A Study Of The Finite Difference Methods And The WENO Schemes In The Resolution Of Schrodinger Equation

The goal of this research was to examine the quantum mechanical phenomenon of tunneling with respect to the Schrödinger Equation. A quantum mechanical wave is said to “tunnel” when it travels (propagates) through a classically forbidden region. In a more physical interpretation, it is when the energy (E) of the wave is lower than the potential at a specific point V(x). [A point where V(x) is greater than E is referred to as a classical turning point.] The phenomenon of a tunneling was examined in a double-well potential. Another goal of this project was to compare two methods for approximating the tunneling time of a wave packet. The two methods used were the finite difference method and WENO approximation scheme In order to compare a potential that experienced tunneling and one that did not, the solution to a quadratic potential was also solved. Index Terms- WENO schemes, Finite difference, Schrodinger equation, Simulation.