Multipole moments using spherical harmonics

[poophead]

Hello,

My question is fairly simple. My instructor solved in class today Laplace's equation in spherical coordinates which resulted in spherical harmonics.

I have not taken any quantum mechanics yet so this is my first exposure to spherical harmonics. What do the "l" and "m" terms in the expressions correspond to exactly in physical reality?

I'm under the impression that l = 0--> monopole, l = 1 --> dipole, etc. But what are the "m" terms for? And how exactly do I use these crazy formulas to find the multipole moments for arbitrary charge distributions?

 

[Tide]

You're basically trying to find a basis set to describe functions in 3-dimensions. Recall back when you described a 2-D function with a simple Fourier transform. There was a single set of "quantum numbers" to represent those functions. In 3-D you have a principle quantum number (n <->corresponding to the radial part of your fields). L and m relate to the dependence of the function on polar angle and azimuthal angle.

Multipole fields refer to the component of the field varying as different powers of the radial coordinate (monopole <-> inverse square, dipole <-> inverse cube, quadrupole <-> inverse fourth power etc.)