Coulomb wave function

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) ,, :[[File:Regular Coulomb wave function F plotted from 0 to 20 with repulsive and attractive interactions in Mathematica.svg|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]]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.

References

References

1. Hill, Robert N.. (2006). "Handbook of atomic, molecular and optical physics". Springer New York.
2. (1977). "Course of theoretical physics III: Quantum mechanics, Non-relativistic theory". Pergamon Press.
3. (1961). "Quantum mechanics". North Holland Publ. Co..
4. (2018). "Connection formulas between Coulomb wave functions". J. Math. Phys..
5. (1961). "Quantum mechanics". North Holland Publ. Co..
6. Formánek, Jiří. (2004). "Introduction to quantum theory I". Academia.
7. (1977). "Course of theoretical physics III: Quantum mechanics, Non-relativistic theory". Pergamon Press.
8. (1977). "Course of theoretical physics III: Quantum mechanics, Non-relativistic theory". Pergamon Press.