The Analysis of the Physical Quantity N Grid, v, and dt in Solving the Schrödinger Equation Using the Crank-Nicolson Method

Abstract

Solving the Schrödinger equation may result in a wave function of a particle in a quantum system, which can afford the information with respect to the particle’s behavior. The Schrödinger equation is useful for examining probability density and determining a wave function of free particles. This research focuses on solving the Schrödinger equation using the Crank-Nicolson method in free particles. The Crank-Nicolson method is a method of solving partial differential equations in the form of the Schrodinger equation, this method is very stable and accurate in giving numerical results. The result indicates that probability density and the form of the wave function of free particles are identified by varying the v, N grid, and dt. When dt = 1, v = 1, and v = 2, and the N grid remains at a score of 100, we acquire the same forms of the wave function and probability density. And yet, when dt = 2, v =2, and the N grid remains at a score of 100, the form of the wave function and probability density is constricted in one area. The N grid and dt are the two most affecting factors on the three variants. Using the Crank-Nicolson method, we can determine the wave function and probability for free particles by varying the value of N grid, dt.