(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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[ 20521, 574]*) (*NotebookOutlinePosition[ 21161, 597]*) (* CellTagsIndexPosition[ 21117, 593]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ In this notebook you will integrate the Schroedinger equation \ numerically, along the lines of problem Q15.A1. First define some useful constants:\ \>", "Text"], Cell[BoxData[ \(\[HBar]\ = 197\ eV\ nm\ /c; \ m\ = \ 511000\ eV\ /\ c^2;\)], "Input"], Cell["Choose a potential:", "Text"], Cell[BoxData[ \(V[x_]\ := \ \((1/2)\)\ m\ \[Omega]^2\ x^2; \ \[Omega]\ = \ 1 eV/\[HBar];\)], "Input"], Cell["\<\ Given a range to look at, and a number of grid points to use, \ define the step size and the locations of the points:\ \>", "Text"], Cell[BoxData[{ \(\(n\ = \ 200;\)\), "\[IndentingNewLine]", \(\(width\ = \ 4\ nm;\)\), "\[IndentingNewLine]", \(xmin\ = \ \(-width\)/2; \ xmax\ = \ \(+width\)/2;\), "\[IndentingNewLine]", \(\(\[Epsilon]\ = \ width/n;\)\), "\[IndentingNewLine]", \(\(x[i_]\ = \ xmin\ + \ i*\[Epsilon];\)\)}], "Input"], Cell["\<\ Next pick an energy and define the mantra to integrate the \ Schroedinger equation. Note the use of recursive function definition where \ we store the results as we compute them. This speeds up the process \ considerably.\ \>", "Text"], Cell[BoxData[{ \(\(E0\ = \ \(( .48)\)\ eV;\)\), "\[IndentingNewLine]", \(\(Clear["\"];\)\), "\[IndentingNewLine]", \(\[Psi][0]\ = \ 0; \ \[Psi][1]\ = \ .01;\), "\[IndentingNewLine]", \(\[Psi][ i_]\ := \ \(\(\[Psi][ i]\)\(=\)\(\ \)\((what\ goes\ here?\(! \ check\ equation\ Q15 .28\))\)\(\[IndentingNewLine]\)\)\)}], \ "Input"], Cell["\<\ To make a list of points, the function Table[] will come in \ handy:\ \>", "Text"], Cell[BoxData[ \(\(points\ = \ Table[{x[i]/nm, \[Psi][i]}, {i, 0, n}];\)\)], "Input"], Cell["And to plot the points, it is as simple as:", "Text"], Cell[BoxData[ \(\(ListPlot[points];\)\)], "Input"], Cell[TextData[{ "Now try varying the energy until you get sensible solutions, i.e. ones \ where the value of \[Psi][xmax] is as close to zero as possible. Note that \ these allowed energies are close to the \"theoretical\" values, but differ \ slightly. Why? If you have time, play with the parameters ", StyleBox["n", FontWeight->"Bold"], " and ", StyleBox["width", FontWeight->"Bold"], " and see how the allowed energies change.\n\nLastly, try the same \ machinery on the following well. 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