Spherical harmonics

• See Spherical harmonics/Catalogs for a table of spherical harmonics through ℓ = 4.

Spherical harmonics are functions arising in physics and mathematics when spherical polar coordinates (coordinates r, θ and φ that locate a point in space) are used in investigating physical problems in three dimensions. The functions appear in physical problems with (near-) spherical symmetry, indeed, in the same kind of physical problems where spherical polar coordinates are preferred over other coordinate systems such as Cartesian or cylinder coordinates.

The name "spherical harmonics" was first used by William Thomson (Lord Kelvin) and Peter Guthrie Tait in their 1867 Treatise on Natural Philosophy. The term harmonic function was coined earlier by William Thomson for solutions of the Laplace equation, ∇²V = 0, and as the spherical harmonic functions appear as the solution of the Laplace equation in spherical polar coordinates, their name followed immediately. In German the functions are called "Kugelfunktionen" (literally sphere functions), and in French they are known as "fonctions harmoniques sphériques", which is equivalent to their English name.

In quantum mechanics spherical harmonics appear as eigenfunctions of (squared) orbital angular momentum. Spherical harmonics are ubiquitous in atomic and molecular physics. Further, they are important in the representation of the gravitational and magnetic fields of planetary bodies, the characterization of the cosmic microwave background radiation, the rotation-invariant description of 3D shapes in computer graphics, the description of electrical potentials due to charge distributions, and in certain types of fluid motion.

It can be shown that the spherical harmonics, almost always written as $Y_{\ell }^{m}(\theta ,\phi )$, form an orthogonal and complete set (a basis of a Hilbert space) of functions of the spherical polar angles, θ and φ, with ℓ and m indicating degree and order of the function. This implies that the harmonics can be used to describe a function of θ and φ in the form of a linear expansion; the expansion coefficients may be used as linear regression parameters, which means that they may be chosen such that the original and expanded function "resemble" each other as closely as possible. The more spherical symmetry the original function possesses, the shorter the expansion and the fewer fit (regression) parameters have to be determined.

Some illustrative images of real spherical harmonics

Polar plots are shown of a few low-order real spherical harmonics (functions of θ and φ) to be defined in this article. The plots show clearly the nodal planes of the functions. The absolute values are meaningless because the functions are not normalized and accordingly the normalization factors are omitted from their definitions.

Definition of complex spherical harmonics

The notation $Y_{\ell }^{m}$ will be reserved for the complex-valued functions normalized to unity. It is convenient to introduce first non-normalized functions that are proportional to the $Y_{\ell }^{m}$. Several definitions are possible, we start with one that is common in quantum mechanically oriented texts. The spherical polar angles are the colatitude angle θ and the longitudinal (azimuthal) angle φ. The numbers ℓ and m are integral numbers and ℓ is positive or zero.

$C_{\ell }^{m}(\theta ,\varphi )\equiv i^{m+|m|}\;\left[{\frac {(\ell -|m|)!}{(\ell +|m|)!}}\right]^{1/2}P_{\ell }^{(|m|)}(\cos \theta )e^{im\varphi },\qquad -\ell \leq m\leq \ell ,$

where $P_{\ell }^{(m)}(\cos \theta )$ is a (phaseless) associated Legendre function. The m dependent phase is known as the Condon & Shortley phase:

$i^{m+|m|}={\begin{cases}(-1)^{m}&\quad {\hbox{if}}\quad m>0\\1&\quad {\hbox{if}}\quad m\leq 0\end{cases}}$

An alternative definition uses the fact that the associated Legendre functions can be defined (via the Rodrigues formula) for negative m,

${\tilde {C}}_{\ell }^{m}(\theta ,\varphi )\equiv (-1)^{m}\left[{\frac {(\ell -m)!}{(\ell +m)!}}\right]^{1/2}P_{\ell }^{(m)}(\cos \theta )e^{im\varphi },\qquad -\ell \leq m\leq \ell ,$

