Plane-wave expansion
Expressing a plane wave as a combination of spherical waves
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
With the spherical-harmonic addition theorem the equation can be rewritten as where
- 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
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
- 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
- v
- t
- e
![]() | This scattering–related article is a stub. You can help Wikipedia by expanding it. |
- v
- t
- e