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Pseudospectral Methods With Various Basis Functions And Applications To Quantum Mechanics

BROWSE_DETAIL_CREATION_DATE: 31-10-2017

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BROWSE_DETAIL_TYPE: Thesis

BROWSE_DETAIL_SUB_TYPE: Masters

BROWSE_DETAIL_PUBLISH_STATE: Unpublished

BROWSE_DETAIL_FORMAT: PDF Document

BROWSE_DETAIL_LANG: English

BROWSE_DETAIL_SUBJECTS: Mathematics,

BROWSE_DETAIL_CREATORS: Wlie, Saeida M. Faraj (Author),

BROWSE_DETAIL_CONTRIBUTERS: Erhan, İnci (Advisor),

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Schr¨odinger equation, orthogonal polynomials, self-adjoint eigenvalueproblems.


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In this thesis, we studied the pseudospectral methods and their application to the solutionof eigenvalue problems associated with ordinary dierential equations. In particular,we considered second order dierential equations and a specific example, theSchr¨odinger equation for quantum dynamical systems with polynomial potentials.After an introduction to self adjoint eigenvalue problems and the Schr¨odinger equationfor particles, in the presence of polynomial potentials, we recollected some importantproperties of Lagrange interpolation and orthogonal polynomials. We presenteda method to compute the zeros of an orthogonal polynomial of arbitrary degree bymeans of a symmetric tridiagonal matrix eigenvalue problem. We constructed theparticular symmetric tridiagonal matrices for computation of the zeros of Hermite,Associated Laguerre, Chebyshev and Legendre polynomials.After that, we explained in details the pseudospectral schemes using Hermite andAssociated Laguerre polynomials by studying some published articles. We also madesubstitutions on the independent variable in order to transform infinite interval to afinite one and derived pseudospectral formulations using Chebyshev and Legendrepolynomials.ivAs a specific example, we applied the pseudospectral methods using the four typesof orthogonal polynomials mentioned above to the Schr¨odinger equation for quantumdynamical systems with polynomial potentials. We compared our numerical resultswith the numerical results obtained previously by other authors and made commentsabout the eciency of our method


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