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Use the Help Browser under the Help menu, too! \ \>", "Subsubtitle", FontColor->RGBColor[1, 0, 0]], Cell[TextData[{ StyleBox["Gaussian momentum (wavenumber) distribution as a function of k.", FontColor->RGBColor[0, 0, 1]], "\nHere, w is a width parameter. ", "The factor ", Cell[BoxData[ \(\@\(\[Pi]\ w\)\%\(-4\) = \(\((\[Pi]\ \[Omega])\)\(^\)\((\(-1\)/ 4)\)\(\ \)\)\)]], "is for normalization of ", Cell[BoxData[ \(\[Phi]\^2\)]], ". " }], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Phi] = \@\(\[Pi]\ w\)\%\(-4\)\ E^\((\(-\(k\^2\/\(2 w\)\)\))\)\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(-\(k\^2\/\(2\ w\)\)\)\/\(\[Pi]\^\(1/4\)\ \ w\^\(1/4\)\)\)], "Output"], Cell[TextData[{ "We can assume this distribution is time independent, and let the planewave \ components \[ExponentialE]^(I k x) carry the free-particle time dependence \ \[ExponentialE]^(-I \[Omega] t) we derived in lecture with \[Omega] = ", StyleBox["E", FontFamily->"Courier"], "/\[HBar] and ", Cell[BoxData[ RowBox[{"E", StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["=", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], FormBox[\(\(\[HBar]\^2\) k\^2/2 m\), "TraditionalForm"]}]]], " the free-particle (kinetic) energy. See below." }], "Subsubsection"], Cell[TextData[StyleBox["Remember\[LongDash] \n\tUse parentheses (...(...)) to \ input algebra.\n\tReserve square brackets for entering functions, say Sin[x] \ or f[x].\n\tReserve curly brackets {...} to enter a list of things.\n\n\ Together, these three conventions allow us to use spaces to denote \ multiplication, rather than *, and thus greatly reduce clutter in the Input \ and Output.", FontColor->RGBColor[0, 0, 1]]], "Subsubsection"], Cell["Check normalization:", "Subsubsection"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\ \ \(\[Phi]\^2\) \(\(\ \[DifferentialD]k\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(If[Re[w] > 0, \@\[Pi]\ \@w, \ Integrate[\[ExponentialE]\^\(-\(k\^2\/w\)\), {k, \(-\[Infinity]\), \ \[Infinity]}, Assumptions \[Rule] Re[w] \[LessEqual] 0]]\/\(\@\[Pi]\ \ \@w\)\)], "Output"] }, Open ]], Cell[TextData[{ "Declare ", StyleBox["w > 0", FontColor->RGBColor[0, 0, 1]], " to obtain convergent integrals" }], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SetOptions", "[", RowBox[{"Integrate", ",", RowBox[{"Assumptions", "\[Rule]", StyleBox[\(w > 0\), FontColor->RGBColor[0, 0, 1]]}]}], "]"}]], "Input"], Cell[BoxData[ \({Assumptions \[Rule] w > 0, GenerateConditions \[Rule] Automatic, PrincipalValue \[Rule] False}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\ \ \(\[Phi]\^2\) \(\(\ \[DifferentialD]k\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell["\<\ Compute mean, or expectation value, and rms deviation of the distribution\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \({\[LeftAngleBracket]k\[RightAngleBracket]\ = \ \[Integral]\_\(-\ \[Infinity]\)\%\[Infinity] k\ \(\[Phi]\^2\) \[DifferentialD]k, \[LeftAngleBracket]k\^2\ \[RightAngleBracket]\ = \ \[Integral]\_\(-\[Infinity]\)\%\[Infinity] k\^2\ \ \(\[Phi]\^2\) \[DifferentialD]k, \ \[Sigma]\_k = \ \@\(\(\[LeftAngleBracket]k\ \^2\[RightAngleBracket]\)\(-\)\(\[LeftAngleBracket]k\[RightAngleBracket]\^2\)\ \(\ \)\)}\)], "Input"], Cell[BoxData[ \({0, w\/2, \@w\/\@2}\)], "Output"] }, Open ]], Cell[TextData[{ "Let's give the distribution a +k0 boost. \nNote, the", StyleBox[" expectation value is changed to k0", FontColor->RGBColor[0, 0, 1]], ", while the rms deviation stays the same." }], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Phi] = \@\(\[Pi]\ w\)\%\(-4\)\ E^\((\(-\(\((k - k0)\)\^2\/\(2 w\)\)\))\)\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(-\(\((k - k0)\)\^2\/\(2\ w\)\)\)\/\(\[Pi]\^\(1/4\)\ \ w\^\(1/4\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "Add ", StyleBox["k0 > 0", FontColor->RGBColor[0, 0, 1]], " to the Integrate list for