Inverse-power potentials with positive-bound energy spectrum from fractal, extended uncertainty principle and position-d
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Inverse-power potentials with positive-bound energy spectrum from fractal, extended uncertainty principle and position-dependent mass arguments Rami Ahmad El-Nabulsia Mathematics and Physics Divisions, Athens Institute for Education and Research, 8 Valaoritou Street, Kolonaki, 10671 Athens, Greece Received: 3 August 2020 / Accepted: 26 August 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this study, we have introduced a new generalized momentum operator which takes simultaneously into account the extended uncertainty principle, the concept of positiondependent mass and fractal gradient arguments. The associated Schrödinger equation is obtained and analyzed for the case of potentials characterized by inverse-power terms with arbitrary strengths. It was observed that, in contrast to what is obtained in the literature, the corresponding quantum systems are characterized by positive-bound energy spectrum similar to what occurred in quasiparticles theory in condensed matter physics.
1 Introduction The concept of differentiation and integration in fractal or non-integer-dimensional spaces is widely used in the description of anisotropic fractal materials [1, 2]. Such a concept is motivated from Mandelbrot works on fractal and self-similarity approach in geometry [3] and has led to important implications in applied mathematics and theoretical physics ranging from materials sciences to quantum mechanics, fluid dynamics, dispersive transport, continuum thermomechanics and thermoelasticity among others [4–35]. In fact, dissimilar methodologies were used in the literature to describe the physics in fractal-dimensional spaces other than the conventional fractal approach known as the “analysis on fractals” [1, 2], e.g., methods based on fractional calculus [36–52], approaches based on fractional-dimensional space characterized by fractional powers of coordinates [34, 35] and approaches based on application of integration and differentiation for non-integer-dimensional spaces [53] among others. These approaches hold several common features although their mathematical structures are different. Besides, they have some advantages and disadvantages (see [54] for a comprehensive discussion). A successful generalization of vector calculus in non-integer-dimensional space using the methodology of product measure was motivated by the works of Stillinger [53] and Ostoja-Starzewski and Li [24, 26–33]. This approach was introduced in [54] as a possibility to describe anisotropic fractal materials in continuum mechanics. The new fractional vector operators are defined as an inverse operation to integration in non-integer-dimensional space. d d In this context, the differential operator is given by ∇α dX c−1 (α, x) dx where c(α, x)
a emails: [email protected]; [email protected] (corresponding author)
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is the density of state and is defined by: c(α, x) π α / 2 Γ −1 ( α2 )|x|α−1
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