Spherical harmonics

In mathematics, spherical harmonics $Y_{\ell }^{m}$ are an orthogonal and complete set of functions of the spherical polar angles θ and φ. The name "spherical harmonics" is due to Lord Kelvin. In quantum mechanics they appear as eigenfunctions of orbital angular momentum. Spherical harmonics are ubiquitous in atomic and molecular physics. They are important in the representation of the gravitational field, geoid, and magnetic field of planetary bodies, characterization of the cosmic microwave background radiation and recognition of 3D shapes in computer graphics.

Definition

The notation $Y_{\ell }^{m}$ will be reserved for 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

Recalling that for m ≠ 0 the associated Legendre function contains the factor (1-x²) and that the ordinary Legendre polynomial P_n(1) = 1, it follows that

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

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 in spherical polar coordinates is given by r → r, θ → π-θ, and φ → π+φ. Hence

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

Also reflection in the x-y plane gives a phase:

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

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 ∇; $\hbar$ 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 $\ell$ is a natural number. From here on we take $\hbar$ 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

$L_{z}=-i{\frac {\partial }{\partial \varphi }}$

we summarize two important relations holding for spherical harmonics:

$L^{2}Y_{\ell }^{m}(\theta ,\varphi )=\ell (\ell +1)Y_{\ell }^{m}(\theta ,\varphi )$

$L_{z}Y_{\ell }^{m}(\theta ,\varphi )=mY_{\ell }^{m}(\theta ,\varphi )$

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 can be solved readily. The solutions are known as solid harmonics. See solid harmonics for more details.

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 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 and find accordingly that the multiplication order of operators and matrices is reverse.

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 it is easily verified that

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

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,

$C_{\ell }^{m}({\hat {\mathbf {r} }})=\sum _{m'=-\ell }^{\ell }\delta _{m',0}D^{(\ell )}(\varphi ,\theta ,0)_{mm'}^{*}\Longrightarrow C_{\ell }^{m}(\theta ,\varphi )^{*}=D^{(\ell )}(\varphi ,\theta ,0)_{m0}.$

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 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.

References

(To be continued: , real spherical harmonics, explicit expressions)