Skip to content
This repository
Browse code

modified: ../../env/.aspell.en.pws

modified:   ../METADATA
modified:   ../TODO.txt
modified:   approx/qmTwoL8.tex
new file:   mathematica/24.4.3.newAttempt.nb
modified:   mathematica/24.4.3_attempt_with_mathematica.nb
modified:   problems/variationHarmonicOscillator.tex
modified:   readings.tex
modified:   thisbook.sty
  • Loading branch information...
commit 4125ff92cfa60c72493225e8daa8623b784e6c47 1 parent 7617e8f
Peeter Joot authored
24 env/.aspell.en.pws
... ... @@ -1,4 +1,4 @@
1   -personal_ws-1.1 en 465
  1 +personal_ws-1.1 en 467
2 2 CDF
3 3 Google
4 4 tuples
@@ -7,8 +7,9 @@ confocal
7 7 ParametericPlot
8 8 Indistinguishability
9 9 Exponentiating
10   -Fick's
  10 +extremum
11 11 Ficks
  12 +Fick's
12 13 cgs
13 14 phasor
14 15 linestyle
@@ -87,8 +88,8 @@ Eikonal
87 88 Goldstein's
88 89 GPS
89 90 colinear
90   -Hmm
91 91 hoc
  92 +Hmm
92 93 Peeter's
93 94 Benard
94 95 ijk
@@ -160,8 +161,8 @@ Hestenes's
160 161 parametrizations
161 162 imaginaries
162 163 reparametrize
163   -nlm
164 164 n'l'm
  165 +nlm
165 166 PDE
166 167 Lut
167 168 quantized
@@ -188,8 +189,8 @@ arctan
188 189 entropic
189 190 invertible
190 191 pion
191   -OuterMorphism
192 192 outermorphism
  193 +OuterMorphism
193 194 QFT
194 195 rescaling
195 196 spinors
@@ -310,8 +311,8 @@ df
310 311 LIGO
311 312 spacetime
312 313 dH
313   -dj
314 314 BT
  315 +dj
315 316 Routhian
316 317 dk
317 318 dL
@@ -339,9 +340,9 @@ Prandtl
339 340 ia
340 341 isync
341 342 inferometer
  343 +eV
342 344 ib
343 345 iB
344   -eV
345 346 orthonormalization
346 347 ic
347 348 ie
@@ -355,16 +356,17 @@ variates
355 356 im
356 357 Dekker's
357 358 ji
358   -KE
  359 +unnormalized
359 360 jj
  361 +KE
360 362 elastostatics
361 363 amino
362   -jk
363 364 ip
  365 +jk
364 366 indices
365 367 kj
366   -kk
367 368 iu
  369 +kk
368 370 mc
369 371 mE
370 372 iz
@@ -377,8 +379,8 @@ Strang's
377 379 xyz
378 380 mk
379 381 resistive
380   -kx
381 382 mn
  383 +kx
382 384 Eulerian
383 385 anticommutes
384 386 n'l
10 notes/METADATA
@@ -4753,6 +4753,11 @@ Generate figures for continuum mechanics problem set II figure 1. Using Show an
4753 4753 path => 'phy452/mathematica/largeTemperatureGaussianFermionDistributionIntegral.nb',
4754 4754 WHAT => qq(Lecture 16, Integral verification for thermal de Broglie lambda calculation.),
4755 4755 },
  4756 +{
  4757 + DATE => 'March 28, 2013',
  4758 + path => 'phy456/mathematica/24.4.3.newAttempt.nb',
  4759 + WHAT => qq(A new attempt at Desai 24.4.3 from scratch. This one has an error, as did the original. The original is now fixed.),
  4760 +},
4756 4761 # not all of these are committed to the repo. Some are, but are not described here.
4757 4762 #blogit/imageProcessingExperimentation.cdf
4758 4763 #blogit/streamSlowCurveFit.cdf
@@ -4901,6 +4906,11 @@ elsif ( $doMathematica )
4901 4906 $filter = "phy454/" ;
4902 4907 }
4903 4908
  4909 + if ( $doPhy456 )
  4910 + {
  4911 + $filter = "phy456/" ;
  4912 + }
  4913 +
4904 4914 if ( $doPhy450 )
4905 4915 {
4906 4916 $filter = "phy450/" ;
3  notes/TODO.txt
@@ -122,6 +122,9 @@ phy456
122 122 Perhaps all the problem
123 123 sets were ungraded and I only did them on paper (I have all those qmII notebooks ...
124 124 maybe I kept them because I intended to review what I'd done).
  125 +
  126 + - have two mathematica references in "phy452/problems/variationalHelium.tex" ... convert to nbref
  127 + - delete most of the initial attempt garbage in "./problems/variationHarmonicOscillator.tex" and just leave the part where it's done right (perhaps with a comment about why the niave attempt doesn't work).
125 128
126 129 ================================================================================================
127 130
3  notes/phy456/approx/qmTwoL8.tex
@@ -75,7 +75,8 @@ \section{Time dependent perturbation}
75 75
76 76 We expect this to have a two lobe Fourier spectrum, with the lobes centered at $\omega = \pm 10$, and width proportional to $1/T$.
