### Modern Physics Problem 10.2.1: Angular Wavefunctions

*Note: Right click on either Physlet to make a copy. **The mouse coordinates may be
observed by left-clicking within the graph.*

#### Description

The two Physlets show a density plot of the Hydrogenic wavefunction and the
solution to the angular, that is, polar, 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 polar solutions used here are the
unnormalized associated Legendre polynomials, P_{lm}(q,f).
Note that the x and z coordinates range from -1 to +1.

#### Question

Make multiple plots of the wavefunctions. On which quantum numbers does the
angular wavefunction depend? Be systematic. Change one number at a time.

#### Question

For any given values of *l* and *m*_{l}, does the angular
wavefunction change when *m*_{l} is changed to -*m*_{l}? Does the
total wavefunction change? Explain.

#### Question

Notice the dependence of the number of lobes on *l* and *m*_{l}.
Obtain a general formula for this dependence.

#### Question

For *l *= 1 and *m*_{l }= 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.

#### Credits

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