The Coulomb wave equation for a single charged particle of mass m is the
Schrödinger equation with
Coulomb potential :\left(-\hbar^2\frac{\nabla^2}{2m}+\frac{Z \hbar c \alpha}{r}\right) \psi_{\vec{k}}(\vec{r}) = \frac{\hbar^2k^2}{2m} \psi_{\vec{k}}(\vec{r}) \,, where Z=Z_1 Z_2 is the product of the charges of the particle and of the field source (in units of the
elementary charge, Z=-1 for the hydrogen atom), \alpha is the
fine-structure constant, and \hbar^2k^2/(2m) is the energy of the particle. The solution, which is the Coulomb wave function, can be found by solving this equation in parabolic coordinates :\xi= r + \vec{r}\cdot\hat{k}, \quad \zeta= r - \vec{r}\cdot\hat{k} \qquad (\hat{k} = \vec{k}/k) \,. Depending on the boundary conditions chosen, the solution has different forms. Two of the solutions are :\psi_{\vec{k}}^{(\pm)}(\vec{r}) = \Gamma(1\pm i\eta) e^{-\pi\eta/2} e^{i\vec{k}\cdot\vec{r}} M(\mp i\eta, 1, \pm ikr - i\vec{k}\cdot\vec{r}) \,, where M(a,b,z) \equiv {}_1\!F_1(a;b;z) is the
confluent hypergeometric function, \eta = Zmc\alpha/(\hbar k) and \Gamma(z) is the
gamma function. The two boundary conditions used here are :\psi_{\vec{k}}^{(\pm)}(\vec{r}) \rightarrow e^{i\vec{k}\cdot\vec{r}} \qquad (\vec{k}\cdot\vec{r} \rightarrow \pm\infty) \,, which correspond to \vec{k}-oriented plane-wave asymptotic states
before or
after its approach of the field source at the origin, respectively. The functions \psi_{\vec{k}}^{(\pm)} are related to each other by the formula :\psi_{\vec{k}}^{(+)} = \psi_{-\vec{k}}^{(-)*} \,.
Partial wave expansion The wave function \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 w_\ell(\eta,\rho). Here \rho=kr. :\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 :\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 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 Y_\ell^m(\hat{r}) :\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 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 M_{-i\eta,\ell+1/2}(-2i\rho) and W_{-i\eta,\ell+1/2}(-2i\rho). The latter can be expressed in terms of the
confluent hypergeometric functions M and U. For \ell\in\mathbb{Z}, one defines the special solutions :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 :\sigma_\ell = \arg \Gamma(\ell+1+i \eta) is called the Coulomb phase shift. One also defines the real functions :F_\ell(\eta,\rho) = \frac{1}{2i} \left(H_\ell^{(+)}(\eta,\rho)-H_\ell^{(-)}(\eta,\rho) \right) \,, :G_\ell(\eta,\rho) = \frac{1}{2} \left(H_\ell^{(+)}(\eta,\rho)+H_\ell^{(-)}(\eta,\rho) \right) \,. In particular one has :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 H_\ell^{(\pm)}(\eta,\rho), F_\ell(\eta,\rho), and G_\ell(\eta,\rho) at large \rho is :H_\ell^{(\pm)}(\eta,\rho) \sim e^{\pm i \theta_\ell(\rho)} \,, :F_\ell(\eta,\rho) \sim \sin \theta_\ell(\rho) \,, :G_\ell(\eta,\rho) \sim \cos \theta_\ell(\rho) \,, where :\theta_\ell(\rho) = \rho - \eta \log(2\rho) -\frac{1}{2} \ell \pi + \sigma_\ell \,. The solutions H_\ell^{(\pm)}(\eta,\rho) correspond to incoming and outgoing spherical waves. The solutions F_\ell(\eta,\rho) and 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 \psi_{\vec{k}}^{(+)}(\vec{r}) :\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}) \,, In the limit \eta\to 0 regular/irregular Coulomb wave functions F_\ell(\eta,\rho),G_\ell(\eta,\rho) are proportional to
Spherical Bessel functions and spherical Coulomb functions H^{(\pm)}_\ell(\eta,\rho) are proportional to
Spherical Hankel functions : F_\ell(0,\rho)/\rho = j_\ell(\rho) : G_\ell(0,\rho)/\rho = - y_\ell(\rho) : H^{(+)}_\ell(0,\rho)/\rho = i\, h^{(1)}_\ell(\rho) : H^{(-)}_\ell(0,\rho)/\rho =-i\, h^{(2)}_\ell(\rho) and are normalized same as
Spherical Bessel functions : \int\limits_0^\infty j_l(k\, r) j_l(k' r)\,r^2 dr = \int\limits_0^\infty \frac{F_\ell\left(\pm \frac{1}{a_0 k},k\, r\right)}{k\, r} \frac{F_\ell\left(\pm \frac{1}{a_0 k'}, k' r\right)}{k' r} \, r^2 d r = \frac{\pi}{2 k^2}\delta(k-k') and similar for other 3. == Properties of the Coulomb function ==