77 77
78   -For reference, as calculated using \href{https://github.com/peeterjoot/physicsplay/tree/master/notes/phy456/qmTwoL8figures.nb}{Mathematica} this Fourier transform is
  78 +For reference, as calculated using \nbref{qmTwoL8figures.nb} this Fourier transform is
  79 +%\href{https://github.com/peeterjoot/physicsplay/tree/master/notes/phy456/qmTwoL8figures.nb}{Mathematica}
79 80
80 81 \begin{equation}\label{eqn:qmTwoL8:130}
81 82 E(\omega) = \frac{e^{-\frac{1}{4} T^2 (\omega_0+\omega )^2}}{2 \sqrt{\frac{2}{T^2}}}+\frac{e^{\omega_0 T^2 \omega -\frac{1}{4} T^2 (\omega_0+\omega )^2}}{2 \sqrt{\frac{2}{T^2}}}
571 notes/phy456/mathematica/24.4.3.newAttempt.nb
... ... @@ -0,0 +1,571 @@
  1 +(* Content-type: application/vnd.wolfram.mathematica *)
  2 +
  3 +(*** Wolfram Notebook File ***)
  4 +(* http://www.wolfram.com/nb *)
  5 +
  6 +(* CreatedBy='Mathematica 9.0' *)
  7 +
  8 +(*CacheID: 234*)
  9 +(* Internal cache information:
  10 +NotebookFileLineBreakTest
  11 +NotebookFileLineBreakTest
  12 +NotebookDataPosition[ 157, 7]
  13 +NotebookDataLength[ 18045, 561]
  14 +NotebookOptionsPosition[ 17292, 532]
  15 +NotebookOutlinePosition[ 17637, 547]
  16 +CellTagsIndexPosition[ 17594, 544]
  17 +WindowFrame->Normal*)
  18 +
  19 +(* Beginning of Notebook Content *)
  20 +Notebook[{
  21 +
  22 +Cell[CellGroupData[{
  23 +Cell[BoxData[
  24 + RowBox[{
  25 + RowBox[{"(*", " ",
  26 + RowBox[{"new", " ", "attempt", " ", "at", " ", "24.4",
  27 + RowBox[{".3", ".", " ", "Get"}], " ", "a", " ", "different", " ",
  28 + "answer"}], " ", "*)"}], "\[IndentingNewLine]",
  29 + RowBox[{
  30 + RowBox[{"Clear", "[", "\"\<Global`*\>\"", "]"}],
  31 + RowBox[{"(*",
  32 + RowBox[{"Setting", " ", "default", " ", "environment"}], "*)"}],
  33 + "\[IndentingNewLine]", "\[IndentingNewLine]",
  34 + RowBox[{"$Assumptions", " ", "=", " ",
  35 + RowBox[{
  36 + RowBox[{"b", " ", ">", " ", "0"}], " ", "&&", " ",
  37 + RowBox[{"\[Omega]", " ", ">", " ", "0"}], " ", "&&", " ",
  38 + RowBox[{"\[HBar]", " ", ">", " ", "0"}], " ", "&&", " ",
  39 + RowBox[{"m", " ", ">", " ", "0"}]}]}], " ", "\[IndentingNewLine]",
  40 + "\[IndentingNewLine]",
  41 + RowBox[{
  42 + RowBox[{
  43 + RowBox[{"psi", "[", "r_", "]"}], " ", "=", " ",
  44 + RowBox[{
  45 + SqrtBox["b"], " ",
  46 + RowBox[{"(",
  47 + RowBox[{
  48 + RowBox[{
  49 + SuperscriptBox["\[ExponentialE]",
  50 + RowBox[{"b", " ", "r"}]], " ",
  51 + RowBox[{"HeavisideTheta", "[",
  52 + RowBox[{"-", "r"}], "]"}]}], "+",
  53 + RowBox[{
  54 + SuperscriptBox["\[ExponentialE]",
  55 + RowBox[{
  56 + RowBox[{"-", "b"}], " ", "r"}]], " ",
  57 + RowBox[{"HeavisideTheta", "[", "r", "]"}]}]}], ")"}]}]}], " ", ";"}],
  58 + "\[IndentingNewLine]",
  59 + RowBox[{
  60 + RowBox[{
  61 + RowBox[{"psiA", "[", "r_", "]"}], " ", "=", " ",
  62 + RowBox[{
  63 + RowBox[{"Sqrt", "[", "b", "]"}], " ",
  64 + RowBox[{"E", "^",
  65 + RowBox[{"(",
  66 + RowBox[{
  67 + RowBox[{"-", "b"}], " ",
  68 + RowBox[{"Abs", "[", "r", "]"}]}], ")"}]}]}]}], " ", ";"}],
  69 + "\[IndentingNewLine]", "\[IndentingNewLine]",
  70 + RowBox[{"(*", " ",
  71 + RowBox[{"check", " ", "normalization"}], " ", "*)"}],
  72 + "\[IndentingNewLine]",
  73 + RowBox[{
  74 + RowBox[{"Integrate", "[", " ",
  75 + RowBox[{
  76 + RowBox[{
  77 + RowBox[{"psi", "[", "r", "]"}], "^", "2"}], ",", " ",
  78 + RowBox[{"{",
  79 + RowBox[{"r", ",", " ",
  80 + RowBox[{"-", "Infinity"}], ",", " ", "Infinity"}], "}"}]}], "]"}],
  81 + " ", ";"}], "\[IndentingNewLine]",
  82 + RowBox[{
  83 + RowBox[{"Integrate", "[", " ",
  84 + RowBox[{
  85 + RowBox[{
  86 + RowBox[{"psiA", "[", "r", "]"}], "^", "2"}], ",", " ",
  87 + RowBox[{"{",
  88 + RowBox[{"r", ",", " ",
  89 + RowBox[{"-", "Infinity"}], ",", " ", "Infinity"}], "}"}]}], "]"}],
  90 + " ", ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]",
  91 + RowBox[{
  92 + RowBox[{"a", " ", "=", " ",
  93 + RowBox[{
  94 + RowBox[{"-",
  95 + RowBox[{"\[HBar]", "^", "2"}]}], "/",
  96 + RowBox[{"(",
  97 + RowBox[{"2", "m"}], ")"}]}]}], " ", ";"}], "\[IndentingNewLine]",
  98 + RowBox[{
  99 + RowBox[{"hPsi", "[", "r_", "]"}], " ", "=", " ",
  100 + RowBox[{
  101 + RowBox[{
  102 + RowBox[{
  103 + RowBox[{"-", "a"}], " ",
  104 + RowBox[{"D", "[", " ",
  105 + RowBox[{
  106 + RowBox[{"D", "[", " ",
  107 + RowBox[{
  108 + RowBox[{"psi", "[", "r", "]"}], ",", " ", "r"}], "]"}], ",", " ",
  109 + "r"}], "]"}]}], " ", "+", " ",
  110 + RowBox[{
  111 + RowBox[{"(",
  112 + RowBox[{"1", "/", "2"}], ")"}], " ", "m", " ",
  113 + RowBox[{"\[Omega]", "^", "2"}], " ",
  114 + RowBox[{"r", "^", "2"}]}]}], " ", "//", " ", "FullSimplify"}]}],
  115 + "\[IndentingNewLine]",
  116 + RowBox[{
  117 + RowBox[{"hPsiA", "[", "r_", "]"}], " ", "=", " ",
  118 + RowBox[{
  119 + RowBox[{"hPsi", "[", "r", "]"}], " ", "/.", " ",
  120 + RowBox[{
  121 + RowBox[{"(",
