{{Short description|In physics, solution to Schrödinger equation}} alt=Irregular Coulomb wave function G plotted from 0 to 20 with repulsive and attractive interactions in Mathematica 13.1|thumb|Irregular Coulomb wave function G plotted from 0 to 20 with repulsive and attractive interactions in Mathematica 13.1 thumb|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 <math>m</math> is the Schrödinger equation with Coulomb potential<ref>{{Citation|first=Robert N.|last=Hill|editor=Drake, Gordon|title=Handbook of atomic, molecular and optical physics|publisher=Springer New York|year=2006|pages=153–155|doi=10.1007/978-0-387-26308-3|isbn=978-0-387-20802-2|url=https://cds.cern.ch/record/882343|url-access=subscription}}</ref> :<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<ref>{{Citation|first1=L. D.|last1=Landau|first2=E. M.|last2=Lifshitz|title=Course of theoretical physics III: Quantum mechanics, Non-relativistic theory|edition=3rd|publisher=Pergamon Press|year=1977|page=569}}</ref><ref>{{Citation|first1=Albert|last1=Messiah|title=Quantum mechanics|publisher=North Holland Publ. Co.|year=1961|page=485}}</ref> :<math>\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}) \,,</math> 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 :<math>\psi_{\vec{k}}^{(\pm)}(\vec{r}) \rightarrow e^{i\vec{k}\cdot\vec{r}} \qquad (\vec{k}\cdot\vec{r} \rightarrow \pm\infty) \,,</math> 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 :<math>\psi_{\vec{k}}^{(+)} = \psi_{-\vec{k}}^{(-)*} \,.</math>
=== Partial wave expansion === 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>. :<math>\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}) \,.</math> A single term of the expansion can be isolated by the scalar product with a specific spherical harmonic :<math>\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.</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> :<math>\frac{d^2 w_\ell}{d\rho^2}+\left(1-\frac{2\eta}{\rho}-\frac{\ell(\ell+1)}{\rho^2}\right)w_\ell=0 \,.</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 <ref>{{Citation|first1=David|last1=Gaspard|title=Connection formulas between Coulomb wave functions|year=2018|journal=J. Math. Phys.|volume=59|issue=11|pages=112104|doi=10.1063/1.5054368|arxiv=1804.10976}}</ref> :<math>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) \,,</math> where :<math>\sigma_\ell = \arg \Gamma(\ell+1+i \eta)</math> is called the Coulomb phase shift. One also defines the real functions :<math>F_\ell(\eta,\rho) = \frac{1}{2i} \left(H_\ell^{(+)}(\eta,\rho)-H_\ell^{(-)}(\eta,\rho) \right) \,,</math> :alt=Regular Coulomb wave function F plotted from 0 to 20 with repulsive and attractive interactions in Mathematica 13.1|thumb|Regular Coulomb wave function F plotted from 0 to 20 with repulsive and attractive interactions in Mathematica 13.1<math>G_\ell(\eta,\rho) = \frac{1}{2} \left(H_\ell^{(+)}(\eta,\rho)+H_\ell^{(-)}(\eta,\rho) \right) \,.</math> In particular one has :<math>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) \,.</math> 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> :<math>F_\ell(\eta,\rho) \sim \sin \theta_\ell(\rho) \,,</math> :<math>G_\ell(\eta,\rho) \sim \cos \theta_\ell(\rho) \,,</math> where :<math>\theta_\ell(\rho) = \rho - \eta \log(2\rho) -\frac{1}{2} \ell \pi + \sigma_\ell \,.</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> <ref>{{Citation|first1=Albert|last1=Messiah|title=Quantum mechanics|publisher=North Holland Publ. Co.|year=1961|page=426}}</ref> :<math>\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}) \,,</math> In the limit <math>\eta\to 0</math> regular/irregular Coulomb wave functions <math>F_\ell(\eta,\rho)</math>,<math>G_\ell(\eta,\rho)</math> are proportional to Spherical Bessel functions and spherical Coulomb functions <math>H^{(\pm)}_\ell(\eta,\rho)</math> are proportional to Spherical Hankel functions :<math> F_\ell(0,\rho)/\rho = j_\ell(\rho) </math> :<math> G_\ell(0,\rho)/\rho = - y_\ell(\rho) </math> :<math> H^{(+)}_\ell(0,\rho)/\rho = i\, h^{(1)}_\ell(\rho) </math> :<math> H^{(-)}_\ell(0,\rho)/\rho =-i\, h^{(2)}_\ell(\rho) </math> and are normalized same as Spherical Bessel functions :<math> \int\limits_0^\infty j_l(k\, r) j_l(k' r)\,r^2 dr = \int\limits_0^\infty \frac{F_\ell\left(\pm \frac{1}{a_0 k},k\, r\right)}{k\, r} \frac{F_\ell\left(\pm \frac{1}{a_0 k'}, k' r\right)}{k' r} \, r^2 d r = \frac{\pi}{2 k^2}\delta(k-k')</math> 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 <ref>{{Citation|first=Jiří|last=Formánek|title=Introduction to quantum theory I|publisher=Academia|location=Prague|year=2004|edition=2nd|language=Czech|pages=128–130}}</ref><ref>{{Citation|first1=L. D.|last1=Landau|first2=E. M.|last2=Lifshitz|title=Course of theoretical physics III: Quantum mechanics, Non-relativistic theory|edition=3rd|publisher=Pergamon Press|year=1977|pages=121}}</ref> :<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> as a consequence of the normalization :<math>\int \psi^{\ast}_{\vec{k}}(\vec{r}) \psi_{\vec{k}'}(\vec{r}) d^3r = (2\pi)^3 \delta(\vec{k}-\vec{k}') \,.</math>
The continuum (or scattering) Coulomb wave functions are also orthogonal to all Coulomb bound states<ref>{{Citation|first1=L. D.|last1=Landau|first2=E. M.|last2=Lifshitz|title=Course of theoretical physics III: Quantum mechanics, Non-relativistic theory|edition=3rd|publisher=Pergamon Press|year=1977|pages=668–669}}</ref> :<math>\int_0^\infty R_{k\ell}^\ast(r) R_{n\ell}(r) r^2 dr = 0 </math> due to being eigenstates of the same hermitian operator (the hamiltonian) with different eigenvalues.
==Further reading== *{{citation|first=Harry|last=Bateman|title=Higher transcendental functions|volume=1|year=1953|publisher=McGraw-Hill|url=http://apps.nrbook.com/bateman/Vol1.pdf|access-date=2011-07-30|archive-date=2011-08-11|archive-url=https://web.archive.org/web/20110811153220/http://apps.nrbook.com/bateman/Vol1.pdf|url-status=dead}}. *{{Citation | last1=Jaeger | first1=J. C. | last2=Hulme | first2=H. R. | title=The Internal Conversion of γ -Rays with the Production of Electrons and Positrons | jstor=96298 | year=1935 | journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences | issn=0080-4630 | volume=148 | issue=865 | pages=708–728 | doi=10.1098/rspa.1935.0043|bibcode = 1935RSPSA.148..708J | doi-access= }} *{{Citation | last1=Slater | first1=Lucy Joan | title=Confluent hypergeometric functions | publisher=Cambridge University Press | mr=0107026 | year=1960}}.
==References== {{Reflist}}
Category:Special hypergeometric functions