Abstract:

Space-time fractional quantum mechanics, described by Schrödinger equation with Caputo derivative and Riesz derivative, is a generalization of quantum mechanics and can depict more extensive quantum phenomena. This paper studies the one-dimensional space-time fractional Schrödinger equation for a particle in the single and double δ-potential well, and gives the wave functions and energy levels of the particle. In addition, the space-time fractional quantum mechanical path integrals kernels of a particle in the δ-potential well are established by using integral transformation, and the corresponding Fox's H-function forms are derived, and the relation between space-time fractional Schrodinger equation and path integrals is constructed. It provides more possibilities to study space-time fractional quantum mechanics from the perspective of path integrals.

Key words: Space-time fractional Schrödinger equation, Quantum mechanical kernel, Fox's H-function