Numerical Solution of Schrödinger Equation by Crank–Nicolson Method

Abstract

In this study, we implemented the well-known Crank–Nicolson scheme for the numerical solution of Schrödinger equation. The numerical results converge to the exact solution because the Crank–Nicolson scheme is unconditionally stable and accurate. We have compared the results for different parameters with analytical solution, and it is found that the Crank–Nicolson scheme is suitable for the numerical solution of Schrödinger equations. Three different problems are included to verify the accuracy, stability, and capability of the Crank–Nicolson scheme.

Stability, convergence and error analysis of B-spline collocation with Crank–Nicolson method and finite element methods for numerical solution of Schrodinger equation arises in quantum mechanics

“…The quadratic B-spline finite element methods employed to solve the time-fractional Schrodinger equation [29] Furthermore, different authors have developed various methods for solving the governing equation (1.1), like multistep and hybrid methods [30] and a two-step hybrid method [31]. The other methods, such as the Crank-Nicolson scheme [32], quintic Hermite scheme [33], and B-spline collocation technique [34][35][36][37][38][39][40][41], are beneficial to the present scheme. The main goal of the proposed scheme is to obtain a better approximate quantum mechanical energy solution following Schrodinger's original solution and to illustrate how it can be applied to a complex environment with the Schrodinger equation using a nonic B-spline collocation method followed by FEM and the Crank-Nicolson scheme.…”

“…According to the initial conditions z x x x , 0 sin , A comparative study of present numerical results with exact solution and results in [32] with the parameters h t 0.8, 0.05 = The exact solution in real and imaginary parts at…”

A Modified Crank-Nicolson Method for Numerical Solution of the Linear Time-Dependent Schrodinger Equation

“…Pathak et al (2022) proposed the Kansa method combined with polyharmonic radial basis functions for solving nonlinear Schrödinger equations in two dimensions. Khan et al (2022) used the Crank-Nicolson scheme with finite difference approximation for solving the Schrödinger equation, demonstrating convergence and effectiveness for non-homogeneous problems. While the aforementioned literature focused on the Schrödinger equation, which neglected the potential energy and jeopardised the equation's physical meaning, our investigation aims to address this problem.…”

“…This paper developed the Modified Crank-Nicolson Method (MCNM) for numerical solutions of the time-dependent Schrödinger equation. While Khan et al (2022) and Kafle et al (2023) worked on the Schrödinger equation, where the potential energy was neglected, resulting in the loss of the equation's physical meaning, this paper aims to address this issue by utilizing the derived (MCNM) to obtain their numerical solutions. The research conducted by Cari and Suparmi (2013) investigated energy eigenvalues and eigenfunctions within the Schrödinger equation.This study was further conducted by Mao and Nakamura (2008), who explored wave front set analysis of solutions to Schrödinger equations involving long-range perturbed harmonic oscillators.…”

Numerical Solution of Schrödinger Equation by using Crank-Nicolson Method

“…These studies have utilized numerical simulations and various numerical approaches to solve the Schrödinger equation and investigate the behavior of particles at the quantum level [4]. Amin Khan (2022) [14] utilized the Crank-Nicolson scheme to solve the Schrödinger equation numerically. The goal was to assess the effectiveness of the method in obtaining accurate results.…”