The two definitions obviously agree for positive and zero m, but for negative m this is less apparent. It is also not immediately clear that the choices of phases yield the same function. However, below we will see that the definitions agree for negative m as well. Hence, for all ℓ ≥ 0,

${\tilde {C}}_{\ell }^{m}(\theta ,\varphi )\equiv C_{\ell }^{m}(\theta ,\varphi ),\quad {\hbox{for}}\quad m=-\ell ,\ldots ,\ell .$

Complex conjugation

Noting that the associated Legendre function is real and that

${\Big (}i^{m+|m|}{\Big )}^{*}=(-1)^{m}\,i^{-m+|m|},\,$

we find for the complex conjugate of the spherical harmonic in the first definition

$C_{\ell }^{m}(\theta ,\varphi )^{*}=(-1)^{m}\,i^{-m+|m|}\;\left[{\frac {(\ell -|m|)!}{(\ell +|m|)!}}\right]^{1/2}P_{\ell }^{(|m|)}(\cos \theta )e^{-im\varphi }=(-1)^{m}C_{\ell }^{-m}(\theta ,\varphi ).$

Complex conjugation gives for the functions of positive m in the second definition

${\tilde {C}}_{\ell }^{|m|}(\theta ,\varphi )^{*}\equiv (-1)^{m}\left[{\frac {(\ell -|m|)!}{(\ell +|m|)!}}\right]^{1/2}P_{\ell }^{(|m|)}(\cos \theta )e^{-i|m|\varphi }.$

Use of the following non-trivial relation (that does not depend on any choice of phase):

$P_{\ell }^{(|m|)}(\cos \theta )=(-1)^{m}{\frac {(\ell +|m|)!}{(\ell -|m|)!}}P_{\ell }^{(-|m|)}(\cos \theta ).$

gives

${\tilde {C}}_{\ell }^{|m|}(\theta ,\varphi )^{*}=\left[{\frac {(\ell +|m|)!}{(\ell -|m|)!}}\right]^{1/2}P_{\ell }^{(-|m|)}(\cos \theta )e^{-i|m|\varphi }=(-1)^{m}{\tilde {C}}_{\ell }^{-|m|}(\theta ,\varphi ).$

Since the two definitions of spherical harmonics coincide for positive m and complex conjugation gives in both definitions the same relation to functions of negative m, it follows that the two definitions agree. From here on we drop the tilde and assume both definitions to be simultaneously valid.

Note

If the m-dependent phase would be dropped in both definitions, the functions would still agree for non-negative m. However, the first definition would satisfy

$C_{\ell }^{m}(\theta ,\varphi )^{*}=C_{\ell }^{-m}(\theta ,\varphi ),$

whereas the second would still satisfy

${\tilde {C}}_{\ell }^{m}(\theta ,\varphi )^{*}=(-1)^{m}{\tilde {C}}_{\ell }^{-m}(\theta ,\varphi ),$

from which follows that the functions would differ in phase for negative m.

Normalization

It can be shown that

$\int _{0}^{\pi }\int _{0}^{2\pi }C_{\ell }^{m}(\theta ,\varphi )^{*}C_{\ell '}^{m'}(\theta ,\varphi )\;\sin \theta \,d\theta \,d\varphi =\delta _{\ell \ell '}\delta _{mm'}{\frac {4\pi }{2\ell +1}}.$

The integral over φ gives 2π and a Kronecker delta on m and m′. Thus, for the integral over θ it suffices to consider the case m = m'. The necessary integral is given here. The (non-unit) normalization of $\,C_{\ell }^{m}$ is known as Racah's normalization or Schmidt's semi-normalization. It is often more convenient than unit normalization. Unit normalized functions are defined as follows

$Y_{\ell }^{m}(\theta ,\varphi )\equiv {\sqrt {\frac {2\ell +1}{4\pi }}}C_{\ell }^{m}(\theta ,\varphi ).$

Condon-Shortley phase

One source of confusion with the definition of the spherical harmonic functions concerns the phase factor. In quantum mechanics the phase, introduced above, is commonly used. It was introduced by Condon and Shortley. In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre functions, or to prefix it to the definition of the spherical harmonic functions, as done above. There is no requirement to use the Condon-Shortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of raising and lowering operators. The geodesy and magnetics communities never include the Condon-Shortley phase factor in their definitions of the spherical harmonic functions.

