Coulomb wave function

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

image of complex plot of regular Coulomb wave function added

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 of mass $m$ is the Schrödinger equation with Coulomb potential

$(-{\hslash }^2\frac{{\mathrm{\nabla }}^2}{2m}+\frac{Z\hslash c\alpha }{r}){\psi }_{\overrightarrow{k}}(\overrightarrow{r})=\frac{{\hslash }^2{k}^2}{2m}{\psi }_{\overrightarrow{k}}(\overrightarrow{r}),$

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 ${\hslash }^2{k}^2/(2m)$ is the energy of the particle. The solution, which is the Coulomb wave function, can be found by solving this equation in parabolic coordinates

$\xi =r+\overrightarrow{r}\cdot \hat{k},\zeta =r-\overrightarrow{r}\cdot \hat{k}(\hat{k}=\overrightarrow{k}/k).$

Depending on the boundary conditions chosen, the solution has different forms. Two of the solutions are

${\psi }_{\overrightarrow{k}}^{(\pm )}(\overrightarrow{r})=\mathrm{\Gamma }(1\pm i\eta )e^{-\pi \eta /2}{e}^{i\overrightarrow{k}\cdot \overrightarrow{r}}M(\mp i\eta ,1,\pm ikr-i\overrightarrow{k}\cdot \overrightarrow{r}),$

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

${\psi }_{\overrightarrow{k}}^{(\pm )}(\overrightarrow{r})\to e^{i\overrightarrow{k}\cdot \overrightarrow{r}}(\overrightarrow{k}\cdot \overrightarrow{r}\to \pm \mathrm{\infty }),$

which correspond to $\overrightarrow{k}$-oriented plane-wave asymptotic states before or after its approach of the field source at the origin, respectively. The functions ${\psi }_{\overrightarrow{k}}^{(\pm )}$ are related to each other by the formula

${\psi }_{\overrightarrow{k}}^{(+)}={\psi }_{-\overrightarrow{k}}^{(-)\ast }.$

Partial wave expansion

The wave function ${\psi }_{\overrightarrow{k}}(\overrightarrow{r})$ 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 $\rho =kr$.

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

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

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

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 $Y_{\ell }^m(\hat{r})$

$\frac{d^2{w}_{\ell }}{d{\rho }^2}+(1-\frac{2\eta }{\rho }-\frac{\ell (\ell +1)}{{\rho }^2})w_{\ell }=0.$

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

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

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

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

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

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

In particular one has

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

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 ${\psi }_{\overrightarrow{k}}^{(+)}(\overrightarrow{r})$

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

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 $H_{\ell }^{(\pm )}(\eta ,\rho )$ are proportional to Spherical Hankel functions

$F_{\ell }(0,\rho )/\rho =j_{\ell }(\rho )$

$G_{\ell }(0,\rho )/\rho =-y_{\ell }(\rho )$

$H_{\ell }^{(+)}(0,\rho )/\rho =i{h}_{\ell }^{(1)}(\rho )$

$H_{\ell }^{(-)}(0,\rho )/\rho =-i{h}_{\ell }^{(2)}(\rho )$

and are normalized same as Spherical Bessel functions

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{j}_l(kr)j_l(k^{\prime }r)r^{2}dr=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{F_{\ell }(\pm \frac{1}{a_{0}k},kr)}{kr}\frac{F_{\ell }(\pm \frac{1}{a_0{k}^{\prime }},k^{\prime }r)}{k^{\prime }r}{r}^{2}dr=\frac{\pi }{2k^2}\delta (k-k^{\prime })$

and similar for other 3.

Properties of the Coulomb function

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

${\int }_0^{\mathrm{\infty }}{R}_{k\ell }^{\ast }(r)R_{k^{\prime }\ell }(r)r^{2}dr=\delta (k-k^{\prime })$

Other common normalizations of continuum wave functions are on the reduced wave number scale ($k/2\pi$-scale),

${\int }_0^{\mathrm{\infty }}{R}_{k\ell }^{\ast }(r)R_{k^{\prime }\ell }(r)r^{2}dr=2\pi \delta (k-k^{\prime }),$

and on the energy scale

${\int }_0^{\mathrm{\infty }}{R}_{E\ell }^{\ast }(r)R_{E^{\prime }\ell }(r)r^{2}dr=\delta (E-E^{\prime }).$

The radial wave functions defined in the previous section are normalized to

${\int }_0^{\mathrm{\infty }}{R}_{k\ell }^{\ast }(r)R_{k^{\prime }\ell }(r)r^{2}dr=\frac{(2\pi {)}^3}{k^2}\delta (k-k^{\prime })$

as a consequence of the normalization

$\int {\psi }_{\overrightarrow{k}}^{\ast }(\overrightarrow{r}){\psi }_{{\overrightarrow{k}}^{\prime }}(\overrightarrow{r})d^{3}r=(2\pi {)}^3\delta (\overrightarrow{k}-{\overrightarrow{k}}^{\prime }).$

The continuum (or scattering) Coulomb wave functions are also orthogonal to all Coulomb bound states

${\int }_0^{\mathrm{\infty }}{R}_{k\ell }^{\ast }(r)R_{n\ell }(r)r^{2}dr=0$

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

Further reading

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