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. In quantum mechanics they appear as eigenfunctions of orbital angular momentum. The name is due to Lord Kelvin. 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 present first 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 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.

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.

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 ∇. From here on we take Planck's reduced constant $\hbar$ equal to unity. The components of L satisfy the angular momentum commutation relations.

$[L_{i},L_{j}]=i\sum _{j=1}^{3}\epsilon _{ijk}L_{k},$

where ε_ijk is the Levi-Civita symbol. In angular momentum theory it is shown that these commutation relations are sufficient to prove that the following eigenvalue equation exists,

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

where $\ell$ is a natural number. 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.