Quantum evolution of free particle with non-periodic initial wavefunction reconstructed by Gegenbauer series

Abstract

The wave function of a confined quantum particle of non-interacting case is numerically evolved through time under non-periodic boundary conditions. The time-evolution of wave functions, however, involves Fourier reconstructions which, under non-periodicity, suffer from Gibbs phenomenon. This undermines the objective to attain accurate point values for the evolving wave function. This paper addresses this problem by utilizing Gegenbauer reconstruction method to the evolving wave function. With the right set of reconstruction parameters, convergent reconstruction is attained and the effect of Gibbs phenomenon is minimized. This technique can be applied to the Confined Quantum Time of Arrival (CTOA) and other quantum systems that involve non-periodic boundary conditions.