Modern Physics Problem 10.2.1: Radial Wavefunctions

n =

 l =

m =

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Description

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.

Question

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.

Question

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.

Question

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.

Credits

Physlet problems authored by Dan Boye.  Script by Wolfgang Christian.