Spherical harmonics

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Visual representations of the first few real spherical harmonics. Blue portions represent regions where the function is positive, and yellow portions represent where it is negative. The distance of the surface from the origin indicates the absolute value of in angular direction .

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 originates from solving Laplace's equation in the spherical domains. Functions that solve 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 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.

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

History[edit]

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 x associated with a set of point masses mi located at points xi 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 r = |x| and r1 = |x1|. He discovered that if rr1 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 memoire, 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.)

In 1867, William Thomson (Lord Kelvin) 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. (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.

The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. The (complex-valued) 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[edit]

Real (Laplace) spherical harmonics Yℓm for = 0, …, 4 (top to bottom) and m = 0, …, (left to right). Zonal, sectoral, and tesseral harmonics are depicted along the left-most column, the main diagonal, and elsewhere, respectively. (The negative order harmonics would be shown rotated about the z axis by with respect to the positive order ones.)
Alternative picture for the real spherical harmonics .

Laplace's equation imposes that the Laplacian of a scalar field f is zero. (Here the scalar field is understood to be complex, i.e. to correspond to a (smooth) function .) In spherical coordinates this is:[2]

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 = ( + 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]

Here the solution was assumed to have the special form Y(θ, φ) = Θ(θ) Φ(φ). For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m. 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 m, is an associated Legendre polynomial, N is a normalization constant, and θ and φ represent colatitude and longitude, respectively. In particular, the colatitude θ, or polar angle, ranges from 0 at the North Pole, to π/2 at the Equator, to π at the South Pole, and the longitude φ, or azimuth, may assume all values with 0 ≤ φ < 2π. For a fixed integer , every solution Y(θ, φ), , of the eigenvalue problem

is a linear combination of Ym. In fact, for any such solution, r 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 2 + 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 r,

where the are constants and the factors r Ym are known as (regular) solid harmonics . Such an expansion is valid in the ball

For , the solid harmonics with negative powers of (the irregular solid 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 .

Orbital angular momentum[edit]

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

The ħ is conventional in quantum mechanics; it is convenient to work in units in which ħ = 1. 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 Y is a joint eigenfunction of L2 and Lz, 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 Lx, Ly, Lz are self-adjoint, it follows that λ ≥ m2.

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

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,mEλ,m+1 (it is a "raising operator") and L : Eλ,mEλ,m−1 (it is a "lowering operator"). In particular, Lk
+
 : Eλ,mEλ,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

Then, since

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.

Harmonic polynomial representation[edit]

See also the section below on spherical harmonics in higher dimensions.

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

,

where is the Laplacian. Then for each , we define

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 .[5] 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 [6] (See the section below on special cases of the spherical harmonics.)

Conventions[edit]

Orthogonality and normalization[edit]

Several different normalizations are in common use for the Laplace spherical harmonic functions . Throughout the section, we use the standard convention that for (see associated Legendre polynomials)

which is the natural normalization given by Rodrigues' formula.

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

while in quantum mechanics:[8][9]

where are associated Legendre polynomials without the Condon–Shortley phase (to avoid counting the phase twice).

In both definitions, the spherical harmonics are orthonormal

where δij is the Kronecker delta and dΩ = sinθ dφ dθ. This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e.

The disciplines of geodesy[10] and spectral analysis use

which possess unit power

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

which have the normalization

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.

Condon–Shortley phase[edit]

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]

Real form[edit]

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.[12] 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.

Use in quantum chemistry[edit]

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.

For example, as can be seen from the table of spherical harmonics, the usual p functions () are complex and mix axis directions, but the real versions are essentially just x, y and z.

Spherical harmonics in Cartesian form[edit]

The Herglotz generating function[edit]

If the quantum mechanical convention is adopted for the , then

Here, is the vector with components , , and

is a vector with complex coefficients. It suffices to take and as real parameters. The essential property of is that it is null:

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.[13] 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 ,[14] 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.

