On the singular position-dependent mass

TL;DR

This paper examines the proper framework for position-dependent mass in quantum systems, emphasizing symmetrization, and explores the effects of singular mass profiles on atomic clusters and quantum states.

Contribution

It clarifies the correct symmetrization approach for PDM using von Roos Hamiltonian and analyzes the quantum behavior near singular mass distributions with Heun functions.

Findings

Singular PDM leads to atomic cluster formation near the singularity.

Heun functions describe quantum states in singular PDM systems.

Constant mass limit recovers harmonic oscillator behavior.

Abstract

Revisiting the issue associated with Position-Dependent Mass (PDM), we reaffirm that the appropriate framework for addressing a generic PDM is the symmetrization proposed by BenDaniel and Duke. To accomplish this result adopts the effective mass Hamiltonian proposed by von Roos, corrected by a symmetrized kinematic term. After verifying the appropriate ordering to approach the PDM issue, one investigates a crystalline lattice with a defect described by a singular PDM. The singular mass profile proves intriguing as it yields an atom's cluster in the neighborhood of the singularity. Considering that a restoring force acts on the atoms, one notes that the confluent Heun function describes the quantum states. Furthermore, one highlights that when the effective mass distribution tends to a constant profile, we recover a system similar to the harmonic oscillator.