In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves:
where
- 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 edit
With the spherical-harmonic addition theorem the equation can be rewritten as
- Yℓm 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 edit
The plane wave expansion is applied in
See also edit
- Helmholtz equation
- Plane wave expansion method in computational electromagnetism
- Weyl expansion
References edit
- 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