Properties

• For m ≠ 0 the associated Legendre function contains the factor (1−x²) and the ordinary Legendre polynomial P_n(1) = 1. So,

$C_{\ell }^{m}(0,0)=\delta _{m,0}\Longrightarrow Y_{\ell }^{m}(0,0)=\delta _{m,0}\,{\sqrt {\frac {2\ell +1}{4\pi }}}.$

• The regular solid harmonics r^ℓY_ℓ^m are homogeneous of degree ℓ in the components x, y, and z of r, so that inversion r → −r gives the factor (−1)^ℓ for the regular solid harmonics. Inversion of spherical polar coordinates: r → r,   θ → π−θ,   and φ → π+φ. So,

$Y_{\ell }^{m}(\pi -\theta ,\pi +\varphi )=(-1)^{\ell }Y_{\ell }^{m}(\theta ,\varphi ).$

• Reflection in the x-y plane:

$Y_{\ell }^{m}(\pi -\theta ,\varphi )=(-1)^{\ell -m}Y_{\ell }^{m}(\theta ,\varphi ).$

Eigenfunctions of orbital angular momentum

In quantum mechanics the following operator, the orbital angular momentum operator, appears frequently

$\mathbf {L} =-i\hbar \mathbf {r} \times \mathbf {\nabla } ,$

where the cross stands for the cross product of the position vector r and the gradient ∇; ℏ is Planck's constant divided by 2π. The components of L satisfy the angular momentum commutation relations.

$[L_{i},L_{j}]=i\hbar \sum _{j=1}^{3}\epsilon _{ijk}L_{k},\qquad i,j,k=x,y,z,$

where ε_ijk is the Levi-Civita symbol. In angular momentum theory it is shown that these commutation relations are sufficient to prove that L² has eigenvalues ℓ(ℓ+1),

$(L_{x}^{2}+L_{y}^{2}+L_{z}^{2})\Psi \equiv L^{2}\Psi =\hbar ^{2}\ell (\ell +1)\Psi ,$

where ℓ is a natural number. From here on we take ℏ equal to unity (this is part of the system of atomic units). The operator L² expressed in spherical polar coordinates is,

$L^{2}=-\left[{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\sin \theta {\frac {\partial }{\partial \theta }}+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}\right].$

The eigenvalue equation can be simplified by separation of variables. We substitute

$\Psi =\Theta (\theta )\Phi (\varphi )$

into the eigenvalue equation. After dividing out Ψ and multiplying with sin²θ we get

$\left[{\frac {1}{\Theta (\theta )}}\sin \theta {\frac {\partial }{\partial \theta }}\sin \theta {\frac {\partial \Theta (\theta )}{\partial \theta }}+\ell (\ell +1)\sin ^{2}\theta \right]+\left[{\frac {1}{\Phi (\varphi )}}{\frac {\partial ^{2}\Phi (\varphi )}{\partial \varphi ^{2}}}\right]=0.$

In the spirit of the method of separation of variables, we put the terms in square brackets equal to plus and minus the same constant, respectively. Without loss of generality we take m² as this constant (m can be complex) and consider

${\frac {\partial ^{2}\Phi (\varphi )}{\partial \varphi ^{2}}}=-m^{2}\Phi (\varphi ).$

This has the solutions

$\Phi (\varphi )=Ne^{\pm im\varphi }$

The requirement that exp[i m (φ + 2π)] = exp[i m φ] gives that m is integral. Substitution of this result into the eigenvalue equation gives

$\left[{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\sin \theta {\frac {\partial \Theta (\theta )}{\partial \theta }}+\ell (\ell +1)-{\frac {m^{2}}{\sin ^{2}\theta }}\right]\Theta (\theta )=0.$

