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The two Physlets show a density plot of the Hydrogenic wavefunction and the solution to the radial equation. The word "density" refers to a method for plotting 3-D information on a two dimensional screen. Here it has nothing to do with the probability density in quantum mechanics. The radial solutions used here are the associated Laguerre polynomials scaled with a0 = 1.
Make multiple plots of the wavefunctions for n = 3. How is the behavior of the radial wavefunction for l = 0 different than for l = 1 or 2? Does the radial wavefunction depend on m? Try this for a few other values for the principal quantum number and see if your conclusion holds.
For n = 3, how many times does the radial wave function cross zero (change signs) for each possible value of l? Try this for a few other values for the principal quantum number and see if your conclusion holds.
For a given principal quantum number, there is a maximum value for l. The graph of the radial wave function in this case should have only one maximum value. Obtain a general formula relating the radius for this max value for all n.
Physlet problems authored by Dan Boye. Script by Wolfgang Christian.