Spherical harmonics

Laplace's 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 ${\displaystyle f:{\mathbb{R}}^3\to \mathbb{C}}$.) In spherical coordinates this is:

$${\displaystyle {\mathrm{\nabla }}^{2}f=\frac{1}{r^2}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r^2\frac{\mathrm{\partial }f}{\mathrm{\partial }r}\right)+\frac{1}{r^2\sin \theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(\sin \theta \frac{\mathrm{\partial }f}{\mathrm{\partial }\theta }\right)+\frac{1}{r^2{\sin }^2\theta }\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{\phi }^2}=0.}$$

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: $${\displaystyle \frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)=\lambda ,\frac{1}{Y}\frac{1}{\sin \theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left(\sin \theta \frac{\mathrm{\partial }Y}{\mathrm{\partial }\theta }\right)+\frac{1}{Y}\frac{1}{{\sin }^2\theta }\frac{{\mathrm{\partial }}^{2}Y}{\mathrm{\partial }{\phi }^2}=-\lambda .}$$ 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

$${\displaystyle \frac{1}{\mathrm{\Phi }}\frac{d^2\mathrm{\Phi }}{d{\phi }^2}=-m^2}$$ $${\displaystyle \lambda {\sin }^2\theta +\frac{\sin \theta }{\mathrm{\Theta }}\frac{d}{d\theta }\left(\sin \theta \frac{d\mathrm{\Theta }}{d\theta }\right)=m^2}$$

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 polynomial P^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 R³ forces B = 0.

Here the solution was assumed to have the special form Y(θ, φ) = Θ(θ) Φ(φ). For a given value of ℓ, there are 2ℓ + 1 independent solutions of this form, one for each integer m with −ℓ ≤ m ≤ ℓ. These angular solutions ${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$ are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:

$${\displaystyle Y_{\ell }^m(\theta ,\phi )=N{e}^{im\phi }{P}_{\ell }^m(\cos \theta )}$$

which fulfill $${\displaystyle r^2{\mathrm{\nabla }}^2{Y}_{\ell }^m(\theta ,\phi )=-\ell (\ell +1)Y_{\ell }^m(\theta ,\phi ).}$$

Here ${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$ is called a spherical harmonic function of degree ℓ and order m, ${\displaystyle P_{\ell }^m:[-1,1]\to \mathbb{R}}$ is an associated Legendre polynomial, N is a normalization constant, and θ and φ represent colatitude and longitude, respectively. In particular, the 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 ℓ, every solution Y(θ, φ), ${\displaystyle Y:S^2\to \mathbb{C}}$, of the eigenvalue problem $${\displaystyle r^2{\mathrm{\nabla }}^{2}Y=-\ell (\ell +1)Y}$$ is a linear combination of ${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$. In fact, for any such solution, r^ℓ Y(θ, φ) is the expression in spherical coordinates of a homogeneous polynomial ${\displaystyle {\mathbb{R}}^3\to \mathbb{C}}$ that is harmonic (see below), and so counting dimensions shows that there are 2ℓ + 1 linearly independent such polynomials.

The general solution ${\displaystyle f:{\mathbb{R}}^3\to \mathbb{C}}$ to Laplace's equation ${\displaystyle \mathrm{\Delta }f=0}$ in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor r^ℓ,

$${\displaystyle f(r,\theta ,\phi )=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }{f}_{\ell }^m{r}^{\ell }{Y}_{\ell }^m(\theta ,\phi ),}$$

where the ${\displaystyle f_{\ell }^m\in \mathbb{C}}$ are constants and the factors r^ℓ Y_ℓ^m are known as (regular) solid harmonics ${\displaystyle {\mathbb{R}}^3\to \mathbb{C}}$. Such an expansion is valid in the ball

$${\displaystyle r<R=\frac{1}{\underset{\ell \to \mathrm{\infty }}{lim sup}|f_{\ell }^m{|}^{1/\ell }}.}$$

For ${\displaystyle r>R}$, the solid harmonics with negative powers of ${\displaystyle r}$ (the irregular solid harmonics ${\displaystyle {\mathbb{R}}^3\setminus \{\mathbf{0}\}\to \mathbb{C}}$) are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about ${\displaystyle r=\mathrm{\infty }}$), instead of the Taylor series (about ${\displaystyle r=0}$) used above, to match the terms and find series expansion coefficients ${\displaystyle f_{\ell }^m\in \mathbb{C}}$.

Orbital angular momentum

In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum $${\displaystyle \mathbf{L}=-i\hslash (\mathbf{x}\times \mathrm{\nabla })=L_x\mathbf{i}+L_y\mathbf{j}+L_z\mathbf{k}.}$$ 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 $${\displaystyle \begin{array}{cc}{\mathbf{L}}^2& =-r^2{\mathrm{\nabla }}^2+\left(r\frac{\mathrm{\partial }}{\mathrm{\partial }r}+1\right)r\frac{\mathrm{\partial }}{\mathrm{\partial }r}\\ & =-\frac{1}{\sin \theta }\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\sin \theta \frac{\mathrm{\partial }}{\mathrm{\partial }\theta }-\frac{1}{{\sin }^2\theta }\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\phi }^2}.\end{array}}$$ 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: $${\displaystyle \begin{array}{cc}{L}_z& =-i\left(x\frac{\mathrm{\partial }}{\mathrm{\partial }y}-y\frac{\mathrm{\partial }}{\mathrm{\partial }x}\right)\\ & =-i\frac{\mathrm{\partial }}{\mathrm{\partial }\phi }.\end{array}}$$

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: $${\displaystyle \frac{1}{(2\pi {)}^{3/2}}{\int }_{{\mathbb{R}}^3}|f(x){|}^2{e}^{-|x{|}^2/2}dx<\mathrm{\infty }.}$$ Furthermore, L² is a positive operator.

