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.

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

Coulomb wave equation

The Coulomb wave equation for a single charged particle of mass $m$ is the Schrödinger equation with Coulomb potential

$\left(-\hbar ^{2}{\frac {\nabla ^{2}}{2m}}+{\frac {Z\hbar c\alpha }{r}}\right)\psi _{\vec {k}}({\vec {r}})={\frac {\hbar ^{2}k^{2}}{2m}}\psi _{\vec {k}}({\vec {r}})\,,$

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

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

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

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

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

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

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

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

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

$\int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{k'\ell }(r)r^{2}dr=\delta (k-k')$

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

$\int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{k'\ell }(r)r^{2}dr=2\pi \delta (k-k')\,,$

and on the energy scale

$\int _{0}^{\infty }R_{E\ell }^{\ast }(r)R_{E'\ell }(r)r^{2}dr=\delta (E-E')\,.$

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

$\int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{k'\ell }(r)r^{2}dr={\frac {(2\pi )^{3}}{k^{2}}}\delta (k-k')$

as a consequence of the normalization

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

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

• Bateman, Harry (1953), Higher transcendental functions (PDF), vol. 1, McGraw-Hill, archived from the original (PDF) on 2011-08-11, retrieved 2011-07-30.
• Jaeger, J. C.; Hulme, H. R. (1935), "The Internal Conversion of γ -Rays with the Production of Electrons and Positrons", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 148 (865): 708–728, Bibcode:1935RSPSA.148..708J, doi:10.1098/rspa.1935.0043, ISSN 0080-4630, JSTOR 96298
• Slater, Lucy Joan (1960), Confluent hypergeometric functions, Cambridge University Press, MR 0107026.