Special mathematical functions defined on the surface of a sphere
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 $Y_{\ell }^{m}(\theta ,\phi )$ in angular direction $(\theta ,\phi )$.
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 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$\ell$ in $(x,y,z)$ that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence $r^{\ell }$ from the above-mentioned polynomial of degree $\ell$; the remaining factor can be regarded as a function of the spherical angular coordinates $\theta$ and $\varphi$ only, or equivalently of the orientationalunit vector$\mathbf {r}$ 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 $Y_{\ell }^{m}(\theta ,\varphi )$ or $Y_{\ell }^{m}({\mathbf {r} })$, 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.
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 r_{1} = |x_{1}|. He discovered that if r ≤ r_{1} then
where γ is the angle between the vectors x and x_{1}. The functions $P_{i}:[-1,1]\to \mathbb {R}$ 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 x_{1} 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.
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 $S^{2}\to \mathbb {C}$ are eigenfunctions of the square of the orbital angular momentum operator
Real (Laplace) spherical harmonics Y_{ℓm} for ℓ = 0, …, 4 (top to bottom) and m = 0, …, ℓ (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 $Y_{\ell (-m)}$ would be shown rotated about the z axis by $90^{\circ }/m$ with respect to the positive order ones.)
Alternative picture for the real spherical harmonics $Y_{\ell m}$.
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 $f:\mathbb {R} ^{3}\to \mathbb {C}$.) 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 = 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 polynomialP^{m} _{ℓ}(cos θ) . Finally, the equation for R has solutions of the form R(r) = A r^{ℓ} + B r^{−ℓ − 1}; requiring the solution to be regular throughout R^{3} 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 ℓ, there are 2ℓ + 1 independent solutions of this form, one for each integer m with −ℓ ≤ m ≤ ℓ. These angular solutions $Y_{\ell }^{m}:S^{2}\to \mathbb {C}$ are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:
Here $Y_{\ell }^{m}:S^{2}\to \mathbb {C}$ is called a spherical harmonic function of degree ℓ and order m, $P_{\ell }^{m}:[-1,1]\to \mathbb {R}$ 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), $Y:S^{2}\to \mathbb {C}$, of the eigenvalue problem
$r^{2}\nabla ^{2}Y=-\ell (\ell +1)Y$
is a linear combination of $Y_{\ell }^{m}:S^{2}\to \mathbb {C}$. In fact, for any such solution, r^{ℓ} Y(θ, φ) is the expression in spherical coordinates of a homogeneous polynomial$\mathbb {R} ^{3}\to \mathbb {C}$ that is harmonic (see below), and so counting dimensions shows that there are 2ℓ + 1 linearly independent such polynomials.
The general solution $f:\mathbb {R} ^{3}\to \mathbb {C}$ to Laplace's equation$\Delta f=0$ in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor r^{ℓ},
where the $f_{\ell }^{m}\in \mathbb {C}$ are constants and the factors r^{ℓ} Y_{ℓ}^{m} are known as (regular) solid harmonics$\mathbb {R} ^{3}\to \mathbb {C}$. Such an expansion is valid in the ball
For $r>R$, the solid harmonics with negative powers of $r$ (the irregularsolid harmonics$\mathbb {R} ^{3}\setminus \{\mathbf {0} \}\to \mathbb {C}$) are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about $r=\infty$), instead of the Taylor series (about $r=0$) used above, to match the terms and find series expansion coefficients $f_{\ell }^{m}\in \mathbb {C}$.
Orbital angular momentum
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:
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
$\mathbf {L} ^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}$
and each of L_{x}, L_{y}, L_{z} are self-adjoint, it follows that λ ≥ m^{2}.
Then L_{+} and L_{−} commute with L^{2}, and the Lie algebra generated by L_{+}, L_{−}, L_{z} is the special linear Lie algebra of order 2, ${\mathfrak {sl}}_{2}(\mathbb {C} )$, with commutation relations
Thus L_{+} : E_{λ,m} → E_{λ,m+1} (it is a "raising operator") and L_{−} : E_{λ,m} → E_{λ,m−1} (it is a "lowering operator"). In particular, L^{k} _{+} : E_{λ,m} → E_{λ,m+k} must be zero for k sufficiently large, because the inequality λ ≥ m^{2} must hold in each of the nontrivial joint eigenspaces. Let Y ∈ E_{λ,m} be a nonzero joint eigenfunction, and let k be the least integer such that
$L_{+}^{k}Y=0.$
Then, since
$L_{-}L_{+}=\mathbf {L} ^{2}-L_{z}^{2}-L_{z}$
it follows that
$0=L_{-}L_{+}^{k}Y=(\lambda -(m+k)^{2}-(m+k))Y.$
Thus λ = ℓ(ℓ + 1) for the positive integer ℓ = m + k.
