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.
Spherical harmonics originate from solvingLaplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are calledharmonics. Despite their name, spherical harmonics take their simplest form inCartesian coordinates, where they can be defined ashomogeneous polynomials ofdegree in that obey Laplace's equation. The connection withspherical 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 theorientationalunit 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 arenot functions on the sphere which are harmonic with respect to theLaplace-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 theMaximum principle. Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (seeHigher dimensions).
A specific set of spherical harmonics, denoted or, are known as Laplace's spherical harmonics, as they were first introduced byPierre Simon de Laplace in 1782.[1] These functions form anorthogonal 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 ofr = |x| andr1 = |x1|. He discovered that ifr ≤r1 then
whereγ is the angle between the vectorsx andx1. The functions are theLegendre 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 angleγ betweenx1 andx. (SeeLegendre polynomials § Applications for more detail.)
The 19th century development ofFourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of theheat equation andwave equation. This could be achieved by expansion of functions in series oftrigonometric functions. Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in astring, the spherical harmonics represent the fundamental modes ofvibration 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 ascomplex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. This was a boon for problems possessingspherical 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 ofquantum mechanics. The (complex-valued) spherical harmonics areeigenfunctions of the square of theorbital angular momentum operatorand therefore they represent the differentquantized configurations ofatomic orbitals.
Real (Laplace) spherical harmonics for (top to bottom) and (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 thez axis by with respect to the positive order ones.). Rotation added for a better visual of the harmonic.Alternative picture for the real spherical harmonics.
Laplace's equation imposes that theLaplacian of a scalar fieldf is zero. (Here the scalar field is understood to be complex, i.e. to correspond to a (smooth) function.) Inspherical coordinates this is:[2]
Consider the problem of finding solutions of the formf(r,θ,φ) =R(r)Y(θ,φ). Byseparation of variables, two differential equations result by imposing Laplace's equation:The second equation can be simplified under the assumption thatY has the formY(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 numberm. A priori,m is a complex constant, but becauseΦ must be aperiodic function whose period evenly divides2π,m is necessarily an integer andΦ is a linear combination of the complex exponentialse±imφ. The solution functionY(θ,φ) is regular at the poles of the sphere, whereθ = 0,π. Imposing this regularity in the solutionΘ of the second equation at the boundary points of the domain is aSturm–Liouville problem that forces the parameterλ to be of the formλ =ℓ (ℓ + 1) for some non-negative integer withℓ ≥ |m|; this is also explainedbelow in terms of theorbital angular momentum. Furthermore, a change of variablest = cosθ transforms this equation into theLegendre equation, whose solution is a multiple of theassociated Legendre polynomialPm ℓ(cosθ) . Finally, the equation forR has solutions of the formR(r) =A rℓ +B r−ℓ − 1; requiring the solution to be regular throughoutR3 forcesB = 0.[3]
Here the solution was assumed to have the special formY(θ,φ) = Θ(θ) Φ(φ). For a given value ofℓ, there are2ℓ + 1 independent solutions of this form, one for each integerm with−ℓ ≤m ≤ℓ. These angular solutions are a product oftrigonometric functions, here represented as acomplex exponential, and associated Legendre polynomials:
which fulfill
Here is called aspherical harmonic function of degreeℓ and orderm, is anassociated Legendre polynomial,N is a normalization constant,[4] andθ andφ represent colatitude and longitude, respectively. In particular, thecolatitudeθ, or polar angle, ranges from0 at the North Pole, toπ/2 at the Equator, toπ at the South Pole, and thelongitudeφ, orazimuth, may assume all values with0 ≤φ < 2π. For a fixed integerℓ, every solutionY(θ,φ),, of the eigenvalue problemis alinear combination of. In fact, for any such solution,rℓ Y(θ,φ) is the expression in spherical coordinates of ahomogeneous polynomial that is harmonic (seebelow), and so counting dimensions shows that there are2ℓ + 1 linearly independent such polynomials.
The general solution toLaplace's equation in a ball centered at the origin is alinear combination of the spherical harmonic functions multiplied by the appropriate scale factorrℓ,
where the are constants and the factorsrℓ Yℓm are known as (regular)solid harmonics. Such an expansion is valid in theball
For, the solid harmonics with negative powers of (theirregularsolid harmonics) are chosen instead. In that case, one needs to expand the solution of known regions inLaurent series (about), instead of theTaylor series (about) used above, to match the terms and find series expansion coefficients.
