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 orientationalunit 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 (see the section Higher dimensions below).
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.
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 r ≤ r1 then
where γ is the angle between the vectors x and x1. 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 γ between x1 and x. (See Applications of Legendre polynomials in physics for a more detailed analysis.)
By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics. (See the section below, "Harmonic polynomial representation".) 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.
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:
The second equation can be simplified under the assumption that Y has the form Y(θ, φ) = Θ(θ) Φ(φ). Applying separation of variables again to the second equation gives way to the pair of differential equations
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 = ℓ (ℓ + 1) for some non-negative integer with ℓ ≥ |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 polynomialPm ℓ(cos θ) . Finally, the equation for R has solutions of the form R(r) = A rℓ + B r−ℓ − 1; requiring the solution to be regular throughout R3 forces B = 0.
Here the solution was assumed to have the special form Y(θ, φ) = Θ(θ) Φ(φ). For a given value of ℓ, there are 2ℓ + 1 independent solutions of this form, one for each integer m with −ℓ ≤ m ≤ ℓ. These angular solutions are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:
Here is called a spherical harmonic function of degree ℓ and order m, is an associated Legendre polynomial, N is a normalization constant, and θ and φ represent colatitude and longitude, respectively. In particular, the colatitudeθ, or polar angle, ranges from 0 at the North Pole, to π/2 at the Equator, to π at the South Pole, and the longitudeφ, or azimuth, may assume all values with 0 ≤ φ < 2π. For a fixed integer ℓ, every solution Y(θ, φ), , of the eigenvalue problem
is a linear combination of . In fact, for any such solution, rℓ Y(θ, φ) is the expression in spherical coordinates of a homogeneous polynomial that is harmonic (see below), and so counting dimensions shows that there are 2ℓ + 1 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 rℓ,
where the are constants and the factors rℓ Yℓm are known as (regular) solid harmonics. Such an expansion is valid in the ball
For , the solid harmonics with negative powers of (the irregularsolid harmonics) are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about ), instead of the Taylor series (about ) used above, to match the terms and find series expansion coefficients .
Then L+ and L− commute with L2, and the Lie algebra generated by L+, L−, Lz is the special linear Lie algebra of order 2, , with commutation relations
Thus L+ : Eλ,m → Eλ,m+1 (it is a "raising operator") and L− : Eλ,m → Eλ,m−1 (it is a "lowering operator"). In particular, Lk + : Eλ,m → Eλ,m+k must be zero for k sufficiently large, because the inequality λ ≥ m2 must hold in each of the nontrivial joint eigenspaces. Let Y ∈ Eλ,m be a nonzero joint eigenfunction, and let k be the least integer such that
it follows that
Thus λ = ℓ(ℓ + 1) for the positive integer ℓ = m + k.
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.
The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions . Specifically, we say that a (complex-valued) polynomial function is homogeneous of degree if
for all real numbers and all . We say that is harmonic if
For example, when , is just the 3-dimensional space of all linear functions , since any such function is automatically harmonic. Meanwhile, when , we have a 5-dimensional space:
For any , the space of spherical harmonics of degree is just the space of restrictions to the sphere of the elements of . As suggested in the introduction, this perspective is presumably the origin of the term “spherical harmonic” (i.e., the restriction to the sphere of a harmonic function).
For example, for any the formula
defines a homogeneous polynomial of degree with domain and codomain , which happens to be independent of . This polynomial is easily seen to be harmonic. If we write in spherical coordinates and then restrict to , we obtain
which can be rewritten as
After using the formula for the associated Legendre polynomial, we may recognize this as the formula for the spherical harmonic  (See the section below on special cases of the spherical harmonics.)
One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of , 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 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.
A real basis of spherical harmonics can be defined in terms of their complex analogues by setting
The Condon–Shortley phase convention is used here for consistency. The corresponding inverse equations defining the complex spherical harmonics in terms of the real spherical harmonics are
The real spherical harmonics are sometimes known as tesseral spherical harmonics. These functions have the same orthonormality properties as the complex ones above. The real spherical harmonics with m > 0 are said to be of cosine type, and those with m < 0 of sine type. The reason for this can be seen by writing the functions in terms of the Legendre polynomials as
The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation.
See here for a list of real spherical harmonics up to and including , which can be seen to be consistent with the output of the equations above.
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.
It turns out that is basis of the space of harmonic and homogeneous polynomials of degree . More specifically, it is the (unique up to normalization) Gelfand-Tsetlin-basis of this representation of the rotational group and an explicit formula for in cartesian coordinates can be derived from that fact.
If the quantum mechanical convention is adopted for the , then
Here, is the vector with components , , and
is a vector with complex coordinates:
The essential property of is that it is null:
It suffices to take and as real parameters.
In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, §VII.7, who credit unpublished notes by him for its discovery.
Essentially all the properties of the spherical harmonics can be derived from this generating function. An immediate benefit of this definition is that if the vector is replaced by the quantum mechanical spin vector operator , such that is the operator analogue of the solid harmonic, one obtains a generating function for a standardized set of spherical tensor operators, :
The parallelism of the two definitions ensures that the 's transform under rotations (see below) in the same way as the 's, which in turn guarantees that they are spherical tensor operators, , with and , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. They are, moreover, a standardized set with a fixed scale or normalization.
The spherical harmonics have definite parity. That is, they are either even or odd with respect to inversion about the origin. Inversion is represented by the operator . Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with being a unit vector,
In terms of the spherical angles, parity transforms a point with coordinates to . The statement of the parity of spherical harmonics is then
(This can be seen as follows: The associated Legendre polynomials gives (−1)ℓ+m and from the exponential function we have (−1)m, giving together for the spherical harmonics a parity of (−1)ℓ.)
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 ℓ changes the sign by a factor of (−1)ℓ.
