Spherical Harmonics - Laplace's Spherical Harmonics

Laplace's Spherical Harmonics

Laplace's equation imposes that the divergence of the gradient of a scalar field f is zero. In spherical coordinates this is:

Consider the problem of finding solutions of the form ƒ(r,θ,φ) = R(r)Y(θ,φ). By separation of variables, two differential equations result by imposing Laplace's equation:

The second equation can be simplified under the assumption that Y has the form Y(θ,φ) = Θ(θ)Φ(φ). Applying separation of variables again to the second equation gives way to the pair of differential equations

for some number m. A priori, m is a complex constant, but because Φ must be a periodic function whose period evenly divides 2π, m is necessarily an integer and Φ is a linear combination of the complex exponentials e±imφ. The solution function Y(θ,φ) is regular at the poles of the sphere, where θ=0,π. Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ(ℓ+1) for some non-negative integer with ℓ ≥ |m|; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables t = cosθ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial . Finally, the equation for R has solutions of the form R(r) = Arℓ + Br−ℓ−1; requiring the solution to be regular throughout R3 forces B = 0.

Here the solution was assumed to have the special form Y(θ,φ) = Θ(θ)Φ(φ). For a given value of ℓ, there are 2ℓ+1 independent solutions of this form, one for each integer m with −ℓ ≤ m ≤ ℓ. These angular solutions are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:

which fulfill

Here is called a spherical harmonic function of degree ℓ and order m, is an associated Legendre polynomial, N is a normalization constant, and θ and φ represent colatitude and longitude, respectively. In particular, the colatitude θ, or polar angle, ranges from 0 at the North Pole to π at the South Pole, assuming the value of π/2 at the Equator, and the longitude φ, or azimuth, may assume all values with 0 ≤ φ < 2π. For a fixed integer ℓ, every solution Y(θ,φ) of the eigenvalue problem

is a linear combination of . In fact, for any such solution, rℓY(θ,φ) is the expression in spherical coordinates of a homogeneous polynomial that is harmonic (see below), and so counting dimensions shows that there are 2ℓ+1 linearly independent such polynomials.

The general solution to Laplace's equation in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor rℓ,

where the are constants and the factors are known as solid harmonics. Such an expansion is valid in the ball