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Plane-wave expansion

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In physics, the plane-wave expansion expresses a plane wave as a linear combination 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

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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

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The plane wave expansion is applied in

See also

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References

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  • Digital Library of Mathematical Functions, Equation 10.60.7, National Institute of Standards and Technology
  • Rami Mehrem (2009), The Plane Wave Expansion, Infinite Integrals and Identities Involving Spherical Bessel Functions, arXiv:0909.0494, Bibcode:2009arXiv0909.0494M