Overview
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 : \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 ) \,, 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^2k^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) \,, : 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 ) \,, 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^ (\pm) _\ell(\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^ (1) _\ell(\rho) : H^ (-) _\ell(0,\rho)/\rho =-i\, h^ (2) _\ell(\rho) and are normalized same as Spherical Bessel functions : \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') 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^\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^ \ast _ \vec k (\vec r ) \psi_ \vec k ' (\vec r ) d^3r = (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.