e²/(4πε₀) 1/r. This changes some of our parameters, i.e., -A model has (nearly) been written in c++ to compute the wavefunction resultant from this potential. The data is output to tab-separated data tables in the format + ρ = κr = r/an and ρ₀ = m e²/(2π ε₀ ħ² κ), -r R₁₀ R₂₀ R₃₀ ... + where a now represents the Bohr Radius, -Still deciding how to output the different values of l. + a = 5.29 × 10⁻² nm. +The rest of the development is essentially the same. +In general, the normalization is given by + ⎛ (2/na)³ (n-l-1)!⎞ 1/2 + ⎜ ────────────── ⎟ + ⎝ 2n [(n+l)!]³ ⎠. +We will notate this normalization as α. If we also notate the generalized Laguerre polynomials of x as L[q,p](x), then our model is + α exp(-r/na) (2r/na)ˡ L[n-l-1,2l+1](2r/na) Yₗᵐ(θ,ϕ). +We have not plotted this, but here Yₗᵐ refers to the spherical harmonics, which have the form (to within a normalization constant) + + Yₗᵐ(θ,ϕ) = exp(ιmϕ) Pₗᵐ (cosθ), + where Pₗᵐ(x) are the Legendre Polynomials in x. +This model is implemented in mathematica, and will be presented to the Physics 520 class on Dec. 8, 2017. diff --git a/project/writeup.mth.ps b/project/writeup.mth.ps new file mode 100644 index 0000000..5c1f444 --- /dev/null +++ b/project/writeup.mth.ps @@ -0,0 +1,3855 @@ +%!PS-Adobe-3.0 +%%Title: Potential Solutions +%%Creator: paps version 0.6.7 by Dov Grobgeld +%%Pages: (atend) +%%BoundingBox: 0 0 595 841 +%%BeginProlog +%%Orientation: Portrait +/papsdict 1 dict def +papsdict begin + +/inch {72 mul} bind def +/mm {1 inch 25.4 div mul} bind def + +% override setpagedevice if it is not defined +/setpagedevice where { + pop % get rid of its dictionary + /setpagesize { + 3 dict begin + /pageheight exch def + /pagewidth exch def + /orientation 0 def + % Exchange pagewidth and pageheight so that pagewidth is bigger + pagewidth pageheight gt { + pagewidth + /pagewidth pageheight def + /pageheight exch def + /orientation 3 def + } if + 2 dict + dup /PageSize [pagewidth pageheight] put + dup /Orientation orientation put + setpagedevice + end + } def +} +{ + /setpagesize { pop pop } def +} ifelse +/duplex { + statusdict /setduplexmode known + { statusdict begin setduplexmode end } {pop} ifelse +} def +/tumble { + statusdict /settumble known + { statusdict begin settumble end } {pop} ifelse +} def +% Turn the page around +/turnpage { + 90 rotate + 0 pageheight neg translate +} def +% User settings +/pagewidth 595 def +/pageheight 841 def +pagewidth pageheight setpagesize +/column_width 523 def +/bodyheight 755 def +/lmarg 36 def +/ytop 791 def +/do_separation_line true def +/do_landscape false def +/do_tumble true def +/do_duplex true def +% Procedures to translate position to first and second column +/lw 20 def % whatever +/setnumcolumns { + /numcolumns exch def + /firstcolumn { /xpos lmarg def /ypos ytop def} def + /nextcolumn { + do_separation_line { + xpos column_width add gutter_width 2 div add % x start + ytop lw add moveto % y start + 0 bodyheight lw add neg rlineto 0 setlinewidth stroke + } if + /xpos xpos column_width add gutter_width add def + /ypos ytop def + } def +} def + +1 setnumcolumns +/showline { + /y exch def + /s exch def + xpos y moveto + column_width 0 rlineto stroke + xpos y moveto /Helvetica findfont 20 scalefont setfont s show +} def +/paps_bop { % Beginning of page definitions + papsdict begin + gsave + do_landscape {turnpage} if + % ps2pdf gets wrong orientation without this! + /Helvetica findfont setfont 100 100 moveto ( ) show + firstcolumn + end +} def + +/paps_eop { % End of page cleanups + grestore +} def +%%BeginProlog +/papsdict 1 dict def +papsdict begin + +/conicto { + /to_y exch def + /to_x exch def + /conic_cntrl_y exch def + /conic_cntrl_x exch def + currentpoint + /p0_y exch def + /p0_x exch def + /p1_x p0_x conic_cntrl_x p0_x sub 2 3 div mul add def + /p1_y p0_y conic_cntrl_y p0_y sub 2 3 div mul add def + /p2_x p1_x to_x p0_x sub 1 3 div mul add def + /p2_y p1_y to_y p0_y sub 1 3 div mul add def + p1_x p1_y p2_x p2_y to_x to_y curveto +} bind def +/start_ol { gsave } bind def +/end_ol { closepath fill grestore } bind def +/draw_char { fontdict begin gsave 0.001000 dup scale last_x last_y translate load exec end grestore} def +/goto_xy { fontdict begin /last_y exch def /last_x exch def end } def +/goto_x { fontdict begin /last_x exch def end } def +/fwd_x { fontdict begin /last_x exch last_x add def end } def +/c /curveto load def +/x /conicto load def +/l /lineto load def +/m /moveto load def +end +/paps_exec { + 1 dict begin + /ps exch def + /len ps length def + /pos 0 def + + % Loop over all the characters of the string + { + pos len eq {exit} if + + % Get character at pos + /ch ps pos 1 getinterval def + + % check for + + (+) ch eq { + /pos 1 pos add def + /xp ps pos 8 getinterval cvi def + /yp ps pos 8 add 8 getinterval cvi def + /pos 16 pos add def + papsdict begin xp yp goto_xy end + } { + (*) ch eq { + /pos 1 pos add def + /xp ps pos 8 getinterval cvi def + /pos 8 pos add def + papsdict begin xp goto_x end + } { (>) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp 2 mul fwd_x end + } { (-) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp neg 2 mul fwd_x end + } { + % Must be a 3 char sym. 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