In quantum mechanics, Laplace's spherical harmonics are understood in terms of theorbital angular momentum[5]Theħ is conventional in quantum mechanics; it is convenient to work in units in whichħ = 1. The spherical harmonics are eigenfunctions of the square of the orbital angular momentumLaplace's spherical harmonics are the joint eigenfunctions of the square of the orbital angular momentum and the generator of rotations about the azimuthal axis:
IfY is a joint eigenfunction ofL2 andLz, then by definitionfor some real numbersm andλ. Herem must in fact be an integer, forY must be periodic in the coordinateφ with period a number that evenly divides 2π. Furthermore, sinceand each ofLx,Ly,Lz are self-adjoint, it follows thatλ ≥m2.
Denote this joint eigenspace byEλ,m, and define theraising and lowering operators byThenL+ andL− commute withL2, and the Lie algebra generated byL+,L−,Lz is thespecial linear Lie algebra of order 2,, with commutation relationsThusL+ :Eλ,m →Eλ,m+1 (it is a "raising operator") andL− :Eλ,m →Eλ,m−1 (it is a "lowering operator"). In particular,Lk + :Eλ,m →Eλ,m+k must be zero fork sufficiently large, because the inequalityλ ≥m2 must hold in each of the nontrivial joint eigenspaces. LetY ∈Eλ,m be a nonzero joint eigenfunction, and letk be the least integer such thatThen, sinceit follows thatThusλ =ℓ(ℓ + 1) for the positive integerℓ =m +k.
The foregoing has been all worked out in the spherical coordinate representation, but may be expressed more abstractly in the complete, orthonormalspherical 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 ishomogeneous of degree iffor all real numbers and all. We say that isharmonic ifwhere is theLaplacian. 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.[6] As suggested in the introduction, this perspective is presumably the origin of the term "spherical harmonic" (i.e., the restriction to the sphere of aharmonic function).
For example, for any the formuladefines 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 obtainwhich can be rewritten asAfter using the formula for theassociated Legendre polynomial, we may recognize this as the formula for the spherical harmonic[7] (SeeSpecial cases.)
Several different normalizations are in common use for the Laplace spherical harmonic functions. Throughout the section, we use the standard convention that for (seeassociated Legendre polynomials)which is the natural normalization given byRodrigues' formula.
Plot of the spherical harmonic with and and in the complex plane from to with colors created with Mathematica 13.1 function ComplexPlot3D
Inacoustics,[8] the Laplace spherical harmonics are generally defined as (this is the convention used in this article)while inquantum mechanics:[9][10]
where are associated Legendre polynomials without the Condon–Shortley phase (to avoid counting the phase twice).
In both definitions, the spherical harmonics are orthonormalwhereδij is theKronecker delta anddΩ = sin(θ)dφdθ. This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e.,
The disciplines ofgeodesy[11] and spectral analysis use
which possess unit power
Themagnetics[11] 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 afterGiulio Racah.
It can be shown that all of the above normalized spherical harmonic functions satisfy
where the superscript* denotescomplex conjugation. Alternatively, this equation follows from the relation of the spherical harmonic functions with theWigner D-matrix.
One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of, commonly referred to as theCondon–Shortley phase in the quantum mechanical literature. In the quantum mechanics community, it is common practice to either include thisphase factor in the definition of theassociated 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 ofraising and lowering operators. The geodesy[12] 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.[13]
A real basis of spherical harmonics can be defined in terms of their complex analogues by settingThe 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 astesseral spherical harmonics.[14] These functions have the same orthonormality properties as the complex ones above. The real spherical harmonics withm > 0 are said to be of cosine type, and those withm < 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.
Seehere 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-relativisticSchrödinger equation without magnetic terms can be made real. This is why the real forms are extensively used in basis functions forquantum chemistry, as the programs don't then need to use complex algebra. Here, the real functions span the same space as the complex ones would.
For example, as can be seen from thetable of spherical harmonics, the usualp functions () are complex and mix axis directions, but thereal versions are essentially justx,y, andz.
The complex spherical harmonics give rise to thesolid harmonics by extending from to all of as ahomogeneous function of degree, i.e. settingIt turns out that is basis of the space of harmonic andhomogeneous polynomials of degree. More specifically, it is the (unique up to normalization)Gelfand-Tsetlin-basis of this representation of the rotational group and anexplicit formula for in cartesian coordinates can be derived from that fact.
If the quantum mechanical convention is adopted for the, thenHere, 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 afterHerglotz, we followCourant & 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.[15] 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 thesolid harmonic,[16] one obtains a generating function for a standardized set ofspherical 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 theClebsch-Gordan composition theorem, and theWigner-Eckart theorem. They are, moreover, a standardized set with a fixed scale or normalization.
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 form = 0:HereandFor this reduces to
The factor is essentially the associated Legendre polynomial, and the factors are essentially.