Consider a rotation about the origin that sends the unit vector to . Under this operation, a spherical harmonic of degree and order transforms into a linear combination of spherical harmonics of the same degree. That is,
where is a matrix of order that depends on the rotation . However, this is not the standard way of expressing this property. In the standard way one writes,
where is the complex conjugate of an element of the Wigner D-matrix. In particular when is a rotation of the azimuth we get the identity,
The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The 's of degree provide a basis set of functions for the irreducible representation of the group SO(3) of dimension . Many facts about spherical harmonics (such as the addition theorem) that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry.
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:
A square-integrable function can also be expanded in terms of the real harmonics above as a sum
The convergence of the series holds again in the same sense, namely the real spherical harmonics form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions. The benefit of the expansion in terms of the real harmonic functions is that for real functions the expansion coefficients are guaranteed to be real, whereas their coefficients in their expansion in terms of the (considering them as functions ) do not have that property.
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(ℓ) and Sfg(ℓ) represent the contributions to the function's variance and covariance for degree ℓ, 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(ℓ) is related to the order of differentiability of f in the next section.
The general technique is to use the theory of Sobolev spaces. Statements relating the growth of the Sff(ℓ) to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. Specifically, if
then f is in the Sobolev space Hs(S2). In particular, the Sobolev embedding theorem implies that f is infinitely differentiable provided that
A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. Given two vectors r and r′, with spherical coordinates and , respectively, the angle between them is given by the relation
in which the role of the trigonometric functions appearing on the right-hand side is played by the spherical harmonics and that of the left-hand side is played by the Legendre polynomials.
where Pℓ is the Legendre polynomial of degree ℓ. This expression is valid for both real and complex harmonics. 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.
In particular, when x = y, this gives Unsöld's theorem
which generalizes the identity cos2θ + sin2θ = 1 to two dimensions.
In the expansion (1), the left-hand side is a constant multiple of the degree ℓzonal spherical harmonic. From this perspective, one has the following generalization to higher dimensions. Let Yj be an arbitrary orthonormal basis of the space Hℓ of degree ℓ spherical harmonics on the n-sphere. Then , the degree ℓ zonal harmonic corresponding to the unit vector x, decomposes as
Furthermore, the zonal harmonic is given as a constant multiple of the appropriate Gegenbauer polynomial:
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
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.
The Laplace spherical harmonics can be visualized by considering their "nodal lines", that is, the set of points on the sphere where , or alternatively where . Nodal lines of are composed of ℓ circles: there are |m| circles along longitudes and ℓ−|m| circles along latitudes. One can determine the number of nodal lines of each type by counting the number of zeros of in the and directions respectively. Considering as a function of , the real and imaginary components of the associated Legendre polynomials each possess ℓ−|m| zeros, each giving rise to a nodal 'line of latitude'. On the other hand, considering as a function of , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'.
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 ℓ = |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.
More general spherical harmonics of degree ℓ are not necessarily those of the Laplace basis , and their nodal sets can be of a fairly general kind.
The classical spherical harmonics are defined as complex-valued functions on the unit sphere inside three-dimensional Euclidean space . Spherical harmonics can be generalized to higher-dimensional Euclidean space as follows, leading to functions . Let Pℓ denote the space of complex-valued homogeneous polynomials of degree ℓ in n real variables, here considered as functions . That is, a polynomial p is in Pℓ provided that for any real , one has
The sum of the spaces Hℓ is dense in the set of continuous functions on with respect to the uniform topology, by the Stone–Weierstrass theorem. As a result, the sum of these spaces is also dense in the space L2(Sn−1) of square-integrable functions on the sphere. Thus every square-integrable function on the sphere decomposes uniquely into a series of spherical harmonics, where the series converges in the L2 sense.
For all f ∈ Hℓ, one has
where ΔSn−1 is the Laplace–Beltrami operator on Sn−1. This operator is the analog of the angular part of the Laplacian in three dimensions; to wit, the Laplacian in n dimensions decomposes as
It follows from the Stokes theorem and the preceding property that the spaces Hℓ are orthogonal with respect to the inner product from L2(Sn−1). That is to say,
for f ∈ Hℓ and g ∈ Hk for k ≠ ℓ.
Conversely, the spaces Hℓ are precisely the eigenspaces of ΔSn−1. In particular, an application of the spectral theorem to the Riesz potential gives another proof that the spaces Hℓ are pairwise orthogonal and complete in L2(Sn−1).
Every homogeneous polynomial p ∈ Pℓ can be uniquely written in the form
where pj ∈ Aj. In particular,
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
where φ is the axial coordinate in a spherical coordinate system on Sn−1. The end result of such a procedure is
where the indices satisfy |ℓ1| ≤ ℓ2 ≤ ⋯ ≤ ℓn−1 and the eigenvalue is −ℓn−1(ℓn−1 + n−2). The functions in the product are defined in terms of the Legendre function
The space Hℓ of spherical harmonics of degree ℓ 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 Hℓ by function composition
The elements of Hℓ arise as the restrictions to the sphere of elements of Aℓ: harmonic polynomials homogeneous of degree ℓ on three-dimensional Euclidean space R3. By polarization of ψ ∈ Aℓ, there are coefficients symmetric on the indices, uniquely determined by the requirement
The condition that ψ be harmonic is equivalent to the assertion that the tensor must be trace free on every pair of indices. Thus as an irreducible representation of SO(3), Hℓ is isomorphic to the space of traceless symmetric tensors of degree ℓ.
More generally, the analogous statements hold in higher dimensions: the space Hℓ of spherical harmonics on the n-sphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric ℓ-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. One is hemispherical functions (HSH), orthogonal and complete on hemisphere. Another is complementary hemispherical harmonics (CHSH).