Position-dependent mass fractal Schrodinger equation from fractal anisotropy and product-like fractal measure and its implications in quantum dots and nanocrystals

Abstract

In this study, the Schrödinger equation with position-dependent mass in fractal dimensions is constructed from fractal anisotropy and product-like fractal measure introduced by Li and Ostoja-Starzewski in their formulation of fractal continuum media and elasticity. The theory is characterized by a fractal uncertainty relation and a generalized fractal momentum operator. The fractal Schrödinger equation is exactly solved for different position-dependent masses and effective potentials. In particular, we discuss the problems of quantum dots and nanocrystals. Modifications in their energies levels are detected which are in agreement with recent studies and experiments.