Quantum Mechanical Wavefunctions

<BODY LANG="EN"> <HR><center> <table border=1> <td><img height=36 width=36 alt="" src="/UCESIcons/tools/previous_fade.gif"></td> <td><A NAME="tex2html2" HREF="Lab2.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html4" HREF="node1.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-Lab2.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="/hamlet-bin/qhandler?exit" target="results"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="/UCES/hamlet/"><img height=36 width=36 alt="HAMLET" src="/UCESIcons/tools/hamlet.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="/UCES/hamlet/usage.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </table> </center> <HR> <P> <H1>Quantum Mechanical Wavefunctions</H1> <P> (references: Atkins ch. 13.3, 13.4, 14.1, 14.2, 18.3 and 24.2; Castellan, ch 20, 21; Eggers et. al. ch 1 and 2 ) <P> In this lab you will do calculations of quantum mechanical wavefunctions by numerically solving the time independent Schrödinger equation. This lab illustrates how a simple finite difference scheme (and, optionally, the more accurate Noumerov method), can be used to calculate wavefunctions for an atom colliding with a surface of a solid, which is first represented simply as a hard wall, but then more realistically with a Morse interaction potential function. Examination of the results of such calculations illustrate the relationship of classical and quantum mechanics through the use of the semi-classical local deBroglie wavelength. The same algorithm is then used to generate wavefunctions for the bound states (in addition to the scattering states above) for the Morse potential but now a search for the allowed values of the energy needs to be carried out. This calculation will actually be carried out in the context of finding vibrational bound states of a diatomic molecule, where the Morse potential is again a good, approximate representation of the interaction potential. You will then choose one of two more sophisticated examples for further explorations. <P> You can use a numerically more stable method for calculating the solution of the Schrödinger equation for the bound states of the Morse potential and implement an algorithm that finds the allowed energy values automatically; or: you may examine solutions of the Schrödinger equation for a particle colliding with a barrier, illustrating `tunneling' of the quantum particle through the barrier. <P> Mathematica Notes: This Lab Unit will require somewhat more independence than the first two labs. After becoming familiar with an initial code, you will then modify the code to include a new form of the potential energy. Use earlier work from your Mathematica Notebooks, as well as the code included here, to aid in development of your codes. Do note that if you open "extra" Notebook windows, that the system slows, so close them when not needed. Also be sure to simply "open" these older notebooks: do not launch multiple Mathematica jobs. It is good practice to start each file with a <B> Clear[...]</B> command to clear all the variables used in subsequent calculations. This prevents problems that can arise if the variables already have assigned values, and/or can prevent "infinite" loops. Remember, if you restart a Notebook, especially in mid stream, earlier commands, including the loading of special libraries, must be explicitly re-executed! To avoid losing work, <B>save</B> your work at regular intervals, and break large notebooks into smaller notebooks. <P> System Note: The Mathematica Notebooks are, we regret to find, sometimes unstable when restarted. An empirical cure: go into the UNIX (Console) window, and enter "ls" following the UNIX prompt. This will give a listing of current files in your directory. If you have saved a Mathematica Notebook as "Mynotes.ma" note that there will also be (automatically created and saved) another file called "Mynotes.mb". This file, which is supposed to facilitate "re-launching" your notebook, is the one that is causing trouble! Remove this startup file by entering "rm Mynotes.mb". The system may prompt you for a yes of no (y or n) before removing (rm) the file. Type another "ls" to see that the deleted file is actually gone. Good luck, and we are very sorry to have found this problem. Mathematica Inc. is hearing from us. <P> <HR><center> <table border=1> <td><img height=36 width=36 alt="" src="/UCESIcons/tools/previous_fade.gif"></td> <td><A NAME="tex2html2" HREF="Lab2.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html4" HREF="node1.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-Lab2.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="/hamlet-bin/qhandler?exit" target="results"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="/UCES/hamlet/"><img height=36 width=36 alt="HAMLET" src="/UCESIcons/tools/hamlet.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="/UCES/hamlet/usage.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </table> </center> <HR><P><ADDRESS> <I>Hannes Jonsson<BR> Modified by Thomas L. Marchioro II<BR> and the <A NAME="tex2html1" HREF="/UCES/">Undergraduate Computational Engineering and Science project</A> </I> </ADDRESS> </BODY>