arXiv:1412.1802 · Physical Review E

Numerical solutions of the Schrödinger equation with source terms or time-dependent potentials

W. van Dijk, F. Masafumi Toyama

Published Dec 17, 2014• 12 citations

Abstract

We develop an approach to solving numerically the time-dependent Schrödinger equation when it includes source terms and time-dependent potentials. The approach is based on the generalized Crank-Nicolson method supplemented with an Euler-MacLaurin expansion for the time-integrated nonhomogeneous term. By comparing the numerical results with exact solutions of analytically solvable models, we find that the method leads to precision comparable to that of the generalized Crank-Nicolson method applied to homogeneous equations. Furthermore, the systematic increase in precision generally permits making estimates of the error.

DOI: 10.1103/physreve.90.063309

Related papers

Accurate numerical solutions of the time-dependent Schrödinger equation

W. van Dijk, F. Masafumi Toyama

We present a generalization of the often-used Crank-Nicolson (CN) method of obtaining numerical solutions of the time-dependent Schrödinger equation. The generalization yields numerical solutions accurate to order (Deltax)2r-1 in space and (Deltat)2M in time for any positive integers r and M, while CN employ r=M=1. We note dramatic improvement in the attainable precision (circa ten or greater orders of magnitude) along with several orders of magnitude reduction of computational time. The improved method is shown to lead to feasible studies of coherent-state oscillations with additional short-range interactions, wave-packet scattering, and long-time studies of decaying systems.

arXiv:physics/0701150Physical Review EMar 23, 2007107 citations

Modified Crank-Nicolson method for solving the time-dependent Schrödinger equation

A R Kurniawan, Evi Suaebah, Z. I. Supardi

Abstract The time-dependent Schrödinger equation is fundamental to quantum mechanics, describing the evolution of quantum systems. This study introduces a novel numerical approach that extends and modifies the Crank-Nicholson method to solve the time-dependent Schrödinger equation. This method is designed to enhance both the precision and performance of computational results, especially in the context of quantum tunneling phenomena and applications that require high accuracy over long periods. To assess the precision and performance of the modified Crank-Nicolson method, we benchmark it against two well-known numerical methods: the Runge-Kutta method and the Split-Step Fourier method. Quantum tunneling is selected as the case study due to its relevance in quantum physics applications. Comparative analysis is based on numerical error metrics. The proposed method can be applied to other physical phenomena, expanding its potential application in physics research and technology industries.

Journal of Physics Conference SeriesNov 1, 20240 citations

Visualization of quantum mechanics on a personal computer

Fredy L. Dubeibe

In recent decades a lot of research has been done on the numerical solution of the time-dependent Schrodinger equation. On the one hand, some of the proposed numerical methods do not need any kind of matrix inversion, but source terms cannot be easily implemented into this schemes; on the other, some methods involving matrix inversion can implement source terms in a natural way, but are not easy to implement into some computational software programs widely used by non-experts in programming (e.g. Mathematica). We present a simple method to solve the time-dependent Schrodinger equation by using a standard Crank-Nicholson method together with a Cayley's form for the finite-difference representation of evolution operator. Here, such standard numerical scheme has been simplified by inverting analytically the matrix of the evolution operator in position representation. The analytical inversion of the N x N matrix let us easily and fully implement the numerical method, with or without source terms, into Mathematica or even into any numerical computing language or computational software used for scientific computing.

arXiv:1005.0044May 1, 20100 citations

Explicit Euler method for solving the time-dependent Schrödinger equation

Ioan Sturzu

Using an explicit Euler substitution, a system of differential equations was obtained, which can be used to find the solution of the time-dependent one-dimensional Schr\"odinger equation for a general form of the time-dependent potential.

arXiv:quant-ph/0108089Physical Review AOct 5, 20015 citations

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

Kabir, Adib

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.

arXiv:2410.10060Oct 14, 20240 citations

Centered-Difference Applications for Schrödinger's Equation

Matthew Thomas Murachver

This project enumerates methods utilizing discretized centered-difference approximations on the second order differential equation for quantum particles known as Schrodinger’s Equation. An eigenvalue-eigenfunction scheme is developed to sieve for valid solutions to The Time Independent Schrodinger Equation. Additionally the Crank-Nicolson method is applied to the Time Dependent Schrodinger Equation to describe wavefunction (eigenfunction) time evolution. The validity of these methods is discussed with applications to several fundamental pedagogical introductory quantum mechanic systems.

Jan 1, 20140 citations