  122 + RowBox[{
  123 + RowBox[{
  124 + SuperscriptBox["\[ExponentialE]",
  125 + RowBox[{"2", "b", " ", "r"}]], " ",
  126 + RowBox[{"HeavisideTheta", "[",
  127 + RowBox[{"-", "r"}], "]"}]}], "+",
  128 + RowBox[{"HeavisideTheta", "[", "r", "]"}]}], " ", ")"}], " ",
  129 + "\[Rule]", " ",
  130 + RowBox[{
  131 + RowBox[{"E", "^",
  132 + RowBox[{"(",
  133 + RowBox[{"b", " ", "r"}], ")"}]}],
  134 + RowBox[{
  135 + RowBox[{"psiA", "[", "r", "]"}], " ", "/",
  136 + RowBox[{"Sqrt", "[", "b", "]"}]}]}]}]}]}], "\[IndentingNewLine]",
  137 + "\[IndentingNewLine]",
  138 + RowBox[{
  139 + RowBox[{"eIntegrand", "[", "r_", "]"}], " ", "=", " ",
  140 + RowBox[{
  141 + RowBox[{
  142 + RowBox[{"psiA", "[", "r", "]"}], " ",
  143 + RowBox[{"hPsiA", "[", "r", "]"}]}], " ", "//", " ", "FullSimplify"}]}],
  144 + "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]",
  145 + RowBox[{
  146 + RowBox[{"e", " ", "=", " ",
  147 + RowBox[{"Integrate", "[", " ",
  148 + RowBox[{
  149 + RowBox[{"eIntegrand", "[", "r", "]"}], ",", " ",
  150 + RowBox[{"{",
  151 + RowBox[{"r", ",", " ",
  152 + RowBox[{"-", "Infinity"}], ",", " ", "Infinity"}], "}"}]}], "]"}]}],
  153 + " ", ";"}], "\[IndentingNewLine]",
  154 + RowBox[{"(*",
  155 + RowBox[{"e", " ", "=", " ",
  156 + RowBox[{"FullSimplify", "[",
  157 + RowBox[{"e", ",", " ",
  158 + RowBox[{"b", " ", ">", " ", "0"}]}], "]"}]}], "*)"}],
  159 + "\[IndentingNewLine]",
  160 + RowBox[{
  161 + RowBox[{"de", "[", "b", "]"}], " ", "=", " ",
  162 + RowBox[{"D", "[",
  163 + RowBox[{"e", ",", "b"}], "]"}]}], "\[IndentingNewLine]",
  164 + RowBox[{"s", "=", " ",
  165 + RowBox[{"Solve", "[",
  166 + RowBox[{
  167 + RowBox[{
  168 + RowBox[{"de", "[", "b", "]"}], "\[Equal]", "0"}], ",", "b"}], "]"}]}],
  169 + "\[IndentingNewLine]", "\[IndentingNewLine]"}]}]], "Input",
  170 + CellChangeTimes->{{3.5735128333044033`*^9, 3.573512884106309*^9}, {
  171 + 3.5735129419046144`*^9, 3.5735130728691053`*^9}, {3.5735131293823376`*^9,
  172 + 3.5735131931979876`*^9}, {3.5735135071969476`*^9, 3.573513553906619*^9}, {
  173 + 3.5735138221519623`*^9, 3.573513940957757*^9}, {3.5735140402384357`*^9,
  174 + 3.5735140471958337`*^9}, 3.5735141116555204`*^9, {3.5735141762802167`*^9,
  175 + 3.573514197805448*^9}, {3.573514400110019*^9, 3.573514448667796*^9}, {
  176 + 3.5735145172117167`*^9, 3.5735145225940247`*^9}, {3.573514651339389*^9,
  177 + 3.5735146848553057`*^9}, {3.573514720491344*^9, 3.573514749932028*^9}, {
  178 + 3.573514822836198*^9, 3.573514825486349*^9}, {3.573514961092106*^9,
  179 + 3.5735149613361197`*^9}, {3.573515742087776*^9, 3.573515767472228*^9}}],
  180 +
  181 +Cell[BoxData[
  182 + RowBox[{
  183 + RowBox[{"b", ">", "0"}], "&&",
  184 + RowBox[{"\[Omega]", ">", "0"}], "&&",
  185 + RowBox[{"\[HBar]", ">", "0"}], "&&",
  186 + RowBox[{"m", ">", "0"}]}]], "Output",
  187 + CellChangeTimes->{{3.573514497710602*^9, 3.5735145256071973`*^9}, {
  188 + 3.573514693568804*^9, 3.573514722717471*^9}, 3.5735147625717506`*^9,
  189 + 3.5735148347958817`*^9}],
  190 +
  191 +Cell[BoxData[
  192 + RowBox[{
  193 + RowBox[{
  194 + FractionBox["1", "2"], " ", "m", " ",
  195 + SuperscriptBox["r", "2"], " ",
  196 + SuperscriptBox["\[Omega]", "2"]}], "+",
  197 + RowBox[{
  198 + FractionBox["1",
  199 + RowBox[{"2", " ", "m"}]],
  200 + RowBox[{
  201 + SqrtBox["b"], " ",
  202 + SuperscriptBox["\[HBar]", "2"], " ",
  203 + RowBox[{"(",
  204 + RowBox[{
  205 + RowBox[{
  206 + RowBox[{"-", "2"}], " ", "b", " ",
  207 + RowBox[{"DiracDelta", "[", "r", "]"}]}], "+",
  208 + RowBox[{
  209 + SuperscriptBox["b", "2"], " ",
  210 + SuperscriptBox["\[ExponentialE]",
  211 + RowBox[{
  212 + RowBox[{"-", "b"}], " ", "r"}]], " ",
  213 + RowBox[{"(",
  214 + RowBox[{
  215 + RowBox[{
  216 + SuperscriptBox["\[ExponentialE]",
  217 + RowBox[{"2", " ", "b", " ", "r"}]], " ",
  218 + RowBox[{"HeavisideTheta", "[",
  219 + RowBox[{"-", "r"}], "]"}]}], "+",
  220 + RowBox[{"HeavisideTheta", "[", "r", "]"}]}], ")"}]}]}],
  221 + ")"}]}]}]}]], "Output",
  222 + CellChangeTimes->{{3.573514497710602*^9, 3.5735145256071973`*^9}, {
  223 + 3.573514693568804*^9, 3.573514722717471*^9}, 3.5735147625717506`*^9,
  224 + 3.5735148356319294`*^9}],
  225 +
  226 +Cell[BoxData[
  227 + RowBox[{
  228 + RowBox[{
  229 + FractionBox["1", "2"], " ", "m", " ",
  230 + SuperscriptBox["r", "2"], " ",
  231 + SuperscriptBox["\[Omega]", "2"]}], "+",
  232 + FractionBox[
  233 + RowBox[{
  234 + SqrtBox["b"], " ",
  235 + SuperscriptBox["\[HBar]", "2"], " ",
  236 + RowBox[{"(",
  237 + RowBox[{
  238 + RowBox[{
  239 + SuperscriptBox["b", "2"], " ",
  240 + SuperscriptBox["\[ExponentialE]",
  241 + RowBox[{
  242 + RowBox[{"-", "b"}], " ",
  243 + RowBox[{"Abs", "[", "r", "]"}]}]]}], "-",
  244 + RowBox[{"2", " ", "b", " ",
  245 + RowBox[{"DiracDelta", "[", "r", "]"}]}]}], ")"}]}],
