LAUSR

Abstract In this study, a path integral approach for an isolated dynamical system which contains a moving body and its surrounding coupled to each other by a fractional dynamical friction is constructed. The dissipative system is characterized by a total Lagrangian L total = L − E d holding the energy term E d = ∫ 0 t f ( x ˙ , x , τ ) d τ which is dissipated by a dynamical fractional friction force f ( x ˙ , x , τ ) . Such a position- and time-dependent friction is in fact motivated from the Brownian motion. Our methodology aims to substitute the standard action by S = ∫ 0 t ( L ( x ˙ , x , τ ) − ∫ 0 τ f ( x ˙ , x , τ ) d μ α ( ξ ) ) d τ where [ μ α ] = − α , 0 α ≤ 1 with the particular choice d μ α ( τ ) = d τ Λ α ( τ ) for some scalar function Λ α ( ξ ) . A modified time-dependent Schrodinger equation is obtained which is characterized by a time-dependent Hamiltonian operator which is suitable to describe for quantum systems characterized by a time-dependent mass. Applications to quantum dots/zero-dimensional nanocrystals in which the size of particles is close to the exciton Bohr radius of the material are studied and explored in details by taking into account both the electron and the hole within the quantum dots besides the energy gap of the bulk. For the case of Cadmium selenide, we have obtained a decrease in the density of state in agreement with recent experimental results. Several consequences were discussed in some details.