If Y is a joint eigenfunction of L² and L_z, then by definition $${\displaystyle \begin{array}{cc}{\mathbf{L}}^{2}Y& =\lambda Y\\ L_{z}Y& =mY\end{array}}$$ 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 $${\displaystyle {\mathbf{L}}^2=L_x^2+L_y^2+L_z^2}$$ and each of L_x, L_y, L_z are self-adjoint, it follows that λ ≥ m².

Denote this joint eigenspace by E_λ,m, and define the raising and lowering operators by $${\displaystyle \begin{array}{cc}{L}_{+}& =L_x+i{L}_y\\ L_{-}& =L_x-i{L}_y\end{array}}$$ Then L₊ and L₋ commute with L², and the Lie algebra generated by L₊, L₋, L_z is the special linear Lie algebra of order 2, ${\displaystyle {\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})}$, with commutation relations $${\displaystyle [L_z,L_{+}]=L_{+},[L_z,L_{-}]=-L_{-},[L_{+},L_{-}]=2L_z.}$$ 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 ≥ m² 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 $${\displaystyle L_{+}^{k}Y=0.}$$ Then, since $${\displaystyle L_{-}{L}_{+}={\mathbf{L}}^2-L_z^2-L_z}$$ it follows that $${\displaystyle 0=L_{-}{L}_{+}^{k}Y=(\lambda -(m+k{)}^2-(m+k))Y.}$$ Thus λ = ℓ(ℓ + 1) for the positive integer ℓ = m + k.

The foregoing has been all worked out in the spherical coordinate representation, ${\displaystyle ⟨\theta ,\phi |lm⟩=Y_l^m(\theta ,\phi )}$ 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 ${\displaystyle {\mathbb{R}}^3\to \mathbb{C}}$. Specifically, we say that a (complex-valued) polynomial function ${\displaystyle p:{\mathbb{R}}^3\to \mathbb{C}}$ is homogeneous of degree ${\displaystyle \ell }$ if $${\displaystyle p(\lambda \mathbf{x})={\lambda }^{\ell }p(\mathbf{x})}$$ for all real numbers ${\displaystyle \lambda \in \mathbb{R}}$ and all ${\displaystyle \mathbf{x}\in {\mathbb{R}}^3}$. We say that ${\displaystyle p}$ is harmonic if $${\displaystyle \mathrm{\Delta }p=0,}$$ where ${\displaystyle \mathrm{\Delta }}$ is the Laplacian. Then for each ${\displaystyle \ell }$, we define $${\displaystyle {\mathbf{A}}_{\ell }=\left\{\text{harmonic polynomials }{\mathbb{R}}^3\to \mathbb{C}\text{ that are homogeneous of degree }\ell \right\}.}$$

For example, when ${\displaystyle \ell =1}$, ${\displaystyle {\mathbf{A}}_1}$ is just the 3-dimensional space of all linear functions ${\displaystyle {\mathbb{R}}^3\to \mathbb{C}}$, since any such function is automatically harmonic. Meanwhile, when ${\displaystyle \ell =2}$, we have a 5-dimensional space: $${\displaystyle {\mathbf{A}}_2={\mathrm{span}}_{\mathbb{C}}(x_1{x}_2,x_1{x}_3,x_2{x}_3,x_1^2-x_2^2,2x_3^2-x_1^2-x_2^2).}$$

For any ${\displaystyle \ell }$, the space ${\displaystyle {\mathbf{H}}_{\ell }}$ of spherical harmonics of degree ${\displaystyle \ell }$ is just the space of restrictions to the sphere ${\displaystyle S^2}$ of the elements of ${\displaystyle {\mathbf{A}}_{\ell }}$. 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 ${\displaystyle c\in \mathbb{C}}$ the formula $${\displaystyle p(x_1,x_2,x_3)=c(x_1+i{x}_2{)}^{\ell }}$$ defines a homogeneous polynomial of degree ${\displaystyle \ell }$ with domain and codomain ${\displaystyle {\mathbb{R}}^3\to \mathbb{C}}$, which happens to be independent of ${\displaystyle x_3}$. This polynomial is easily seen to be harmonic. If we write ${\displaystyle p}$ in spherical coordinates ${\displaystyle (r,\theta ,\phi )}$ and then restrict to ${\displaystyle r=1}$, we obtain $${\displaystyle p(\theta ,\phi )=c\sin (\theta {)}^{\ell }(\cos (\phi )+i\sin (\phi ){)}^{\ell },}$$ which can be rewritten as $${\displaystyle p(\theta ,\phi )=c{\left(\sqrt{1-{\cos }^2(\theta )}\right)}^{\ell }{e}^{i\ell \phi }.}$$ After using the formula for the associated Legendre polynomial ${\displaystyle P_{\ell }^{\ell }}$, we may recognize this as the formula for the spherical harmonic ${\displaystyle Y_{\ell }^{\ell }(\theta ,\phi ).}$ (See Special cases.)

Conventions

Condon–Shortley phase

One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of ${\displaystyle (-1{)}^m}$, commonly referred to as the Condon–Shortley phase in the quantum mechanical literature. This phase factor may be included either in the definition of the associated Legendre polynomials or in the definition of the spherical harmonic functions, but it should not be counted twice.

