In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves:
ℓ = 0
{\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 In the special case where k is aligned with the z axis,
i k r cos θ
( cos θ ) ,
{\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
= 4 π
{\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 Note that the complex conjugation can be interchanged between the two spherical harmonics due to symmetry.
The plane wave expansion is applied in
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