Time-fractional Schrödinger equation from path integral and its implications in quantum dots and semiconductors

A new fractional Schrödinger equation is constructed from path integral based on the notions of fractional velocity recently introduced in literature and the concept of fractional actionlike variational approach motivated from fractal arguments. The new equation is characterized by an emergent position-dependent mass and a time-dependent effective potential where both have important implications in semiconductors and molecular physics. Based on this equation, the problem of the quantum particle in a box is analyzed where the consequent outcomes are applied to semiconductors. It was observed that quantum size effects in nanostructures are significant and enhancement in the ground energy state for specific values of the fractional parameter is obtained. The fractional density of states for the case of bulk materials with no confinement is derived and it was revealed that a significant number of available states may be occupied even for lengths of the order of picometers. Additional results were obtained and discussed accordingly.