Eur. Phys. J. Plus (2021) 136: 96
https://doi.org/10.1140/epjp/s13360-021-01088-6

Regular Article

Probability density correlation for PDM-Hamiltonians and superstatistical PDM-partition functions

¹ Department of Physics, PUC-Rio, Rua Marquês de São Vicente, 225, 22451-900, Rio de Janeiro, Brazil
² Instituto de Física, Universidade Federal da Bahia, Campus Universitário de Ondina, 40170-115, Salvador, Bahia, Brazil
³ Instituto Federal de Educação, Ciência e Tecnologia do Sertão Pernambucano, Rua Maria Luiza de Araújo Gomes Cabral, 56316-686, Petrolina, Pernambuco, Brazil
⁴ Department of Physics, Eastern Mediterranean University, G. Magusa, North Cyprus, Mersin 10, Turkey

^b ignacio.sebastian@ufba.br

Received: 22 November 2020
Accepted: 8 January 2021
Published online: 18 January 2021

Abstract

Schrödinger equation with position-dependent mass (PDM) allows the identification of quantum wave functions in a complex environment. Following the progress of this investigation field, in this work, we consider the non-Hermitian kinetic operators associated with the PDM Schrödinger equation. We provide a simplified picture for PDM quantum systems that admit exact solutions in confining potentials. First, we investigate the solutions for a sinusoidal and an exponential PDM distributions in an infinite potential well. Next, we consider the solutions for a PDM harmonic oscillator potential associated with a power-law PDM distribution. The results presented in this work offer a way to approach new classes of solutions for PDM quantum systems in confining potential (bound states). Complementarily, we interpret the quantum partition function of the canonical ensemble of a PDM system in the context of the superstatistics, which, in turn, allows us to express the inhomogeneity of the PDM in terms of beta distribution $f(\beta )$, Dirac delta distributions for $f(\beta )$, and effective temperatures. Our results are, hereby, reported for the sinusoidal and the exponential PDM distributions.