(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 701036, 18460]*) (*NotebookOutlinePosition[ 701988, 18489]*) (* CellTagsIndexPosition[ 701944, 18485]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Set-Up", "Subsection"], Cell["\<\ Begin with the constants, in reduced variables to make things \ simple. Also, set the barrier height to 1 and offset V by 1e-3 for \ convenience. \ \>", "Text"], Cell[BoxData[{ \(hbar := 1\), "\[IndentingNewLine]", \(me := 1\), "\[IndentingNewLine]", \(V0 := 1\), "\[IndentingNewLine]", \(En =. \)}], "Input"], Cell["Enter the values for the wavenumber, following CTDL pg. 84", "Text"], Cell[BoxData[{ \(k1 =. \), "\[IndentingNewLine]", \(k2 =. \)}], "Input"], Cell[BoxData[{ \(K0 := \@\(\(2*me*V0\)\/\((hbar^2)\)\)\), "\[IndentingNewLine]", \(k1[k_] := k\), "\[IndentingNewLine]", \(k2[k_] := \@\(k\^2 - K0\^2\)\)}], "Input"], Cell["\<\ The coefficients for reflection have already been calculated for \ this problem - set them up as recommended in CTDL. Notice that this makes A1 \ and A2 functions of k.\ \>", "Text"], Cell[BoxData[{ \(A1 =. \), "\[IndentingNewLine]", \(A2 =. \)}], "Input"], Cell[BoxData[{ \(A1[k_] := \((k1[k] - k2[k])\)\/\((k1[k] + k2[k])\)\), "\ \[IndentingNewLine]", \(A2[k_] := \(2*k1[k]\)\/\((k1[k] + k2[k])\)\)}], "Input"], Cell["\<\ Define \[CapitalDelta]k, from the problem set. From this, construct \ g(k). Also throw in \[Omega](k) - this will be convenient later on.\ \>", \ "Text"], Cell[BoxData[{ \(deltak := \@\(0.1*V0\)\), "\[IndentingNewLine]", \(a := \@2/deltak\), "\[IndentingNewLine]", \(g[k_] := Exp[\(-\((\((a^2)\)/ 4)\)\)*\((\((k - k0)\)^2)\)]\), "\[IndentingNewLine]", \(\[Omega][k_] := \(hbar*\((k^2)\)\)\/\(2*me\)\)}], "Input"], Cell["\<\ Define the last parameters, k0, x0, and the location of the step.\ \ \>", "Text"], Cell[BoxData[{ \(k0 := \@2.1\), "\[IndentingNewLine]", \(x0 := \(-10\)\), "\[IndentingNewLine]", \(steploc := 0\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Wave functions", "Subsection"], Cell[TextData[{ "Now: the wavefunctions. I separated the incident wave from the reflected \ and transmitted waves (maybe it's inappropriate to call it the 'initial' \ wave, but that's what I started with). ", StyleBox["Mathematica", FontSlant->"Italic"], " starts making some inconvenient assumptions when the 'Conj' function is \ called, so it is usually more convenient to flip the signs in front of the \ i's manually and name a new function. Let me know if you have an easier way \ to do this." }], "Text"], Cell[BoxData[{ \(\[CapitalPsi]initial[x_, t_, k_] := 1\/\@\(2*\[Pi]\)*g[k]* Exp[\[ImaginaryI]*\((\((k*x)\) - \[Omega][k]*t)\)]* Exp[\(-\[ImaginaryI]\)*\((k - k0)\)*x0]\), "\[IndentingNewLine]", \(\[CapitalPsi]initialconj[x_, t_, k_] := 1\/\@\(2*\[Pi]\)*g[k]* Exp[\(-\[ImaginaryI]\)*\((\((k*x - \[Omega][k]*t)\))\)]* Exp[\[ImaginaryI]*\((k - k0)\)*x0]\), "\[IndentingNewLine]", \(\[CapitalPsi]reflected[x_, t_, k_] := 1\/\@\(2*\[Pi]\)*g[k]*A1[k]* Exp[\(-\[ImaginaryI]\)*\((\((k*x)\) + \[Omega][k]*t)\)]* Exp[\(-\[ImaginaryI]\)*\((k - k0)\)*x0]\), "\[IndentingNewLine]", \(\[CapitalPsi]reflectedconj[x_, t_, k_] := 1\/\@\(2*\[Pi]\)*g[k]*A1[k]* Exp[\[ImaginaryI]*\((\((k*x)\) + \[Omega][k]*t)\)]* Exp[\[ImaginaryI]*\((k - k0)\)*x0]\), "\[IndentingNewLine]", \(\[CapitalPsi]transmitted[x_, t_, k_] := 1\/\@\(2*\[Pi]\)*g[k]*A2[k]* Exp[\[ImaginaryI]*\((\((\@\(k\^2 - K0\^2\)*x)\) - \[Omega][k]*t)\)]* Exp[\(-\[ImaginaryI]\)*\((k - k0)\)*x0]\), "\[IndentingNewLine]", \(\[CapitalPsi]transmittedconj[x_, t_, k_] := 1\/\@\(2*\[Pi]\)*g[k]*A2[k]* Exp[\(-\[ImaginaryI]\)*\((\((\@\(k\^2 - K0\^2\)*x)\) - \[Omega][k]* t)\)]*Exp[\[ImaginaryI]*\((k - k0)\)*x0]\)}], "Input"], Cell["\<\ I ended up summing k over nine locations because this made for \ better pictures. Again, it looks cluttered due to manual complex \ conjugate-ing. The function names are long, but I wanted to make them \ clear.