In this section, unless otherwise stated, ${\displaystyle P_{\ell }^m}$ denotes the associated Legendre function with the Condon–Shortley phase included, so that for ${\displaystyle m>0}$ $${\displaystyle P_{\ell }^m(x)=(-1{)}^m(1-x^2{)}^{m/2}\frac{d^m}{d{x}^m}{P}_{\ell }(x),}$$ and $${\displaystyle P_{\ell }^{-m}=(-1{)}^m\frac{(\ell -m)!}{(\ell +m)!}{P}_{\ell }^m.}$$ This is the convention used, for example, by the NIST Digital Library of Mathematical Functions and in many mathematical references.

Some authors instead define associated Legendre functions without this phase factor. If these are denoted here by ${\displaystyle {\tilde{P}}_{\ell }^m}$, then $${\displaystyle {\tilde{P}}_{\ell }^m(x)=(1-x^2{)}^{m/2}\frac{d^m}{d{x}^m}{P}_{\ell }(x),P_{\ell }^m(x)=(-1{)}^m{\tilde{P}}_{\ell }^m(x).}$$ With this notation, a formula with an external factor ${\displaystyle (-1{)}^m}$ multiplying ${\displaystyle {\tilde{P}}_{\ell }^m}$ gives the same spherical harmonic as a formula using ${\displaystyle P_{\ell }^m}$ without an external phase factor.

There is no requirement to use the Condon–Shortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of raising and lowering operators. The geodesy and magnetics communities commonly use conventions in which the Condon–Shortley phase factor is not included in the associated Legendre functions or in the spherical harmonics.

Orthogonality and normalization

Several different normalizations are in common use for the Laplace spherical harmonic functions ${\displaystyle S^2\to \mathbb{C}}$. These normalization choices are independent of the placement of the Condon–Shortley phase.

With the Condon–Shortley phase included in ${\displaystyle P_{\ell }^m}$, the orthonormal complex spherical harmonics are $${\displaystyle Y_{\ell }^m(\theta ,\phi )=\sqrt{\frac{(2\ell +1)}{4\pi }\frac{(\ell -m)!}{(\ell +m)!}}{P}_{\ell }^m(\cos \theta )e^{im\phi }.}$$ This is the convention used in this article, and is common in acoustics.

The same orthonormal functions are often written in quantum mechanics as $${\displaystyle Y_{\ell }^m(\theta ,\phi )=(-1{)}^m\sqrt{\frac{(2\ell +1)}{4\pi }\frac{(\ell -m)!}{(\ell +m)!}}{\tilde{P}}_{\ell }^m(\cos \theta )e^{im\phi },}$$ where ${\displaystyle {\tilde{P}}_{\ell }^m}$ denotes the associated Legendre function without the Condon–Shortley phase. This avoids counting the phase twice.

With either placement of the Condon–Shortley phase, the orthonormal spherical harmonics satisfy $${\displaystyle {\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{Y}_{\ell }^m{Y}_{{\ell }^{\prime }}^{m^{\prime }}{}^{\ast }d\mathrm{\Omega }={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }},}$$ where δ_ij is the Kronecker delta and dΩ = sin(θ) dφ dθ. This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e., $${\displaystyle \int |Y_{\ell }^m{|}^{2}d\mathrm{\Omega }=1.}$$

The disciplines of geodesy and spectral analysis often use the corresponding ${\displaystyle 4\pi }$-normalized harmonics. Written with the same phase convention as above, these are $${\displaystyle Y_{\ell }^m(\theta ,\phi )=\sqrt{(2\ell +1)\frac{(\ell -m)!}{(\ell +m)!}}{P}_{\ell }^m(\cos \theta )e^{im\phi },}$$ which possess unit power $${\displaystyle \frac{1}{4\pi }{\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{Y}_{\ell }^m{Y}_{{\ell }^{\prime }}^{m^{\prime }}{}^{\ast }d\mathrm{\Omega }={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}.}$$ In geodetic applications this normalization is commonly combined with omission of the Condon–Shortley phase.

The magnetics community, in contrast, often uses Schmidt semi-normalized harmonics. Written with the same phase convention as above, these are $${\displaystyle Y_{\ell }^m(\theta ,\phi )=\sqrt{\frac{(\ell -m)!}{(\ell +m)!}}{P}_{\ell }^m(\cos \theta )e^{im\phi },}$$ which have the normalization $${\displaystyle {\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{Y}_{\ell }^m{Y}_{{\ell }^{\prime }}^{m^{\prime }}{}^{\ast }d\mathrm{\Omega }=\frac{4\pi }{(2\ell +1)}{\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}.}$$ In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah.

For the phase convention used in this article, the complex spherical harmonics satisfy $${\displaystyle Y_{\ell }^m{}^{\ast }(\theta ,\phi )=(-1{)}^m{Y}_{\ell }^{-m}(\theta ,\phi ),}$$ where the superscript * denotes complex conjugation. Equivalently, since ${\displaystyle (-1{)}^m=(-1{)}^{-m}}$ for integer ${\displaystyle m}$, this may also be written with ${\displaystyle (-1{)}^{-m}}$. This equation also follows from the relation of the spherical harmonic functions with the Wigner D-matrix.