  246 + RowBox[{"2", " ", "m"}]]}]], "Output",
  247 + CellChangeTimes->{{3.573514497710602*^9, 3.5735145256071973`*^9}, {
  248 + 3.573514693568804*^9, 3.573514722717471*^9}, 3.5735147625717506`*^9,
  249 + 3.5735148356339293`*^9}],
  250 +
  251 +Cell[BoxData[
  252 + RowBox[{
  253 + SqrtBox["b"], " ",
  254 + SuperscriptBox["\[ExponentialE]",
  255 + RowBox[{
  256 + RowBox[{"-", "b"}], " ",
  257 + RowBox[{"Abs", "[", "r", "]"}]}]], " ",
  258 + RowBox[{"(",
  259 + RowBox[{
  260 + RowBox[{
  261 + FractionBox["1", "2"], " ", "m", " ",
  262 + SuperscriptBox["r", "2"], " ",
  263 + SuperscriptBox["\[Omega]", "2"]}], "+",
  264 + FractionBox[
  265 + RowBox[{
  266 + SqrtBox["b"], " ",
  267 + SuperscriptBox["\[HBar]", "2"], " ",
  268 + RowBox[{"(",
  269 + RowBox[{
  270 + RowBox[{
  271 + SuperscriptBox["b", "2"], " ",
  272 + SuperscriptBox["\[ExponentialE]",
  273 + RowBox[{
  274 + RowBox[{"-", "b"}], " ",
  275 + RowBox[{"Abs", "[", "r", "]"}]}]]}], "-",
  276 + RowBox[{"2", " ", "b", " ",
  277 + RowBox[{"DiracDelta", "[", "r", "]"}]}]}], ")"}]}],
  278 + RowBox[{"2", " ", "m"}]]}], ")"}]}]], "Output",
  279 + CellChangeTimes->{{3.573514497710602*^9, 3.5735145256071973`*^9}, {
  280 + 3.573514693568804*^9, 3.573514722717471*^9}, 3.5735147625717506`*^9,
  281 + 3.57351483563593*^9}],
  282 +
  283 +Cell[BoxData[
  284 + RowBox[{
  285 + RowBox[{"-",
  286 + FractionBox[
  287 + RowBox[{"5", " ", "m", " ",
  288 + SuperscriptBox["\[Omega]", "2"]}],
  289 + SuperscriptBox["b",
  290 + RowBox[{"7", "/", "2"}]]]}], "-",
  291 + FractionBox[
  292 + RowBox[{"b", " ",
  293 + SuperscriptBox["\[HBar]", "2"]}], "m"]}]], "Output",
  294 + CellChangeTimes->{{3.573514497710602*^9, 3.5735145256071973`*^9}, {
  295 + 3.573514693568804*^9, 3.573514722717471*^9}, 3.5735147625717506`*^9,
  296 + 3.5735148359649487`*^9}],
  297 +
  298 +Cell[BoxData[
  299 + RowBox[{"{",
  300 + RowBox[{
  301 + RowBox[{"{",
  302 + RowBox[{"b", "\[Rule]",
  303 + FractionBox[
  304 + RowBox[{
  305 + SuperscriptBox[
  306 + RowBox[{"(",
  307 + RowBox[{"-", "5"}], ")"}],
  308 + RowBox[{"2", "/", "9"}]], " ",
  309 + SuperscriptBox["m",
  310 + RowBox[{"4", "/", "9"}]], " ",
  311 + SuperscriptBox["\[Omega]",
  312 + RowBox[{"4", "/", "9"}]]}],
  313 + SuperscriptBox["\[HBar]",
  314 + RowBox[{"4", "/", "9"}]]]}], "}"}], ",",
  315 + RowBox[{"{",
  316 + RowBox[{"b", "\[Rule]",
  317 + FractionBox[
  318 + RowBox[{
  319 + SuperscriptBox["5",
  320 + RowBox[{"2", "/", "9"}]], " ",
  321 + SuperscriptBox["m",
  322 + RowBox[{"4", "/", "9"}]], " ",
  323 + SuperscriptBox["\[Omega]",
  324 + RowBox[{"4", "/", "9"}]]}],
  325 + SuperscriptBox["\[HBar]",
  326 + RowBox[{"4", "/", "9"}]]]}], "}"}], ",",
  327 + RowBox[{"{",
  328 + RowBox[{"b", "\[Rule]",
  329 + RowBox[{"-",
  330 + FractionBox[
  331 + RowBox[{
  332 + SuperscriptBox[
  333 + RowBox[{"(",
  334 + RowBox[{"-", "1"}], ")"}],
  335 + RowBox[{"1", "/", "9"}]], " ",
  336 + SuperscriptBox["5",
  337 + RowBox[{"2", "/", "9"}]], " ",
  338 + SuperscriptBox["m",
  339 + RowBox[{"4", "/", "9"}]], " ",
  340 + SuperscriptBox["\[Omega]",
  341 + RowBox[{"4", "/", "9"}]]}],
  342 + SuperscriptBox["\[HBar]",
  343 + RowBox[{"4", "/", "9"}]]]}]}], "}"}], ",",
  344 + RowBox[{"{",
  345 + RowBox[{"b", "\[Rule]",
  346 + RowBox[{"-",
  347 + FractionBox[
  348 + RowBox[{
  349 + SuperscriptBox[
  350 + RowBox[{"(",
  351 + RowBox[{"-", "1"}], ")"}],
  352 + RowBox[{"1", "/", "3"}]], " ",
  353 + SuperscriptBox["5",
  354 + RowBox[{"2", "/", "9"}]], " ",
  355 + SuperscriptBox["m",
  356 + RowBox[{"4", "/", "9"}]], " ",
  357 + SuperscriptBox["\[Omega]",
  358 + RowBox[{"4", "/", "9"}]]}],
  359 + SuperscriptBox["\[HBar]",
  360 + RowBox[{"4", "/", "9"}]]]}]}], "}"}], ",",
  361 + RowBox[{"{",
  362 + RowBox[{"b", "\[Rule]",
  363 + FractionBox[
  364 + RowBox[{
  365 + SuperscriptBox[
  366 + RowBox[{"(",
  367 + RowBox[{"-", "1"}], ")"}],
  368 + RowBox[{"4", "/", "9"}]], " ",
  369 + SuperscriptBox["5",
  370 + RowBox[{"2", "/", "9"}]], " ",
  371 + SuperscriptBox["m",
  372 + RowBox[{"4", "/", "9"}]], " ",
  373 + SuperscriptBox["\[Omega]",
  374 + RowBox[{"4", "/", "9"}]]}],
  375 + SuperscriptBox["\[HBar]",
  376 + RowBox[{"4", "/", "9"}]]]}], "}"}], ",",
  377 + RowBox[{"{",
  378 + RowBox[{"b", "\[Rule]",
  379 + RowBox[{"-",
  380 + FractionBox[
  381 + RowBox[{
  382 + SuperscriptBox[
  383 + RowBox[{"(",
  384 + RowBox[{"-", "1"}], ")"}],
  385 + RowBox[{"5", "/", "9"}]], " ",
  386 + SuperscriptBox["5",
  387 + RowBox[{"2", "/", "9"}]], " ",
  388 + SuperscriptBox["m",
  389 + RowBox[{"4", "/", "9"}]], " ",
  390 + SuperscriptBox["\[Omega]",
  391 + RowBox[{"4", "/", "9"}]]}],
  392 + SuperscriptBox["\[HBar]",
  393 + RowBox[{"4", "/", "9"}]]]}]}], "}"}], ",",
  394 + RowBox[{"{",
  395 + RowBox[{"b", "\[Rule]",
  396 + FractionBox[
  397 + RowBox[{
  398 + SuperscriptBox[