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:
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: Theassociated 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 apoint reflection to a spherical harmonic of degreeℓ changes the sign by a factor of(−1)ℓ.
The rotation of a real spherical function withm = 0 andℓ = 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 theWigner 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.
The Laplace spherical harmonics form a complete set of orthonormal functions and thus form anorthonormal basis of theHilbert space ofsquare-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 inL2 of the sphere — which is to say that
The expansion coefficients are the analogs ofFourier 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 anorthonormal basis of theHilbert space ofsquare-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 functionf is defined in thesignal processing literature as the integral of the function squared, divided by the area of its domain. Using theorthonormality 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 ofParseval'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 aswhere
is defined as the cross-power spectrum. If the functionsf andg have a zero mean (i.e., the spectral coefficientsf00 andg00 are zero), thenSff(ℓ) andSfg(ℓ) represent the contributions to the function's variance and covariance for degreeℓ, respectively. It is common that the (cross-)power spectrum is well approximated by a power law of the form
Whenβ = 0, the spectrum is "white" as each degree possesses equal power. Whenβ < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, whenβ > 0, the spectrum is termed "blue". The condition on the order of growth ofSff(ℓ) is related to the order of differentiability off in the next section.
The general technique is to use the theory ofSobolev spaces. Statements relating the growth of theSff(ℓ) to differentiability are then similar to analogous results on the growth of the coefficients ofFourier series. Specifically, ifthenf is in the Sobolev spaceHs(S2). In particular, theSobolev embedding theorem implies thatf is infinitely differentiable provided thatfor alls.
A mathematical result of considerable interest and use is called theaddition theorem for spherical harmonics. Given two vectorsr andr′, with spherical coordinates and, respectively, the angle between them is given by the relationin 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 theLegendre polynomials.
wherePℓ is theLegendre polynomial of degreeℓ. This expression is valid for both real and complex harmonics.[18] The result can be proven analytically, using the properties of thePoisson kernel in the unit ball, or geometrically by applying a rotation to the vectory so that it points along thez-axis, and then directly calculating the right-hand side.[19]
In particular, whenx =y, this gives Unsöld's theorem[20]which generalizes the identitycos2θ + sin2θ = 1 to two dimensions.
In the expansion (1), the left-hand side is a constant multiple of the degreeℓzonal spherical harmonic. From this perspective, one has the following generalization to higher dimensions. LetYj be an arbitrary orthonormal basis of the spaceHℓ of degreeℓ spherical harmonics on then-sphere. Then, the degreeℓ zonal harmonic corresponding to the unit vectorx, decomposes as[21]
2
Furthermore, the zonal harmonic is given as a constant multiple of the appropriateGegenbauer polynomial:
3
Combining (2) and (3) gives (1) in dimensionn = 2 whenx andy are represented in spherical coordinates. Finally, evaluating atx =y gives the functional identitywhereωn−1 is the volume of the (n−1)-sphere.
Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics[22]Many of the terms in this sum are trivially zero. The values of and that result in non-zero terms in this sum are determined by the selection rules for the3j-symbols.
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 Wigner3-jm symbol, theRacah coefficients, and theSlater integrals. Abstractly, the Clebsch–Gordan coefficients express thetensor product of twoirreducible representations of therotation group as a sum of irreducible representations: suitably normalized, the coefficients are then the multiplicities.
Schematic representation of on the unit sphere and its nodal lines. is equal to 0 alongmgreat circles passing through the poles, and alongℓ−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 degreen = 5. Note thatn =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 ofℓ circles: there are|m| circles along longitudes andℓ−|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 possessℓ−|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'.[23]
When the spherical harmonic orderm is zero (upper-left in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to aszonal. Such spherical harmonics are a special case ofzonal spherical functions. Whenℓ = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to assectoral. For the other cases, the functionschecker the sphere, and they are referred to astesseral.
More general spherical harmonics of degreeℓ are not necessarily those of the Laplace basis, and their nodal sets can be of a fairly general kind.[24]
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.[25] LetPℓ denote thespace of complex-valuedhomogeneous polynomials of degreeℓ inn real variables, here considered as functions. That is, a polynomialp is inPℓ provided that for any real, one has
LetAℓ denote the subspace ofPℓ consisting of allharmonic polynomials:These are the (regular)solid spherical harmonics. LetHℓ denote the space of functions on the unit sphereobtained by restriction fromAℓ
The following properties hold:
The sum of the spacesHℓ isdense in the set of continuous functions on with respect to theuniform topology, by theStone–Weierstrass theorem. As a result, the sum of these spaces is also dense in the spaceL2(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 theL2 sense.
