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 z-component 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 px, py, dxz, dxy, etc.

Exercise 2. Try the following to get some physical insight from these wave functions: Keeping m at 0 (no angular momentum around the z-axis) 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 z-axis to the negative z-axis 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, Ycos. As you increase m from 0 to 6, note how the number of oscillations from pole to pole (the z-axis) 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 z-axis, and this corresponds to a large value of m (the z-component of the angular momemtum).