  399 + RowBox[{"(",
  400 + RowBox[{"-", "1"}], ")"}],
  401 + RowBox[{"2", "/", "3"}]], " ",
  402 + SuperscriptBox["5",
  403 + RowBox[{"2", "/", "9"}]], " ",
  404 + SuperscriptBox["m",
  405 + RowBox[{"4", "/", "9"}]], " ",
  406 + SuperscriptBox["\[Omega]",
  407 + RowBox[{"4", "/", "9"}]]}],
  408 + SuperscriptBox["\[HBar]",
  409 + RowBox[{"4", "/", "9"}]]]}], "}"}], ",",
  410 + RowBox[{"{",
  411 + RowBox[{"b", "\[Rule]",
  412 + RowBox[{"-",
  413 + FractionBox[
  414 + RowBox[{
  415 + SuperscriptBox[
  416 + RowBox[{"(",
  417 + RowBox[{"-", "1"}], ")"}],
  418 + RowBox[{"7", "/", "9"}]], " ",
  419 + SuperscriptBox["5",
  420 + RowBox[{"2", "/", "9"}]], " ",
  421 + SuperscriptBox["m",
  422 + RowBox[{"4", "/", "9"}]], " ",
  423 + SuperscriptBox["\[Omega]",
  424 + RowBox[{"4", "/", "9"}]]}],
  425 + SuperscriptBox["\[HBar]",
  426 + RowBox[{"4", "/", "9"}]]]}]}], "}"}], ",",
  427 + RowBox[{"{",
  428 + RowBox[{"b", "\[Rule]",
  429 + FractionBox[
  430 + RowBox[{
  431 + SuperscriptBox[
  432 + RowBox[{"(",
  433 + RowBox[{"-", "1"}], ")"}],
  434 + RowBox[{"8", "/", "9"}]], " ",
  435 + SuperscriptBox["5",
  436 + RowBox[{"2", "/", "9"}]], " ",
  437 + SuperscriptBox["m",
  438 + RowBox[{"4", "/", "9"}]], " ",
  439 + SuperscriptBox["\[Omega]",
  440 + RowBox[{"4", "/", "9"}]]}],
  441 + SuperscriptBox["\[HBar]",
  442 + RowBox[{"4", "/", "9"}]]]}], "}"}]}], "}"}]], "Output",
  443 + CellChangeTimes->{{3.573514497710602*^9, 3.5735145256071973`*^9}, {
  444 + 3.573514693568804*^9, 3.573514722717471*^9}, 3.5735147625717506`*^9,
  445 + 3.5735148360429535`*^9}]
  446 +}, Open ]],
  447 +
  448 +Cell[CellGroupData[{
  449 +
  450 +Cell[BoxData[
  451 + RowBox[{"\[IndentingNewLine]",
  452 + RowBox[{
  453 + RowBox[{
  454 + RowBox[{"psiA", "[", "r", "]"}], " ", "/.", " ",
  455 + RowBox[{"b", " ", "\[Rule]", " ",
  456 + FractionBox[
  457 + RowBox[{
  458 + SuperscriptBox["5",
  459 + RowBox[{"2", "/", "9"}]], " ",
  460 + SuperscriptBox["m",
  461 + RowBox[{"4", "/", "9"}]], " ",
  462 + SuperscriptBox["\[Omega]",
  463 + RowBox[{"4", "/", "9"}]]}],
  464 + SuperscriptBox["\[HBar]",
  465 + RowBox[{"4", "/", "9"}]]]}]}], "\[IndentingNewLine]"}]}]], "Input",
  466 + CellChangeTimes->{{3.573514855134045*^9, 3.5735148623224564`*^9}, {
  467 + 3.573514911464267*^9, 3.573514986590564*^9}}],
  468 +
  469 +Cell[BoxData[
  470 + RowBox[{
  471 + SuperscriptBox["5",
  472 + RowBox[{"1", "/", "9"}]], " ",
  473 + SuperscriptBox["\[ExponentialE]",
  474 + RowBox[{"-",
  475 + FractionBox[
  476 + RowBox[{
  477 + SuperscriptBox["5",
  478 + RowBox[{"2", "/", "9"}]], " ",
  479 + SuperscriptBox["m",
  480 + RowBox[{"4", "/", "9"}]], " ",
  481 + SuperscriptBox["\[Omega]",
  482 + RowBox[{"4", "/", "9"}]], " ",
  483 + RowBox[{"Abs", "[", "r", "]"}]}],
  484 + SuperscriptBox["\[HBar]",
  485 + RowBox[{"4", "/", "9"}]]]}]], " ",
  486 + SqrtBox[
  487 + FractionBox[
  488 + RowBox[{
  489 + SuperscriptBox["m",
  490 + RowBox[{"4", "/", "9"}]], " ",
  491 + SuperscriptBox["\[Omega]",
  492 + RowBox[{"4", "/", "9"}]]}],
  493 + SuperscriptBox["\[HBar]",
  494 + RowBox[{"4", "/", "9"}]]]]}]], "Output",
  495 + CellChangeTimes->{3.5735148763352575`*^9}]
  496 +}, Open ]],
  497 +
  498 +Cell[CellGroupData[{
  499 +
  500 +Cell[BoxData[
  501 + RowBox[{
  502 + RowBox[{"e", " ", "/.", " ",
  503 + RowBox[{"b", " ", "\[Rule]", " ",
  504 + FractionBox[
  505 + RowBox[{
  506 + SuperscriptBox["5",
  507 + RowBox[{"2", "/", "9"}]], " ",
  508 + SuperscriptBox["m",
  509 + RowBox[{"4", "/", "9"}]], " ",
  510 + SuperscriptBox["\[Omega]",
  511 + RowBox[{"4", "/", "9"}]]}],
  512 + SuperscriptBox["\[HBar]",
  513 + RowBox[{"4", "/", "9"}]]]}]}], " ", "//", " ",
  514 + "FullSimplify"}]], "Input",
  515 + CellChangeTimes->{{3.5735149928069196`*^9, 3.5735150201984863`*^9}}],
  516 +
  517 +Cell[BoxData[
  518 + RowBox[{"-",
  519 + FractionBox[
  520 + SuperscriptBox[
  521 + RowBox[{"(",
  522 + FractionBox[
  523 + RowBox[{
  524 + SuperscriptBox["\[Omega]", "8"], " ",
  525 + SuperscriptBox["\[HBar]", "10"]}], "m"], ")"}],
  526 + RowBox[{"1", "/", "9"}]],
  527 + RowBox[{"2", " ",
  528 + SuperscriptBox["5",
  529 + RowBox[{"5", "/", "9"}]]}]]}]], "Output",
  530 + CellChangeTimes->{{3.5735150001343384`*^9, 3.5735150208635244`*^9}}]
  531 +}, Open ]]
  532 +},
  533 +WindowSize->{707, 637},
  534 +WindowMargins->{{Automatic, 393}, {Automatic, 10}},
  535 +FrontEndVersion->"9.0 for Microsoft Windows (64-bit) (January 25, 2013)",
  536 +StyleDefinitions->"Default.nb"
  537 +]
  538 +(* End of Notebook Content *)
  539 +
  540 +(* Internal cache information *)
  541 +(*CellTagsOutline
  542 +CellTagsIndex->{}
  543 +*)
  544 +(*CellTagsIndex
  545 +CellTagsIndex->{}
  546 +*)
  547 +(*NotebookFileOutline
  548 +Notebook[{
  549 +Cell[CellGroupData[{
  550 +Cell[579, 22, 5972, 156, 780, "Input"],
