A Rayleigh-Ritz method adapted to some q-difference equations of hypergeometric type

Date

2023-7-20

Author

Doğan Çalışır, Ayşe

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

219
views

639
downloads

Cite This

In this thesis, the spectrum of the q-deformed Schrödinger equations with different forms of the q-Hamiltonians is obtained by using the Rayleigh-Ritz variational method which is adapted to system under investigation. The discrete q-Hermite I and discrete q-Hermite II polynomials are taken as the basis in this method. The q-deformed Schrödinger equations are considered with symmetric and asymmetric potentials not only in polynomial case but also in non-polynomial case. The eigenvalue problem is reduced to a matrix eigenvalue problem for which some recursive relations have been obtained to evaluate of matrix elements. As applications, the energy spectrum of the symmetric polynomial potentials such as q-harmonic, purely q-quartic, q-quartic, q-sextic oscillators, and the q-analogue of Gaussian potential for an example of a non-polynomial symmetric potential have been investigated. As typical examples of asymmetric potentials, q-versions of an asymmetric double well potential (ADWP) and a Morse potential (MP) are investigated among polynomial and non-polynomial potentials, respectively. For each potential, energy levels corresponding to various q-values are obtained. Several numerical examples including convergence studies are presented that show the efficiency and accuracy of the method by comparing the derived numerical results with the results available in the literature. It is seen that, in the limiting case as q→1^-, the eigenvalues in the discrete problem approach to those in the continuous one.

Subject Keywords

Discrete Schrödinger equation, q-harmonic oscillator, purely q-quartic oscillator, Rayleigh-Ritz method, Discrete q-Hermite I polynomials, Discrete q-Hermite II polynomials

URI

https://hdl.handle.net/11511/104858

Collections

Graduate School of Natural and Applied Sciences, Thesis