Nonlocal quantum system with fractal distribution of states

Abstract

In this paper, we generalize simple model of quantum mechanics and statistics to take into account power-law nonlocality by using fractional differential equations, and fractal distribution of states. These equations with derivatives of non-integer orders are widely used in describing such complex properties of systems as power-law spatial nonlocality and spatial dispersion, long-range interactions, as well as the openness of quantum systems (interactions with the environment). We suggest a generalization of quantum statistical model of an ideal gas of particles for the case of spatial nonlocality, long-range interactions and fractal distribution of permitted states. To take into account the power-law nonlocality and long-range interactions, we use the fractional Laplacian in the Riesz–Trujillo form in the Schrodinger equation. To take into account the fractal distribution of states, we use non-integer dimensional space approach. Expression for the fractal density of states for fractal distribution of permitted quantum states is proposed. The equation to calculate energy and total number of particles for quantum systems with power-law non-locality and fractal distribution of quantum states are derived. Examples of these calculations for degenerate quantum system are suggested in the framework of the proposed quantum statistical model.