Note: Right click on any applet to make a copy of the image.  Choosing the Show Multiple option and Resetting allows up to 4 plots to be viewed simultaneously.  The mouse coordinates may be observed by left-clicking within the graph.

Polar Wave Functions

The polar solutions graphed here are the unnormalized associated Legendre polynomials, Plm(q,f).  Refer to the convention accepted for spherical coordinates.  A positive angle q is defined to be the angle down from the z-axis toward the positive x-axis.  The length of a vector from the origin to the wave function is the magnitude of the wave function at that angle.


  1. For any given values of l and ml, observe that the plot does not change when ml is changed to -ml.  Explain.

  2. Notice the dependence of the number of lobes on l and ml.  Obtain a general formula for this dependence.

  3. For l = 1 and ml = 0, determine the angles for which the wave function is a maximum and a minimum.  Explain your results in terms of the formula for the wave function for this state.  Also do this for l = 2 and ml = 0 and l = 2 and ml = 1.