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.

The Coulomb wave equation for a single charged particle of mass ${\displaystyle m}$ is the Schrödinger equation with Coulomb potential^[1]

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

where ${\displaystyle Z=Z_{1}Z_{2}}$ is the product of the charges of the particle and of the field source (in units of the elementary charge, ${\displaystyle Z=-1}$ for the hydrogen atom), ${\displaystyle \alpha }$ is the fine-structure constant, and ${\displaystyle \hbar ^{2}k^{2}/(2m)}$ is the energy of the particle. The solution, which is the Coulomb wave function, can be found by solving this equation in parabolic coordinates

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

Depending on the boundary conditions chosen, the solution has different forms. Two of the solutions are^[2][3]

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

where ${\displaystyle M(a,b,z)\equiv {}_{1}\!F_{1}(a;b;z)}$ is the confluent hypergeometric function, ${\displaystyle \eta =Zmc\alpha /(\hbar k)}$ and ${\displaystyle \Gamma (z)}$ is the gamma function. The two boundary conditions used here are

${\displaystyle \psi _{\vec {k}}^{(\pm )}({\vec {r}})\rightarrow e^{i{\vec {k}}\cdot {\vec {r}}}\qquad ({\vec {k}}\cdot {\vec {r}}\rightarrow \pm \infty )\,,}$

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

${\displaystyle \psi _{\vec {k}}^{(+)}=\psi _{-{\vec {k}}}^{(-)*}\,.}$

The wave function ${\displaystyle \psi _{\vec {k}}({\vec {r}})}$ can be expanded into partial waves (i.e. with respect to the angular basis) to obtain angle-independent radial functions ${\displaystyle w_{\ell }(\eta ,\rho )}$. Here ${\displaystyle \rho =kr}$.

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

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

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

The equation for single partial wave ${\displaystyle w_{\ell }(\eta ,\rho )}$ can be obtained by rewriting the laplacian in the Coulomb wave equation in spherical coordinates and projecting the equation on a specific spherical harmonic ${\displaystyle Y_{\ell }^{m}({\hat {r}})}$

${\displaystyle {\frac {d^{2}w_{\ell }}{d\rho ^{2}}}+\left(1-{\frac {2\eta }{\rho }}-{\frac {\ell (\ell +1)}{\rho ^{2}}}\right)w_{\ell }=0\,.}$

The solutions are also called Coulomb (partial) wave functions or spherical Coulomb functions. Putting ${\displaystyle z=-2i\rho }$ changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments ${\displaystyle M_{-i\eta ,\ell +1/2}(-2i\rho )}$ and ${\displaystyle W_{-i\eta ,\ell +1/2}(-2i\rho )}$. The latter can be expressed in terms of the confluent hypergeometric functions ${\displaystyle M}$ and ${\displaystyle U}$. For ${\displaystyle \ell \in \mathbb {Z} }$, one defines the special solutions ^[4]

${\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )=\mp 2i(-2)^{\ell }e^{\pi \eta /2}e^{\pm i\sigma _{\ell }}\rho ^{\ell +1}e^{\pm i\rho }U(\ell +1\pm i\eta ,2\ell +2,\mp 2i\rho )\,,}$

where

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

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

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

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

In particular one has

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

The asymptotic behavior of the spherical Coulomb functions ${\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}$, ${\displaystyle F_{\ell }(\eta ,\rho )}$, and ${\displaystyle G_{\ell }(\eta ,\rho )}$ at large ${\displaystyle \rho }$ is

${\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )\sim e^{\pm i\theta _{\ell }(\rho )}\,,}$

${\displaystyle F_{\ell }(\eta ,\rho )\sim \sin \theta _{\ell }(\rho )\,,}$

${\displaystyle G_{\ell }(\eta ,\rho )\sim \cos \theta _{\ell }(\rho )\,,}$

where

${\displaystyle \theta _{\ell }(\rho )=\rho -\eta \log(2\rho )-{\frac {1}{2}}\ell \pi +\sigma _{\ell }\,.}$

The solutions ${\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}$ correspond to incoming and outgoing spherical waves. The solutions ${\displaystyle F_{\ell }(\eta ,\rho )}$ and ${\displaystyle G_{\ell }(\eta ,\rho )}$ 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 ${\displaystyle \psi _{\vec {k}}^{(+)}({\vec {r}})}$ ^[5]

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

In the limit ${\displaystyle \eta \to 0}$ regular/irregular Coulomb wave functions ${\displaystyle F_{\ell }(\eta ,\rho )}$,${\displaystyle G_{\ell }(\eta ,\rho )}$ are proportional to Spherical Bessel functions and spherical Coulomb functions ${\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}$ are proportional to Spherical Hankel functions

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

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

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

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

and are normalized same as Spherical Bessel functions

${\displaystyle \int \limits _{0}^{\infty }j_{l}(k\,r)j_{l}(k'r)\,r^{2}dr=\int \limits _{0}^{\infty }{\frac {F_{\ell }\left(\pm {\frac {1}{a_{0}k}},k\,r\right)}{k\,r}}{\frac {F_{\ell }\left(\pm {\frac {1}{a_{0}k'}},k'r\right)}{k'r}}\,r^{2}dr={\frac {\pi }{2k^{2}}}\delta (k-k')}$

and similar for other 3.

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 ^[6][7]

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

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

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

and on the energy scale

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

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

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

as a consequence of the normalization

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

The continuum (or scattering) Coulomb wave functions are also orthogonal to all Coulomb bound states^[8]

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

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

• 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.