Spherical harmonics

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics contains a list of common spherical harmonics.
Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, certain functions defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO, the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO.
Spherical harmonics originate from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree in that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence from the above-mentioned polynomial of degree ; the remaining factor can be regarded as a function of the spherical angular coordinates and only, or equivalently of the orientational unit vector specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition. Notice, however, that spherical harmonics are not functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the Maximum principle. Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator.
A specific set of spherical harmonics, denoted or, are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above.
Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation. In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting and modelling of 3D shapes.

History

Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. In 1782, Pierre-Simon de Laplace had, in his Mécanique Céleste, determined that the gravitational potential at a point associated with a set of point masses located at points was given by
Each term in the above summation is an individual Newtonian potential for a point mass. Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of and. He discovered that if then
where is the angle between the vectors and. The functions are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between and.
In 1867, William Thomson and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise on Natural Philosophy, and also first introduced the name of "spherical harmonics" for these functions. The solid harmonics were homogeneous polynomial solutions of Laplace's equation
By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics. The term "Laplace's coefficients" was employed by William Whewell to describe the particular system of solutions introduced along these lines, whereas others reserved this designation for the zonal spherical harmonics that had properly been introduced by Laplace and Legendre.
The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the heat equation and wave equation. This could be achieved by expansion of functions in series of trigonometric functions. Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a string, the spherical harmonics represent the fundamental modes of vibration of a sphere in much the same way. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre.
The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. The spherical harmonics are eigenfunctions of the square of the orbital angular momentum operator
and therefore they represent the different quantized configurations of atomic orbitals.

Laplace's spherical harmonics

imposes that the Laplacian of a scalar field is zero. In spherical coordinates this is:
Consider the problem of finding solutions of the form. By separation of variables, two differential equations result by imposing Laplace's equation:
The second equation can be simplified under the assumption that has the form. Applying separation of variables again to the second equation gives way to the pair of differential equations
for some number. A priori, is a complex constant, but because must be a periodic function whose period evenly divides, is necessarily an integer and is a linear combination of the complex exponentials. The solution function is regular at the poles of the sphere, where. Imposing this regularity in the solution of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter to be of the form for some non-negative integer with ; this is also explained [|below] in terms of the orbital angular momentum. Furthermore, a change of variables transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial . Finally, the equation for has solutions of the form ; requiring the solution to be regular throughout forces.
Here the solution was assumed to have the special form. For a given value of, there are independent solutions of this form, one for each integer with. 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 and order , is an associated Legendre polynomial, is a normalization constant, and and represent colatitude and longitude, respectively. In particular, the colatitude, or polar angle, ranges from at the North Pole, to at the Equator, to at the South Pole, and the longitude, or azimuth, may assume all values with. For a fixed integer, every solution,, of the eigenvalue problem
is a linear combination of. In fact, for any such solution, is the expression in spherical coordinates of a homogeneous polynomial that is harmonic, and so counting dimensions shows that there are 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,
where the are constants and the factors are known as solid harmonics. Such an expansion is valid in the ball
For, the solid harmonics with negative powers of are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series, instead of the Taylor series used above, to match the terms and find series expansion coefficients.

Orbital angular momentum

In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum
The is conventional in quantum mechanics; it is convenient to work in units in which. The spherical harmonics are eigenfunctions of the square of the orbital angular momentum
Laplace's spherical harmonics are the joint eigenfunctions of the square of the orbital angular momentum and the generator of rotations about the azimuthal axis:
These operators commute, and are densely defined self-adjoint operators on the weighted Hilbert space of functions f square-integrable with respect to the normal distribution as the weight function on R³:
Furthermore, L² is a positive operator.
If is a joint eigenfunction of and, then by definition
for some real numbers m and λ. Here m must in fact be an integer, for Y must be periodic in the coordinate φ with period a number that evenly divides 2π. Furthermore, since
and each of L_x, L_y, L_z are self-adjoint, it follows that.
Denote this joint eigenspace by, and define the raising and lowering operators by
Then and commute with, and the Lie algebra generated by,, is the special linear Lie algebra of order 2,, with commutation relations
Thus and . In particular, must be zero for k sufficiently large, because the inequality must hold in each of the nontrivial joint eigenspaces. Let be a nonzero joint eigenfunction, and let be the least integer such that
Then, since
it follows that
Thus for the positive integer.
The foregoing has been all worked out in the spherical coordinate representation, but may be expressed more abstractly in the complete, orthonormal spherical ket basis.