An Application of Gröbner Basis in Differential Equations of Physics

Abstract

We apply the Gröbner basis to the ansatz method in quantum mechanics to obtain the energy eigenvalues and the wave functions in a very simple manner. There are important physical potentials such as the Cornell interaction which play significant roles in particle physics and can be treated via this technique. As a typical example, the algorithm is applied to the semi-relativistic spinless Salpeter equation under the Cornell interaction. Many other applications of the idea in a wide range of physical fields are listed as well.

Exact Solution of a Fully General Non-Local-Thermodynamic-Equilibrium Two-Level Atom

“…Polynomial system solving is ubiquitous, as many models in the sciences and engineering can be described by non-linear polynomials. This includes: algebraic statistics, algebraic biology, chemical reaction networks, coding theory, computer vision, cryptography, networks modelling, neuroscience, robotics, string theory or topological data analysis via (multivariate) persistent homology [12].…”