The Canonical Function Method and its applications in quantum physics

Abstract

a b s t r a c tThe Canonical Function Method (CFM) is a powerful method that solves the radial Schrödinger equation for the eigenvalues directly, without having to evaluate the eigenfunctions. It is applied to various quantum mechanical problems in Atomic and Molecular physics in the presence of regular or singular potentials. It has also been developed to handle single and multiple channel scattering problems, where phaseshift is required for the evaluation of the scattering cross-section. Its controllable accuracy makes it a valuable tool for the evaluation of vibrational levels of cold molecules, a sensitive test of the Bohr correspondence principle and a powerful method for tackling local and non-local spin dependent problems.

Marching toward the eigenvalues: The Canonical Function Method and the Schrödinger equation

“…The CFM has been tested succesfully in a variety of potentials [3] and gives accurate results for bound and free states. The tunable accuracy of our method allows to evaluate eigenvalues close to the ground state as well as close to highly excited states near the continuum limit to a large number of digits without any extrapolation.…”

“…The CFM can handle a large variety of Quantum problem problems [3] besides the eigenvalue problem making it an extremely versatile, fast and highly accurate. The evaluation of the Schrödinger operator spectrum is done without performing diagonalization, bypassing the evaluation of the eigenfunctions.…”

“…The CFM method is superior to many standard techniques that have been used to solve the Schrödinger equation such as Numerov [5] or relaxation methods that are particularly tailored for solving TPBV problems (see the Physics Reports Review [3]).…”

Recursive determination of phase shifts for screened Coulomb potentials

“…In the theory of collisions, phase shifts determine the scattering cross-sections [2,3]. For instance, the elastic scattering cross-section of particles is The phase shift plays a major role in phase-amplitude methods [4][5][6][7][8]; but whatever the technique chosen for its computation (solving Schrödinger equation or using semi-classical methods [9]), the difficult point is the determination of the potential V(r). Due to configuration mixing, the number of involved radial integrals can be very large, and analytical potentials can be an alternative to self-consistent field methods.…”

Solving Schrodinger equation specializing to the Stark effect in linear potential by the canonical function method

“…The Canonical Function Method (CFM) can handle a large variety of quantum problems [1], also the eigenvalues problem making it an extremely versatile, fast and highly accurate.…”