Matlab package for the Schrödinger equation

Trif, Damian

doi:10.1007/s10910-007-9266-2

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…Low order eigenvalues have accurately computed, using Runge-Kutta type methods, for instance in [34,35]. In [36], the author used Hermite collocation in order to find only the first eigenvalues for some Schrödinger eigenproblem. With respect to the departure from orthogonality, i.e., the distance from zero of the scalar product of two eigenvectors, we display in panel b of Figure 13, in a log linear plot, the absolute values of the of the scalar products of u 1 and u j , j = 2, .…”

Section: A Singular Schrödinger Eigenproblem On the Real Line-the Anh...mentioning

confidence: 99%

Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations

Gheorghiu¹

2020

Computation

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We are concerned with the study of some classical spectral collocation methods, mainly Chebyshev and sinc as well as with the new software system Chebfun in computing high order eigenpairs of singular and regular Schrödinger eigenproblems. We want to highlight both the qualities as well as the shortcomings of these methods and evaluate them in conjunction with the usual ones. In order to resolve a boundary singularity, we use Chebfun with domain truncation. Although it is applicable with spectral collocation, a special technique to introduce boundary conditions as well as a coordinate transform, which maps an unbounded domain to a finite one, are the special ingredients. A challenging set of “hard”benchmark problems, for which usual numerical methods (f. d., f. e. m., shooting, etc.) fail, were analyzed. In order to separate “good”and “bad”eigenvalues, we have estimated the drift of the set of eigenvalues of interest with respect to the order of approximation and/or scaling of domain parameter. It automatically provides us with a measure of the error within which the eigenvalues are computed and a hint on numerical stability. We pay a particular attention to problems with almost multiple eigenvalues as well as to problems with a mixed spectrum.

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Section: A Singular Schrödinger Eigenproblem On the Real Line-the Anh...mentioning

confidence: 99%

Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations

Gheorghiu¹

2020

Computation

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show abstract

“…Low order eigenvalues have accurately computed using Runge-Kutta type methods for instance in [32] and [33]. In [34] the author uses Hermite collocation in order to find only the first eigenvalues for some Schrödinger eigenproblem. With respect to the departure from orthogonality, i.e.…”

Section: A Singular Schrödinger Eigenproblem On the Real Linementioning

confidence: 99%

Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations

Gheorghiu¹

2020

Preprint

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We are concerned with the study of some classical spectral collocation methods as well as with the new software system Chebfun in computing high order eigenpairs of singular and regular Schrödinger eigenproblems. We want to highlight both the qualities as well as the shortcomings of these methods and evaluate them in conjunction with the usual ones. In order to resolve a boundary singularity we use Chebfun with domain truncation. Although it is applicable with spectral collocation, a special technique to introduce boundary conditions as well as a coordinate transform, which maps an unbounded domain to a finite one, are the special ingredients. A challenging set of "hard"benchmark problems, for which usual numerical methods (f. d., f. e. m., shooting etc.) fail, are analyzed. In order to separate "good"and "bad"eigenvalues we estimate the drift of the set of eigenvalues of interest with respect to the order of approximation and/or scaling of domain parameter. It automatically provides us with a measure of the error within which the eigenvalues are computed and a hint on numerical stability. We pay a particular attention to problems with almost multiple eigenvalues as well as to problems with a mixed spectrum.

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“…The promotion of interdisciplinary activities between chemistry and computational thinking is something we felt worth exploring as it can enable our students to make the leap from learning the subject as an individual silo to making vital connections to see the bigger picture. The use of MATLAB and Mathematica to teach quantum physics have been employed in numerous courses and resources. − However, none has attempted to evaluate the student’s acceptance or the efficacy of such assignments. Many courses or studies employed computers as a tool for understanding or visualizing science concepts and do not develop the computational thinking that is vital to today’s society.…”

Section: Introductionmentioning

confidence: 99%

Helping Students Connect Interdisciplinary Concepts and Skills in Physical Chemistry and Introductory Computing: Solving Schrödinger’s Equation for the Hydrogen Atom

et al. 2019

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Integrating knowledge across disciplines has been shown to be a challenging, and yet it is a necessary skill that university students need to develop. Students who are able to connect different concepts, perspectives, and angles of a given topic are generally more engaged and have better understanding. However, designing lesson plans or assignments that integrate knowledge across disciplines is a challenge to most instructors. This paper describes our effort to provide an interdisciplinary and independent assignment that couples topics from a Physical Chemistry course with skills taught in an Introductory Computing course. It was shown that such an interdisciplinary assignment, though done independently, can help to boost students' learning. However, it helps only in the understanding of certain aspects of the topics. We present the design of the assignment and the lessons learned for other instructors willing to adopt this kind of interdisciplinary approach.

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Matlab package for the Schrödinger equation

Cited by 4 publications

References 8 publications

Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations

Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations

Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations

Helping Students Connect Interdisciplinary Concepts and Skills in Physical Chemistry and Introductory Computing: Solving Schrödinger’s Equation for the Hydrogen Atom

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