convergence" }], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SetOptions", "[", RowBox[{"Integrate", ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"{", StyleBox[\(w > 0, \ k0 > 0\), FontColor->RGBColor[0, 0, 1]], StyleBox["}", FontColor->RGBColor[0, 0, 1]]}]}]}], "]"}]], "Input"], Cell[BoxData[ \({Assumptions \[Rule] {w > 0, k0 > 0}, GenerateConditions \[Rule] Automatic, PrincipalValue \[Rule] False}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\ \(\[Phi]\^2\) \ \[DifferentialD]k\), ",", RowBox[{ StyleBox[\(\[LeftAngleBracket]k\[RightAngleBracket]\), FontColor->RGBColor[0, 0, 1]], " ", "=", " ", \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity] k\ \(\[Phi]\^2\) \[DifferentialD]k\)}], ",", \(\[LeftAngleBracket]k\^2\[RightAngleBracket]\ = \ \ \[Integral]\_\(-\[Infinity]\)\%\[Infinity] k\^2\ \(\[Phi]\^2\) \ \[DifferentialD]k\), ",", " ", \(\[Sigma]\_k = \ \@\(\(\[LeftAngleBracket]k\^2\ \[RightAngleBracket]\)\(-\)\(\[LeftAngleBracket]k\[RightAngleBracket]\^2\)\(\ \ \)\)\)}], "}"}], "//", StyleBox["PowerExpand", FontColor->RGBColor[0, 0, 1]]}], " ", "//", "Simplify", StyleBox[ RowBox[{" ", StyleBox[" ", FontColor->RGBColor[0, 0, 1]]}]], StyleBox[\( (*\ PowerExpand\ collapses\ powers, \ click\ on\ More\ below\ *) \), FontColor->RGBColor[0, 0, 1]]}]], "Input"], Cell[BoxData[ \({1, k0, k0\^2 + w\/2, \@w\/\@2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(?PowerExpand\), FontColor->RGBColor[0, 0, 1]]], "Input"], Cell[BoxData[ RowBox[{"\<\"PowerExpand[expr] expands all powers of products and powers.\ \"\>", " ", ButtonBox[ StyleBox["More\[Ellipsis]", "SR"], ButtonData:>"PowerExpand", Active->True, ButtonStyle->"RefGuideLink"]}]], "Print", CellTags->"Info3286175672-1083832"] }, Open ]], Cell[TextData[{ StyleBox["Coordinate-space ", FontColor->RGBColor[0, 0, 1]], StyleBox["initial", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" wavepacket ", FontColor->RGBColor[0, 0, 1]], "\n(no time dependence yet)" }], "Subsection"], Cell[TextData[{ "Form linear superposition of planewaves. Note, the result, ", StyleBox["obtained by an integration over k", FontColor->RGBColor[0, 0, 1]], ", is another gaussian but now a function of x boosted by the planewave \ factor E^(I k0 x). " }], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Psi]0", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{\(1\/\@\(2 \[Pi]\)\), " ", RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\), " ", RowBox[{"\[Phi]", " ", RowBox[{"Exp", "[", RowBox[{"I", " ", StyleBox["k", FontColor->RGBColor[0, 0, 1]], " ", "x"}], "]"}], StyleBox[\(\[DifferentialD]k\), FontColor->RGBColor[0, 0, 1]]}]}]}], " ", "//", "PowerExpand"}], " ", "//", "Simplify"}]}]], "Input"], Cell[BoxData[ \(\(\[ExponentialE]\^\(\[ImaginaryI]\ k0\ x - \(w\ x\^2\)\/2\)\ \ w\^\(1/4\)\)\/\[Pi]\^\(1/4\)\)], "Output"] }, Open ]], Cell[TextData[{ "\[Phi] is of course the Fourier transform of \[Psi]0, and \[Psi]0 is the \ inverse Fourier transform of \[Phi]. Functions related in this way are called \ Fourier transform pairs. \n\nLet's check that we can get back \[Phi]. Note \ the complex-conjugate PW factor ", StyleBox["Exp[-I k x] ", FontColor->RGBColor[0, 0, 1]], "and the integration over ", StyleBox["x", FontColor->RGBColor[0, 0, 1]], ". Here, ", StyleBox["\[Equal]", FontFamily->"Courier", FontColor->RGBColor[1, 0, 0]], " (double equal) is a logical equal, and checks that the LHS and RHS of the \ equation are form equivalent." }], "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Phi]", " ", StyleBox["\[Equal]", FontColor->RGBColor[1, 0, 0]], " ", RowBox[{\(1\/\@\(2 \[Pi]\)\), " ", RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\), RowBox[{" ", "\[Psi]0", StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox[\(Exp[\(-I\)\ k\ x]\), FontColor->RGBColor[0, 0, 1]], RowBox[{ StyleBox["\[DifferentialD]", FontColor->RGBColor[0, 0, 1]], StyleBox["x", FontColor->RGBColor[0, 0, 1]], " "}]}]}]}]}]], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "Because