Real form

Assuming the complex spherical harmonics are normalized and phased so that $${\displaystyle Y_{\ell }^m{}^{\ast }=(-1{)}^m{Y}_{\ell }^{-m},}$$ a real basis of spherical harmonics ${\displaystyle Y_{\ell m}:S^2\to \mathbb{R}}$ can be defined in terms of their complex analogues ${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$ by setting $${\displaystyle \begin{array}{cc}{Y}_{\ell m}& =\{\begin{array}{ll}{\displaystyle \frac{i}{\sqrt{2}}}\left(Y_{\ell }^m-(-1{)}^m{Y}_{\ell }^{-m}\right)& \text{if}m<0\\ Y_{\ell }^0& \text{if}m=0\\ {\displaystyle \frac{1}{\sqrt{2}}}\left(Y_{\ell }^{-m}+(-1{)}^m{Y}_{\ell }^m\right)& \text{if}m>0.\end{array}\\ & =\{\begin{array}{ll}{\displaystyle \frac{i}{\sqrt{2}}}\left(Y_{\ell }^{-|m|}-(-1{)}^m{Y}_{\ell }^{|m|}\right)& \text{if}m<0\\ Y_{\ell }^0& \text{if}m=0\\ {\displaystyle \frac{1}{\sqrt{2}}}\left(Y_{\ell }^{-|m|}+(-1{)}^m{Y}_{\ell }^{|m|}\right)& \text{if}m>0.\end{array}\\ & =\{\begin{array}{ll}\sqrt{2}(-1{)}^m\mathrm{\Im }[Y_{\ell }^{|m|}]& \text{if}m<0\\ Y_{\ell }^0& \text{if}m=0\\ \sqrt{2}(-1{)}^m\mathrm{\Re }[Y_{\ell }^m]& \text{if}m>0.\end{array}\end{array}}$$ The Condon–Shortley phase convention is used here for consistency. The corresponding inverse equations defining the complex spherical harmonics ${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$ in terms of the real spherical harmonics ${\displaystyle Y_{\ell m}:S^2\to \mathbb{R}}$ are $${\displaystyle Y_{\ell }^m=\{\begin{array}{ll}{\displaystyle \frac{1}{\sqrt{2}}}\left(Y_{\ell |m|}-i{Y}_{\ell ,-|m|}\right)& \text{if}m<0\\ Y_{\ell 0}& \text{if}m=0\\ {\displaystyle \frac{(-1{)}^m}{\sqrt{2}}}\left(Y_{\ell |m|}+i{Y}_{\ell ,-|m|}\right)& \text{if}m>0.\end{array}}$$

The real spherical harmonics ${\displaystyle Y_{\ell m}:S^2\to \mathbb{R}}$ are sometimes known as tesseral spherical harmonics. These functions have the same orthonormality properties as the complex ones ${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$ above. The real spherical harmonics ${\displaystyle Y_{\ell m}}$ 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 $${\displaystyle Y_{\ell m}=\{\begin{array}{ll}{\left(-1\right)}^m\sqrt{2}\sqrt{{\displaystyle \frac{2\ell +1}{4\pi }}{\displaystyle \frac{(\ell -|m|)!}{(\ell +|m|)!}}}{P}_{\ell }^{|m|}(\cos \theta )\sin (|m|\phi )& \text{if }m<0\\ \sqrt{{\displaystyle \frac{2\ell +1}{4\pi }}}{P}_{\ell }^0(\cos \theta )& \text{if }m=0\\ {\left(-1\right)}^m\sqrt{2}\sqrt{{\displaystyle \frac{2\ell +1}{4\pi }}{\displaystyle \frac{(\ell -m)!}{(\ell +m)!}}}{P}_{\ell }^m(\cos \theta )\cos (m\phi )& \text{if }m>0.\end{array}}$$ Here ${\displaystyle P_{\ell }^m}$ again denotes the associated Legendre function with the Condon–Shortley phase included. With a different placement of that phase, the displayed signs must be changed consistently.

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 ${\displaystyle \ell =4}$, which can be seen to be consistent with the output of the equations above.

Use in quantum chemistry

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 do not 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 the table of spherical harmonics, the usual p functions (${\displaystyle \ell =1}$) 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 ${\displaystyle Y_{\ell }^m}$ give rise to the solid harmonics by extending from ${\displaystyle S^2}$ to all of ${\displaystyle {\mathbb{R}}^3}$ as a homogeneous function of degree ${\displaystyle \ell }$, i.e. setting $${\displaystyle R_{\ell }^m(v):=\Vert v{\Vert }^{\ell }{Y}_{\ell }^m\left(\frac{v}{\Vert v\Vert }\right)}$$ It turns out that ${\displaystyle R_{\ell }^m}$ is basis of the space of harmonic and homogeneous polynomials of degree ${\displaystyle \ell }$. More specifically, it is the (unique up to normalization) Gelfand-Tsetlin-basis of this representation of the rotational group ${\displaystyle SO(3)}$ and an explicit formula for ${\displaystyle R_{\ell }^m}$ in cartesian coordinates can be derived from that fact.

The Herglotz generating function

If the quantum mechanical convention is adopted for the ${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$, then $${\displaystyle e^{v\mathbf{a}\cdot \mathbf{r}}=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }\sqrt{\frac{4\pi }{2\ell +1}}\frac{r^{\ell }{v}^{\ell }{\lambda }^m}{\sqrt{(\ell +m)!(\ell -m)!}}{Y}_{\ell }^m(\mathbf{r}/r).}$$ Here, ${\displaystyle \mathbf{r}}$ is the vector with components ${\displaystyle (x,y,z)\in {\mathbb{R}}^3}$, ${\displaystyle r=|\mathbf{r}|}$, and $${\displaystyle \mathbf{a}=\hat{\mathbf{z}}-\frac{\lambda }{2}\left(\hat{\mathbf{x}}+i\hat{\mathbf{y}}\right)+\frac{1}{2\lambda }\left(\hat{\mathbf{x}}-i\hat{\mathbf{y}}\right).}$$ ${\displaystyle \mathbf{a}}$ is a vector with complex coordinates:

${\displaystyle \mathbf{a}=[\frac{1}{2}(\frac{1}{\lambda }-\lambda ),-\frac{i}{2}(\frac{1}{\lambda }+\lambda ),1].}$

The essential property of ${\displaystyle \mathbf{a}}$ is that it is null: $${\displaystyle \mathbf{a}\cdot \mathbf{a}=0.}$$

It suffices to take ${\displaystyle v}$ and ${\displaystyle \lambda }$ as real parameters. In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, §VII.7, who credit unpublished notes by him for its discovery.

