A Dynamic Adaptive Wavelet Method for Solution of the Schrodinger Equation

2015 ◽  
Vol 06 (01) ◽  
pp. 1450001 ◽  
Author(s):  
Ratikanta Behera ◽  
Mani Mehra
Keyword(s):  
Schrödinger Equation ◽  
Computational Efficiency ◽  
Schrodinger Equation ◽  
Computational Cost ◽  
Two Dimensional ◽  
Wavelet Method ◽  
Adaptive Wavelet ◽  
One Dimensional ◽  
Grid Adaptation ◽  
Adaptive Wavelet Method

In this paper, we present a dynamically adaptive wavelet method for solving Schrodinger equation on one-dimensional, two-dimensional and on the sphere. Solving one-dimensional and two-dimensional Schrodinger equations are based on Daubechies wavelet with finite difference method on an arbitrary grid, and for spherical Schrodinger equation is based on spherical wavelet over an optimal spherical geodesic grid. The method is applied to the solution of Schrodinger equation for computational efficiency and achieve accuracy with controlling spatial grid adaptation — high resolution computations are performed only in regions where a solution varies greatly (i.e., near steep gradients, or near-singularities) and a much coarser grid where the solution varies slowly. Thereupon the dynamic adaptive wavelet method is useful to analyze local structure of solution with very less number of computational cost than any other methods. The prowess and computational efficiency of the adaptive wavelet method is demonstrated for the solution of Schrodinger equation on one-dimensional, two-dimensional and on the sphere.

The Application of Adaptive Wavelet Method to Multi-Dimensional Limiting Process for Enhancement of Computational Efficiency

2011 ◽  
Author(s):  
Hyungmin Kang ◽  
Kyunghyun Park ◽  
Dongho Lee ◽  
Kyuhong Kim ◽  
Seunghwan Park ◽  
...  
Keyword(s):  
Computational Efficiency ◽  
Interpolation Method ◽  
Higher Order ◽  
High Order ◽  
Computational Time ◽  
Numerical Accuracy ◽  
Wavelet Method ◽  
Adaptive Wavelet ◽  
Computational Fluid Dynamics Cfd ◽  
Adaptive Wavelet Method

An adaptive wavelet method is applied in order to enhance the computational efficiency of enhanced Multi-dimensional Limiting Process (e-MLP) without deterioration of the numerical accuracy of original Computational Fluid Dynamics (CFD) scheme. For this purpose, higher order of adaptive wavelet method is constructed including higher order of wavelet decomposition and modified thresholding. Besides, the locations of crucial features such as shock, vortex core, etc. are automatically and accurately searched in the CFD dataset through wavelet transformation. Only on these locations, high order spatial interpolation scheme with e-MLP are performed; in the other locations, interpolation method is utilized to compute residual values, which reduces the computational time of flux evaluation. This high order adaptive wavelet method was applied to unsteady Euler flow computations such as shock-vortex interaction problem. Throughout these processes, it was verified that computational efficiency was enhanced with preservation of numerical accuracy of CFD solver.

scholarly journals Algebraic approach to non-separable two-dimensional Schrödinger equation: Ground states for polynomial and Morse-like potentials

Open Physics ◽  
2014 ◽  
Vol 12 (10) ◽  
Author(s):  
Vladimír Tichý ◽  
Lubomír Skála ◽  
René Hudec
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Ground States ◽  
Algebraic Approach ◽  
Algebraic Method ◽  
Analytic Solutions ◽  
Wave Functions ◽  
Two Dimensional ◽  
One Dimensional

AbstractThis paper presents a direct algebraic method of searching for analytic solutions of the two-dimensional time-independent Schrödinger equation that is impossible to separate into two one-dimensional ones. As examples, two-dimensional polynomial and Morse-like potentials are discussed. Analytic formulas for the ground state wave functions and the corresponding energies are presented. These results cannot be obtained by another known method.

scholarly journals Excitonic Eigenstates of Disordered Semiconductor Quantum Wires: Adaptive Wavelet Computation of Eigenvalues for the Electron-Hole Schrödinger Equation

2013 ◽  
Vol 14 (1) ◽  
pp. 21-47 ◽  
Author(s):  
Christian Mollet ◽  
Angela Kunoth ◽  
Torsten Meier
Keyword(s):  
Schrödinger Equation ◽  
Quantum Wire ◽  
Schrodinger Equation ◽  
Quantum Wires ◽  
Exciton State ◽  
Weak Form ◽  
Electron Hole ◽  
Adaptive Wavelet ◽  
One Dimensional ◽  
The Wire

AbstractA novel adaptive approach to compute the eigenenergies and eigenfunctions of the two-particle (electron-hole) Schrödinger equation including Coulomb attraction is presented. As an example, we analyze the energetically lowest exciton state of a thin one-dimensional semiconductor quantum wire in the presence of disorder which arises from the non-smooth interface between the wire and surrounding material. The eigenvalues of the corresponding Schrödinger equation, i.e., the one-dimensional exciton Wannier equation with disorder, correspond to the energies of excitons in the quantum wire. The wavefunctions, in turn, provide information on the optical properties of the wire.We reformulate the problem of two interacting particles that both can move in one dimension as a stationary eigenvalue problem with two spacial dimensions in an appropriate weak form whose bilinear form is arranged to be symmetric, continuous, and coercive. The disorder of the wire is modelled by adding a potential in the Hamiltonian which is generated by normally distributed random numbers. The numerical solution of this problem is based on adaptive wavelets. Our scheme allows for a convergence proof of the resulting scheme together with complexity estimates. Numerical examples demonstrate the behavior of the smallest eigenvalue, the ground state energies of the exciton, together with the eigenstates depending on the strength and spatial correlation of disorder.

scholarly journals Classical fluid aspects of nonlinear Schrödinger equations and solitons

1987 ◽  
Vol 1 (2) ◽  
pp. 99-114
Author(s):  
James G. Gilson
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Point Of View ◽  
Alternative Theory ◽  
Two Dimensional ◽  
One Dimensional ◽  
Nonlinear Schrödinger ◽  
Time Region ◽  
The One

The author extends his alternative theory for Schrödinger quantum mechanics by introducing the idea of energy reference strata over configuration space. It is then shown that the view from various such strata defines, the content of the system of interest and enables a variety of different descriptions of events in the same space time region. Thus according to “the point of view” or energy stratum chosen so the type of Schrödinger equation, linear or otherwise, appropriate to describe the system is determined. A nonlinear information channel between two dimensional fluid action in hyperspace into two dimensional energy hyperspace is shown to exist generally as a background to nonlinear Schrödinger structures. In addition it is shown how soliton solutions of the one dimensional Schrödinger equation are related to two dimensional vortex fields in hyperspace.

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