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

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

It is proportional to the confluent hypergeometric function $M(a,b,z)\equiv {}_{1}\!F_{1}(a;b;z)$ and has the form

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

where $\eta =-Z/k$ and $\Gamma (x)$ is the gamma function. Alternative expression can be obtained from the Kummer transform

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

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 $\rho =kr$.

$\psi _{\vec {k}}({\vec {r}})={\frac {1}{(2\pi )^{3/2}}}{\frac {1}{\rho }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }w_{\ell }(\eta ,\rho )Y_{\ell }^{m}({\hat {r}})Y_{\ell }^{m\ast }({\hat {k}})$

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

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

Further reading

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 14". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 538. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
• Bateman, Harry (1953), Higher transcendental functions (PDF), vol. 1, McGraw-Hill.
• 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), The Royal Society: 708–728, doi:10.1098/rspa.1935.0043, ISSN 0080-4630, JSTOR 96298
• Slater, Lucy Joan (1960), Confluent hypergeometric functions, Cambridge University Press, MR 0107026.