Behavior of a free quantum particle in the Poincare upper half-plane geometry

Abstract

Inspired by the recent work of Filho et al. (2016) a Hermitian momentum operator is introduced in a general curved space with diagonal metric. The modified Hamiltonian associated with this new momentum is calculated and discussed. Furthermore, granting the validity of the Heisenberg equation in a curved space, the Ehrenfest theorem is generalized and interpreted with the new position-dependent differential operator in a curved space. The modified Hamiltonian leads to a modified time-independent Schrodinger equation, which is solved explicitly for a free particle in the Poincare upper half-plane geometry. It is shown that a free particle'' does not behave as it is totally free due to the curved background geometry. (C) 2020 Elsevier Inc. All rights reserved.