Complex Spherical Harmonics

l

0

1

2

3

4

m

0

Spherical harmonic functions arise for central force problems in quantum mechanics as the angular part of the Schrödinger equation in spherical polar coordinates. They are given by

m

Y

ℓ

(θ,ϕ)=

(2ℓ+l)

4π

(ℓ-|m|)!

(ℓ+|m|)!

m

P

ℓ

(cosθ)

mϕ

e

, where

m

P

ℓ

are associated Legendre polynomials and

l

and

m

are the orbital and magnetic quantum numbers, respectively. The allowed values of the quantum numbers, which follow from the boundary conditions of the problem, are

ℓ=0,1,2,...,∞,m=0,±1,±2,...,±ℓ

. The complex function

m

Y

ℓ

(θ,ϕ)

is shown on the left, where the shape is its modulus and the coloring corresponds to its argument, the range 0 to

2π

corresponding to colors from red to magenta. The center and right graphics show the corresponding real and imaginary parts.