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,
• j_ℓ are spherical Bessel functions,
• P_ℓ 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.

With the spherical-harmonic addition theorem the equation can be rewritten as where

• Y_ℓ^m 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.

The plane wave expansion is applied in

• Acoustics
• Optics
• S-matrix
• Quantum mechanics

• Helmholtz equation
• Plane wave expansion method in computational electromagnetism
• Weyl expansion

• 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