The foregoing has been all worked out in the spherical coordinate representation, $\langle \theta ,\phi |lm\rangle =Y_{l}^{m}(\theta ,\phi )$ but may be expressed more abstractly in the complete, orthonormal spherical ket basis.
Harmonic polynomial representation
The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions $\mathbb {R} ^{3}\to \mathbb {C}$. Specifically, we say that a (complex-valued) polynomial function $p:\mathbb {R} ^{3}\to \mathbb {C}$ is homogeneous of degree $\ell$ if
for all real numbers $\lambda \in \mathbb {R}$ and all $x\in \mathbb {R} ^{3}$. We say that $p$ is harmonic if
$\Delta p=0,$
where $\Delta$ is the Laplacian. Then for each $\ell$, we define
$\mathbf {A} _{\ell }=\left\{{\text{harmonic polynomials }}\mathbb {R} ^{3}\to \mathbb {C} {\text{ that are homogeneous of degree }}\ell \right\}.$
For example, when $\ell =1$, $\mathbf {A} _{1}$ is just the 3-dimensional space of all linear functions $\mathbb {R} ^{3}\to \mathbb {C}$, since any such function is automatically harmonic. Meanwhile, when $\ell =2$, we have a 5-dimensional space:
For any $\ell$, the space $\mathbf {H} _{\ell }$ of spherical harmonics of degree $\ell$ is just the space of restrictions to the sphere $S^{2}$ of the elements of $\mathbf {A} _{\ell }$.^{[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 $c\in \mathbb {C}$ the formula
$p(x_{1},x_{2},x_{3})=c(x_{1}+ix_{2})^{\ell }$
defines a homogeneous polynomial of degree $\ell$ with domain and codomain $\mathbb {R} ^{3}\to \mathbb {C}$, which happens to be independent of $x_{3}$. This polynomial is easily seen to be harmonic. If we write $p$ in spherical coordinates $(r,\theta ,\phi )$ and then restrict to $r=1$, we obtain
After using the formula for the associated Legendre polynomial$P_{\ell }^{\ell }$, we may recognize this as the formula for the spherical harmonic $Y_{\ell }^{\ell }(\theta ,\phi ).$^{[6]} (See the section below on special cases of the spherical harmonics.)
Conventions
Orthogonality and normalization
Several different normalizations are in common use for the Laplace spherical harmonic functions $S^{2}\to \mathbb {C}$. Throughout the section, we use the standard convention that for $m>0$ (see associated Legendre polynomials)
where δ_{ij} is the Kronecker delta and dΩ = sin(θ) dφdθ. This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e.,
$\int {|Y_{\ell }^{m}|^{2}d\Omega }=1.$
The disciplines of geodesy^{[10]} and spectral analysis use
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
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
A real basis of spherical harmonics $Y_{\ell m}:S^{2}\to \mathbb {R}$ can be defined in terms of their complex analogues $Y_{\ell }^{m}:S^{2}\to \mathbb {C}$ by setting
The Condon–Shortley phase convention is used here for consistency. The corresponding inverse equations defining the complex spherical harmonics $Y_{\ell }^{m}:S^{2}\to \mathbb {C}$ in terms of the real spherical harmonics $Y_{\ell m}:S^{2}\to \mathbb {R}$ are
The real spherical harmonics $Y_{\ell m}:S^{2}\to \mathbb {R}$ are sometimes known as tesseral spherical harmonics.^{[12]} These functions have the same orthonormality properties as the complex ones $Y_{\ell }^{m}:S^{2}\to \mathbb {C}$ above. The real spherical harmonics $Y_{\ell m}$ 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 $\ell =4$, which can be seen to be consistent with the output of the equations above.
Use in quantum chemistry
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 ($\ell =1$) are complex and mix axis directions, but the real versions are essentially just x, y, and z.
