Solution of the One-Dimensional Schroedinger Equation by the Combined Use of Symbolic and Numerical Computation

1995 ◽  
Vol 99 (42) ◽  
pp. 15608-15610 ◽  
Author(s):  
Ota Bludsky ◽  
Vladimir Spirko ◽  
Jiri Cizek
Keyword(s):  
Numerical Computation ◽  
Schroedinger Equation ◽  
One Dimensional ◽  
Combined Use ◽  
The One

Effect of the Spatial Extent of the Control in a Bilinear Control Problem for the Schroedinger Equation

2009 ◽  
Author(s):  
Katherine A. Kime
Keyword(s):  
Differential Geometry ◽  
Finite Difference ◽  
Lie Algebras ◽  
Grid Point ◽  
Schroedinger Equation ◽  
Time Varying ◽  
One Dimensional ◽  
Bilinear Control ◽  
The One ◽  
Bilinear Control Problem

We consider control of the one-dimensional Schroedinger equation through a time-varying potential. Using a finite difference semi-discretization, we consider increasing the extent of the potential from a single central grid-point in space to two or more gridpoints. With the differential geometry package in Maple 8, we compute and compare the corresponding Control Lie Algebras, identifying a trend in the number of elements which span the Control Lie Algebras.

scholarly journals On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2137
Author(s):  
Huizeng Qin ◽  
Youmin Lu
Keyword(s):  
Boundary Value Problem ◽  
Numerical Computation ◽  
Unique Solution ◽  
Boundary Value ◽  
Boundary Values ◽  
Three Solutions ◽  
One Dimensional ◽  
The One ◽  
Gelfand Problem ◽  
Existing Result

We investigate the well-known one-dimensional perturbed Gelfand boundary value problem and approximate the values of α0,λ* and λ* such that this problem has a unique solution when 0<α<α0 and λ>0, and has three solutions when α>α0 and λ*<λ<λ*. The solutions of this problem are always even functions due to its symmetric boundary values and autonomous characteristics. We use numerical computation to show that 4.0686722336<α0<4.0686722344. This result improves the existing result for α0≈4.069 and increases the accuracy of α0 to 10−8. We developed an algorithm that reduces errors and increases efficiency in our computation. The interval of λ for this problem to have three solutions for given values of α is also computed with accuracy up to 10−14.

scholarly journals KAJIAN GELOMBANG SATU DIMENSI BERDASARKAN HASIL KOMPUTASI NUMERIK

2014 ◽  
Vol 2 (2) ◽  
pp. 217
Author(s):  
Dwi Sabda Budi Prasetya ◽  
Indira Puteri Kinasih
Keyword(s):  
Wave Equation ◽  
Numerical Computation ◽  
Wave Problem ◽  
One Dimensional ◽  
Programming Software ◽  
Mathematical Equations ◽  
The Finite Difference Method ◽  
The One ◽  
Matlab Programming ◽  
Dimensional Wave

One-dimensional wave problem research has been done based on numerical computation reviews. The one-dimensional wave equation has been completed using the finite difference method. Results of completion based on the results of numerical computation reviews are simulated using MATLAB programming software. The simulation suggests that numerical computation is reliable for solving mathematical equations.

Numerical Approximation of Bilinear Control of the Schroedinger Equation

2005 ◽  
Author(s):  
Katherine A. Kime
Keyword(s):  
Control Problem ◽  
Numerical Approximation ◽  
Schroedinger Equation ◽  
Quantum Systems ◽  
One Dimensional ◽  
Bilinear Control ◽  
The One ◽  
Localized Initial Data ◽  
Bilinear Control Problem ◽  
Crank Nicolson

We consider the one-dimensional Schroedinger equation in which the control is a time-dependent rectangular potential barrier/well. This is a bilinear control problem, as the potential multiplies the state. Differential geometric methods have been used to treat the bilinear control of systems of finitely many ODEs, and have been applied to the Schroedinger equation (quantum systems). In this paper we will calculate, using MATLAB, explicit controls which steer localized initial data to localized terminal data. These will be obtained using the Crank-Nicolson approximation, in which both space and time are discretized. If one semi-discretizes, in space, one obtains a bilinear control problem for a system of finitely many ODEs. One may pass from the semi-discretized system to Crank-Nicolson using the trapezoid rule. Thus the controls we calculate may be used to construct approximations to controls for the system of ODEs.