For allf ∈Hℓ, one has whereΔSn−1 is theLaplace–Beltrami operator onSn−1. This operator is the analog of the angular part of the Laplacian in three dimensions; to wit, the Laplacian inn dimensions decomposes as
It follows from theStokes theorem and the preceding property that the spacesHℓ are orthogonal with respect to the inner product fromL2(Sn−1). That is to say, forf ∈Hℓ andg ∈Hk fork ≠ℓ.
Conversely, the spacesHℓ are precisely the eigenspaces ofΔSn−1. In particular, an application of thespectral theorem to theRiesz potential gives another proof that the spacesHℓ are pairwise orthogonal and complete inL2(Sn−1).
Every homogeneous polynomialp ∈Pℓ can be uniquely written in the form[26] wherepj ∈Aj. In particular,
An orthogonal basis of spherical harmonics in higher dimensions can be constructedinductively by the method ofseparation of variables, by solving the Sturm-Liouville problem for the spherical Laplacianwhereφ is the axial coordinate in a spherical coordinate system onSn−1. The end result of such a procedure is[27]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 theLegendre function
The spaceHℓ of spherical harmonics of degreeℓ is arepresentation of the symmetrygroup of rotations around a point (SO(3)) and its double-coverSU(2). Indeed, rotations act on the two-dimensionalsphere, and thus also onHℓ by function compositionforψ a spherical harmonic andρ a rotation. The representationHℓ is anirreducible representation of SO(3).[28]
The elements ofHℓ arise as the restrictions to the sphere of elements ofAℓ: harmonic polynomials homogeneous of degreeℓ on three-dimensional Euclidean spaceR3. Bypolarization ofψ ∈Aℓ, there are coefficients symmetric on the indices, uniquely determined by the requirementThe condition thatψ be harmonic is equivalent to the assertion that thetensor must betrace free on every pair of indices. Thus as an irreducible representation ofSO(3),Hℓ is isomorphic to the space of tracelesssymmetric tensors of degreeℓ.
More generally, the analogous statements hold in higher dimensions: the spaceHℓ of spherical harmonics on then-sphere is the irreducible representation ofSO(n+1) corresponding to the traceless symmetricℓ-tensors. However, whereas every irreducible tensor representation ofSO(2) andSO(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 additionalspin representations that are not tensor representations, and aretypically not spherical harmonics. An exception are thespin representation of SO(3): strictly speaking these are representations of thedouble cover SU(2) of SO(3). In turn, SU(2) is identified with the group of unitquaternions, and so coincides with the3-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 sets of functions.[29] One is hemispherical harmonics (HSH), orthogonal and complete on hemisphere. Another is complementary hemispherical harmonics (CHSH).
Theangle-preserving symmetries of thetwo-sphere are described by the group ofMöbius transformations PSL(2,C). With respect to this group, the sphere is equivalent to the usualRiemann 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 thecelestial sphere inMinkowski space. The analog of the spherical harmonics for the Lorentz group is given by thehypergeometric series; furthermore, the spherical harmonics can be re-expressed in terms of the hypergeometric series, asSO(3) = PSU(2) is asubgroup ofPSL(2,C).
More generally, hypergeometric series can be generalized to describe the symmetries of anysymmetric space; in particular, hypergeometric series can be developed for anyLie group.[30][31][32][33]
^A historical account of various approaches to spherical harmonics in three dimensions can be found in Chapter IV ofMacRobert 1967. The term "Laplace spherical harmonics" is in common use; seeCourant & Hilbert 1962 andMeijer & Bauer 2004.
^The approach to spherical harmonics taken here is found in (Courant & Hilbert 1962, §V.8, §VII.5).
^Physical applications often take the solution that vanishes at infinity, makingA = 0. This does not affect the angular portion of the spherical harmonics.
^Williams, Earl G. (1999).Fourier acoustics: sound radiation and nearfield acoustical holography. San Diego, Calif.: Academic Press.ISBN0-08-050690-9.OCLC181010993.
^Messiah, Albert (1999).Quantum mechanics: two volumes bound as one (Two vol. bound as one, unabridged reprint ed.). Mineola, NY: Dover. pp. 520–523.ISBN0-486-40924-4.
^Claude Cohen-Tannoudji; Bernard Diu; Franck Laloë (1996).Quantum mechanics. Translated by Susan Reid Hemley; et al. Wiley-Interscience: Wiley.ISBN978-0-471-56952-7.
D. A. Varshalovich, A. N. Moskalev, V. K. KhersonskiiQuantum Theory of Angular Momentum,(1988) World Scientific Publishing Co., Singapore,ISBN9971-5-0107-4
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