Essentially all the properties of the spherical harmonics can be derived from this generating function. An immediate benefit of this definition is that if the vector ${\displaystyle \mathbf{r}}$ is replaced by the quantum mechanical spin vector operator ${\displaystyle \mathbf{J}}$, such that ${\displaystyle {\mathcal{Y}}_{\ell }^m(\mathbf{J})}$ is the operator analogue of the solid harmonic ${\displaystyle r^{\ell }{Y}_{\ell }^m(\mathbf{r}/r)}$, one obtains a generating function for a standardized set of spherical tensor operators, ${\displaystyle {\mathcal{Y}}_{\ell }^m(\mathbf{J})}$:

$${\displaystyle e^{v\mathbf{a}\cdot \mathbf{J}}=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }\sqrt{\frac{4\pi }{2\ell +1}}\frac{v^{\ell }{\lambda }^m}{\sqrt{(\ell +m)!(\ell -m)!}}{\mathcal{Y}}_{\ell }^m(\mathbf{J}).}$$

The parallelism of the two definitions ensures that the ${\displaystyle {\mathcal{Y}}_{\ell }^m}$'s transform under rotations (see below) in the same way as the ${\displaystyle Y_{\ell }^m}$'s, which in turn guarantees that they are spherical tensor operators, ${\displaystyle T_q^{(k)}}$, with ${\displaystyle k=\ell }$ and ${\displaystyle q=m}$, 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 ${\displaystyle z}$ and another of ${\displaystyle x}$ and ${\displaystyle y}$, as follows (Condon–Shortley phase): $${\displaystyle r^{\ell }\left(\begin{array}{c}{Y}_{\ell }^m\\ Y_{\ell }^{-m}\end{array}\right)={\left[\frac{2\ell +1}{4\pi }\right]}^{1/2}{\overline{\mathrm{\Pi }}}_{\ell }^m(z)\left(\begin{array}{c}{\left(-1\right)}^m(A_m+i{B}_m)\\ (A_m-i{B}_m)\end{array}\right),m>0.}$$ and for m = 0: $${\displaystyle r^{\ell }{Y}_{\ell }^0\equiv \sqrt{\frac{2\ell +1}{4\pi }}{\overline{\mathrm{\Pi }}}_{\ell }^0.}$$ Here $${\displaystyle A_m(x,y)=\sum _{p=0}^m(\genfrac{}{}{0ex}{}{m}{p})x^p{y}^{m-p}\cos \left((m-p)\frac{\pi }{2}\right),}$$ $${\displaystyle B_m(x,y)=\sum _{p=0}^m(\genfrac{}{}{0ex}{}{m}{p})x^p{y}^{m-p}\sin \left((m-p)\frac{\pi }{2}\right),}$$ and $${\displaystyle {\overline{\mathrm{\Pi }}}_{\ell }^m(z)={\left[\frac{(\ell -m)!}{(\ell +m)!}\right]}^{1/2}\sum _{k=0}^{\lfloor (\ell -m)/2\rfloor }(-1{)}^k{2}^{-\ell }(\genfrac{}{}{0ex}{}{\ell }{k})(\genfrac{}{}{0ex}{}{2\ell -2k}{\ell })\frac{(\ell -2k)!}{(\ell -2k-m)!}{r}^{2k}{z}^{\ell -2k-m}.}$$ For ${\displaystyle m=0}$ this reduces to $${\displaystyle {\overline{\mathrm{\Pi }}}_{\ell }^0(z)=\sum _{k=0}^{\lfloor \ell /2\rfloor }(-1{)}^k{2}^{-\ell }(\genfrac{}{}{0ex}{}{\ell }{k})(\genfrac{}{}{0ex}{}{2\ell -2k}{\ell })r^{2k}{z}^{\ell -2k}.}$$

The factor ${\displaystyle {\overline{\mathrm{\Pi }}}_{\ell }^m(z)}$ is essentially the associated Legendre polynomial ${\displaystyle P_{\ell }^m(\cos \theta )}$, and the factors ${\displaystyle (A_m\pm i{B}_m)}$ are essentially ${\displaystyle e^{\pm im\phi }}$.