Control Lie Algebras of Semi-Discretizations of the Schroedinger Equation

2007 ◽  
Author(s):  
Katherine A. Kime
Keyword(s):  
Lie Algebras ◽  
Space Variable ◽  
Single Point ◽  
Schroedinger Equation ◽  
Time Dependent ◽  
Full Width ◽  
One Dimensional ◽  
Space And Time ◽  
The One

We consider control of the one-dimensional Schroedinger equation via a time-dependent rectangular potential. We discretize the equation in the space variable, obtaining a system of ODEs in which the control is bilinear. We find Control Lie Algebras for several cases, including single point and full width potentials. We use full discretizations, in space and time, to examine the effect of the number of inputs.

AN AUDIO-FREQUENCY CIRCUIT MODEL OF THE ONE-DIMENSIONAL SCHROEDINGER EQUATION AND ITS SOURCES OF ERROR

1955 ◽  
Vol 33 (8) ◽  
pp. 483-491
Author(s):  
J. H. Blackwell ◽  
D. R. Fewer ◽  
L. J. Allen ◽  
R. S. Cass
Keyword(s):  
Finite Difference ◽  
Schroedinger Equation ◽  
Circuit Model ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Audio Frequency ◽  
Previous Theory ◽  
One Dimensional ◽  
The Past ◽  
The One

An audio-frequency circuit model of the one-dimensional Schroedinger equation has been constructed, based on original suggestions by G. Kron. The previous theory of such devices has been re-examined thoroughly and errors due to the basic finite-difference approximation separated from those due to electrical causes. It is found that the former type of error is likely to be dominant in practice and that in the past discrepancies due to errors of this kind have actually been ascribed to experimental causes.

scholarly journals Deep ice layer formation in an alpine snowpack: monitoring and modeling

2020 ◽  
Vol 14 (10) ◽  
pp. 3449-3464
Author(s):  
Louis Quéno ◽  
Charles Fierz ◽  
Alec van Herwijnen ◽  
Dylan Longridge ◽  
Nander Wever
Keyword(s):  
Preferential Flow ◽  
Richards Equation ◽  
One Dimensional ◽  
Dimensional Modeling ◽  
Combined Use ◽  
Formation And Evolution ◽  
The One ◽  
Complex Phenomena ◽  
Snow Microstructure ◽  
Ground Penetrating

Abstract. Ice layers may form deep in the snowpack due to preferential water flow, with impacts on the snowpack mechanical, hydrological and thermodynamical properties. This detailed study at a high-altitude alpine site aims to monitor their formation and evolution thanks to the combined use of a comprehensive observation dataset at a daily frequency and state-of-the-art snow-cover modeling with improved ice formation representation. In particular, daily SnowMicroPen penetration resistance profiles enabled us to better identify ice layer temporal and spatial heterogeneity when associated with traditional snowpack profiles and measurements, while upward-looking ground penetrating radar measurements enabled us to detect the water front and better describe the snowpack wetting when associated with lysimeter runoff measurements. A new ice reservoir was implemented in the one-dimensional SNOWPACK model, which enabled us to successfully represent the formation of some ice layers when using Richards equation and preferential flow domain parameterization during winter 2017. The simulation of unobserved melt-freeze crusts was also reduced. These improved results were confirmed over 17 winters. Detailed snowpack simulations with snow microstructure representation associated with a high-resolution comprehensive observation dataset were shown to be relevant for studying and modeling such complex phenomena despite limitations inherent to one-dimensional modeling.

Exploring the Multidimensional Facets of Authoritarianism: Authoritarian Aggression and Social Dominance Orientation

2008 ◽  
Vol 67 (1) ◽  
pp. 51-60 ◽  
Author(s):  
Stefano Passini
Keyword(s):  
Social Dominance ◽  
Factor Model ◽  
Three Dimensional ◽  
Social Dominance Orientation ◽  
Model Fit ◽  
Regression Analyses ◽  
One Dimensional ◽  
Confirmatory Factor ◽  
The One ◽  
Better Than

The relation between authoritarianism and social dominance orientation was analyzed, with authoritarianism measured using a three-dimensional scale. The implicit multidimensional structure (authoritarian submission, conventionalism, authoritarian aggression) of Altemeyer’s (1981, 1988) conceptualization of authoritarianism is inconsistent with its one-dimensional methodological operationalization. The dimensionality of authoritarianism was investigated using confirmatory factor analysis in a sample of 713 university students. As hypothesized, the three-factor model fit the data significantly better than the one-factor model. Regression analyses revealed that only authoritarian aggression was related to social dominance orientation. That is, only intolerance of deviance was related to high social dominance, whereas submissiveness was not.

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