The Spherical Harmonics, Y_{ℓ,m}(θ, φ), are functions defined on the
sphere. They are used to describe the wave function of the electron in a
hydrogen atom, oscillations of a soap bubble, etc. The spherical harmonics
describe non-symmetric solutions to problems with spherical symmetry.

The Y_{ℓ,m}’s are complex valued. The radius of the figure is the
magnitude, and the color shows the phase, of Y_{ℓ,m}(θ, φ). These are
the numbers on the unit circle: 1 is red, i is purple, -1 is cyan (light blue),
and -i is yellow-green.

For each value of ℓ, there are 2ℓ + 1 linearly independent functions Y_{ℓ,m},
where m = -ℓ, -ℓ+1, ... , ℓ-1, ℓ. I have chosen a different set of 2ℓ + 1
functions, as you see below.

Y _{0,0} |
||||||

Re(Y _{1,1}) |
Y _{1,0} |
Y _{1,1} |
||||

Re(Y _{2,2}) |
Re(Y _{2,1}) |
Y _{2,0} |
Y _{2,1} |
Y _{2,2} |
||

Re(Y _{3,3}) |
Re(Y _{3,2}) |
Re(Y _{3,1}) |
Y _{3,0} |
Y _{3,1} |
Y _{3,2} |
Y _{3,3} |

The following figure is called “inside Y_{2,2}”. My son, Michael, made
this by holding down the “Page Up” key until the viewpoint gets *inside*
the surface. (He suggests that you set the figure rotating continuously, and
move the viewpoint a bit down before zooming in.)

**Oscillations of a Soap Bubble**

The volume of the bubble is constant, so Y_{0,0} is not used. The center
of mass of the bubble is constant, so Y_{1,m} is not used. The lowest
frequency oscillations of a soap bubble are ℓ = 2. The radius of the soap film
is
r = 1 + ε Y_{2,m}
(θ, φ). The oscillations with different m all have the same frequency. The shape
of the oscillations with
m = 1
and
m = 2
are the same up to a rotation, but the
m = 0
oscillation is different.

**Physics and Math notation**

WARNING: Spherical coordinates are different in physics and mathematics. The
symbols θ and φ are switched! The math notation makes r and θ the same in
cylincrical and spherical coordinates. DPGraph uses math notation.

arccos(z/r) = θ (physics) = φ (math)

arctan(y/x) + n π = φ (physics) = θ (math)

These graphs were produced by
DPGraph, which is a **
fast** program for viewing 3D objects.

e-mail: Jim.Swift@nau.edu