Plane wave expansion

 In physics, the plane wave expansion expresses a plane wave as a linear combination of spherical waves,

${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }({\hat {\mathbf {k} }}\cdot {\hat {\mathbf {r} }}),}$

where

• i is the imaginary unit,
• k is a wave vector of length k,
• r is a position vector of length r,
• j_ℓ are spherical Bessel functions,
• P_ℓ are Legendre polynomials, and
• the hat ^ denotes the unit vector.

In the special case where k is aligned with the z-axis,

${\displaystyle e^{ikr\cos \theta }=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }(\cos \theta ),}$

where θ is the spherical polar angle of r.

Expansion in spherical harmonicsEdit

With the spherical harmonic addition theorem the equation can be rewritten as

${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=4\pi \sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }i^{\ell }j_{\ell }(kr)Y_{\ell }^{m}{}({\hat {\mathbf {k} }})Y_{\ell }^{m*}({\hat {\mathbf {r} }}),}$

where

• Y_ℓ^m are the spherical harmonics and
• the superscript * denotes complex conjugation.

Note that the complex conjugation can be interchanged between the two spherical harmonics due to symmetry.

This article uses material from the Wikipedia article  Metasyntactic variable, which is released under the  Creative Commons Attribution-ShareAlike 3.0 Unported License.