  551 +Cell[6554, 180, 348, 8, 40, "Output"],
  552 +Cell[6905, 190, 1108, 33, 89, "Output"],
  553 +Cell[8016, 225, 787, 23, 67, "Output"],
  554 +Cell[8806, 250, 1005, 30, 106, "Output"],
  555 +Cell[9814, 282, 458, 13, 64, "Output"],
  556 +Cell[10275, 297, 4592, 147, 272, "Output"]
  557 +}, Open ]],
  558 +Cell[CellGroupData[{
  559 +Cell[14904, 449, 628, 17, 127, "Input"],
  560 +Cell[15535, 468, 779, 26, 73, "Output"]
  561 +}, Open ]],
  562 +Cell[CellGroupData[{
  563 +Cell[16351, 499, 512, 15, 71, "Input"],
  564 +Cell[16866, 516, 410, 13, 76, "Output"]
  565 +}, Open ]]
  566 +}
  567 +]
  568 +*)
  569 +
  570 +(* End of internal cache information *)
  571 +
696 notes/phy456/mathematica/24.4.3_attempt_with_mathematica.nb
@@ -10,10 +10,10 @@
10 10 NotebookFileLineBreakTest
11 11 NotebookFileLineBreakTest
12 12 NotebookDataPosition[ 157, 7]
13   -NotebookDataLength[ 18040, 490]
14   -NotebookOptionsPosition[ 16681, 450]
15   -NotebookOutlinePosition[ 17028, 465]
16   -CellTagsIndexPosition[ 16985, 462]
  13 +NotebookDataLength[ 18122, 486]
  14 +NotebookOptionsPosition[ 17441, 461]
  15 +NotebookOutlinePosition[ 17864, 477]
  16 +CellTagsIndexPosition[ 17821, 474]
17 17 WindowFrame->Normal*)
18 18
19 19 (* Beginning of Notebook Content *)
@@ -21,83 +21,122 @@ Notebook[{
21 21
22 22 Cell[CellGroupData[{
23 23 Cell[BoxData[{
24   - RowBox[{"Clear", "[",
25   - RowBox[{
26   - "harmonicOscHamiltonian", ",", " ", "f", ",", " ", "g", ",", " ", "h", ",",
27   - " ", "hF", ",", " ", "v", ",", " ", "x", ",", " ", "braket", ",", " ",
28   - "norm", ",", " ", "energy", ",", " ", "energyUnnorm", ",", " ", "\[Alpha]",
29   - ",", " ", "\[Beta]", ",", " ", "\[Omega]", ",", " ", "\[HBar]", ",", " ",
30   - "m"}], "]"}], "\n",
31 24 RowBox[{
32   - RowBox[{"braket", "[",
33   - RowBox[{"g_", ",", " ", "h_"}], "]"}], " ", ":=", " ",
34   - RowBox[{"Integrate", "[",
35   - RowBox[{
36   - RowBox[{"h", "*",
37   - RowBox[{"Conjugate", "[", "g", "]"}]}], ",", " ",
38   - RowBox[{"{",
39   - RowBox[{"x", ",", " ",
40   - RowBox[{"-", "Infinity"}], ",", " ", "Infinity"}], "}"}]}],
41   - "]"}]}], "\n",
  25 + RowBox[{"Clear", "[", "\"\<Global`*\>\"", "]"}],
  26 + RowBox[{"(*",
  27 + RowBox[{"Setting", " ", "default", " ", "environment"}], "*)"}],
  28 + "\[IndentingNewLine]"}], "\[IndentingNewLine]",
42 29 RowBox[{
43   - RowBox[{"\[Alpha]", " ", ":=", " ",
44   - RowBox[{"Sqrt", "[",
  30 + RowBox[{
  31 + RowBox[{"$Assumptions", " ", "=", " ",
45 32 RowBox[{
46   - RowBox[{"(",
47   - RowBox[{"m", "*", "\[Omega]"}], ")"}], "/", "\[HBar]"}], "]"}]}],
48   - "\[IndentingNewLine]",
49   - RowBox[{"(*", "\n",
  33 + RowBox[{
  34 + RowBox[{"{",
  35 + RowBox[{
  36 + "\[HBar]", ",", " ", "\[Omega]", ",", " ", "m", ",", " ", "\[Beta]"}],
  37 + "}"}], " ", "\[Element]", " ", "Reals"}], " ", "&&", " ",
  38 + RowBox[{"m", " ", ">", " ", "0"}], " ", "&&", " ",
  39 + RowBox[{"\[Omega]", " ", ">", " ", "0"}], " ", "&&",
  40 + RowBox[{"\[HBar]", ">", "0"}], " ", "&&", " ",
  41 + RowBox[{"\[Beta]", " ", ">", " ", "0"}]}]}], " ", ";"}],
  42 + "\[IndentingNewLine]"}], "\[IndentingNewLine]",
  43 + RowBox[{
  44 + RowBox[{
  45 + RowBox[{"braket", "[",
  46 + RowBox[{"g_", ",", " ", "h_"}], "]"}], " ", ":=", " ",
  47 + RowBox[{"Integrate", "[",
  48 + RowBox[{
  49 + RowBox[{"h", "*",
  50 + RowBox[{"Conjugate", "[", "g", "]"}]}], ",", " ",
  51 + RowBox[{"{",
  52 + RowBox[{"x", ",", " ",
  53 + RowBox[{"-", "Infinity"}], ",", " ", "Infinity"}], "}"}]}], "]"}]}],
  54 + "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]",
  55 + RowBox[{"(*", " ",
50 56 RowBox[{
51 57 RowBox[{"f", "[", "x_", "]"}], " ", ":=", " ",
52 58 RowBox[{
53 59 RowBox[{
54   - RowBox[{
55   - RowBox[{"(",
56   - RowBox[{
57   - RowBox[{"(",
58   - RowBox[{"m", "*", "\[Omega]"}], ")"}], "/", " ",
  60 + RowBox[{"(",
  61 + RowBox[{
  62 + RowBox[{"(",
  63 + RowBox[{"m", "*", "\[Omega]"}], ")"}], "/", " ",
  64 + RowBox[{"(",
  65 + RowBox[{"Pi", " ", "\[HBar]"}], ")"}]}], ")"}], "^",
  66 + RowBox[{"(",
  67 + RowBox[{"1", "/", "4"}], ")"}]}], " ",
  68 + RowBox[{"E", "^",
  69 + RowBox[{"(",
  70 + RowBox[{
  71 + RowBox[{"-",
59 72 RowBox[{"(",
60   - RowBox[{"Pi", " ", "\[HBar]"}], ")"}]}], ")"}], "^",
61   - RowBox[{"(",
62   - RowBox[{"1", "/", "4"}], ")"}]}], " ",
63   - RowBox[{"E", "^",
64   - RowBox[{"(",
  73 + RowBox[{"m", "*",
  74 + RowBox[{"x", "^", "2"}], "*", "\[Omega]"}], ")"}]}], "/",
  75 + RowBox[{"(",
  76 + RowBox[{"2", "*", "\[HBar]"}], ")"}]}], ")"}]}]}]}], " ", "*)"}],
  77 + "\[IndentingNewLine]", "\n",
  78 + RowBox[{"(*", " ",
  79 + RowBox[{
  80 + RowBox[{"f", "[", "x_", "]"}], " ", ":=", " ",
  81 + RowBox[{
  82 + RowBox[{"Sqrt", "[", "\[Beta]", "]"}], " ",
  83 + RowBox[{"Exp", "[", " ",
  84 + RowBox[{
  85 + RowBox[{"-", " ", "\[Beta]"}], " ",
  86 + RowBox[{"Sqrt", "[",