\ \>", "Text"], Cell[BoxData[{ \(\[CapitalPsi]initial[x_, t_] := UnitStep[\(-\((x)\)\)]*\((\[CapitalPsi]initial[x, t, k0]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 + deltak/2]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 - deltak/2]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 + deltak]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 - deltak]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 + 3*deltak/2]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 - 3*deltak/2]\[IndentingNewLine] + \[CapitalPsi]initial[ x, t, k0 + 2*deltak]\[IndentingNewLine] + \[CapitalPsi]initial[ x, t, k0 - 2*deltak])\)\), "\[IndentingNewLine]", \(\[CapitalPsi]initialconj[x_, t_] := UnitStep[\(-\((x)\)\)]*\((\[CapitalPsi]initialconj[x, t, k0]\[IndentingNewLine] + \[CapitalPsi]initialconj[x, t, k0 + deltak/2]\[IndentingNewLine] + \[CapitalPsi]initialconj[x, t, k0 - deltak/2]\[IndentingNewLine] + \[CapitalPsi]initialconj[ x, t, k0 + deltak]\[IndentingNewLine] + \[CapitalPsi]initialconj[x, t, k0 - deltak]\[IndentingNewLine] + \[CapitalPsi]initialconj[x, t, k0 + 3* deltak/2]\[IndentingNewLine] + \[CapitalPsi]initialconj[x, t, k0 - 3* deltak/2]\[IndentingNewLine] + \[CapitalPsi]initialconj[x, t, k0 + 2*deltak]\[IndentingNewLine] + \[CapitalPsi]initialconj[ x, t, k0 - 2*deltak])\)\), "\[IndentingNewLine]", \(\[CapitalPsi]reflected[x_, t_] := UnitStep[\(-\((x)\)\)]*\((\[CapitalPsi]reflected[x, t, k0]\[IndentingNewLine] + \[CapitalPsi]reflected[x, t, k0 + deltak/2]\[IndentingNewLine] + \[CapitalPsi]reflected[x, t, k0 - deltak/2]\[IndentingNewLine] + \[CapitalPsi]reflected[ x, t, k0 + deltak]\[IndentingNewLine] + \[CapitalPsi]reflected[ x, t, k0 - deltak]\[IndentingNewLine] + \[CapitalPsi]reflected[ x, t, k0 + 3*deltak/2]\[IndentingNewLine] + \[CapitalPsi]reflected[x, t, k0 - 3*deltak/2]\[IndentingNewLine] + \[CapitalPsi]reflected[x, t, k0 + 2*deltak]\[IndentingNewLine] + \[CapitalPsi]reflected[ x, t, k0 - 2*deltak])\)\), "\[IndentingNewLine]", \(\[CapitalPsi]reflectedconj[x_, t_] := UnitStep[\(-\((x)\)\)]*\((\[CapitalPsi]reflectedconj[x, t, k0]\[IndentingNewLine] + \[CapitalPsi]reflectedconj[x, t, k0 + deltak/2]\[IndentingNewLine] + \[CapitalPsi]reflectedconj[ x, t, k0 - deltak/2]\[IndentingNewLine] + \[CapitalPsi]reflectedconj[x, t, k0 + deltak]\[IndentingNewLine] + \[CapitalPsi]reflectedconj[ x, t, k0 - deltak]\[IndentingNewLine] + \[CapitalPsi]reflectedconj[x, t, k0 + 3*deltak/ 2]\[IndentingNewLine] + \[CapitalPsi]reflectedconj[x, t, k0 - 3*deltak/ 2]\[IndentingNewLine] + \[CapitalPsi]reflectedconj[x, t, k0 + 2*deltak]\[IndentingNewLine] + \[CapitalPsi]reflectedconj[ x, t, k0 - 2*deltak])\)\), "\[IndentingNewLine]", \(\[CapitalPsi]transmitted[x_, t_] := UnitStep[\((x)\)]*\[IndentingNewLine]\((\[CapitalPsi]transmitted[x, t, k0] + \[IndentingNewLine]\[CapitalPsi]transmitted[x, t, k0 + deltak/2] + \[IndentingNewLine]\[CapitalPsi]transmitted[x, t, k0 - deltak/2] + \[IndentingNewLine]\[CapitalPsi]transmitted[ x, t, k0 + deltak] + \[IndentingNewLine]\[CapitalPsi]transmitted[x, t, k0 - deltak] + \[IndentingNewLine]\[CapitalPsi]transmitted[x, t, k0 + 3* deltak/2] + \[IndentingNewLine]\[CapitalPsi]transmitted[x, t, k0 - 3* deltak/2] + \[IndentingNewLine]\[CapitalPsi]transmitted[x, t, k0 + 2*deltak] + \[IndentingNewLine]\[CapitalPsi]transmitted[ x, t, k0 - 