Upon writing x = cos θ the equation becomes the associated Legendre equation

$(1-x^{2}){\frac {d^{2}\Theta }{dx^{2}}}-2x{\frac {d\Theta }{dx}}+\left[\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}}\right]\Theta =0.$

This equation has two classes of solutions: the associated Legendre functions of the first and second kind. The functions of the second kind are non-regular for x = ±1 and do not concern us further. The functions of the first kind are the associated Legendre functions:

$\Theta (\theta )\propto P_{\ell }^{(\pm m)}(\cos \theta ).$

It follows that

$L^{2}\Psi =\ell (\ell +1)\Psi \Longrightarrow \Psi =P_{\ell }^{(\pm m)}(\cos \theta )e^{\pm im\varphi }.$

The eigenvalue equation does not establish phase and normalization, so that these must be imposed separately. This was done earlier in this article.

Finally, noting that

${\begin{aligned}L_{z}&=-i{\frac {\partial }{\partial \varphi }}\\L_{\pm }&=L_{x}\pm iL_{y},\end{aligned}}$

we summarize the action of the components of orbital angular momentum on spherical harmonics:

${\begin{aligned}L^{2}Y_{\ell }^{m}(\theta ,\varphi )&=\ell (\ell +1)Y_{\ell }^{m}(\theta ,\varphi )\\L_{z}Y_{\ell }^{m}(\theta ,\varphi )&=mY_{\ell }^{m}(\theta ,\varphi )\\L_{\pm }Y_{\ell }^{m}(\theta ,\varphi )&={\sqrt {\ell (\ell +1)-m(m\pm 1)}}Y_{\ell }^{m\pm 1}(\theta ,\varphi )\\\end{aligned}}$

Laplace equation

The Laplace equation ∇² Ψ = 0 reads in spherical polar coordinates

${\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}r^{2}{\frac {\partial \Psi }{\partial r}}+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\sin \theta {\frac {\partial \Psi }{\partial \theta }}+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}\Psi }{\partial \varphi ^{2}}}=0.$

Clearly, this can be rewritten as

${\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}r^{2}{\frac {\partial \Psi }{\partial r}}-{\frac {L^{2}}{r^{2}}}\Psi =0.$

Making the Ansatz Ψ = R(r) Y^m_ℓ   the equation becomes

${\frac {\partial }{\partial r}}r^{2}{\frac {\partial R}{\partial R}}=\ell (\ell +1)R,$

where we divided out $Y_{\ell }^{m}/r^{2}$. Inserting the following functions

$R_{1}(r)=r^{\ell }\quad {\hbox{and}}\quad R_{2}(r)={\frac {1}{r^{\ell +1}}}.$

shows that these functions are solutions. They give rise to functions known as regular and irregular solid harmonics. See solid harmonics for more details.

Finally, it is evident that

$\nabla ^{2}Y_{\ell }^{m}(\theta ,\phi )=-{\frac {\ell (\ell +1)}{r^{2}}}Y_{\ell }^{m}(\theta ,\phi ),$

because $\partial Y_{\ell }^{m}/\partial r=0$.

Lie algebra

Consider the infinitesimal rotation from which finite rotations can be generated. For example, a rotation by angle α about the z-axis is described by the matrix:

${\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}\ {\underset {\overset {\alpha \rightarrow 0}{}}{\to }}\ {\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}+i\alpha {\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}+i\alpha R_{z}\ ,$

where the form following the arrow applies for very small angles α. The matrix R_z is called the generator of the z-rotation. The factor i is introduced so the finite rotation can be expressed in terms of this generator as a simple exponential:

${\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}=e^{i\alpha R_{z}}\ ,$

as can be verified using the Taylor series:

$e^{i\alpha R_{z}}=1+i\alpha R_{z}+{\frac {1}{2}}\left(i\alpha R_{z}\right)^{2}+...\ .$

If the three coordinate axes are labeled {i, j, k } and the infinitesimal rotations about each of these axes are labeled {R_i, R_j, R_k}, then these generators of infinitesimal rotations obey the commutation relations:

$R_{i}R_{j}-R_{j}R_{i}=i\varepsilon _{ijk}R_{k}\ ,$

for any choices of subscripts. Here ε_ijk is the Levi-Civita symbol which equals one if ijk = xyz or any permutation that keeps the same cyclic order, or minus one if the order is different, or zero if any two of the indices are the same. The commutation relations express the fact that the order (sequence) of rotations matters.

