q-Deformed Statistics from Position-Dependent Mass Schrödinger Equationa

Abstract

An algebraic approach is used to obtain the canonical form of the position-dependent mass Schrödinger equation from where a couple of canonical quantum variables, the q-deformed operators for the position x_q, and the hermitian linear momentum operator p_q are derived. In this q-deformed coordinate space, the commutator remains invariant namely [x_q, p_q] = iħ. By taking advantage of these q-deformed variables, one gets to a q-deformed exponential function exp_q(x) as well as its corresponding q-deformed logarithm function ln_q(x). From these q-deformed mathematical relations and from the fact that thermodynamic properties such as the internal energy U, entropy S, free energy F, heat capacity C, and others are related to the partition function Z and ln(Z), it is proposed their generalizations in terms of the q-deformed exponential and q-deformed logarithmic functions. As a result, the structure of Legendre transformations between these statistical properties remains invariant. The usefulness of the proposal is exemplified by considering two specific position-dependent mass distributions. In the same way, other possibilities could be used to generalize the statistical properties straightforwardly.