Plane wave expansion
In physics, the plane wave expansion expresses a plane wave as a sum of spherical waves,
- i is the imaginary unit,
- k is a wave vector of length k,
- r is a position vector of length r,
- jℓ are spherical Bessel functions,
- Pℓ are Legendre polynomials, and
- the hat ^ denotes the unit vector.
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
With the spherical harmonic addition theorem the equation can be rewritten as
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: