Since the plane waves depend only on the absolute values of position and momentum, \( \vert\mathbf{k}\vert \) and \( \vert\mathbf{r}\vert \), and the angle between them, \( \theta_{kr} \), they may be expanded in terms of bipolar harmonics of zero rank, i.e. $$ \exp{(i \mathbf{k}\cdot \mathbf{r})} = 4\pi\sum_{l=0}^{\infty} i^l j_l(kr)\left( Y_l(\hat{k}) \cdot Y_l(\hat{r}) \right)= \sum_{l=0}^{\infty} (2l+1)i^l j_l(kr) P_l(\cos \theta_{kr}) $$ where the addition theorem for spherical harmonics has been used in order to write the expansion in terms of Legendre polynomials. The spherical Bessel functions, \( j_l(z) \), are given in terms of Bessel functions of the first kind with half integer orders, $$ j_l(z) = \sqrt{\pi \over 2 z} J_{l+1/2}(z). $$