# Plane wave expansion

Jump to navigation Jump to search

In physics, the plane wave expansion expresses a plane wave as a sum 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

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 harmonics

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

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

## Applications

The plane wave expansion is applied in

## References

• 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, Bibcode:2009arXiv0909.0494M