Plane-wave expansion

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Short description: 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: [math]\displaystyle{ 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}), }[/math] where

  • i is the imaginary unit,
  • k is a wave vector of length k,
  • r is a position vector of length r,
  • j are spherical Bessel functions,
  • P are Legendre polynomials, and
  • the hat ^ denotes the unit vector.

In the special case where k is aligned with the z axis, [math]\displaystyle{ e^{i k r \cos \theta} = \sum_{\ell = 0}^\infty (2 \ell + 1) i^\ell j_\ell(k r) P_\ell(\cos \theta), }[/math] 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 [math]\displaystyle{ 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}), }[/math] where

Note that the complex conjugation can be interchanged between the two spherical harmonics due to symmetry.

Applications

The plane wave expansion is applied in

See also

References