Partial wave expansion

The wave function ${\displaystyle \psi _{\vec {k}}({\vec {r}})}$ can be expanded into partial waves (i.e. with respect to the angular basis) to obtain angle-independent radial functions ${\displaystyle w_{\ell }(\eta ,\rho )}$. Here ${\displaystyle \rho =kr}$.

${\displaystyle \psi _{\vec {k}}({\vec {r}})={\frac {4\pi }{r}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }i^{\ell }w_{\ell }(\eta ,\rho )Y_{\ell }^{m}({\hat {r}})Y_{\ell }^{m\ast }({\hat {k}})\,.}$

A single term of the expansion can be isolated by the scalar product with a specific spherical harmonic

${\displaystyle \psi _{k\ell m}({\vec {r}})=\int \psi _{\vec {k}}({\vec {r}})Y_{\ell }^{m}({\hat {k}})d{\hat {k}}=R_{k\ell }(r)Y_{\ell }^{m}({\hat {r}}),\qquad R_{k\ell }(r)=4\pi i^{\ell }w_{\ell }(\eta ,\rho )/r.}$

The equation for single partial wave ${\displaystyle w_{\ell }(\eta ,\rho )}$ can be obtained by rewriting the laplacian in the Coulomb wave equation in spherical coordinates and projecting the equation on a specific spherical harmonic ${\displaystyle Y_{\ell }^{m}({\hat {r}})}$

${\displaystyle {\frac {d^{2}w_{\ell }}{d\rho ^{2}}}+\left(1-{\frac {2\eta }{\rho }}-{\frac {\ell (\ell +1)}{\rho ^{2}}}\right)w_{\ell }=0\,.}$

The solutions are also called Coulomb (partial) wave functions or spherical Coulomb functions. Putting ${\displaystyle z=-2i\rho }$ changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments ${\displaystyle M_{-i\eta ,\ell +1/2}(-2i\rho )}$ and ${\displaystyle W_{-i\eta ,\ell +1/2}(-2i\rho )}$. The latter can be expressed in terms of the confluent hypergeometric functions ${\displaystyle M}$ and ${\displaystyle U}$. For ${\displaystyle \ell \in \mathbb {Z} }$, one defines the special solutions ^[4]

${\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )=\mp 2i(-2)^{\ell }e^{\pi \eta /2}e^{\pm i\sigma _{\ell }}\rho ^{\ell +1}e^{\pm i\rho }U(\ell +1\pm i\eta ,2\ell +2,\mp 2i\rho )\,,}$

where

${\displaystyle \sigma _{\ell }=\arg \Gamma (\ell +1+i\eta )}$

is called the Coulomb phase shift. One also defines the real functions

${\displaystyle F_{\ell }(\eta ,\rho )={\frac {1}{2i}}\left(H_{\ell }^{(+)}(\eta ,\rho )-H_{\ell }^{(-)}(\eta ,\rho )\right)\,,}$

Regular Coulomb wave function F plotted from 0 to 20 with repulsive and attractive interactions in Mathematica 13.1

${\displaystyle G_{\ell }(\eta ,\rho )={\frac {1}{2}}\left(H_{\ell }^{(+)}(\eta ,\rho )+H_{\ell }^{(-)}(\eta ,\rho )\right)\,.}$

In particular one has

${\displaystyle F_{\ell }(\eta ,\rho )={\frac {2^{\ell }e^{-\pi \eta /2}|\Gamma (\ell +1+i\eta )|}{(2\ell +1)!}}\rho ^{\ell +1}e^{i\rho }M(\ell +1+i\eta ,2\ell +2,-2i\rho )\,.}$

The asymptotic behavior of the spherical Coulomb functions ${\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}$, ${\displaystyle F_{\ell }(\eta ,\rho )}$, and ${\displaystyle G_{\ell }(\eta ,\rho )}$ at large ${\displaystyle \rho }$ is

${\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )\sim e^{\pm i\theta _{\ell }(\rho )}\,,}$

${\displaystyle F_{\ell }(\eta ,\rho )\sim \sin \theta _{\ell }(\rho )\,,}$

${\displaystyle G_{\ell }(\eta ,\rho )\sim \cos \theta _{\ell }(\rho )\,,}$

where

${\displaystyle \theta _{\ell }(\rho )=\rho -\eta \log(2\rho )-{\frac {1}{2}}\ell \pi +\sigma _{\ell }\,.}$

The solutions ${\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}$ correspond to incoming and outgoing spherical waves. The solutions ${\displaystyle F_{\ell }(\eta ,\rho )}$ and ${\displaystyle G_{\ell }(\eta ,\rho )}$ are real and are called the regular and irregular Coulomb wave functions. In particular one has the following partial wave expansion for the wave function ${\displaystyle \psi _{\vec {k}}^{(+)}({\vec {r}})}$ ^[5]

${\displaystyle \psi _{\vec {k}}^{(+)}({\vec {r}})={\frac {4\pi }{\rho }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }i^{\ell }e^{i\sigma _{\ell }}F_{\ell }(\eta ,\rho )Y_{\ell }^{m}({\hat {r}})Y_{\ell }^{m\ast }({\hat {k}})\,,}$