Spherical harmonics

In mathematics, spherical harmonics Y^m_l 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

Several definitions are possible, we present one that is the most usual in quantum mechanical applications. The spherical polar angles are the colatitude angle θ and the longitudinal (azimuthal) angle φ.

$Y_{\ell }^{m}(\theta ,\varphi )\equiv i^{m+|m|}\;\Theta _{\ell }^{|m|}(\cos \theta )e^{im\varphi },\qquad -\ell \leq m\leq \ell ,$

with

$\Theta _{\ell }^{m}(\cos \theta )\equiv \left[{\frac {(\ell -m)!}{(\ell +m)!}}\right]^{1/2}P_{\ell }^{m}(\cos \theta ),\quad m\geq 0,$

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

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