Method for Numerical Solution of the Stationary Schrödinger Equation

Knyazev, S. Yu.; Shcherbakova, E. E.

doi:10.1007/s11182-017-0953-6

Cited by 10 publications

(4 citation statements)

References 5 publications

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“…To solve this problem, there are some approximate approaches that can analytically provide accepted solutions such as the WKB, the variational method (VM) and the perturbation theory. Moreover, many numerical methods, such as the Airy function approach, the asymptotic iteration, the Numerov method (NM) and the finite element method [2][3][4][5][6][7][8][9][10][11] have been suggested as solutions. Finding exact solutions to the Schrödinger equation for potentials that prove useful in the modelling of physical phenomena is a very important challenge for a deep understanding of the structures and interactions in such systems.…”

Section: Introductionmentioning

confidence: 99%

Etingof’s conjecture for quantized quiver varieties

Bezrukavnikov

Losev

2020

Invent. math.

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We propose the Whittaker function approach as a theoretical method for finding exact solutions of a quantum mechanical system placed in the Kratzer potential. We then show that the effect of an external uniform magnetic field on this system can be satisfactorily determined using variational method. By following the onestep treatment suggested in this study, we increase the reliability and the accuracy of the solutions of Schrödinger equation for a quantum mechanical system placed in potential energy and perturbed by a uniform magnetic field that proves to be useful in modelling physical phenomena. We find that the achieved numerical and analytical results agree very well with those already published and those calculated using the Numerov method.

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Section: Introductionmentioning

confidence: 99%

Etingof’s conjecture for quantized quiver varieties

Bezrukavnikov

Losev

2020

Invent. math.

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“…As another test example, we consider the determination of the Dirichlet problem eigenvalues for the homogeneous Helmholtz equation in a cube domain with side L = 2. For this problem, the eigenvalues are [21]  …”

Section: № 1 / 2017mentioning

confidence: 99%

“…PSM advantage is its simplicity, and a much smaller amount of computation in comparison with traditional numerical methods for solving boundary value problems, such as, for example, a finite element method (FEM). Application of PSM can be justified as in the problems solution of eigenvalues for elliptic equations, for example, the Helmholtz equation [18,19], the Schrödinger equation [20,21].…”

Section: Introductionmentioning

confidence: 99%

Solving the eigenvalues and eigenfunctions problems for the Helmholtz equation by the point-sources method

Shcherbakova¹

2017

CMIT

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The paper provides the developed approach to solve the eigenvalues and eigenfunctions problems for the Helmholtz equation in domains with an arbitrary configuration. In developing the approach for numerical solution of problems, the point-sources method (PSM) was used. The proposed method is based on the analysis of the condition number of the PSM system or the error of the numerical solution of problems. The concept of "eigenvalues criteria" is introduced. The research result is a developed effective method -an algorithm for solving problems of eigenvalues and eigenfunctions for the Helmholtz equation. It is shown that at the approach of the Helmholtz parameter to the problem eigenvalue, the condition number of the PSM system and the error of the numerical solution rise sharply. Therefore, we calculate the dependence of the condition number of the PSM system or error of the problem numerical solution on the Helmholtz parameter. Then, according to position of the maximum of the received dependences we find the eigenvalues of the Helmholtz equation in a given domain.. After finding the eigenvalues, it is possible to proceed to the determination of the eigenfunctions. At that, if the eigenvalue appears degenerate, that is some eigenfunctions correspond to it, then it is possible to find all the eigenfunctions taking into account the symmetry of the solution domain. The two-dimensional and three-dimensional test problems are solved. Upon the results obtained, the conclusion about the efficiency of the proposed method is made.

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“…If the system consists of two particles or more, the solutions become complicated. Hence, approximation and numerical approaches like WKB and the variational methods, the perturbation theory, the airy function approach, the asymptotic iteration, and the finite element methods [4][5][6][7][8][9][10][11] are considered to solve such problems.…”

Section: Introductionmentioning

confidence: 99%

Analytic solutions for vibrational energy levels of the pseudoharmonic potential

Maiz¹

2020

Phys. Scr.

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The non-relativistic analytic solutions of a quantum mechanical system have been calculated for the case of pseudoharmonic potential by using the Whittaker functions approach. A diatomic quantum system is placed in a Pseudoharmonic potential and perturbed by an external magnetic field. The resulting Schrodinger’s equation has been solved exactly to obtain the analytic expressions of vibrational energy levels and associated wave functions. In this work, we have compared the results of six diatomic molecules with those available in the literature.

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Method for Numerical Solution of the Stationary Schrödinger Equation

Cited by 10 publications

References 5 publications

Etingof’s conjecture for quantized quiver varieties

Etingof’s conjecture for quantized quiver varieties

Solving the eigenvalues and eigenfunctions problems for the Helmholtz equation by the point-sources method

Analytic solutions for vibrational energy levels of the pseudoharmonic potential

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