Examples

Using the expressions for ${\displaystyle {\overline{\mathrm{\Pi }}}_{\ell }^m(z)}$, ${\displaystyle A_m(x,y)}$, and ${\displaystyle B_m(x,y)}$ listed explicitly above we obtain: $${\displaystyle Y_3^1=-\frac{1}{r^3}{\left[\frac{7}{4\pi }\cdot \frac{3}{16}\right]}^{1/2}\left(5z^2-r^2\right)\left(x+iy\right)=-{\left[\frac{7}{4\pi }\cdot \frac{3}{16}\right]}^{1/2}\left(5{\cos }^2\theta -1\right)\left(\sin \theta e^{i\phi }\right)}$$

$${\displaystyle Y_4^{-2}=\frac{1}{r^4}{\left[\frac{9}{4\pi }\cdot \frac{5}{32}\right]}^{1/2}\left(7z^2-r^2\right){\left(x-iy\right)}^2={\left[\frac{9}{4\pi }\cdot \frac{5}{32}\right]}^{1/2}\left(7{\cos }^2\theta -1\right)\left({\sin }^2\theta e^{-2i\phi }\right)}$$ 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 ${\displaystyle m>0}$ only the ${\displaystyle A_m}$ terms (cosines) are included, and for ${\displaystyle m<0}$ only the ${\displaystyle B_m}$ terms (sines) are included:

$${\displaystyle r^{\ell }\left(\begin{array}{c}{Y}_{\ell m}\\ Y_{\ell -m}\end{array}\right)=\sqrt{\frac{2\ell +1}{2\pi }}{\overline{\mathrm{\Pi }}}_{\ell }^m(z)\left(\begin{array}{c}{A}_m\\ B_m\end{array}\right),m>0.}$$ and for m = 0: $${\displaystyle r^{\ell }{Y}_{\ell 0}\equiv \sqrt{\frac{2\ell +1}{4\pi }}{\overline{\mathrm{\Pi }}}_{\ell }^0.}$$

Special cases and values

1. When ${\displaystyle m=0}$, the spherical harmonics ${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$ reduce to the ordinary Legendre polynomials: $${\displaystyle Y_{\ell }^0(\theta ,\phi )=\sqrt{\frac{2\ell +1}{4\pi }}{P}_{\ell }(\cos \theta ).}$$
2. When ${\displaystyle m=\pm \ell }$, $${\displaystyle Y_{\ell }^{\pm \ell }(\theta ,\phi )=\frac{(\mp 1{)}^{\ell }}{2^{\ell }\ell !}\sqrt{\frac{(2\ell +1)!}{4\pi }}{\sin }^{\ell }\theta e^{\pm i\ell \phi },}$$ or more simply in Cartesian coordinates, $${\displaystyle r^{\ell }{Y}_{\ell }^{\pm \ell }(\mathbf{r})=\frac{(\mp 1{)}^{\ell }}{2^{\ell }\ell !}\sqrt{\frac{(2\ell +1)!}{4\pi }}(x\pm iy{)}^{\ell }.}$$
3. At the north pole, where ${\displaystyle \theta =0}$, and ${\displaystyle \phi }$ is undefined, all spherical harmonics except those with ${\displaystyle m=0}$ vanish: $${\displaystyle Y_{\ell }^m(0,\phi )=Y_{\ell }^m(\mathbf{z})=\sqrt{\frac{2\ell +1}{4\pi }}{\delta }_{m0}.}$$

Symmetry properties

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 ${\displaystyle P\mathrm{\Psi }(\mathbf{r})=\mathrm{\Psi }(-\mathbf{r})}$. Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with ${\displaystyle \mathbf{r}}$ being a unit vector, $${\displaystyle Y_{\ell }^m(-\mathbf{r})=(-1{)}^{\ell }{Y}_{\ell }^m(\mathbf{r}).}$$

In terms of the spherical angles, parity transforms a point with coordinates ${\displaystyle \{\theta ,\phi \}}$ to ${\displaystyle \{\pi -\theta ,\pi +\phi \}}$. The statement of the parity of spherical harmonics is then $${\displaystyle Y_{\ell }^m(\theta ,\phi )\to Y_{\ell }^m(\pi -\theta ,\pi +\phi )=(-1{)}^{\ell }{Y}_{\ell }^m(\theta ,\phi )}$$ (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

Consider a rotation ${\displaystyle \mathcal{R}}$ about the origin that sends the unit vector ${\displaystyle \mathbf{r}}$ to ${\displaystyle {\mathbf{r}}^{\prime }}$. Under this operation, a spherical harmonic of degree ${\displaystyle \ell }$ and order ${\displaystyle m}$ transforms into a linear combination of spherical harmonics of the same degree. That is, $${\displaystyle Y_{\ell }^m({\mathbf{r}}^{\prime })=\sum _{m^{\prime }=-\ell }^{\ell }{A}_{m{m}^{\prime }}{Y}_{\ell }^{m^{\prime }}(\mathbf{r}),}$$ where ${\displaystyle A_{m{m}^{\prime }}}$ is a matrix of order ${\displaystyle (2\ell +1)}$ that depends on the rotation ${\displaystyle \mathcal{R}}$. However, this is not the standard way of expressing this property. In the standard way one writes,

$${\displaystyle Y_{\ell }^m({\mathbf{r}}^{\prime })=\sum _{m^{\prime }=-\ell }^{\ell }[D_{m{m}^{\prime }}^{(\ell )}(\mathcal{R}){]}^{\ast }{Y}_{\ell }^{m^{\prime }}(\mathbf{r}),}$$ where ${\displaystyle D_{m{m}^{\prime }}^{(\ell )}(\mathcal{R}{)}^{\ast }}$ is the complex conjugate of an element of the Wigner D-matrix. In particular when ${\displaystyle {\mathbf{r}}^{\prime }}$ is a ${\displaystyle {\varphi }_0}$ rotation of the azimuth we get the identity,

$${\displaystyle Y_{\ell }^m({\mathbf{r}}^{\prime })=Y_{\ell }^m(\mathbf{r})e^{im{\varphi }_0}.}$$