Spherical harmonics in Cartesian form
The complex spherical harmonics $Y_{\ell }^{m}$ give rise to the solid harmonics by extending from $S^{2}$ to all of $\mathbb {R} ^{3}$ as a homogeneous function of degree $\ell$, i.e. setting
It turns out that $R_{\ell }^{m}$ is basis of the space of harmonic and homogeneous polynomials of degree $\ell$. More specifically, it is the (unique up to normalization) Gelfand-Tsetlin-basis of this representation of the rotational group $SO(3)$ and an explicit formula for $R_{\ell }^{m}$ in cartesian coordinates can be derived from that fact.
The Herglotz generating function
If the quantum mechanical convention is adopted for the $Y_{\ell }^{m}:S^{2}\to \mathbb {C}$, then
is a vector with complex coefficients. It suffices to take $v$ and $\lambda$ as real parameters. The essential property of $\mathbf {a}$ is that it is null:
$\mathbf {a} \cdot \mathbf {a} =0.$
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 $\mathbf {r}$ is replaced by the quantum mechanical spin vector operator $\mathbf {J}$, such that ${\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })$ is the operator analogue of the solid harmonic$r^{\ell }Y_{\ell }^{m}(\mathbf {r} /r)$,^{[14]} one obtains a generating function for a standardized set of spherical tensor operators, ${\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })$:
The parallelism of the two definitions ensures that the ${\mathcal {Y}}_{\ell }^{m}$'s transform under rotations (see below) in the same way as the $Y_{\ell }^{m}$'s, which in turn guarantees that they are spherical tensor operators, $T_{q}^{(k)}$, with $k={\ell }$ and $q=m$, 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
The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of $z$ and another of $x$ and $y$, as follows (Condon–Shortley phase):
The factor ${\bar {\Pi }}_{\ell }^{m}(z)$ is essentially the associated Legendre polynomial $P_{\ell }^{m}(\cos \theta )$, and the factors $(A_{m}\pm iB_{m})$ are essentially $e^{\pm im\varphi }$.
Examples
Using the expressions for ${\bar {\Pi }}_{\ell }^{m}(z)$, $A_{m}(x,y)$, and $B_{m}(x,y)$ listed explicitly above we obtain:
It may be verified that this agrees with the function listed here and here.
Real forms
Using the equations above to form the real spherical harmonics, it is seen that for $m>0$ only the $A_{m}$ terms (cosines) are included, and for $m<0$ only the $B_{m}$ terms (sines) are included:
The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation.
Parity
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 $P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )$. Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with $\mathbf {r}$ being a unit vector,
In terms of the spherical angles, parity transforms a point with coordinates $\{\theta ,\phi \}$ to $\{\pi -\theta ,\pi +\phi \}$. 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}.
Rotations
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 ${\mathcal {R}}$ about the origin that sends the unit vector $\mathbf {r}$ to $\mathbf {r} '$. Under this operation, a spherical harmonic of degree $\ell$ and order $m$ transforms into a linear combination of spherical harmonics of the same degree. That is,
where $A_{mm'}$ is a matrix of order $(2\ell +1)$ that depends on the rotation ${\mathcal {R}}$. However, this is not the standard way of expressing this property. In the standard way one writes,
where $D_{mm'}^{(\ell )}({\mathcal {R}})^{*}$ is the complex conjugate of an element of the Wigner D-matrix. In particular when $\mathbf {r} '$ is a $\phi _{0}$ 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 $Y_{\ell }^{m}$'s of degree $\ell$ provide a basis set of functions for the irreducible representation of the group SO(3) of dimension $(2\ell +1)$. 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
The Laplace spherical harmonics $Y_{\ell }^{m}:S^{2}\to \mathbb {C}$ form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions$L_{\mathbb {C} }^{2}(S^{2})$. On the unit sphere $S^{2}$, any square-integrable function $f:S^{2}\to \mathbb {C}$ can thus be expanded as a linear combination of these:
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 $f:S^{2}\to \mathbb {R}$ can also be expanded in terms of the real harmonics $Y_{\ell m}:S^{2}\to \mathbb {R}$ above as a sum
The convergence of the series holds again in the same sense, namely the real spherical harmonics $Y_{\ell m}:S^{2}\to \mathbb {R}$ form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions$L_{\mathbb {R} }^{2}(S^{2})$. The benefit of the expansion in terms of the real harmonic functions $Y_{\ell m}$ is that for real functions $f:S^{2}\to \mathbb {R}$ the expansion coefficients $f_{\ell m}$ are guaranteed to be real, whereas their coefficients $f_{\ell }^{m}$ in their expansion in terms of the $Y_{\ell }^{m}$ (considering them as functions $f:S^{2}\to \mathbb {C} \supset \mathbb {R}$) do not have that property.
