WOLFRAM|DEMONSTRATIONS PROJECT

Complex Spherical Harmonics

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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.