Suppose these commutation relations now are viewed as applying in general, and while still considered as connected to rotations in three dimensional space, the question is opened as to what general mathematical objects might satisfy these rules. A set of symbols with a defined sum and a product taken as a commutator of the symbols is called a Lie algebra. In particular, one can construct sets of square matrices of various dimensions that satisfy these commutation rules; each set is a so-called representation of the rules. One finds that there are many such sets, but they can be sorted into two kinds: irreducible and reducible. The reducible sets of matrices can be shown to be equivalent to matrices with smaller irreducible matrices down the diagonal, so that the rules are satisfied within these smaller constituent matrices, and the entire matrix is not essential. The irreducible sets cannot be arranged this way.

The matrices can be viewed as acting upon vectors in an abstract space. For example, a space with an odd number of dimensions (2ℓ+1) can be constructed from the spherical harmonics Y_ℓ^m, and their transformations under infinitesimal rotations. The Y_ℓ^m depend upon the angles θ,φ describing orientation in ordinary three-dimensional space, but infinitesimal rotations of these arguments mix up the Y_ℓ^m in a fashion described by irreducible matrices of dimension (2ℓ+1) that satisfy the commutation relations.

The construction of irreducible matrices of any dimension at all is done as follows. If the generator of an infinitesimal rotation is labeled J, then the basis vectors in this space can be labeled by the integers j and m where m is restricted to the values { −j, −j+1, ... , j−1, j }. Denoting a basis vector by |j, m⟩, one finds:

$J^{2}|j,\ m\rangle =j(j+1)|j,\ m\rangle \ ,$

$J_{z}|j,\ m\rangle =m|j,\ m\rangle \ .$

Here J_z generates an infinitesimal rotation about a direction chosen as the z-axis, and J² = J_x² + J_y² + J_z² is the so-called Casimir operator. In particular, these equations recover the Pauli spin matrices in two dimensions and the infinitesimal transformations of the Y_ℓ^m in (2ℓ+1) dimensions. See the development in Eigenfunctions of orbital angular momentum earlier in this article.

Connection with 3D full rotation group

The group of proper (no reflections) rotations in three dimensions is SO(3). It consists of all 3 x 3 orthogonal matrices with unit determinant. A unit vector is uniquely determined by two spherical polar angles and conversely. Hence we write

$Y_{\ell }^{m}({\hat {\mathbf {r} }})\quad {\hbox{with}}\quad {\hat {\mathbf {r} }}\equiv {\frac {\mathbf {r} }{|\mathbf {r} |}}.$

Let R be a unimodular (unit determinant) orthogonal matrix, then we define a rotation operator by

${\mathcal {R}}Y_{\ell }^{m}({\hat {\mathbf {r} }})\equiv Y_{\ell }^{m}(\mathbf {R} ^{-1}{\hat {\mathbf {r} }}).$

The inverse matrix appears here (acting on a column vector) in order to assure that this map of rotation matrices to rotation operators is a group homomorphism. Since this point was discussed at some length in Wigner's famous book on group theory, it is known as Wigner's convention. Some authors omit the inverse on the rotation and find accordingly that the map from matrices to operators is antihomomorphic (i.e., multiplication of operators and matrices is in mutually reversed order).

