Spherical harmonics

The spherical harmonics are a complete and orthonormal set of eigenfunctions of the angular part of the Laplace operator. This angle component is shown when the Laplacian is written in spherical coordinates. The eigenvalue equation is:

The eigenfunctions are the spherical harmonics, there are normalization factors and the associated Legendre polynomials ( see details below):

Especially in theoretical physics, the spherical harmonics are of great importance for the solution of partial differential equations. They occur, for example in the calculation of atomic orbitals, as the descriptive time-independent Schrödinger equation contains the Laplace operator and can best solve the problem in spherical coordinates. The boundary value problems arising in electrostatics can be elegantly solved by the expansion in spherical harmonics. In geophysics and geodesy, the spherical harmonics are used in the approximation of the geoid and the magnetic field.

Conjunction with the Laplace operator

The angular part of the Laplace operator becomes apparent when it is written in spherical coordinates:

The right, bracketed part is referred to as angle component. He is directly proportional to the square of the angular momentum operator.

The Laplace's equation in spherical coordinates

Has in addition to the trivial solution, a variety of solutions with many technical applications.

To solve the following product approach is used, depends only on the radius and only polar and azimuthal angle:

This is used:

Multiplication and division supplies by:

This equation can only be fulfilled if independently radius and angle can be varied in two summands. Both summands must therefore assume the same constant value that is to be selected ( this setting later turns out to be useful ):

By this method, which is called separation of variables, so was the original problem, namely the solution of the Laplace equation ( partial differential equation with three independent variables) on the simpler problem of solving an ordinary differential equation (radial equation)

And a partial differential equation with independent variables, two ( angle-dependent equation), which is being filled with the spherical harmonics is reduced.

Now it can be shown due to the orthogonality and completeness of the spherical harmonics that every square integrable function from these special functions can be composed as the sum of:

Due to the linearity of the Laplace operator can thus be calculated by adding the solutions of the radial equation by the spherical harmonics, construct any number of solutions of the Laplace equation. This automatically results in a representation of the solution space of the Laplace equation.

The spherical harmonics have been treated particularly of Legendre ( spherical functions of the first kind ), Laplace ( spherical functions of the second kind ) and Carl Gottfried Neumann ( spherical functions with several variables ).

Solution of the eigenvalue equation

The eigenvalue equation

Is separated with the following product approach:

Rearranging yields:

To both side separately to vary from each other, both sides must assume the same constant value. This separation constant is declared elected. This results in two ordinary differential equations, the polar equation

And the Azimutalgleichung.

The Azimutalgleichung is achieved by, the regions are on the ball surface due to the additional condition the uniqueness of integers. With one obtains the normalized solution of the Azimutalgleichung:

The polar equation can be solved by a power series approach. The solutions are only finite -valued and continuous, if

Then, the solutions are the associated Legendre polynomials and obtained the normalized solution of the polar equation:

The total solution of the angular part is the product of the two solutions obtained, namely the spherical harmonics.

Representation

The representation of the spherical harmonics is obtained as the solution of the above eigenvalue equation. The concrete calculation gives:

Here are

The associated Legendre polynomials, and

Are normalization factors. Sometimes the calculation is about:

With

Advantageous (), as -fold deriving omitted.

Another definition goes beyond homogeneous harmonic polynomials. These are uniquely determined by its value on the sphere. Each homogeneous harmonic polynomial of degree n can be written with and vice versa will be multiplied by a linear combination of spherical harmonics. If one chooses, for example, the function that is constant 1, as the basis of the one-dimensional vector space of 0 - homogeneous harmonic polynomials and x, y and z as the basis of the three-dimensional vector space of 1- homogeneous, we obtain in spherical coordinates by dividing the functions

For the homogeneous polynomials of degree 2 one can readily see in the list below and the terms again, only with a wrong prefactor.

Properties

The spherical harmonics have the following properties:

• Orthogonality: ( is the Kronecker delta)

• Completeness: (the delta function )

• Parity: The transition looks in spherical coordinates as follows:. Under this transformation, the spherical harmonics behave as follows:

• Complex conjugation: the respective obtained from the by:

Expansion in spherical harmonics

The spherical harmonics form a complete functional system. Therefore, all square integrable functions can be developed according to the spherical harmonics (in terms of spherical coordinates with and ):

The expansion coefficients are calculated as follows:

It must be the complex conjugate. The representation of a function - and function as a Fourier series is an analogue to the development of a two-dimensional function on a spherical surface.

Addition theorem

An interesting result for the spherical harmonics is the addition theorem. This purpose are presented and two unit vectors and spherical coordinates. Then for the angle between these two vectors

The addition theorem for spherical harmonics states now

The theorem can also be written instead of the spherical harmonic functions with the associated Legendrefunktionen

For obtained from the addition theorem

This can be viewed as a generalization of the identity of three dimensions.

The first spherical harmonics

Applications

Quantum mechanics

As eigenfunctions of the angular part of the Laplace operator, the spherical harmonics are also eigenfunctions of the angular momentum operator for the quantum number as an intrinsic value. Therefore, they play a major role in the description of nuclear states. Furthermore, it is

Solution of the Laplace 's equation

For each of the function is a solution of the Laplace equation in three dimensions, because the function just met above equation

Each solution of the Laplace equation can now be clearly identified as

Represent. Thus can be with the spherical harmonics the Laplace equation solved with spherical Dirichlet boundary conditions: Insert the boundary conditions the value of the solution to be defined on the closed unit ball, to a certain square-integrable function on the unit sphere fixed, so can be developed by spherical harmonics whereby the coefficients and thus in a unique way entirely devoted. Based on this knowledge of the solubility with spherical boundary conditions can the general solvability of the Dirichlet problem of the Laplace equation show for sufficiently smooth boundary conditions, this proof goes back to Oskar Perron. The Dirichlet problem has applications in electrostatics and magnetostatics. To solve the Laplace equation in which a function is sought which is defined outside of a sphere and vanishes at infinity, for given boundary conditions, the approach is a decomposition

Possible, which is also always provides a solution of the Laplace equation for the given boundary conditions.

Nomenclature in Geophysics

Spherical harmonics are used in geophysics. One differentiates between here:

• Zonal (): independent of longitude
• Sectoral ():

• Tesseral (otherwise ): length and latitude dependent