The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The ${\displaystyle Y_{\ell }^m}$'s of degree ${\displaystyle \ell }$ provide a basis set of functions for the irreducible representation of the group SO(3) of dimension ${\displaystyle (2\ell +1)}$. 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 ${\displaystyle Y_{\ell }^m:S^2\to \mathbb{C}}$ form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions ${\displaystyle L_{\mathbb{C}}^2(S^2)}$. On the unit sphere ${\displaystyle S^2}$, any square-integrable function ${\displaystyle f:S^2\to \mathbb{C}}$ can thus be expanded as a linear combination of these:

$${\displaystyle f(\theta ,\phi )=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }{f}_{\ell }^m{Y}_{\ell }^m(\theta ,\phi ).}$$

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

$${\displaystyle \underset{N\to \mathrm{\infty }}{lim}{\int }_0^{2\pi }{\int }_0^{\pi }{\left|f(\theta ,\phi )-\sum _{\ell =0}^N\sum _{m=-\ell }^{\ell }{f}_{\ell }^m{Y}_{\ell }^m(\theta ,\phi )\right|}^2\sin \theta d\theta d\phi =0.}$$

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:

$${\displaystyle f_{\ell }^m={\int }_{\mathrm{\Omega }}f(\theta ,\phi )Y_{\ell }^{m\ast }(\theta ,\phi )d\mathrm{\Omega }={\int }_0^{2\pi }d\phi {\int }_0^{\pi }d\theta \sin \theta f(\theta ,\phi )Y_{\ell }^{m\ast }(\theta ,\phi ).}$$

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

A square-integrable function ${\displaystyle f:S^2\to \mathbb{R}}$ can also be expanded in terms of the real harmonics ${\displaystyle Y_{\ell m}:S^2\to \mathbb{R}}$ above as a sum

$${\displaystyle f(\theta ,\phi )=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }{f}_{\ell m}{Y}_{\ell m}(\theta ,\phi ).}$$

The convergence of the series holds again in the same sense, namely the real spherical harmonics ${\displaystyle Y_{\ell m}:S^2\to \mathbb{R}}$ form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions ${\displaystyle L_{\mathbb{R}}^2(S^2)}$. The benefit of the expansion in terms of the real harmonic functions ${\displaystyle Y_{\ell m}}$ is that for real functions ${\displaystyle f:S^2\to \mathbb{R}}$ the expansion coefficients ${\displaystyle f_{\ell m}}$ are guaranteed to be real, whereas their coefficients ${\displaystyle f_{\ell }^m}$ in their expansion in terms of the ${\displaystyle Y_{\ell }^m}$ (considering them as functions ${\displaystyle f:S^2\to \mathbb{C}\supset \mathbb{R}}$) do not have that property.

Completeness Relation

The completeness relation is a distributional equality between a sum over all indices and the Dirac delta function. For the spherical harmonics, the Dirac delta is the tensor product of two Dirac delta functions, one for the azimuthal angle on the sphere, and one for the polar angle. The relation reads,

$${\displaystyle \sum _{l=0}^{\mathrm{\infty }}\sum _{m=-l}^l{Y}_{lm}^{\ast }(\theta {}^{\prime },\varphi {}^{\prime })Y_{lm}(\theta ,\varphi )=\delta (\varphi -\varphi {}^{\prime })\delta (\cos \theta -\cos \theta {}^{\prime }).}$$

Spectrum analysis

Power spectrum in signal processing

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):

$${\displaystyle \frac{1}{4\pi }{\int }_{\mathrm{\Omega }}|f(\mathrm{\Omega }){|}^{2}d\mathrm{\Omega }=\sum _{\ell =0}^{\mathrm{\infty }}{S}_{ff}(\ell ),}$$ where $${\displaystyle S_{ff}(\ell )=\frac{1}{2\ell +1}\sum _{m=-\ell }^{\ell }|f_{\ell m}{|}^2}$$

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 $${\displaystyle \frac{1}{4\pi }{\int }_{\mathrm{\Omega }}f(\mathrm{\Omega })g^{\ast }(\mathrm{\Omega })d\mathrm{\Omega }=\sum _{\ell =0}^{\mathrm{\infty }}{S}_{fg}(\ell ),}$$ where $${\displaystyle S_{fg}(\ell )=\frac{1}{2\ell +1}\sum _{m=-\ell }^{\ell }{f}_{\ell m}{g}_{\ell m}^{\ast }}$$

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 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

$${\displaystyle S_{ff}(\ell )=C{\ell }^{\beta }.}$$

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.

Approximation and smoothness

Spherical harmonics are used to approximate functions on the sphere, and the order of decay of the power governs smoothness. Let ${\displaystyle {\mathrm{\Pi }}_n(S^2)}$ denote the space spanned by spherical harmonics of degree at most ${\displaystyle n}$. For a function ${\displaystyle f}$ on the sphere, the error of best approximation by spherical polynomials of degree at most ${\displaystyle n}$ is

${\displaystyle E_n(f{)}_p=\underset{P\in {\mathrm{\Pi }}_n(S^2)}{inf}\Vert f-P{\Vert }_p,}$

with the norm taken in ${\displaystyle L^p(S^2)}$, or in the uniform norm when ${\displaystyle p=\mathrm{\infty }}$. The asymptotic behavior of ${\displaystyle E_n(f{)}_p}$ as ${\displaystyle n\to \mathrm{\infty }}$ gives a quantitative measure of how well ${\displaystyle f}$ can be approximated by low-degree spherical harmonics. This is analogous to approximation by trigonometric polynomials in ordinary Fourier analysis.