It can be shown that the rotation operator is an exponential operator in the components of the orbital angular momentum operator L. It can also be shown that the action of these operators on the spherical harmonics do no change ℓ. That is, the linear space spanned by 2ℓ+1 spherical harmonics of same ℓ and different m is invariant under L, and therefore also under rotations,

${\mathcal {R}}Y_{\ell }^{m}({\hat {\mathbf {r} }})=\sum _{m'=-\ell }^{\ell }Y_{\ell }^{m'}({\hat {\mathbf {r} }})D^{(\ell )}(\mathbf {R} )_{m'm}.$

The square 2ℓ+1 dimensional matrix that appears here is known as Wigner's D-matrix. Obviously, the set of matrices of fixed ℓ form a representation of the group SO(3). It can be shown that they form an irreducible representation of this group. The rotation operator is unitary and the spherical harmonics are orthonormal, hence the Wigner rotation matrix is a unitary matrix:

$\left(\mathbf {D} ^{(\ell )}\right)^{\dagger }\mathbf {D} ^{(\ell )}=\mathbf {E} _{\ell }\Longleftrightarrow \sum _{m=-\ell }^{\ell }{\big (}D_{mm'}^{(\ell )}{\big )}^{*}D_{mm''}^{(\ell )}=\delta _{m'm''},$

where E_ℓ is the 2ℓ+1 dimensional identity matrix. From this unitarity follows the following useful invariance

$\sum _{m=-\ell }^{\ell }Y_{\ell }^{m}({\hat {\mathbf {r} }})^{*}\;Y_{\ell }^{m}({\hat {\mathbf {r} }}')=\sum _{m=-\ell }^{\ell }Y_{\ell }^{m}(\mathbf {R} {\hat {\mathbf {r} }})^{*}\;Y_{\ell }^{m}(\mathbf {R} {\hat {\mathbf {r} }}')\quad {\hbox{for any}}\quad \mathbf {R} \in \mathrm {SO(3)} .$

Connection with Wigner D-matrices

The rotation of spherical harmonics may be rewritten as follows (where we introduce the Racah normalized functions):

$C_{\ell }^{m}({\hat {\mathbf {r} }})=\sum _{m'=-\ell }^{\ell }C_{\ell }^{m'}(\mathbf {R} {\hat {\mathbf {r} }})D^{(\ell )}(\mathbf {R} )_{m'm}.$

Let θ and φ be the spherical polar angles of r, then as is shown here,

$\left[{\begin{pmatrix}\cos \varphi &-\sin \varphi &0\\\sin \varphi &\cos \varphi &0\\0&0&1\\\end{pmatrix}}{\begin{pmatrix}\cos \theta &0&\sin \theta \\0&1&0\\-\sin \theta &0&\cos \theta \\\end{pmatrix}}\right]^{-1}{\begin{pmatrix}\cos \varphi \sin \theta \\\sin \varphi \sin \theta \\\cos \theta \end{pmatrix}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}$

Substitution of this rotation, use of group homomorphism and unitarity of D-matrices,

$D^{(\ell )}(\mathbf {R} ^{-1})_{m'm}=D^{(\ell )}(\mathbf {R} )_{m'm}^{-1}=D^{(\ell )}(\mathbf {R} )_{mm'}^{*},$

and the fact that spherical harmonics of zero θ give a Kronecker delta on m, we get a relation between spherical harmonics and Wigner D-matrices,

$C_{\ell }^{m}({\hat {\mathbf {r} }})=\sum _{m'=-\ell }^{\ell }\delta _{m',0}D^{(\ell )}(\varphi ,\theta ,0)_{mm'}^{*}.$

Hence,

$C_{\ell }^{m}(\theta ,\varphi )^{*}=D^{(\ell )}(\varphi ,\theta ,0)_{m0}.$

Completeness of spherical harmonics

The spherical harmonics are orthogonal and it can be shown that they are complete in the least squares sense for functions f of θ and φ. That is, the square of the "distance" between f and the expansion

$\int _{0}^{\pi }\int _{0}^{2\pi }\left|f(\theta ,\varphi )-\sum _{\ell =0}^{N}\sum _{m=-\ell }^{\ell }c_{\ell ,m}Y_{\ell }^{m}(\theta ,\varphi )\right|^{2}\;\sin \theta \;d\theta \,d\varphi$

can be made arbitrarily small for sufficiently large N. It is common to write somewhat loosely

