Associated Legendre Polynomials - Applications in Physics: Spherical Harmonics

Applications in Physics: Spherical Harmonics

In many occasions in physics, associated Legendre polynomials in terms of angles occur where spherical symmetry is involved. The colatitude angle in spherical coordinates is the angle used above. The longitude angle, appears in a multiplying factor. Together, they make a set of functions called spherical harmonics. These functions express the symmetry of the two-sphere under the action of the Lie group SO(3).

What makes these functions useful is that they are central to the solution of the equation on the surface of a sphere. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is

When the partial differential equation

is solved by the method of separation of variables, one gets a φ-dependent part or for integer m≥0, and an equation for the θ-dependent part

for which the solutions are with and .

Therefore, the equation

has nonsingular separated solutions only when, and those solutions are proportional to

and

For each choice of ℓ, there are 2ℓ + 1 functions for the various values of m and choices of sine and cosine. They are all orthogonal in both ℓ and m when integrated over the surface of the sphere.

The solutions are usually written in terms of complex exponentials:

The functions are the spherical harmonics, and the quantity in the square root is a normalizing factor. Recalling the relation between the associated Legendre functions of positive and negative m, it is easily shown that the spherical harmonics satisfy the identity

The spherical harmonic functions form a complete orthonormal set of functions in the sense of Fourier series. It should be noted that workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here (see spherical harmonics).

When a 3-dimensional spherically symmetric partial differential equation is solved by the method of separation of variables in spherical coordinates, the part that remains after removal of the radial part is typically of the form

and hence the solutions are spherical harmonics.