Nonlocal Thermodynamics Properties of Position-Dependent Mass Particle in Magnetic and Aharonov-Bohm Flux Fields
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Rami Ahmad El-Nabulsi
Nonlocal Thermodynamics Properties of Position-Dependent Mass Particle in Magnetic and Aharonov-Bohm Flux Fields
Received: 20 August 2020 / Accepted: 17 September 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this study, we have constructed a generalized momentum operator based on the notion of backward–forward coordinates characterized by a low dynamical nonlocality decaying exponentially with position. We have derived the associated Schrödinger equation and we have studied the dynamics of a particle characterized by an exponentially decreasing position-dependent mass following the arguments of von Roos. In the absence of magnetic fields, it was observed that the dynamics of the particle is similar to the harmonic oscillator with damping and its energy state is affected by nonlocality. We have also studied the dynamics of a charged particle in the presence of Morse–Coulomb potentials and external magnetic and Aharonov-Bohm flux fields. Both the energy states and the thermodynamical properties were obtained. It was observed that all these physical quantities are affected by nonlocality and that for small magnetic fields and high quantum magnetic numbers, the entropy of the system decreases with increasing temperature unless the nonlocal parameter is negative. For positive value of the nonlocal parameter, it was found that the entropy increases with temperature and tends toward an asymptotically stable value similar to an isolated system.
1 Introduction There are generic and broad arguments that the Heisenberg uncertainty principle must be generalized in order to acquire minimal and maximal lengths in nature [1]. The generalization of the uncertainty principle would have several important observable consequences ranging from nanoscales to large scales. Such a generalization leads in reality to a nonzero minimal uncertainty in position measurements which has important implications in the dynamics of spinless particle in a magnetic field [2], radiation physics and quantum electromagnetic theory [3,4]. Various kinds of generalized uncertainty principle were addressed in literature (see [5–7] and references therein) which lead to several predictions such as energy shifts in the spectrum of the hydrogen atom, the Landau levels, Scanning Tunneling Microscope, charmonium levels, etc [8]. Analytically, the generalization of the uncertainty principle for minimum position and momentum has been presented in the form of the extended uncertainty principle [9] which has lead to a generalized momentum operator (GMO) and a generalized commutation relation of position and momentum operators of the form [x, ˆ p] ˆ = i h¯ (1 + μ(x)). Here √ xˆ and pˆ = −i h¯ ∂∂x are respectively the position and momentum operator, h¯ the Planck’s constant, i = −1the complex number and μ(x) is a real well-defined function of position. As a result, the momentum operator is generalized and takes the form pˆ = −i h¯ (1 + μ(x)) ∂∂x − i2h¯ dμ(x) d x . Such a GMO satisfies the generalized extended unc
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