2*deltak])\)\), "\[IndentingNewLine]", \(\[CapitalPsi]transmittedconj[x_, t_] := UnitStep[\((x)\)]*\((\[CapitalPsi]transmittedconj[x, t, k0]\[IndentingNewLine] + \[CapitalPsi]transmittedconj[x, t, k0 + deltak/ 2]\[IndentingNewLine] + \[CapitalPsi]transmittedconj[x, t, k0 - deltak/ 2]\[IndentingNewLine] + \[CapitalPsi]transmittedconj[x, t, k0 + deltak]\[IndentingNewLine] + \[CapitalPsi]transmittedconj[ x, t, k0 - deltak]\[IndentingNewLine] + \[CapitalPsi]transmittedconj[x, t, k0 + 3*deltak/ 2]\[IndentingNewLine] + \[CapitalPsi]transmittedconj[x, t, k0 - 3* deltak/2]\[IndentingNewLine] + \[CapitalPsi]transmittedconj[ x, t, k0 + 2*deltak]\[IndentingNewLine] + \[CapitalPsi]transmittedconj[ x, t, k0 - 2*deltak])\)\)}], "Input"], Cell["\<\ Test to see how the transmission and reflection coefficients (or \ their precursors) look.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(A1[k0]\), "\[IndentingNewLine]", \(A2[k0]\)}], "Input"], Cell[BoxData[ \(0.868225531212423`\)], "Output"], Cell[BoxData[ \(1.868225531212423`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Movies", "Subsection"], Cell["A movie of the incident wave and its reflection.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(For[time = 0, time < 15, \(time++\), \ \[IndentingNewLine]Plot[\[IndentingNewLine]Re[\((\[CapitalPsi]initial[x, time] + \[CapitalPsi]reflected[x, time])\)*\((\[CapitalPsi]initialconj[x, time] + \[CapitalPsi]reflectedconj[x, time])\)], \[IndentingNewLine]{x, \(-20\), 0}, \[IndentingNewLine]PlotRange \[Rule] {0, 4}]]\)], "Input"], Cell[CellGroupData[{ 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Below is a movie of that approximation vs. the incedent \ wave in the absence of a barrier. Notice the periodicity in the transmitted \ wave because we approximated the integration with a sum.\ \>", "Text"], Cell[BoxData[{ \(\[CapitalPsi]initialnostep[x_, t_] := \((\[CapitalPsi]initial[x, t, k0]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 + deltak/2]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 - deltak/2]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 + deltak]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 - deltak]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 + 3*deltak/2]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 - 3*deltak/2]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 + 2*deltak]\[IndentingNewLine] + \[CapitalPsi]initial[x, t, k0 - 2*deltak])\)\), "\[IndentingNewLine]", \(\[CapitalPsi]initialnostepconj[x_, t_] := \((\[CapitalPsi]initialconj[x, t, k0]\[IndentingNewLine] + \[CapitalPsi]initialconj[x, t, k0 + deltak/2]\[IndentingNewLine] + \[CapitalPsi]initialconj[x, t, k0 - deltak/2]\[IndentingNewLine] + \[CapitalPsi]initialconj[ x, t, k0 + deltak]\[IndentingNewLine] + \[CapitalPsi]initialconj[ x, t, k0 - deltak]\[IndentingNewLine] + \[CapitalPsi]initialconj[ x, t, k0 + 3*deltak/2]\[IndentingNewLine] + \[CapitalPsi]initialconj[x, t, k0 - 3*deltak/2]\[IndentingNewLine] + \[CapitalPsi]initialconj[x, t, k0 + 2*deltak]\[IndentingNewLine] + \[CapitalPsi]initialconj[ x, t, k0 - 2*deltak])\)\)}], "Input"], Cell["\<\ The approximate width of the transmitted wave, and its reciporical: \ the coefficient of x in the wave equations.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(deltatransmit = \@\(k0\^2 - K0\^2\)/k0\), "\[IndentingNewLine]", \(1/deltatransmit\)}], "Input"], Cell[BoxData[ \(0.21821789023599245`\)], "Output"], Cell[BoxData[ \(4.582575694955839`\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(For[time = 10, time < 18, \(time++\), \[IndentingNewLine]Plot[\[IndentingNewLine]{Re[ A2[k0]^2*\[CapitalPsi]initialnostep[x*4.5, time]*\[CapitalPsi]initialnostepconj[x*4.5, time]], 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