Spectrum of theq-Schrödinger equation by means of the variational method based on the discreteq-Hermite I polynomials

Abstract

In this work, the [Formula: see text]-Schrödinger equations with symmetric polynomial potentials are considered. The spectrum of the model is obtained for several values of [Formula: see text], and the limiting case as [Formula: see text] is considered. The Rayleigh–Ritz variational method is adopted to the system. The discrete [Formula: see text]-Hermite I polynomials are handled as basis in this method. Furthermore, the following potentials with numerous results are presented as applications: [Formula: see text]-harmonic, purely [Formula: see text]-quartic and [Formula: see text]-quartic oscillators. It is also shown that the obtained results confirm the ones that exist in the literature for the continuous case.

Spectrum of a q‐deformed Schrödinger equation by means of the variational method

“…Tables 1 and 2 show the ground‐ and excited‐state energies of $$$ q $$$‐Schrödinger equation for the purely $$$ q $$$‐quartic oscillator as a function of $$$ q, $$$ respectively. Also in these tables, ground‐ and excited‐state energies of the purely quartic oscillator are compared with the results in [5]. In Table 3, the convergence of ground‐state energy levels of the purely $$$ q $$$‐quartic oscillator for $$$ q=1-1{0}^{-8} $$$ are compared with the results in [5].…”

“…Also in these tables, ground‐ and excited‐state energies of the purely quartic oscillator are compared with the results in [5]. In Table 3, the convergence of ground‐state energy levels of the purely $$$ q $$$‐quartic oscillator for $$$ q=1-1{0}^{-8} $$$ are compared with the results in [5].…”

“…In this paper, we compared the results with those in [5]. A noticeable difference of this study is that the exact eigenvalues of the $$$ q $$$‐Schrödinger equation for discrete $$$ q $$$‐harmonic oscillator are derived.…”

“…Schrödinger equations have been studied in several fields of physics, chemistry, and mathematics by using several methods [1, 2]. The $$$ q $$$‐analogues of such equations have been also considered by many authors [3–5].…”

“…In [5], the authors considered the $$$ q $$$‐Schrödinger equation of the form $$$$ \left[-{D}_q{D}_{q^{-1}}+V(x)\right]\Psi (x)=\mathcal{E}\Psi (x), $$$$ where the first term on the left side stands for a $$$ q $$$‐analogue of the kinetic energy and the second one for the potential energy at the position $$$ x, $$$ and as such, $$$ \mathcal{E} $$$ is the total energy. There, they have studied this equation with symmetric polynomial potentials such as $$$ q $$$‐harmonic, purely $$$ q $$$‐quartic , and $$$ q $$$‐quartic potentials, and the spectrum of the model has been obtained by using the Rayleigh‐Ritz variational method.…”