  87 + RowBox[{"x", "^", "2"}], "]"}]}], " ", "]"}]}]}], " ", "*)"}],
  88 + "\[IndentingNewLine]",
  89 + RowBox[{"(*",
  90 + RowBox[{
  91 + RowBox[{"f", "[", "x_", "]"}], " ", ":=", " ",
  92 + RowBox[{
  93 + SqrtBox["2"], " ",
  94 + SuperscriptBox["\[ExponentialE]",
  95 + FractionBox[
  96 + RowBox[{
65 97 RowBox[{
66 98 RowBox[{"-",
67   - RowBox[{"(",
68   - RowBox[{"m", "*",
69   - RowBox[{"x", "^", "2"}], "*", "\[Omega]"}], ")"}]}], "/",
70   - RowBox[{"(",
71   - RowBox[{"2", "*", "\[HBar]"}], ")"}]}], ")"}]}],
72   - "\[IndentingNewLine]", "\n",
73   - RowBox[{"f", "[", "x_", "]"}]}], " ", ":=", " ",
74   - RowBox[{
75   - RowBox[{"Sqrt", "[", "\[Beta]", "]"}], " ",
76   - RowBox[{"Exp", "[", " ",
  99 + SuperscriptBox["x", "2"]}], " ",
  100 + SuperscriptBox["\[Alpha]", "4"]}], "+",
  101 + SuperscriptBox["\[Beta]", "2"]}],
  102 + RowBox[{"2", " ",
  103 + SuperscriptBox["\[Alpha]", "2"]}]]], " ",
  104 + SqrtBox["\[Beta]"], " ",
  105 + RowBox[{"Erfc", "[",
  106 + FractionBox["\[Beta]",
77 107 RowBox[{
78   - RowBox[{"-", " ", "\[Beta]"}], " ",
79   - RowBox[{"Sqrt", "[",
80   - RowBox[{"x", "^", "2"}], "]"}]}], " ", "]"}]}]}]}],
81   - "\[IndentingNewLine]", "*)"}]}], "\[IndentingNewLine]",
  108 + SqrtBox["2"], " ", "\[Alpha]"}]], "]"}]}]}], "*)"}],
  109 + "\[IndentingNewLine]"}], "\n",
82 110 RowBox[{
83   - RowBox[{"f", "[", "x_", "]"}], " ", ":=", " ",
84 111 RowBox[{
85   - SqrtBox["2"], " ",
86   - SuperscriptBox["\[ExponentialE]",
87   - FractionBox[
  112 + RowBox[{"f", "[", "x_", "]"}], " ", "=", " ",
  113 + RowBox[{
  114 + RowBox[{"Sqrt", "[", "\[Beta]", "]"}], " ",
  115 + RowBox[{"(",
88 116 RowBox[{
89 117 RowBox[{
90   - RowBox[{"-",
91   - SuperscriptBox["x", "2"]}], " ",
92   - SuperscriptBox["\[Alpha]", "4"]}], "+",
93   - SuperscriptBox["\[Beta]", "2"]}],
94   - RowBox[{"2", " ",
95   - SuperscriptBox["\[Alpha]", "2"]}]]], " ",
96   - SqrtBox["\[Beta]"], " ",
97   - RowBox[{"Erfc", "[",
98   - FractionBox["\[Beta]",
99   - RowBox[{
100   - SqrtBox["2"], " ", "\[Alpha]"}]], "]"}]}]}], "\[IndentingNewLine]",
  118 + SuperscriptBox["\[ExponentialE]",
  119 + RowBox[{"\[Beta]", " ", "x"}]], " ",
  120 + RowBox[{"HeavisideTheta", "[",
  121 + RowBox[{"-", "x"}], "]"}]}], "+",
  122 + RowBox[{
  123 + SuperscriptBox["\[ExponentialE]",
  124 + RowBox[{
  125 + RowBox[{"-", "\[Beta]"}], " ", "x"}]], " ",
  126 + RowBox[{"HeavisideTheta", "[", "x", "]"}]}]}], " ", ")"}]}]}], " ",
  127 + ";"}], "\[IndentingNewLine]",
  128 + RowBox[{
  129 + RowBox[{
  130 + RowBox[{
  131 + RowBox[{"f2", "[", "x_", "]"}], " ", "=", " ",
  132 + RowBox[{
  133 + RowBox[{"Sqrt", "[", "\[Beta]", "]"}], " ",
  134 + RowBox[{"E", "^",
  135 + RowBox[{"(",
  136 + RowBox[{
  137 + RowBox[{"-", "\[Beta]"}], " ",
  138 + RowBox[{"Abs", "[", "x", "]"}]}], ")"}]}]}]}], " ", ";"}],
  139 + "\[IndentingNewLine]", "\[IndentingNewLine]"}], "\[IndentingNewLine]",
101 140 RowBox[{
102 141 RowBox[{"harmonicOscHamiltonian", "[", "v_", "]"}], " ", ":=", " ",
103 142 RowBox[{
@@ -117,60 +156,54 @@ Cell[BoxData[{
117 156 RowBox[{
118 157 RowBox[{
119 158 RowBox[{"Derivative", "[", "2", "]"}], "[", "v", "]"}], "[", "x",
120   - "]"}]}]}]}], "\n",
121   - RowBox[{"Assuming", "[",
  159 + "]"}]}]}]}], "\[IndentingNewLine]",
  160 + RowBox[{
122 161 RowBox[{
123   - RowBox[{
124   - RowBox[{"Element", "[",
  162 + RowBox[{"norm", " ", "=", " ",
  163 + RowBox[{"braket", "[",
125 164 RowBox[{
126   - RowBox[{"{",
127   - RowBox[{
128   - "\[HBar]", ",", " ", "\[Omega]", ",", " ", "m", ",", " ", "\[Beta]"}],
129   - "}"}], ",", " ", "Reals"}], "]"}], " ", "&&", " ",
130   - RowBox[{
131   - RowBox[{"{",
132   - RowBox[{
133   - "\[HBar]", ",", " ", "\[Omega]", ",", " ", "m", ",", " ", "\[Beta]",
134   - ",", " ",
135   - RowBox[{"\[HBar]", "*", "Pi"}], ",", " ",
136   - RowBox[{"m", "*", "\[Omega]"}], ",", " ",
137   - RowBox[{"m", "*", "\[Omega]", "*", "\[HBar]"}]}], " ", "}"}], " ", ">",
138   - " ", "0"}]}], ",", " ",
139   - RowBox[{"FullSimplify", "[",
140   - RowBox[{"norm", " ", "=", " ",
141   - RowBox[{"braket", "[",
142   - RowBox[{
143   - RowBox[{"f", "[", "x", "]"}], ",", " ",
144   - RowBox[{"f", "[", "x", "]"}]}], "]"}]}], "]"}]}], "]"}], "\n",
145   - RowBox[{"hF", " ", "=", " ",
146   - RowBox[{"harmonicOscHamiltonian", "[", "f", "]"}]}], "\[IndentingNewLine]",
  165 + RowBox[{"f", "[", "x", "]"}], ",", " ",
  166 + RowBox[{"f", "[", "x", "]"}]}], "]"}]}], " ", ";"}],
  167 + "\[IndentingNewLine]"}], "\n",
147 168 RowBox[{
148   - RowBox[{"Assuming", "[", " ",
  169 + RowBox[{
  170 + RowBox[{"hF", "[", "x_", "]"}], " ", "=", " ",
149 171 RowBox[{
  172 + RowBox[{"harmonicOscHamiltonian", "[", "f", "]"}], " ", "//", " ",
  173 + "FullSimplify"}]}], " "}], "\[IndentingNewLine]",
  174 + RowBox[{
  175 + RowBox[{
  176 + RowBox[{"hF", "[", "x_", "]"}], " ", "=", " ",
  177 + RowBox[{
  178 + RowBox[{"hF", "[", "x", "]"}], "/.", " ",
