Plane-wave expansion

In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves: $${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }({\hat {\mathbf {k} }}\cdot {\hat {\mathbf {r} }}),}$${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }({\hat {\mathbf {k} }}\cdot {\hat {\mathbf {r} }}),} 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, $${\displaystyle e^{ikr\cos \theta }=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }(\cos \theta ),}$${\displaystyle e^{ikr\cos \theta }=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }(\cos \theta ),} where θ is the spherical polar angle of r.

With the spherical-harmonic addition theorem the equation can be rewritten as $${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=4\pi \sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }i^{\ell }j_{\ell }(kr)Y_{\ell }^{m}{}({\hat {\mathbf {k} }})Y_{\ell }^{m*}({\hat {\mathbf {r} }}),}$${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=4\pi \sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }i^{\ell }j_{\ell }(kr)Y_{\ell }^{m}{}({\hat {\mathbf {k} }})Y_{\ell }^{m*}({\hat {\mathbf {r} }}),} 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

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