Plane-wave expansion

In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves: $$e^{i \mathbf k \cdot \mathbf r} = \sum_{\ell = 0}^\infty (2 \ell + 1) i^\ell j_\ell(k r) P_\ell(\hat{\mathbf k} \cdot \hat{\mathbf r}),$$ 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, $$e^{i k r \cos \theta} = \sum_{\ell = 0}^\infty (2 \ell + 1) i^\ell j_\ell(k r) P_\ell(\cos \theta),$$ 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 $$e^{i \mathbf{k} \cdot \mathbf{r}} = 4 \pi \sum_{\ell = 0}^\infty \sum_{m = -\ell}^\ell i^\ell j_\ell(k r) Y_\ell^m{}(\hat{\mathbf k}) Y_\ell^{m*}(\hat{\mathbf r}),$$ 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.

Applications
The plane wave expansion is applied in
 * Acoustics
 * Optics
 * S-matrix
 * Quantum mechanics