The program is an implementation of the Noumerov method for solving differential equations. The Noumerov method is described in textbooks on numerical methods and its application to simple quantum mechanical problems has been discussed. The problems to which the the methodology is applied are discussed here.

The program exisits as a Java applet. That means the program resides on a distant computer, has been sent to your computer because you clicked on a link requesting it, and, finally, is run on your computer. It will run in a browser that can provide the required services (the Java Virtual Machine). If you obtain, immediately below, a panel with buttons then you have such a browser.

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Click on the potential of your choice. Be prepared for a short wait, however, even if the download is complete, as the program needs to be initialized in your machine. When using the program, keep in mind that y refers to the wave function. Your goal is to obtain energies that will yield well behaved wave functons. These may be recognized by their becoming asymptotic to the x axis as x goes to positive and negative infinity. Note that because you are using a numerical method on a computer that an initially chosen perfect solution will lose that perfection as the numerical methods do their work. Thus, even a "correct" choice for E will usually become tangent to the x axis for awhile and then diverge to infinity. Such, alas, are the results of limited precision in a computer.

Here are some hints to make working with the program easier. Some screen captures are also available here.

• You will have to enter one or more parameters for the potential. On each of the parameter dialogs there is an "Next" button. An erroneous value becomes highlighted in yellow and should be re-entered.
• The size of the area being plotted may be adjusted in the final dialog. Large limits to the plot allow for a convenient overall view. The "Graph" button draws the plot. The window may be maximized for viewing ease.
• The most negative (stable) acceptable energy solution will have no crossings of the x axis. The next more positive acceptable energy solution will have one crossing, followed by one with two, etc. For the symmetric polynomial cases the solutions of Even symmetry will cross the x axis an even (0,2,4,...) number of times whereas those of Odd symmetry cross an odd (1,3,...) number of times.
• The energy associated with a function is related to how many times it crosses the x axis. If your function is not reaching the x axis so as to cross it, then you should make the energy more positive. Conversely, if it crosses the x axis instead of becoming asymptotic then you should make the energy more negative.
• An assymetric potential, where the function should become asymptotic at both positive and negative x, is more difficult. An assymetric polynomial and the Morse potential are examples. For this type of potential you must obtain asymptotic behavior of the wave function at both ends. This is done by adjusting both the energy and the initial slope. If the wave function crosses the x axis for positive x but does not reach it for negative x or vice versa then you have to adjust the initial slope so that both ends of the function show the same behavior, both not reaching or both crossing. Having done that, then adjust the energy to obtain asymptotic behavior. Most likely, several cycles of adjustments will have to be made to both slope and energy.
• The step parameter is made smaller to obtain a better, but more slowly drawn plot. You may experiment with step size (as everything else). A good value seems to about 0.01.

Click here to return to the page describing the kinds of problems that may be solved.

For earlier discussions of this approach see:

• Blukis, U., and Howell, J.M., J. Chem. Ed., 60, 207 (1983).
• Kubach, C., J. Chem. Ed., 60, 212 (1983).
• Boleman, J.S., Amer. J. Phys., 40, 1511 (1972).

For the original work see:

• Noumerov, B., Publ. Central Astrophys. Obs. Russia, vol. 7, 1023, Moscow.

A textbook reference to the Noumerov method:

Acton, F.S., "Numerical Methods That (Usually) Work", Harper & Row, New York, 1970, pp. 130ff.