Harmonic tensors
Formula
As a rule, harmonic functions are useful in theoretical physics to consider fields in the far field when the distance from charges is much farther than the size of their location. In this case, the radius R is constant and coordinates (θ,φ) are convenient to use. Theoretical physics considers many problems when a solution of Laplace's equation is needed as a function of Сartesian coordinates. At the same time, it is important to get an invariant form of solutions relative to the rotation of space or, generally speaking, relative 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 harmonic functions. The total set of tensors is defined by the Taylor series of a point charge field potential for $r_{0}<r$:
where tensor $x_{0i}\ldots x_{0k}$ is denoted by symbol $\otimes {\boldsymbol {r_{0}}}^{l}$ and contraction of the tensors is in the brackets [...].
Therefore, the tensor ${\boldsymbol {T}}^{(l)}$ is defined by the $l$-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 the following properties that repeat mainly those of solid and spherical functions.
The tensor is the harmonic polynomial i. e. $\Delta {\boldsymbol {T}}^{(l)}=0$.
The trace over each pair of indices is zero, as far as ${\textstyle \Delta {\frac {1}{r}}=0}$.
The tensor is a homogeneous polynomial of degree $l$ i.e. summed degree of variables x, y, z of each item is equal to $l$.
The tensor has invariant form under rotations of variables x,y,z i.e. of vector $\mathbf {r}$.
The total set of potentials $\mathbf {T} ^{(l)}$ is complete.
Contraction of $\mathbf {T} ^{(l)}(\mathbf {r} )$ with a tensor $\otimes {\boldsymbol {\rho }}^{l}$ is proportional to contraction of two harmonic potentials:
The formula for a harmonic invariant tensor was found in paper.^{[21]} A detailed description is given in the monograph.^{[22]} 4-D harmonic tensors are important in Fock symmetry found in quantum the Coulomb problem.^{[23]}
The formula contains products of tensors $x_{i}\ldots x_{k}=\mathbf {\otimes } r^{m}$ and Kronecker symbols$\delta _{ik}$:
The number of Kronecker symbols is increased by two in the product of each following item when the range of tensors $x_{i}\dots x_{k}$ is reduced by two accordingly. The operation $\left\langle \cdot \right\rangle$ symmetrizes a tensor by means of summing all independent permutations of indices. Particularly, each $\delta _{ik}$ does not need to be transformed into $\delta _{ki}$ and tensors $x_{i}x_{k}$ do not become $x_{k}x_{i}$.
These tensors are convenient to substitute into Laplace's equation:
The last relation is Euler's formula for homogeneous polynomials. The Laplace operator $\Delta$ does not affect the index symmetry of tensors. The two relations allow substitution of a tensor into Laplace's equation to check directly that the tensor is a harmonic function:
$\Delta \mathbf {T} ^{(l)}=0.$
Simplified moments
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 ${f}(\mathbf {r} )$ is the charge density. The convolution is applied to tensors in the formula naturally. Integrals in the sum are called multipole moments in physics. Three of them are used actively while others are applied less often as their structure (or that of spherical functions) is more complicated. Nevertheless, the last property gives the way to simplify calculations in theoretical physics by using integrals with tensor ${\boldsymbol {\otimes }}r^{l}$ instead of harmonic tensor $\mathbf {T} ^{(l)}$. 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
Spherical functions have a few recurrent formulas.^{[24]} In quantum mechanics recurrent formulas plays a role when they connect $\psi -$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 harmonic states found in paper^{[21]} and detailed in.^{[22]}
For that purpose, transformation of ${\boldsymbol {\rho }}$-space is applied that conserves form of Laplace equation:
Operator ${\boldsymbol {\nabla }}$ applying to the harmonic tensor potential in ${\boldsymbol {\rho }}$-space goes into Efimov's ladder operator acting on transformed tensor in $\mathbf {r}$-space:
Operator ${\hat {l}}$ multiplies harmonic tensor by its degree i.e. by $l$ if to recall according spherical function for quantum numbers$l$, $m$.