$f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }c_{\ell ,m}Y_{\ell }^{m}(\theta ,\varphi ).$

It is known from Hilbert space theory that the expansion (Fourier) coefficients are given by

$c_{\ell ,m}=\int _{0}^{\pi }\int _{0}^{2\pi }Y_{\ell }^{m}(\theta ,\varphi )^{*}f(\theta ,\varphi )\;\sin \theta \;d\theta \,d\varphi .$

The proof of the completeness follows from the facts that the exponential functions of φ are complete, as is known from Fourier theory and that the associated Legendre differential equation is of the Sturm-Liouville type. In quantum mechanics one expresses this by stating that the associated Legendre equation is an eigenvalue equation of a Hermitian operator.

Alternatively one can invoke the Peter-Weyl theorem, from which follows that the Wigner D-matrices are complete, as the rotation group SO(3) is compact. In general Wigner D-matrices depend on three rotation angles (for instance Euler angles). Application of the completeness of the D-matrices to functions that do not depend on one of the three angles proves the completeness of spherical harmonics, while noting the relation between the spherical harmonics and the D-matrices pointed out earlier in this article.

Spherical harmonic addition theorem

The spherical harmonic addition theorem reads

$P_{\ell }({\hat {\mathbf {r} }}_{1}\cdot {\hat {\mathbf {r} }}_{2})=\sum _{m=-\ell }^{\ell }C_{\ell }^{m}({\hat {\mathbf {r} }}_{1})^{*}\;C_{\ell }^{m}({\hat {\mathbf {r} }}_{2})={\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }Y_{\ell }^{m}({\hat {\mathbf {r} }}_{1})^{*}\;Y_{\ell }^{m}({\hat {\mathbf {r} }}_{2}).$

There are two proofs: a short one, referred to by Whittaker and Watson (p. 395) as a "physical proof", and a long analytic proof.

We skip the analytic proof and outline the physical proof. Under a simultaneous rotation R of two vectors the angle between them is not changed,

${\begin{aligned}\cos \gamma \,'\equiv {\hat {\mathbf {r} }}'_{1}\cdot {\hat {\mathbf {r} }}'_{2}=&(\mathbf {R} {\hat {\mathbf {r} }}_{1})\cdot (\mathbf {R} {\hat {\mathbf {r} }}_{2})\\=&\;{\hat {\mathbf {r} }}_{1}^{T}\,\mathbf {R} ^{T}\;\mathbf {R} \,{\hat {\mathbf {r} }}_{2}={\hat {\mathbf {r} }}_{1}\cdot {\hat {\mathbf {r} }}_{2}\equiv \cos \gamma ,\end{aligned}}$

because R^TR is equal to the 3 × 3 identity matrix. Choose the rotation R such that the rotated unit vector ${\hat {\mathbf {r} }}_{2}$ coincides with the z-axis, and use that the sum over m in the following is a rotation invariant (see earlier in this article)

$\sum _{m=-\ell }^{\ell }C_{\ell }^{m}({\hat {\mathbf {r} }}_{1})^{*}\;C_{\ell }^{m}({\hat {\mathbf {r} }}_{2})=\sum _{m=-\ell }^{\ell }C_{\ell }^{m}(\mathbf {R} {\hat {\mathbf {r} }}_{1})^{*}\;C_{\ell }^{m}({\hat {\mathbf {R} \mathbf {r} }}_{2})=\sum _{m=-\ell }^{\ell }C_{\ell }^{m}({\hat {\mathbf {r} }}_{1})^{*}\;\delta _{m,0}=P_{\ell }(\cos \theta _{1}).$

The angle θ₁ is the colatitude (polar) angle of the rotated vector r₁ and hence is the angle with the rotated vector r₂, which lies along the z-axis. Since the angle between the two vectors is invariant under rotation we have

$\cos \theta _{1}=\cos \gamma ={\hat {\mathbf {r} }}_{1}\cdot {\hat {\mathbf {r} }}_{2},\,$

which proves the spherical harmonic addition theorem.

