Particle on a Sphere  Spherical Harmonics
This applet displays the wave
functions associated with a particle confined to
the surface of a sphere. These wave functions are
also the solutions for a rigidly rotating
diatomic molecule. They are also the angular
parts of the hydrogen atom wave functions!
Exercise 1. Change the values of l (the angular momentum
quantum number) and m (the quantum number for
the zcomponent of the angular momentum, also
called the magnetic quantum number). Rotate
the sphere by clicking and dragging. The
value of the wave function is illustrated by
color: positive real values are red, negative
real values are blue, positive imaginary
values are yellow, and negative imaginary
values are green. Try looking at the cosine
and sine forms of the wave functions: these
correspond to the angular parts of the atomic
orbitals p_{x}, p_{y}, d_{xz},
d_{xy}, etc.
Exercise 2. Try
the following to get some physical insight
from these wave functions: Keeping m at 0 (no
angular momentum around the zaxis) change l
from 0 to 1 to 2 and so on. Note how the wave
functions increase the number of oscillations
from pole to pole as l increases. This
corresponds to the particle moving from pole
to pole, that is, from the positive zaxis to
the negative zaxis and then back to the
positive. As l increases the motion gets
faster (more angular momentum). But the wave
function rings indicate that we don't know
anything about the motion relative to the x
and y axes.
Exercise 3. Now,
select an l value of 6. Then move from m = 0
to m = 6. Note how, for m > 0, there are
both real and imaginary parts of the wave
function. The real parts (red and blue) are
"out of phase" with the imaginary
parts (green and yellow). That is, the real
parts are large in magnitude where the
imaginary parts are small and vice versa. To
simplify things, select the cosine version of
the wave functions, Y_{cos}. As you
increase m from 0 to 6, note how the number
of oscillations from pole to pole (the
zaxis) decreases while the oscillations
while going around the equator increases.
This means that as m is increased the
particle's motion moves closer to orbiting
the equator. This type of motion has all the
angular momentum about the zaxis, and this
corresponds to a large value of m (the
zcomponent of the angular momemtum).
