Numerical Simulation of the Time-Dependent Schrodinger Equation Using the Crank-Nicolson Method

TL;DR

This paper demonstrates the application of the Crank-Nicolson method to numerically simulate the time evolution of a quantum electron in a potential well, capturing wave function dynamics consistent with quantum theory.

Contribution

It introduces a stable and accurate numerical approach for simulating time-dependent quantum systems using the Crank-Nicolson method.

Findings

Wave function evolution matches quantum predictions

Method remains stable over multiple time steps

Simulations reveal quantum superposition and interference

Abstract

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.