Plane wave expansion

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In physics, the plane wave expansion expresses a plane wave as a sum 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}),


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[edit]

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}),


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


The plane wave expansion is applied in

See also[edit]