To check action of the ladder operator $\mathbf {\hat {D}}$, one can apply it to dipole and quadrupole tensors:
The operator $\mathbf {\hat {D}}$ analogous to the oscillatorladder operator. To trace relation with a quantum operator it is useful to multiply it by $i\hbar$ to go to reversed space:
The property gives zero trace of the harmonic tensor $\mathbf {T} ^{l}$ 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.^{[25]} 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.^{[26]}
Invariant form of spherical harmonics
Spherical harmonics accord with the system of coordinates. Let be $\mathbf {n} _{x},\mathbf {n} _{y},\mathbf {n} _{z}$ the unit vectors along axes X, Y, Z. Denote following unit vectors as $\mathbf {n} _{+}$ and $\mathbf {n} _{-}$:
Angular momentum $\mathbf {\hat {L}}$ is defined by the rotational group. The mechanical momentum $\mathbf {\hat {p}}$ 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 $i$ to leave others:
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 f_{00} and g_{00} are zero), then S_{ff}(ℓ) and S_{fg}(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
$S_{f\!f}(\ell )=C\,\ell ^{\beta }.$
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 S_{ff}(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 S_{ff}(ℓ) to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. Specifically, if
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 $(r,\theta ,\varphi )$ and $(r',\theta ',\varphi ')$, respectively, the angle $\gamma$ 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 P_{ℓ} is the Legendre polynomial of degree ℓ. This expression is valid for both real and complex harmonics.^{[28]} 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.^{[29]}
In particular, when x = y, this gives Unsöld's theorem^{[30]}
which generalizes the identity cos^{2}θ + sin^{2}θ = 1 to two dimensions.
In the expansion (1), the left-hand side P_{ℓ}(x⋅y) is a constant multiple of the degree ℓzonal spherical harmonic. From this perspective, one has the following generalization to higher dimensions. Let Y_{j} be an arbitrary orthonormal basis of the space H_{ℓ} of degree ℓ spherical harmonics on the n-sphere. Then $Z_{\mathbf {x} }^{(\ell )}$, the degree ℓ zonal harmonic corresponding to the unit vector x, decomposes as^{[31]}
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 the values of $c$ and $\gamma$ are determined by the selection rules for the 3j-symbols.
Clebsch–Gordan coefficients
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
Schematic representation of $Y_{\ell m}$ on the unit sphere and its nodal lines. $\Re [Y_{\ell m}]$ is equal to 0 along mgreat 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 $Y_{\ell }^{m}$ can be visualized by considering their "nodal lines", that is, the set of points on the sphere where $\Re [Y_{\ell }^{m}]=0$, or alternatively where $\Im [Y_{\ell }^{m}]=0$. Nodal lines of $Y_{\ell }^{m}$ 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 $Y_{\ell }^{m}$ in the $\theta$ and $\varphi$ directions respectively. Considering $Y_{\ell }^{m}$ as a function of $\theta$, 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 $Y_{\ell }^{m}$ as a function of $\varphi$, 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 $Y_{\ell }^{m}$, and their nodal sets can be of a fairly general kind.^{[33]}
List of spherical harmonics
Analytic expressions for the first few orthonormalized Laplace spherical harmonics $Y_{\ell }^{m}:S^{2}\to \mathbb {C}$ that use the Condon–Shortley phase convention:
The classical spherical harmonics are defined as complex-valued functions on the unit sphere $S^{2}$ inside three-dimensional Euclidean space $\mathbb {R} ^{3}$. Spherical harmonics can be generalized to higher-dimensional Euclidean space $\mathbb {R} ^{n}$ as follows, leading to functions $S^{n-1}\to \mathbb {C}$.^{[34]} Let P_{ℓ} denote the space of complex-valued homogeneous polynomials of degree ℓ in n real variables, here considered as functions $\mathbb {R} ^{n}\to \mathbb {C}$. That is, a polynomial p is in P_{ℓ} provided that for any real $\lambda \in \mathbb {R}$, one has
$\mathbf {H} _{\ell }:=\left\{f:S^{n-1}\to \mathbb {C} \,\mid \,{\text{ for some }}p\in \mathbf {A} _{\ell },\,f(\mathbf {x} )=p(\mathbf {x} ){\text{ for all }}\mathbf {x} \in S^{n-1}\right\}.$
The following properties hold:
The sum of the spaces H_{ℓ} is dense in the set $C(S^{n-1})$ of continuous functions on $S^{n-1}$ 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 L^{2}(S^{n−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 L^{2} sense.
