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EPJ Web of Conferences

Numerical Solution of the Time Dependent 3D Schrödinger Equation Describing Tunneling of Atoms from Anharmonic Traps

I. S. Ishmukhamedov, Altay Ishmukhamedov, Vladimir S. Melezhik

Published Jan 1, 2018• 3 citations
Publisher

Abstract

We present an efficient numerical method for the integration of the 3D Schrödinger equation. A tunneling problem of two interacting bosonic atoms confined in a 1D anharmonic trap has been successfully solved by means of this method. We demonstrate fast convergence of the final results with respect to spatial and temporal grid steps. The computational scheme is based on the operator-splitting technique with the implicit Crank-Nicolson algorithm on spatial sixth-order finite-differences. The computational time is proportional to the number of spatial grid points.

DOI
10.1051/epjconf/201817303011

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This study presents a numerical simulation of a quantum electron confined in a 10 nm potential well, using the Crank-Nicolson numerical technique to solve the time-dependent Schrodinger equation. The results capture the evolution of the electron's wave function at the 2000th time step, illustrating distinct standing wave patterns and probability densities that align with quantum mechanical predictions. Additionally, both 2D and 3D simulations across multiple time steps reveal the dynamic nature of quantum superposition and interference within the well. These findings highlight the method's stability and accuracy, offering a valuable tool for exploring quantum phenomena in constrained quantum systems.
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