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Wikipedia / Vector spherical harmonics

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

Definition

Several conventions have been used to define the VSH. We follow that of Barrera et al.. Given a scalar spherical harmonic , we define three VSH:

being the unit vector along the radial direction in spherical coordinates. The radial factors are included to guarantee that the dimensions of the VSH are the same as those of the ordinary spherical harmonics and that the VSH do not depend on the radial spherical coordinate.

The interest of these new vector fields is to separate the radial dependence from the angular one when using spherical coordinates, so that a vector field admits a multipole expansion

).

In physics

are defined as spin eigenfunctions of the angular momentum operators is the total angular momentum.using the Clebsch-Gordan coefficients.

Because vector bosons such as the photon are spin-one, the vector spherical harmonics are commonly used in physics to describe vector and pseudovector) case of the spin spherical harmonics.

To derive these relations, one begins with the plane-wave expansion for plane waves with vector polarization.

Main properties

Symmetry

Like the scalar spherical harmonics, the VSH satisfy

which cuts the number of independent functions roughly in half. The star indicates complex conjugation.

Orthogonality

The VSH are orthogonal:

They are also orthogonal in Hilbert space:

,

Vector multipole moments

The orthogonality relations allow one to compute the spherical multipole moments of a vector field as

The gradient of a scalar field

Given the multipole expansion of a scalar field

we can express its gradient in terms of the VSH as

Divergence

For any multipole field we have

By superposition we obtain the divergence of any vector field:

We see that the component on is always solenoidal.

Curl

For any multipole field we have

By superposition we obtain the curl of any vector field:

Laplacian

The action of the Laplace operator separates as follows:

and

Also note that this action becomes symmetric, for properly normalized VSH.

Examples

First vector spherical harmonics

Expressions for negative values of are obtained by applying the symmetry relations.

Applications

Electrodynamics

The VSH are especially useful in the study of multipole radiation fields. For instance, a magnetic multipole is due to an oscillating current with angular frequency and complex amplitude

and the corresponding electric and magnetic fields, can be written as

Substituting into Maxwell equations, Gauss's law is automatically satisfied

while Faraday's law decouples as

Gauss' law for the magnetic field implies

and Ampère–Maxwell's equation gives

In this way, the partial differential equations have been transformed into a set of ordinary differential equations.

Alternative definition

In many applications, vector spherical harmonics are defined as fundamental set of the solutions of vector Helmholtz equation in spherical coordinates.

are the associated Legendre polynomials are any of the spherical Bessel functions.

Vector spherical harmonics are defined as:

, but complex functions can be introduced in the same way.

electric and magnetic harmonics vectors are equal in value and perpendicular to each other.

Orthogonality

The solutions of the Helmholtz vector equation obey the following orthogonality relations:

All other integrals over the angles between different functions or functions with different indices are equal to zero.

Rotation and inversion

Under rotation, vector spherical harmonics are transformed through each other in the same way as the corresponding scalar spherical functions, which are generating for a specific type of vector harmonics. For example, if the generating functions are the usual spherical harmonics, then the vector harmonics will also be transformed through the Wigner D-matrices The behavior under rotations is the same for electrical, magnetic and longitudinal harmonics.

Fluid dynamics

In the calculation of the Stokes' law for the drag that a viscous fluid exerts on a small spherical particle, the velocity distribution obeys Navier–Stokes equations neglecting inertia, i.e.,

with the boundary conditions

where U is the relative velocity of the particle to the fluid far from the particle. In spherical coordinates this velocity at infinity can be written as

The last expression suggests an expansion in spherical harmonics for the liquid velocity and the pressure

Substitution in the Navier–Stokes equations produces a set of ordinary differential equations for the coefficients.

Integral relations

Here the following definitions are used:

are spherical Bessel functions, with help of plane wave expansion one can obtain the following integral relations:

are spherical Hankel functions, one should use the different formulae. For vector spherical harmonics the following relations are obtained:

means, that spherical Hankel functions are used.

See also

References