For all f ∈ H_{ℓ}, one has
$\Delta _{S^{n-1}}f=-\ell (\ell +n-2)f.$
where Δ_{Sn−1} is the Laplace–Beltrami operator on S^{n−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_{ℓ} are orthogonal with respect to the inner product from L^{2}(S^{n−1}). That is to say,
Conversely, the spaces H_{ℓ} are precisely the eigenspaces of Δ_{Sn−1}. In particular, an application of the spectral theorem to the Riesz potential$\Delta _{S^{n-1}}^{-1}$ gives another proof that the spaces H_{ℓ} are pairwise orthogonal and complete in L^{2}(S^{n−1}).
Every homogeneous polynomial p ∈ P_{ℓ} can be uniquely written in the form^{[35]}
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 the indices satisfy |ℓ_{1}| ≤ ℓ_{2} ≤ ⋯ ≤ ℓ_{n−1} and the eigenvalue is −ℓ_{n−1}(ℓ_{n−1} + n−2). The functions in the product are defined in terms of the Legendre function
The space H_{ℓ} 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_{ℓ} by function composition
$\psi \mapsto \psi \circ \rho ^{-1}$
for ψ a spherical harmonic and ρ a rotation. The representation H_{ℓ} is an irreducible representation of SO(3).^{[37]}
The elements of H_{ℓ} arise as the restrictions to the sphere of elements of A_{ℓ}: harmonic polynomials homogeneous of degree ℓ on three-dimensional Euclidean space R^{3}. By polarization of ψ ∈ A_{ℓ}, there are coefficients $\psi _{i_{1}\dots i_{\ell }}$ symmetric on the indices, uniquely determined by the requirement
The condition that ψ be harmonic is equivalent to the assertion that the tensor$\psi _{i_{1}\dots i_{\ell }}$ must be trace free on every pair of indices. Thus as an irreducible representation of SO(3), H_{ℓ} is isomorphic to the space of traceless symmetric tensors of degree ℓ.
More generally, the analogous statements hold in higher dimensions: the space H_{ℓ} 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
Spherical harmonics can be separated into two set of functions.^{[38]} One is hemispherical functions (HSH), orthogonal and complete on hemisphere. Another is complementary hemispherical harmonics (CHSH).
Generalizations
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 {{{1}}} 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.^{[39]}^{[40]}^{[41]}^{[42]}
See also
Wikimedia Commons has media related to Spherical harmonics.
Cubic harmonic (often used instead of spherical harmonics in computations)
^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.
^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, making A = 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. ISBN 0080506909. OCLC 181010993.
^Messiah, Albert (1999). Quantum mechanics : two volumes bound as one (Two vol. bound as one, unabridged reprint ed.). Mineola, NY: Dover. ISBN 9780486409245.
^Claude Cohen-Tannoudji; Bernard Diu; Franck Laloë (1996). Quantum mechanics. Translated by Susan Reid Hemley; et al. Wiley-Interscience: Wiley. ISBN 9780471569527.
^ ^{a}^{b}Blakely, Richard (1995). Potential theory in gravity and magnetic applications. Cambridge England New York: Cambridge University Press. p. 113. ISBN 978-0521415088.
^Heiskanen and Moritz, Physical Geodesy, 1967, eq. 1-62
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^Edmonds, A. R. (1996). Angular Momentum In Quantum Mechanics. Princeton University Press. p. 63.
^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.
^Cf. Corollary 1.8 of Axler, Sheldon; Ramey, Wade (1995), Harmonic Polynomials and Dirichlet-Type Problems
^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.
^Zheng Y, Wei K, Liang B, Li Y, Chu X (2019-12-23). "Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering". Optics Express (in English). 27 (26): 37180–37195. Bibcode:2019OExpr..2737180Z. doi:10.1364/OE.27.037180. ISSN 1094-4087. PMID31878503.
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