this wpkt is complex valued, we need to a rule for taking the \ complex conjugate to calculate the square of the distribution. Define complex \ conjugation rule using the asterisk box ", Cell[BoxData[ \(TraditionalForm\`\(\[Placeholder]\^*\)\)]], " or the 'raise-to-power' box ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Placeholder]\^\[Placeholder]\)\(\ \)\)\)]], "from the palette. ", "The symbol combination ", StyleBox["/.", FontColor->RGBColor[1, 0, 0]], " (slash dot) = 'replace', ", StyleBox["Complex", FontColor->RGBColor[0, 0, 1]], " is the built-in function for a + I b." }], "Subsubsection"], Cell[BoxData[ RowBox[{\(\[Psi]_\^*\), " ", ":=", " ", RowBox[{"\[Psi]", StyleBox["/.", FontColor->RGBColor[1, 0, 0]], RowBox[{ RowBox[{ StyleBox["Complex", FontColor->RGBColor[0, 0, 1]], "[", \(a_, b_\), "]"}], "\[Rule]", " ", \(a\ - \ I\ b\)}]}]}]], "Input"], Cell[TextData[{ "The symbol combination ", StyleBox["/.", FontFamily->"Courier", FontColor->RGBColor[1, 0, 0]], " (slash dot) ", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "means 'replace', and \[Psi]_, a_ and b_ are dummy variables.", StyleBox[" Complex ", FontColor->RGBColor[0, 0, 1]], "is the ", StyleBox["Mathematica", FontSlant->"Italic"], " internal form of complex-valued quantities. Note, all parameters must be \ real or written explicitly in terms of their real and imaginary parts for \ this to work, which is what we generally do in QM." }], "Subsubsection"], Cell["Check", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \({\(\((x + I\ y)\)\^*\), \(\[Psi]0\^*\) \[Psi]0}\ \ \ // Simplify\)], "Input"], Cell[BoxData[ \({x - \[ImaginaryI]\ y, \(\[ExponentialE]\^\(\(-w\)\ x\^2\)\ \@w\)\/\@\ \[Pi]}\)], "Output"] }, Open ]], Cell["\<\ Check norm, and calculate expectation value and rms deviation.\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\({\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\ \(\[Psi]0\^*\)\ \[Psi]0 \ \[DifferentialD]x, \[LeftAngleBracket]x\[RightAngleBracket]\ = \ \[Integral]\ \_\(-\[Infinity]\)\%\[Infinity] x\ \(\[Psi]0\^*\)\ \[Psi]0 \[DifferentialD]x, \ \[LeftAngleBracket]x\^2\[RightAngleBracket]\ = \ \[Integral]\_\(-\[Infinity]\ \)\%\[Infinity] x\^2\ \(\[Psi]0\^*\)\ \[Psi]0 \[DifferentialD]x, \ \ \[Sigma]\_x = \ \@\(\(\[LeftAngleBracket]x\^2\[RightAngleBracket]\)\(-\)\(\ \[LeftAngleBracket]x\[RightAngleBracket]\^2\)\(\ \)\)} // PowerExpand\)\ // Simplify\)], "Input"], Cell[BoxData[ \({1, 0, 1\/\(2\ w\), 1\/\(\@2\ \@w\)}\)], "Output"] }, Open ]], Cell["\<\ Gaussians are minimum uncertainty wavepackets. (Here \[HBar]=hbar from the palette but an ordinary h would work!)\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\[Sigma]\_x\ \[HBar]\ \[Sigma]\_k // PowerExpand\)\ // Simplify\)], "Input"], Cell[BoxData[ \(\[HBar]\/2\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Moving coordinate wavepacket", FontColor->RGBColor[0, 0, 1]]], "Subsection"], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox["Add", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], StyleBox["time", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], StyleBox["dependence", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], StyleBox["to", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontFamily->"Times New Roman", FontColor->RGBColor[0, 0, 1]], 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(ii) The initial width of the wavepacket is determined by w: when \ the coordinate wavepacket \[Psi] is fat, the momentum wavepacket \[Phi] is \ thin, and vice versus. This behavior is typical of Fourier transforms pairs \ and is the key to the Heisenberg Uncertainty Principle. (iii) The spreading \ with time is a natural consequence of the quantum wave equation: even if k0 = \ 0, the wavepacket still moves with simple spreading." }], "Subsubsection", FontColor->RGBColor[0, 0, 1]], Cell["\<\ Check the wpkt remains normalized independent of t. 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