Coulomb wave function

In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument.

Coulomb wave equation

The Coulomb wave equation for a single charged particle is the Schrödinger equation with Coulomb potential

$\left(-\frac{\nabla^2}{2}+\frac{Z}{r}\right) \psi_{\vec{k}}(\vec{r}) = \frac{k^2}{2} \psi_{\vec{k}}(\vec{r}) \,,$

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 hydrogen atom) and $k^2$ is proportional to the asymptotic energy of the particle. The solution – 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

$\psi_{\vec{k}}^{(\pm)}(\vec{r}) = \frac{1}{(2\pi)^{3/2}}\Gamma(1\pm i\eta) e^{-\pi\eta/2} e^{i\vec{k}\cdot\vec{r}} M(\mp i\eta, 1, \pm ikr - i\vec{k}\cdot\vec{r}) \,,$

where $M(a,b,z) \equiv {}_1\!F_1(a;b;z)$ is the confluent hypergeometric function, $\eta = Z/k$ and $\Gamma(z)$ is the gamma function. The two boundary conditions used here are

$\psi_{\vec{k}}^{(\pm)}(\vec{r}) \rightarrow \frac{1}{(2\pi)^{3/2}} e^{i\vec{k}\cdot\vec{r}} \qquad (\vec{r}\cdot\vec{k} \rightarrow \mp\infty) \,,$

which correspond to $\vec{k}$-oriented plane-wave asymptotic state 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 $\rho=kr$.

$\psi_{\vec{k}}(\vec{r}) = \frac{1}{(2\pi)^{3/2}} \frac{1}{r} \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell 4\pi(-i)^\ell w_{\ell}(\eta,\rho) Y_\ell^m (\hat{r}) Y_{\ell}^{m\ast} (\hat{k}) \,.$

A single term of the expansion can be isolated by the scalar product with a specific angular state

$\psi_{k\ell m}(\vec{r}) = \int \psi_{\vec{k}}(\vec{r}) Y_\ell^m (\vec{k}) d\hat{k} = R_{k\ell}(r) Y_\ell^m(\hat{r}), \qquad R_{k\ell}(r) = \sqrt{\frac{2}{\pi}} (-i)^\ell \frac{1}{r} w_\ell(\eta,\rho).$

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. Putting $x=2i\rho$ changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments. Two special solutions called the regular and irregular Coulomb wave functions are denoted by $F_\ell(\eta,\rho)$ and $G_\ell(\eta,\rho)$, and defined in terms of the confluent hypergeometric function by

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

The two possible sets of signs are related to each other by the Kummer transform.

Properties of the Coulomb function

The radial parts for a given angular momentum are orthonormal,

$\int_0^\infty R_{k\ell}^\ast(r) R_{k'\ell}(r) r^2 dr = \delta(k-k')$

and for $Z=-1$ they are also orthogonal to all hydrogen bound states

$\int_0^\infty R_{k\ell}^\ast(r) R_{n\ell}(r) r^2 dr = 0$

due to being eigenstates of the same hermitian operator (the hamiltonian) with different eigenvalues.

Further reading

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