Separated Cartesian form[edit]

The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of and another of and , as follows (Condon-Shortley phase):

and for m = 0:

Here

and

For this reduces to

The factor is essentially the associated Legendre polynomial , and the factors are essentially .

Examples[edit]

Using the expressions for , , and listed explicitly above we obtain:

It may be verified that this agrees with the function listed here and here.

Real forms[edit]

Using the equations above to form the real spherical harmonics, it is seen that for only the terms (cosines) are included, and for only the terms (sines) are included:

and for m = 0:

Special cases and values[edit]

1. When , the spherical harmonics reduce to the ordinary Legendre polynomials:

2. When ,

or more simply in Cartesian coordinates,

3. At the north pole, where , and is undefined, all spherical harmonics except those with vanish:

Symmetry properties[edit]

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

Parity[edit]

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).)

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).

Rotations[edit]

The rotation of a real spherical function with m = 0 and l = 3. The coefficients are not equal to the Wigner D-matrices, since real functions are shown, but can be obtained by re-decomposing the complex functions

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.

Spherical harmonics expansion[edit]

The Laplace spherical harmonics form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions . On the unit sphere , any square-integrable function can thus be expanded as a linear combination of these:

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 ℓ sufficiently rapidly — for instance, exponentially — then the series also converges uniformly to f.

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.

Harmonical tensors[edit]

Formula[edit]

As a rule, harmonic functions are useful in theoretical physics to consider fields in far-zone when distance from charges is much further than size of their location. In that case, radius R is constant and coordinates (θ,φ) are convenient to use. Theoretical physics considers many problems when solution of Laplace's equation is needed as a function of Сartesian coordinates. At the same time, it is important to get invariant form of solutions relatively to rotation of space or generally speaking, relatively to group transformations.[15][16][17][18] The simplest tensor solutions- dipole, quadrupole and octupole potentials are fundamental concepts of general physics:

, ,.

It is easy to verify that they are the harmonical functions. Total set of tensors is defined by Taylor series of point charge field potential for :

,

where tensor is denoted by symbol and contraction of the tensors is in the brackets [...]. Therefore, the tensor is defined by -th tensor derivative:

James Clerk Maxwell used similar considerations without tensors naturally.[19] E. W. Hobson analysed Maxwell's method as well.[20] One can see from the equation following properties that repeat mainly those of solid and spherical functions.

  • Tensor is the harmonic polynomial i. e. .
  • Trace over each two indices is zero, as far as .
  • Tensor is homogeneous polynomial of degree i.e. summed degree of variables x, y, z of each item is equal to .
  • Tensor has invariant form under rotations of variables x,y,z i.e. of vector .
  • Total set of potentials is complete.
  • Contraction of with tensor is proportional to contraction of two harmonic potentials:

Formula for harmonical invariant tensor was found in paper.[21] Detailed description is given in monography.[22] Formula contains products of tensors and Kronecker symbols :

.

Quantity of Kronecker symbols is increased by two in the product of each following item when rang of tensor is reduced by two accordingly. Operation symmetrizes tensor by means of all independent permutations of indices with following summing of got items. Particularly, don't need to be transformed into and tensor don't go into .

Regarded tensors are convenient to substitute to Laplace equation:

.

The last relation is Euler formula for homogeneous polynomials actually. Laplace operator leaves the indices symmetry of tensors. The two relations allows to substitute found tensor into Laplace equation and to check straightly that tensor is the harmonical function:

.

Simplified moments[edit]

The last property is important for theoretical physics for the following reason. Potential of charges outside of their location is integral to be equal to the sum of multipole potentials:

,

where is the charge density. The convolution is applied to tensors in the formula naturally. Integrals in the sum are called in physics as multipole moments. Three of them are used actively while others applied less often as their structure (or that of spherical functions) is more complicated. Nevertheless, last property gives the way to simplify calculations in theoretical physics by using integrals with tensor instead of harmonical tensor . Therefore, simplified moments give the same result and there is no need to restrict calculations for dipole, quadrupole and octupole potentials only. It is the advantage of the tensor point of view and not the only that.

