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CRC Press eBooks

Numerical solution of the Schrödinger equation for two-dimensional double-well oscillators

Amlan K. Roy, Ajit J. Thakkar, Bimalendu Deb

Published Apr 29, 2019• 0 citations
Publisher

Abstract

Wave functions, energies and selected expectation values of the low-lying stationary states of two-dimensional double well potentials are obtained from the long-time solutions of the corresponding time-dependent Schrödinger equation. The latter is transformed to a diffusion-like equation which is then solved by an alternating-direction, implicit, finite-difference method.

DOI
10.1201/9780429081385-264

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Wave functions, energies and selected expectation values of the low-lying stationary states of two-dimensional double well potentials are obtained from the long-time solutions of the corresponding time-dependent Schrödinger equation. The latter is transformed to a diffusion-like equation which is then solved by an alternating-direction, implicit, finite-difference method.
Wavefunctions, energies and selected expectation values of the low-lying stationary states of two-dimensional double-well potentials are obtained from the long-time solutions of the corresponding time-dependent Schrödinger equation. The numerical method consists of transformation to a diffusion-like equation which is then solved by an alternating-direction, implicit, finite-difference method. The method is tested for cases in which the energies have been obtained by other methods. Then the dependence of the energies and other properties on the potential parameters is discussed on the basis of results for four states of 35 different sets of potential parameters including some that lead to pseudo-degeneracies.
This project looks into the time evolution of a wave function within a two-dimensional quantum well. We start by solving the time-dependent Schrödinger equation for stationary states in a quantum well. Next, we express any wave function as a linear combination of stationary states, allowing us to understand their time evolution. Two methods are presented: one relies on decomposing the wave function into a basis of stationary states and the other on discretisation of the time-dependent Schrödinger equation, incorporating three-point formulas for derivatives. These approaches necessitate confronting intricate boundary conditions and require maintaining energy conservation for numerical accuracy. We further demonstrate the methods using a wave packet, revealing fundamental phenomena in quantum physics. Our results demonstrate the utility of these methods in understanding quantum systems, despite the challenges in determining stationary states for a given potential. This study enhances our comprehension of the dynamics of quantum states in constrained systems, essential for fields like quantum computing and nanotechnology.
The Schrödinger equation serves as a pivotal differential equation, capturing the fundamental duality of particles as both matter and waves. Upon solving it, we derive a particle's wave function within a given potential, delineating the confines within which the particle operates. This wave function enables us to ascertain the "most probable" values for parameters such as position, momentum, and energy within the specified potential. In this study, we focus on the double-well potential in two dimensions, resembling a bowl-like structure. Specifically tailored equations, termed Equation 1 and Equation 2, stem from the general Schrödinger equation when applied to the prescribed conditions. This research aims to utilize these equations to describe a particle’s behavior within the ground and the first excited states of the double well potential, employing two distinct methodologies. The first methodology involves leveraging the variational method on Equations 1 and 2 to obtain approximate solutions, providing crucial inputs for our subsequent approach. The second methodology serves as entails an analytical solution of Equations 1 and 2. A comparative analysis of the outcomes from both methods is anticipated to ensure the coherence and validity of the variational method. This project integrates key principles and proficiencies from both physics and mathematics disciplines, offering a platform for honing practical skills in applying these methodologies within a real-world context.
Ground- and excited-state energies as well as probability densities of quantum-mechanical anharmonic and double-well oscillators are obtained by numerically solving the time-dependent Schr\"odinger equation (TDSE). The methodology employed is exact in principle and is based on solving the TDSE in imaginary time, coupled with the minimization of energy expectation value. The present method works very well for deep wells and for double-well oscillators with pseudodegenerate energy levels where time-independent methods experience difficulties. The dependence of energy on anharmonicity parameter and the splitting of pseudodegeneracy by an asymmetry parameter as well as probability-density plots are discussed. Several potentials, especially those with pseudodegenerate energy levels, are reported here.
We study the quantum behaviour of a particle moving in a one-dimensional double well potential. This double well is obtained by gluing together, at the origin, two shifted harmonic oscillator potentials. The Schrödinger equation is exactly solvable. The requirement that discontinuities, in the wavefunction and its first derivative, are absent at the origin, leads to the quantisation of the energy eigenvalues. We also show that oscillations in time take place between two nearby single harmonic oscillator ground states. Finally, the double well potential is augmented by a Dirac delta-function potentials at the origin and the corresponding Schrödinger equation is solved.