Quantum dynamics in low-dimensional systems with position-dependent mass and product-like fractal geometry

Abstract

In this study, a quantum mechanical system characterized by the product-like fractal geometry constructed by Li and Ostoja-Starzewski in order to explore physical properties of porous and anisotropic media and a position-dependent mass is constructed. We have derived the modified Schrödinger equation and we have discussed some of its main consequences for different forms of potentials. Due to the particular forms of fractal gradient and Laplacian operators in Li and Ostoja-Starzewski approach, the potentials are characterized by fractional powers. For particular choices of the free parameters introduced in the theory, it was observed that both the commutation relation and the uncertainty principle are not altered although the momentum operator is modified. We have discussed a number of particular effective potentials and we have derived their associated energy eigenvalues which hold a number of new features. For a particular value of the fractal parameter, we have analyzed the case of low-dimensional nanocrystals of semiconductor materials characterized by particles sizes close to the exciton Bohr radius of the material. The results obtained are strongly affected by the anisotropic properties of Li and Ostoja-Starzewski approach. We have considered, as an application, the Cadmium selenide and we have evaluated the corresponding density of states and it is found to be extremely less than the standard result. Besides, the group velocity is enhanced and such an enhancement is detected in a homogenized dielectric medium in its component material phases due to anisotropy.