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.

Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function 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 (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) 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(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).

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 (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and modelling of 3D shapes.

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 r = |x| and r1 = |x1|. He discovered that if rr1 then

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.

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

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 = ( + 1) for some non-negative integer with ≥ |m|; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial Pm
(cos θ)
. Finally, the equation for R has solutions of the form R(r) = A r + B r − 1; requiring the solution to be regular throughout R3 forces B = 0.[3]

In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[4]

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:

Denote this joint eigenspace by Eλ,m, and define the raising and lowering operators by

In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article)

The magnetics[10] community, in contrast, uses Schmidt semi-normalized harmonics

In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah.

It can be shown that all of the above normalized spherical harmonic functions satisfy

where the superscript * denotes complex conjugation. Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix.

One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of (−1)m, commonly referred to as the Condon–Shortley phase in the quantum mechanical literature. In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre polynomials, or to append it to the definition of the spherical harmonic functions. There is no requirement to use the Condon–Shortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of raising and lowering operators. The geodesy[11] and magnetics communities never include the Condon–Shortley phase factor in their definitions of the spherical harmonic functions nor in the ones of the associated Legendre polynomials.[citation needed]

The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation.

As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. However, the solutions of the non-relativistic Schrödinger equation without magnetic terms can be made real. This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. Here, it is important to note that the real functions span the same space as the complex ones would.

In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, §VII.7, who credit unpublished notes by him for its discovery.

The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation.

Parity continues to hold for real spherical harmonics, and for spherical harmonics in higher dimensions: applying a point reflection to a spherical harmonic of degree changes the sign by a factor of (−1).

This expansion holds in the sense of mean-square convergence — convergence in L2 of the sphere — which is to say that

The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle Ω, and utilizing the above orthogonality relationships. This is justified rigorously by basic Hilbert space theory. For the case of orthonormalized harmonics, this gives:

If the coefficients decay in sufficiently rapidly — for instance, exponentially — then the series also converges uniformly to f.

The total power of a function f is defined in the signal processing literature as the integral of the function squared, divided by the area of its domain. Using the orthonormality properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its spectral coefficients by a generalization of Parseval's theorem (here, the theorem is stated for Schmidt semi-normalized harmonics, the relationship is slightly different for orthonormal harmonics):

is defined as the angular power spectrum (for Schmidt semi-normalized harmonics). In a similar manner, one can define the cross-power of two functions as

is defined as the cross-power spectrum. If the functions f and g have a zero mean (i.e., the spectral coefficients f00 and g00 are zero), then Sff() and Sfg() represent the contributions to the function's variance and covariance for degree , respectively. It is common that the (cross-)power spectrum is well approximated by a power law of the form

When β = 0, the spectrum is "white" as each degree possesses equal power. When β < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when β > 0, the spectrum is termed "blue". The condition on the order of growth of Sff() is related to the order of differentiability of f in the next section.

One can also understand the differentiability properties of the original function f in terms of the asymptotics of Sff(). In particular, if Sff() decays faster than any rational function of as → ∞, then f is infinitely differentiable. If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere.

The general technique is to use the theory of Sobolev spaces. Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. Specifically, if

where P is the Legendre polynomial of degree . This expression is valid for both real and complex harmonics.[16] The result can be proven analytically, using the properties of the Poisson kernel in the unit ball, or geometrically by applying a rotation to the vector y so that it points along the z-axis, and then directly calculating the right-hand side.[17]

Combining (2) and (3) gives (1) in dimension n = 2 when x and y are represented in spherical coordinates. Finally, evaluating at x = y gives the functional identity

Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics[20]

The Clebsch–Gordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. A variety of techniques are available for doing essentially the same calculation, including the Wigner 3-jm symbol, the Racah coefficients, and the Slater integrals. Abstractly, the Clebsch–Gordan coefficients express the tensor product of two irreducible representations of the rotation group as a sum of irreducible representations: suitably normalized, the coefficients are then the multiplicities.

When the spherical harmonic order m is zero (upper-left in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. Such spherical harmonics are a special case of zonal spherical functions. When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. For the other cases, the functions checker the sphere, and they are referred to as tesseral.

Let A denote the subspace of P consisting of all harmonic polynomials:

An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of separation of variables, by solving the Sturm-Liouville problem for the spherical Laplacian

The space H of spherical harmonics of degree is a representation of the symmetry group of rotations around a point (SO(3)) and its double-cover SU(2). Indeed, rotations act on the two-dimensional sphere, and thus also on H by function composition

More generally, the analogous statements hold in higher dimensions: the space H of spherical harmonics on the n-sphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric -tensors. However, whereas every irreducible tensor representation of SO(2) and SO(3) is of this kind, the special orthogonal groups in higher dimensions have additional irreducible representations that do not arise in this manner.

The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). In turn, SU(2) is identified with the group of unit quaternions, and so coincides with the 3-sphere. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication.

Spherical harmonics can be separated into two set of functions.[26] One is hemispherical functions (HSH), orthogonal and complete on hemisphere. Another is complementary hemispherical harmonics (CHSH).

The angle-preserving symmetries of the two-sphere are described by the group of Möbius transformations PSL(2,C). With respect to this group, the sphere is equivalent to the usual Riemann sphere. The group PSL(2,C) is isomorphic to the (proper) Lorentz group, and its action on the two-sphere agrees with the action of the Lorentz group on the celestial sphere in Minkowski space. The analog of the spherical harmonics for the Lorentz group is given by the hypergeometric series; furthermore, the spherical harmonics can be re-expressed in terms of the hypergeometric series, as SO(3) = PSU(2) is a subgroup of PSL(2,C).

More generally, hypergeometric series can be generalized to describe the symmetries of any symmetric space; in particular, hypergeometric series can be developed for any Lie group.[27][28][29][30]