Hydrogen wave functions – class activity October 31.
Now we have talked for a while about the radial wave functions and are starting to think
about the spherical harmonics. Open a web browser and go to www.vis.uni-
stuttgart.de/~kraus/LiveGraphics3D/java_script/SphericalHarmonics.html. This applet
will allow you to look at the spherical harmonic functions for values of l and ml up to 5.
Before you start, click “enlarge” once or twice to make a larger picture, and increase the
resolution in both the angles to about 30 or 40. Also, under “Spherical polar plot of”
choose “absolute value (colored by complex phase)”. Now choose l = 1, and ml = 0 and
plot it. You will now see a 3D object. For each value of θ and φ, the absolute value of the
spherical harmonic function for those coordinates is represented by the distance from the
origin (so a large value, like right on the z-axis, is represented by a long distance from the
origin to the top of the object, and a small value (like zero along the xy-plane for which z
is zero) is represented by a short distance from the axis.
1) Look at the formula for Y10(θ,φ) – does that help you understand the plot of the
funtion? Try to in particular pay attention to the angles for which the Y10 function should
have its maximum, minimum, and zero. Can you guess what the two different colors
represent?
2) Do the same for Y1,1(θ,φ) and Y1,-1(θ,φ) – look at the formula and plot the function
and try to make sense of what you see. Explain why there is no difference in the plots
with ml = +1 and ml = -1.
3) Next let us try to understand why the electron angular distributions would look like
this for the l = 1, ml = 0 cases, and the l = 1, ml = ±1 cases. Start by sketching the
(classical) motion that would give rise to an angular momentum vector that points
straight upwards, along the positive z-direction. Which motion would give rise to an
angular momentum vector pointing in the positive x-direction?
