Plane wave expansion

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

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

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

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 harmonics

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

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

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

The plane wave expansion is applied in

Digital Library of Mathematical Functions, Equation 10.60.7, National Institute of Standards and Technology
Rami Mehrem, The Plane Wave Expansion, Infinite Integrals and Identities Involving Spherical Bessel Functions, arXiv:0909.0494

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