In physics, the plane-wave expansion or Rayleigh expansion expresses a plane wave as a linear combination of spherical waves:
where is the imaginary unit, is a real or complex wave vector of length , is a position vector of length , are spherical Bessel functions, are Legendre polynomials, and the hat denotes the unit vector.
In the special case where is aligned with the z axis,
where is the spherical polar angle of .
For proof, expand in Legendre polynomials , and evaluate the coeffient integrals.
Expansion in spherical harmonics
With the spherical-harmonic addition theorem the equation can be rewritten as
where are the spherical harmonics and the superscript denotes complex conjugation.
Note that the complex conjugation can be interchanged between the two spherical harmonics due to symmetry.
Applications
The plane wave expansion is applied in Acoustics Optics S-matrix Quantum mechanics
See also
Helmholtz equation Plane wave expansion method in computational electromagnetism Weyl expansion
References