On the circle, smoothness can be measured by the decay of Fourier coefficients. On the sphere, smoother functions admit faster approximation by spherical polynomials, while conversely, sufficiently rapid decay of the approximation error implies smoothness. More precisely, Sobolev-type smoothness can also be described spectrally, since spherical harmonics of degree ${\displaystyle \ell }$ are eigenfunctions of the Laplace–Beltrami operator with eigenvalue ${\displaystyle -\ell (\ell +1)}$.

For ${\displaystyle p=2}$ this relation can be stated especially explicitly. If

${\displaystyle f=\sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }{f}_{\ell m}{Y}_{\ell m}}$

is the spherical harmonic expansion of ${\displaystyle f}$, then the Sobolev norm of order ${\displaystyle s}$ is equivalent, up to the normalization convention for the harmonics, to

${\displaystyle \Vert f{\Vert }_{H^s(S^2)}^2=\sum _{\ell =0}^{\mathrm{\infty }}(1+\ell (\ell +1){)}^s{S}_{ff}(\ell ).}$

Thus ${\displaystyle f\in H^s(S^2)}$ precisely when this weighted sum is finite. Since ${\displaystyle \ell (\ell +1)}$ grows like ${\displaystyle {\ell }^2}$, higher Sobolev smoothness requires faster decay of the power in high degrees.

In ${\displaystyle L^2}$, the best approximation by spherical harmonics of degree at most ${\displaystyle n}$ is obtained by truncating the spherical harmonic expansion, so that

${\displaystyle E_n(f{)}_2^2=\sum _{\ell >n}{S}_{ff}(\ell ).}$

Consequently, if ${\displaystyle f\in H^s(S^2)}$, then

${\displaystyle E_n(f{)}_2\le C_s{n}^{-s}\Vert f{\Vert }_{H^s(S^2)},}$

which bounds the approximation error by the Sobolev norm.

With the above power normalization, a decay estimate of the form ${\displaystyle S_{ff}(\ell )=O({\ell }^{-\alpha })}$ implies membership in ${\displaystyle H^s(S^2)}$ whenever ${\displaystyle \alpha >2s+1}$. By the Sobolev embedding theorem, ${\displaystyle H^s(S^2)}$ embeds into ${\displaystyle C^k(S^2)}$ when ${\displaystyle s>k+1}$. Thus sufficiently fast polynomial decay of the power spectrum implies differentiability, and decay faster than every power implies smoothness of all orders.

Algebraic properties

Addition theorem

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 ${\displaystyle (r,\theta ,\phi )}$ and ${\displaystyle (r,{\theta }^{\prime },{\phi }^{\prime })}$, respectively, the angle ${\displaystyle \gamma }$ between them is given by the relation $${\displaystyle \cos \gamma =\cos {\theta }^{\prime }\cos \theta +\sin \theta \sin {\theta }^{\prime }\cos (\phi -{\phi }^{\prime })}$$ 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

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

In particular, when x = y, this gives Unsöld's theorem $${\displaystyle \sum _{m=-\ell }^{\ell }{Y}_{\ell }^m{}^{\ast }(\mathbf{x})Y_{\ell }^m(\mathbf{x})=\frac{2\ell +1}{4\pi }}$$ which generalizes the identity cos²θ + sin²θ = 1 to two dimensions.

In the expansion (1), the left-hand side ${\displaystyle P_{\ell }(\mathbf{x}\cdot \mathbf{y})}$ is a constant multiple of the degree ℓ zonal 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 ${\displaystyle Z_{\mathbf{x}}^{(\ell )}}$, the degree ℓ zonal harmonic corresponding to the unit vector x, decomposes as

Furthermore, the zonal harmonic ${\displaystyle Z_{\mathbf{x}}^{(\ell )}(\mathbf{y})}$ is given as a constant multiple of the appropriate Gegenbauer polynomial:

Combining (2) and (3) gives (1) in dimension n = 2 when x and y are represented in spherical coordinates. Finally, evaluating at x = y gives the functional identity $${\displaystyle \frac{\dim {\mathbf{H}}_{\ell }}{{\omega }_{n-1}}=\sum _{j=1}^{\dim ({\mathbf{H}}_{\ell })}|Y_j(\mathbf{x}){|}^2}$$ where ω_n−1 is the volume of the (n−1)-sphere.

Contraction rule

Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics $${\displaystyle \begin{array}{cc}{Y}_a^{\alpha }\left(\theta ,\phi \right)Y_b^{\beta }\left(\theta ,\phi \right)& =\sqrt{\frac{\left(2a+1\right)\left(2b+1\right)}{4\pi }}\sum _{c=0}^{\mathrm{\infty }}\sum _{\gamma =-c}^c{\left(-1\right)}^{\gamma }\sqrt{2c+1}\left(\begin{array}{ccc}a& b& c\\ \alpha & \beta & -\gamma \end{array}\right)\left(\begin{array}{ccc}a& b& c\\ 0& 0& 0\end{array}\right)Y_c^{\gamma }\left(\theta ,\phi \right)\\ & =\sqrt{\frac{\left(2a+1\right)\left(2b+1\right)}{4\pi }}\sum _{c=|a-b|}^{a+b}\frac{1}{\sqrt{2c+1}}⟨a\alpha b\beta |c\alpha +\beta ⟩⟨a0b0|c0⟩Y_c^{\gamma }\left(\theta ,\phi \right)\end{array}}$$ Many of the terms in this sum are trivially zero. The values of ${\displaystyle c}$ and ${\displaystyle \gamma }$ that result in non-zero terms in this sum are determined by the selection rules for the 3j-symbols.