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Jurnal Penelitian Fisika dan Aplikasinya (JPFA)

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

Sri Purwaningsih, Ramacos Fardela

Published Jun 30, 2022• 1 citation

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.

DOI: 10.26740/jpfa.v12n1.p1-13

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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.

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.

Research has been carried out to determine the physical meaning and derivation of the Schrodinger equation with the literature study research method. Based on the research results obtained through data processing in the form of information obtained from various literature sources, it can be concluded that the general solution of the Schrodinger equation is a wave function. In general, based on mathematical calculations by applying the technique of solving differential equations to the Schrodinger equation, it can be seen that this equation has many solutions, but not all of these solutions can be said to be functions that meet the wave function criteria. This wave function must be able to interpret the real physical criteria of a particle.

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.

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.

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.

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.