Efimov's ladder operator[edit]

Spherical functions have a few recurrent formulas.[23] In quantum mechanics recurrent formulas plays a role when they connect functions of quantum states by means of a ladder operator.The property is occurred due to symmetry group of considered system. The vector ladder operator for the invariant harmonical states found in paper[21] and detailed in.[22]

For that purpose, transformation of -space is applied that conserves form of Laplace equation:
.

Operator applying to the harmonical tensor potential in -space goes into Efimov's ladder operator acting on transformed tensor in -space:

,

where is operator of module of angular momentum:

.

Operator multiplies harmonic tensor by its degree i.e. by if to recall according spherical function for quantum numbers , . To check action of the ladder operator , one can apply it to dipole and quadrupole tensors:

,
.

Applying successively to we get general form of invariant harmonic tensors:

.

The operator analogous to the oscillator ladder operator. To trace relation with a quantum operator it is useful to multiply it by to go to reversed space:

.

As a result, operator goes into the operator of momentum in -space :

.

It is useful to apply the following properties of .

.

The property is utterly convenient for calculations.

  • The scalar operator product is zero in the space of harmonical functions:
.

The property gives zero trace of the harmonical tensor over each two indices.

The ladder operator is analogous for that in problem of the quantum oscillator. It generates Glauber states those are created in the quantum theory of electromagnetic radiation fields. [24] It was shown later as theoretical result that the coherent states are intrinsic for any quantum system with a group symmetry to include the rotational group.[25]

Invariant form of spherical harmonics[edit]

Spherical harmonics accord with the system of coordinates. Let be the unit vectors along axises X, Y, Z. Denote following unit vectors as and :

.

Using the vectors, the solid harmonics are equal to:

=

where is the constant:

Angular momentum is defined by the rotational group. The mechanical momentum is related to the translation group. The ladder operator is the mapping of momentum upon inversion 1/r of 3-d space. It is raising operator. Lowering operator here is the gradient naturally together with partial contraction over pair indices to leave others:

Spectrum analysis[edit]

Power spectrum in signal processing[edit]

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):

where

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

where

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.

Differentiability properties[edit]

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

for all s.

Algebraic properties[edit]

Addition theorem[edit]

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.

The addition theorem states[26]

 

 

 

 

(1)

where P is the Legendre polynomial of degree ℓ. This expression is valid for both real and complex harmonics.[27] 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.[28]

In particular, when x = y, this gives Unsöld's theorem[29]

which generalizes the identity cos2θ + sin2θ = 1 to two dimensions.

In the expansion (1), the left-hand side P(x·y) is a constant multiple of the degree ℓ zonal spherical harmonic. From this perspective, one has the following generalization to higher dimensions. Let Yj be an arbitrary orthonormal basis of the space H of degree ℓ spherical harmonics on the n-sphere. Then , the degree ℓ zonal harmonic corresponding to the unit vector x, decomposes as[30]

 

 

 

 

(2)

Furthermore, the zonal harmonic is given as a constant multiple of the appropriate Gegenbauer polynomial:

 

 

 

 

(3)

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

where ωn−1 is the volume of the (n−1)-sphere.

Contraction rule[edit]

Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics[31]

where the values of and are determined by the selection rules for the 3j-symbols.

Clebsch–Gordan coefficients[edit]

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.

Visualization of the spherical harmonics[edit]

Schematic representation of on the unit sphere and its nodal lines. is equal to 0 along m great circles passing through the poles, and along ℓ−m circles of equal latitude. The function changes sign each time it crosses one of these lines.
3D color plot of the spherical harmonics of degree n = 5. Note that n = ℓ.

