Effect of confining potential on information entropy measures in hydrogen atom: extensive and non-extensive entropy
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ORIGINAL PAPER
Effect of confining potential on information entropy measures in hydrogen atom: extensive and non-extensive entropy R Khordad1*, A Ghanbari1 and A Ghaffaripour2 1
Department of Physics, College of Sciences, Yasouj University, Yasouj, Iran
2
Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran Received: 23 May 2019 / Accepted: 16 September 2019
Abstract: A spherically confined hydrogen atom has been considered using two different confined potentials such as modified Kratzer and non-spherical oscillator potentials. First, the Schro¨dinger equation in r-space has been solved and the exact analytical wave functions and energy levels have been obtained. Then, the Renyi entropy and Shannon entropy in rspace have been calculated. In addition, we have numerically calculated the wave functions in p-space by performing Fourier transform. The calculations have been done for different quantum numbers n and l and also quantum size r0 . The Bialynicki–Birula–Mycielski inequality is also tested. It is found that this inequality depends on non-extensive parameter q. Keywords: Shannon entropy; Renyi entropy; Onicescu energy; Hydrogen atom PACS Nos.: 05.70.–a; 65.40.Gr; 71.90.?q
1. Introduction Quantum confinement of charge carriers, atoms or molecules inside a nanostructure has an important role in physics, chemistry and engineering [1–3]. Examples of the nanostructures are quantum wells, quantum wires, quantum dots, atoms under plasma environment and impurities in semiconductor materials [4]. The quantum confinement has a noticeable role in chemical, electronic, optical and thermodynamics properties [5–7]. Parabolic confining potential is very attractive because the energy levels and wave functions of the potential have a simple analytical form. Therefore, it is possible to derive explicit expressions for other physical properties [8–10]. There are several confining potentials such as Gaussian, modified Gaussian, Kratzer, modified Kratzer and nonspherical models [11–17]. Such models can be employed to study the high-pressure environment inside the core of planets and various astrophysical phenomena [18]. Hitherto, many authors have tried to investigate effect of quantum confinement on physical properties of different systems. For instance, Chen and Takeo [19] have studied
confinement of an atom or molecule inside an impenetrable cavity. Michels et al. [20] have theoretically studied a hydrogen atom within an infinite spherical cavity. The Michels’s work has provided an important insight into the consequences of confinement in atomic electronic structure. After Michels’s work, other authors have studied trapping of hydrogen atom in a penetrable spherical box, impenetrable walls and inside a hard box of different geometrical shapes and sizes [1, 21–24]. Recently, Mukherjee and Roy [25] have studied information entropy measures in free and confined hydrogen atom. It is fully known that confined hydrogen atom is an interesting, important and attractive system in the realm of a
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