150 179 RowBox[{
151 180 RowBox[{
152   - RowBox[{"{",
153   - RowBox[{
154   - "\[HBar]", ",", " ", "\[Omega]", ",", " ", "m", ",", " ", "\[Beta]"}],
155   - "}"}], " ", "\[Element]", " ", "Reals"}], " ", "&&", " ",
156   - RowBox[{
157   - RowBox[{"{",
158   - RowBox[{
159   - "\[HBar]", ",", " ", "\[Omega]", ",", " ", "m", ",", " ", "\[Beta]"}],
160   - "}"}], " ", ">", " ", "0"}], " ", "&&", " ",
  181 + RowBox[{
  182 + SuperscriptBox["\[ExponentialE]",
  183 + RowBox[{"2", " ", "x", " ", "\[Beta]"}]], " ",
  184 + RowBox[{"HeavisideTheta", "[",
  185 + RowBox[{"-", "x"}], "]"}]}], "+",
  186 + RowBox[{"HeavisideTheta", "[", "x", "]"}]}], " ", "\[Rule]", " ",
161 187 RowBox[{
162   - RowBox[{"m", " ", "\[Omega]", " ", "\[HBar]"}], ">", "0"}]}], ",", " ",
163   - RowBox[{"FullSimplify", "[",
164   - RowBox[{"energyUnnorm", " ", "=", " ",
165   - RowBox[{"inner", "[", " ",
166   - RowBox[{
167   - RowBox[{"f", "[", "x", "]"}], ",", " ", "hF"}], "]"}]}], " ",
168   - "]"}]}], "]"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]",
169   - RowBox[{
170   - RowBox[{"FullSimplify", "[",
171   - RowBox[{"energy", " ", "=", " ",
172   - RowBox[{"energyUnnorm", "/", "norm"}]}], "]"}],
173   - "\[IndentingNewLine]"}], "\[IndentingNewLine]"}], "Input",
  188 + RowBox[{"f2", "[", "x", "]"}], " ",
  189 + RowBox[{
  190 + RowBox[{"E", "^",
  191 + RowBox[{"(",
  192 + RowBox[{"\[Beta]", " ", "x"}], ")"}]}], "/",
  193 + RowBox[{"Sqrt", "[", "\[Beta]", "]"}]}]}]}]}]}],
  194 + "\[IndentingNewLine]"}], "\[IndentingNewLine]",
  195 + RowBox[{"energy", " ", "=", " ",
  196 + RowBox[{"braket", "[",
  197 + RowBox[{
  198 + RowBox[{"f2", "[", "x", "]"}], ",", " ",
  199 + RowBox[{"hF", "[", "x", "]"}]}], "]"}]}], "\[IndentingNewLine]",
  200 + RowBox[{"de", " ", "=", " ",
  201 + RowBox[{"D", "[",
  202 + RowBox[{"energy", ",", " ", "\[Beta]"}], "]"}]}], "\[IndentingNewLine]",
  203 + RowBox[{"Solve", "[",
  204 + RowBox[{
  205 + RowBox[{"de", " ", "\[Equal]", " ", "0"}], ",", " ", "\[Beta]"}],
  206 + "]"}], "\[IndentingNewLine]"}], "Input",
174 207 CellChangeTimes->{{3.526594587842564*^9, 3.5265948205918765`*^9}, {
175 208 3.5265948820523915`*^9, 3.5265949149362726`*^9}, {3.526595058029457*^9,
176 209 3.526595103098035*^9}, 3.5265962137515607`*^9, 3.5265962743510265`*^9, {
@@ -213,244 +246,223 @@ Cell[BoxData[{
213 246 3.5269465974886255`*^9, 3.526946604052001*^9}, {3.5269467051337824`*^9,
214 247 3.5269467162764196`*^9}, {3.5269468031213865`*^9, 3.526946818495266*^9}, {
215 248 3.5269468687871428`*^9, 3.5269468846910524`*^9}, {3.5269470131534*^9,
216   - 3.526947028736291*^9}, 3.5269471120620575`*^9},
217   - FormatType->"InputForm"],
218   -
219   -Cell[BoxData[
220   - RowBox[{"ConditionalExpression", "[",
221   - RowBox[{
222   - FractionBox[
223   - RowBox[{"2", " ",
224   - SuperscriptBox["\[ExponentialE]",
225   - FractionBox[
226   - RowBox[{
227   - SuperscriptBox["\[Beta]", "2"], " ", "\[HBar]"}],
228   - RowBox[{"m", " ", "\[Omega]"}]]], " ",
229   - SqrtBox["\[Pi]"], " ",
230   - RowBox[{"Abs", "[", "\[Beta]", "]"}], " ",
231   - SuperscriptBox[
232   - RowBox[{"Erfc", "[",
233   - FractionBox["\[Beta]",
234   - RowBox[{
235   - SqrtBox["2"], " ",
236   - SqrtBox[
237   - FractionBox[
238   - RowBox[{"m", " ", "\[Omega]"}], "\[HBar]"]]}]], "]"}], "2"]}],
239   - SqrtBox[
240   - FractionBox[
241   - RowBox[{"m", " ", "\[Omega]"}], "\[HBar]"]]], ",",
242   - RowBox[{
243   - RowBox[{"m", " ", "\[Omega]", " ", "\[HBar]"}], ">", "0"}]}],
244   - "]"}]], "Output",
245   - CellChangeTimes->{3.5269471308811336`*^9}],
  249 + 3.526947028736291*^9}, 3.5269471120620575`*^9, {3.573515473928438*^9,
  250 + 3.5735154823569202`*^9}, {3.573515649132459*^9, 3.573515679371189*^9}, {
  251 + 3.573515783717157*^9, 3.5735157881294093`*^9}, {3.573515879136615*^9,
  252 + 3.573515894009465*^9}, {3.573515985071674*^9, 3.57351599620131*^9},
  253 + 3.573516187406247*^9, {3.573516238493169*^9, 3.5735163246260953`*^9}, {
  254 + 3.5735164962949142`*^9, 3.573516531215912*^9}, {3.573516566960956*^9,
  255 + 3.5735166271103964`*^9}, {3.5735166651235704`*^9,
  256 + 3.5735166949492764`*^9}, {3.573516736811671*^9, 3.5735167818242455`*^9}, {
  257 + 3.5735168410946355`*^9, 3.573516896348796*^9}, {3.5735169496338434`*^9,
  258 + 3.5735169664648066`*^9}, {3.5735169978396006`*^9,
  259 + 3.5735171326223097`*^9}, {3.573517163279063*^9, 3.5735171786899447`*^9}, {
  260 + 3.5735172522711535`*^9, 3.573517358277217*^9}, 3.573517453080639*^9}],
246 261
247 262 Cell[BoxData[
248 263 RowBox[{
  264 + RowBox[{
  265 + FractionBox["1", "2"], " ",
  266 + SuperscriptBox["\[ExponentialE]",
  267 + RowBox[{
  268 + RowBox[{"-", "x"}], " ", "\[Beta]"}]], " ", "m", " ",
  269 + SuperscriptBox["x", "2"], " ",
  270 + SqrtBox["\[Beta]"], " ",
  271 + SuperscriptBox["\[Omega]", "2"], " ",
  272 + RowBox[{"(",
  273 + RowBox[{