Plane wave expansion

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

,

where

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

,

where θ is the spherical polar angle of r.

Expansion in spherical harmonics[edit]

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

,

where

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

Applications[edit]

The plane wave expansion is applied in

See also[edit]

References[edit]

  • 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