<math>\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}) \,,</math>
where <math>Z=Z_1 Z_2</math> is the product of the charges of the particle and of the field source (in units of the elementary charge, <math>Z=-1</math> for the hydrogen atom), <math>\alpha</math> is the fine-structure constant, and <math>\hbar^2k^2/(2m)</math> is the energy of the particle. The solution, which is the Coulomb wave function, can be found by solving this equation in parabolic coordinates
<math>\xi= r + \vec{r}\cdot\hat{k}, \quad \zeta= r - \vec{r}\cdot\hat{k} \qquad (\hat{k} = \vec{k}/k) \,.</math>
Depending on the boundary conditions chosen, the solution has different forms. Two of the solutions are[2][3]
where <math>M(a,b,z) \equiv {}_1\!F_1(a;b;z)</math> is the confluent hypergeometric function, <math>\eta = Zmc\alpha/(\hbar k)</math> and <math>\Gamma(z)</math> is the gamma function. The two boundary conditions used here are
which correspond to <math>\vec{k}</math>-oriented plane-wave asymptotic states before or after its approach of the field source at the origin, respectively. The functions <math>\psi_{\vec{k}}^{(\pm)}</math> are related to each other by the formula
The wave function <math>\psi_{\vec{k}}(\vec{r})</math> can be expanded into partial waves (i.e. with respect to the angular basis) to obtain angle-independent radial functions <math>w_\ell(\eta,\rho)</math>. Here <math>\rho=kr</math>.
The equation for single partial wave <math>w_\ell(\eta,\rho)</math> can be obtained by rewriting the laplacian in the Coulomb wave equation in spherical coordinates and projecting the equation on a specific spherical harmonic <math>Y_\ell^m(\hat{r})</math>
The solutions are also called Coulomb (partial) wave functions or spherical Coulomb functions. Putting <math>z=-2i\rho</math> changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments <math>M_{-i\eta,\ell+1/2}(-2i\rho)</math> and <math>W_{-i\eta,\ell+1/2}(-2i\rho)</math>. The latter can be expressed in terms of the confluent hypergeometric functions <math>M</math> and <math>U</math>. For <math>\ell\in\mathbb{Z}</math>, one defines the special solutions [4]
The asymptotic behavior of the spherical Coulomb functions <math>H_\ell^{(\pm)}(\eta,\rho)</math>, <math>F_\ell(\eta,\rho)</math>, and <math>G_\ell(\eta,\rho)</math> at large <math>\rho</math> is
<math>H_\ell^{(\pm)}(\eta,\rho) \sim e^{\pm i \theta_\ell(\rho)} \,,</math>
The solutions <math>H_\ell^{(\pm)}(\eta,\rho)</math> correspond to incoming and outgoing spherical waves. The solutions <math>F_\ell(\eta,\rho)</math> and <math>G_\ell(\eta,\rho)</math> 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 <math>\psi_{\vec{k}}^{(+)}(\vec{r})</math> [5]
The radial parts for a given angular momentum are orthonormal. When normalized on the wave number scale (k-scale), the continuum radial wave functions satisfy [6][7]
<math>\int_0^\infty R_{k\ell}^\ast(r) R_{k'\ell}(r) r^2 dr = \delta(k-k')</math>
Other common normalizations of continuum wave functions are on the reduced wave number scale (<math>k/2\pi</math>-scale),
<math>\int_0^\infty R_{k\ell}^\ast(r) R_{k'\ell}(r) r^2 dr = 2\pi \delta(k-k') \,,</math>
and on the energy scale
<math>\int_0^\infty R_{E\ell}^\ast(r) R_{E'\ell}(r) r^2 dr = \delta(E-E') \,.</math>
The radial wave functions defined in the previous section are normalized to
<math>\int_0^\infty R_{k\ell}^\ast(r) R_{k'\ell}(r) r^2 dr = \frac{(2\pi)^3}{k^2} \delta(k-k') </math>