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.[32]

List of spherical harmonics[edit]

Analytic expressions for the first few orthonormalized Laplace spherical harmonics that use the Condon-Shortley phase convention:

Higher dimensions[edit]

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 .[33] Let P 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 P provided that for any real , one has

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

These are the (regular) solid spherical harmonics. Let H denote the space of functions on the unit sphere

obtained by restriction from A

The following properties hold:

  • The sum of the spaces H is dense in the set C(Sn−1) of continuous functions on Sn−1 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 a spherical harmonics, where the series converges in the L2 sense.
  • For all f ∈ H, 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 H are orthogonal with respect to the inner product from L2(Sn−1). That is to say,
for f ∈ H and g ∈ Hk for k ≠ ℓ.
  • Conversely, the spaces H 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 H are pairwise orthogonal and complete in L2(Sn−1).
  • Every homogeneous polynomial p ∈ P can be uniquely written in the form[34]
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[35]

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

Connection with representation theory[edit]

The space H 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 H by function composition

for ψ a spherical harmonic and ρ a rotation. The representation H is an irreducible representation of SO(3).[36]

The elements of H arise as the restrictions to the sphere of elements of A: harmonic polynomials homogeneous of degree ℓ on three-dimensional Euclidean space R3. By polarization of ψ ∈ A, 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), H is isomorphic to the space of traceless symmetric tensors of degree ℓ.

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 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.

Connection with hemispherical harmonics[edit]

Spherical harmonics can be separated into two set of functions.[37] One is hemispherical functions (HSH), orthogonal and complete on hemisphere. Another is complementary hemispherical harmonics (CHSH).

Generalizations[edit]

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.[38][39][40][41]

See also[edit]

Notes[edit]