As a corollary we find Unsöld's theorem

$1={\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }Y_{\ell }^{m}({\hat {\mathbf {r} }})^{*}\;Y_{\ell }^{m}({\hat {\mathbf {r} }}),$

by simply taking $\scriptstyle {\hat {\mathbf {r} }}_{1}={\hat {\mathbf {r} }}_{2}={\hat {\mathbf {r} }}$.

Gaunt series

Since the spherical harmonics are complete and orthonormal, we can expand a binary product of spherical harmonics again in spherical harmonics. This gives the Gaunt series,

${\begin{aligned}Y_{\ell }^{m}(\theta ,\varphi )Y_{\ell '}^{m'}(\theta ,\varphi )=&\sum _{L,M}Y_{L}^{M}(\theta ,\varphi )\;G_{L\ell \ell '}^{Mmm'}\\\end{aligned}}$

with

$G_{L\ell \ell '}^{Mmm'}=\int _{0}^{\pi }\int _{0}^{2\pi }Y_{L}^{M}(\theta ,\varphi )^{*}Y_{\ell }^{m}(\theta ,\varphi )Y_{\ell '}^{m'}(\theta ,\varphi )\;\sin \theta \;d\theta d\varphi .$

This double integral is called a Gaunt coefficient. By the Wigner-Eckart theorem it is proportional to the 3j-symbol

${\begin{pmatrix}L&\ell &\ell '\\-M&m&m'\\\end{pmatrix}}.$

The 3j-symbol is zero unless

$|\ell -\ell '|\leq L\leq \ell +\ell '\quad {\hbox{and}}\quad M=m+m'.$

These conditions constrain the sum over L in the Gaunt series and remove the sum over M. In total the Gaunt coefficient is

$G_{L\ell \ell '}^{Mmm'}=(-1)^{M}{\sqrt {\frac {(2L+1)(2\ell +1)(2\ell '+1)}{4\pi }}}{\begin{pmatrix}L&\ell &\ell '\\0&0&0\\\end{pmatrix}}{\begin{pmatrix}L&\ell &\ell '\\-M&m&m'\\\end{pmatrix}},$

where the quantity with three zeros in the bottom row is also a 3j-symbol.

Real form

The following expression defines real spherical harmonics of cosine and sine type respectively:

${\begin{pmatrix}^{c}Y_{\ell }^{|m|}\\^{s}Y_{\ell }^{|m|}\end{pmatrix}}\equiv {\frac {1}{\sqrt {2}}}{\begin{pmatrix}(-1)^{m}&\quad 1\\-(-1)^{m}i&\quad i\end{pmatrix}}{\begin{pmatrix}Y_{\ell }^{|m|}\\Y_{\ell }^{-|m|}\end{pmatrix}},\qquad m\neq 0.$

and for m = 0:

$^{c}Y_{\ell }^{0}\equiv Y_{\ell }^{0}.$

Since the transformation is by a unitary matrix the normalization of the real and the complex spherical harmonics is the same. By definition, for m > 0 we have the phaseless expressions

${\begin{aligned}^{c}Y_{\ell }^{m}=&{\sqrt {\frac {2\ell +1}{2\pi }}}\left[{\frac {(\ell -m)!}{(\ell +m)!}}\right]^{1/2}P_{\ell }^{(m)}(\cos \theta )\cos m\varphi ,\\^{s}Y_{\ell }^{m}=&{\sqrt {\frac {2\ell +1}{2\pi }}}\left[{\frac {(\ell -m)!}{(\ell +m)!}}\right]^{1/2}P_{\ell }^{(m)}(\cos \theta )\sin m\varphi .\\\end{aligned}}$

The real functions are sometimes referred to as tesseral harmonics, see Whittaker and Watson p. 392 for an explanation of this name.