Estimation of Energy Separation in Tunelling States in Symmetric-Hyperbolicus Double-Well Potential Problem using Filter Method

Abstract

Double well potential (DWP) plays an important role in quantum physics but its analytical exact solutions are incomplete, except for some limited quasi-exact solvable (QES) and therefore numerical solutions are still required. This paper is aimed to obtain numerical solution of the recently proposed symnetric-hyperbolicus DWP  and  by using our newly developed filter method. The filter method is able to produce eigen-energies ​​and eigenfunctions those are in accordance with the exact analytical results. Next, we obtain the dependence of the ground state energy and the energy separation between the 2 lowest states on the DWP parameter k. For large k, the 2 lowest energy levels are below the potential barrier so the eigenfunctions penetrate the barrier, so the 2 lowest energy levels can be considered as the result of a single energy split. In this case, we observe that the energy separation  is an exponential function k, as opposed to a linear function for smaller k in the non tunelling region. The exponential trend of the numerical results is in good agreement with the theoretical approach and allows one to estimate the 2 lowest energies for a given k. HIGHLIGHTS For small , the two lowest energy levels lie above the barrier potential, where both ground state and the energy separation  is a linier function of For large , the two lowest energy levels lie below the barrier potential so that the eigenfunction penetrates the barrier. As a result, the two lowest energy levels can be thought of as splitting energy states. In this case, we observe that the energy separation is an exponential function . Since  also depends on , it is possible to estimate the two lowest energy levels The exponential trend of the numerical results for large agrees well with the theoretical approach, shown by the linear relationship between  and the classical actions in the classical forbidden regions, GRAPHICAL ABSTRACT