  1. ^ A historical account of various approaches to spherical harmonics in three dimensions can be found in Chapter IV of MacRobert 1967. The term "Laplace spherical harmonics" is in common use; see Courant & Hilbert 1962 and Meijer & Bauer 2004.
  2. ^ The approach to spherical harmonics taken here is found in (Courant & Hilbert 1962, §V.8, §VII.5).
  3. ^ Physical applications often take the solution that vanishes at infinity, making A = 0. This does not affect the angular portion of the spherical harmonics.
  4. ^ Edmonds 1957, §2.5
  5. ^ Hall 2013 Section 17.6
  6. ^ Hall 2013 Lemma 17.16
  7. ^ George), Williams, Earl G. (Earl (1999). Fourier acoustics : sound radiation and nearfield acoustical holography. San Diego, Calif.: Academic Press. ISBN 0080506909. OCLC 181010993.
  8. ^ Messiah, Albert (1999). Quantum mechanics : two volumes bound as one (Two vol. bound as one, unabridged reprint ed.). Mineola, NY: Dover. ISBN 9780486409245.
  9. ^ al.], Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë; transl. from the French by Susan Reid Hemley ... [et (1996). Quantum mechanics. Wiley-Interscience: Wiley. ISBN 9780471569527.
  10. ^ a b Blakely, Richard (1995). Potential theory in gravity and magnetic applications. Cambridge England New York: Cambridge University Press. p. 113. ISBN 978-0521415088.
  11. ^ Heiskanen and Moritz, Physical Geodesy, 1967, eq. 1-62
  12. ^ Watson & Whittaker 1927, p. 392.
  13. ^ See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012).
  14. ^ Li, Feifei; Braun, Carol; Garg, Anupam, The Weyl-Wigner-Moyal Formalism for Spin (PDF)
  15. ^ Efimov Sergei P.; Muratov Rodes Z. (1990). "Theory of multipole representation of the potentialsod an ellipsoid. Tensor porentials". Astron. Zh. 67 (2): 152–157. Bibcode:1990SvA....34..152E.
  16. ^ Efimov Sergei P., Muratov Rodes Z. (1990). "Theory of multipole representation of the potentials of an ellipsoid. Moments". Astron. Zh. 67 (2): 157–162. Bibcode:1990SvA....34..157E.
  17. ^ Buchbinder I.L. and Shapiro I.L. (1990). "On the renormalization group equations in curved spacetime with the torsion". Classical and Quantum Gravity. 7 (7): 1197. doi:10.1088/0264-9381/7/7/015.
  18. ^ Kalmykov M. Yu., Pronin P.I. (1991). "One-loop effective action in gauge gravitational theory". Il Nuovo Cimento B, Series 11. 106 (12): 1401. Bibcode:1991NCimB.106.1401K. doi:10.1007/BF02728369.
  19. ^ Maxwell, James Clerk (1892). A treatise on Electricity & Magnetism. N. Y.: Dover Publications Inc. 1954. pp. ch.9.
  20. ^ Hobson, E. W. (2012). The Theory of Spherical and Ellipsoidal Harmonics. Cambridge: Cambridge Academ. ISBN 978-1107605114.
  21. ^ a b Efimov, Sergei P. (1979). "Transition operator between multipole states and their tensor structure". Theoretical and Mathematical Physics. 39 (2): 425–434. Bibcode:1979TMP....39..425E. doi:10.1007/BF01014921.
  22. ^ a b Muratov, Rodes Z. (2015). Multipoles and Fields of Ellipsoid. Moscow: Izd. Dom MISIS. pp. 142–155. ISBN 978-5-600-01057-4.
  23. ^ Vilenkin, N. Ja. (1968). Special functions and the theory of Group Representations. Am. Math. Society. ISBN 9780821815724.
  24. ^ Glauber, Roy J. (1963). "Coherent and Incoherent States of the Radiation Field". Physical Review. 131 (6): 2766–2788. Bibcode:1963PhRv..131.2766G. doi:10.1103/physrev.131.2766.
  25. ^ Perelomov, A. M. (1972). "Coherent states for arbitrary Lie groups". Communications in Mathematical Physics. 26 (3): 222–236. arXiv:math-ph/0203002. Bibcode:1972CMaPh..26..222P. doi:10.1007/BF01645091.
  26. ^ Edmonds, A. R. (1996). Angular Momentum In Quantum Mechanics. Princeton University Press. p. 63.
  27. ^ This is valid for any orthonormal basis of spherical harmonics of degree ℓ. For unit power harmonics it is necessary to remove the factor of 4π.
  28. ^ Watson & Whittaker 1927, p. 395
  29. ^ Unsöld 1927
  30. ^ Stein & Weiss 1971, §IV.2
  31. ^ Brink, D. M.; Satchler, G. R. Angular Momentum. Oxford University Press. p. 146.
  32. ^ Eremenko, Jakobson & Nadirashvili 2007
  33. ^ Solomentsev 2001; Stein & Weiss 1971, §Iv.2
  34. ^ Cf. Corollary 1.8 of Axler, Sheldon; Ramey, Wade (1995), Harmonic Polynomials and Dirichlet-Type Problems
  35. ^ Higuchi, Atsushi (1987). "Symmetric tensor spherical harmonics on the N-sphere and their application to the de Sitter group SO(N,1)". Journal of Mathematical Physics. 28 (7): 1553–1566. Bibcode:1987JMP....28.1553H. doi:10.1063/1.527513.
  36. ^ Hall 2013 Corollary 17.17
  37. ^ Zheng, Yi; Wei, Kai; Wei, Kai; Liang, Bin; Liang, Bin; Li, Ying; Li, Ying; Chu, Xinhui; Chu, Xinhui (2019-12-23). "Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering". Optics Express. 27 (26): 37180–37195. doi:10.1364/OE.27.037180. ISSN 1094-4087. Missing |author2= (help)
  38. ^ N. Vilenkin, Special Functions and the Theory of Group Representations, Am. Math. Soc. Transl.,vol. 22, (1968).
  39. ^ J. D. Talman, Special Functions, A Group Theoretic Approach, (based on lectures by E.P. Wigner), W. A. Benjamin, New York (1968).
  40. ^ W. Miller, Symmetry and Separation of Variables, Addison-Wesley, Reading (1977).
  41. ^ A. Wawrzyńczyk, Group Representations and Special Functions, Polish Scientific Publishers. Warszawa (1984).

References[edit]

Cited references
General references