Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem

Abstract

We show that there exist some intimate connections between three unconventional Schrödinger equations based on the use of deformed canonical commutation relations, of a position-dependent effective mass or of a curved space, respectively. This occurs whenever a specific relation between the deforming function, the position-dependent mass and the (diagonal) metric tensor holds true. We illustrate these three equivalent approaches by considering a new Coulomb problem and solving it by means of supersymmetric quantum mechanical and shape invariance techniques. We show that in contrast with the conventional Coulomb problem, the new one gives rise to only a finite number of bound states.

A study of the bound states for square potential wells with position-dependent mass

“…In recent times a good number of articles have been published [10,11,12,13,14,15,16,17,18,19] in this field. The Schrödinger equation with position-dependent mass has been studied in the contexts of supersymmetry, shapeinvariance, Lie algebra, point-canonical transformation, etc.…”

Point Canonical Transformation versus Deformed Shape Invariance for Position-Dependent Mass Schrödinger Equations

“…In one-dimensional nonrelativistic quantum mechanics, one may deform the conventional canonical commutation relation [x, p] = i, where p = −id/dx and = 1, into [7] […”

“…with a mass and an effective potential given by 5) respectively [7,15,16]. Here a prime stands for derivative with respect to x.…”

“…Position-dependent masses (PDM) also hold out to deformation in the quantum canonical commutation relations or curvature of the underlying space [7,8,9]. Furthermore, they may also appear in Hermitian Hamiltonians equivalent to PT -symmetric or pseudo-Hermitian ones [10,11], whose study is a topic of current considerable interest [12].…”

“…On building on the previously noted equivalence between the presence of a PDM and a deformation of the commutation relations [7], we have recently devised a method for generating pairs of potential and mass for which the Schrödinger equation is exactly solvable [15]. For such a purpose, we have used an approach inspired by a branch of supersymmetric quantum mechanics [17,18,19], whose development dates back to that of quantum groups and q-algebras [20,21,22,23,24,25,26].…”