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Position-dependent mass momentum operator and minimal coupling: point canonical transformation and isospectrality

  • Omar MustafaEmail author
  • Zeinab Algadhi
Regular Article
  • 18 Downloads

Abstract.

The classical and quantum-mechanical correspondence for constant mass settings is used, along with some point canonical transformation, to find the position-dependent mass (PDM) classical and quantum Hamiltonians. The comparison between the resulting quantum PDM-Hamiltonian and the von Roos PDM-Hamiltonian implied that the ordering ambiguity parameters of von Roos are strictly determined. Eliminating, in effect, the ordering ambiguity associated with the von Roos PDM-Hamiltonian. This, consequently, played a vital role in the construction and identification of the PDM-momentum operator. The same recipe is followed to identify the form of the minimal coupling of electromagnetic interactions for the classical and quantum PDM-Hamiltonians. It turned out that whilst the minimal coupling may very well inherit the usual form in classical mechanics (i.e., \(p_{j} (\vec{x}) \rightarrow p_{j}(\vec{x}) -e A_{j}(\vec{x})\), where \(p_{j}(\vec{x})\) is the j-th component of the classical PDM-canonical-momentum), it admits a necessarily different and vital form in quantum mechanics (i.e., \(\hat{p}_{j}(\vec{x})/\sqrt{m(\vec{x})} \rightarrow (\hat{p}_{j}(\vec{x})\)\( -e A_{j}(\vec{x}))/\sqrt{m(\vec{x})}\), where \( \hat{p}_{j}(\vec{x})\) is the j-th component of the quantum PDM-momentum operator). Under our point transformation settings, only one of the two commonly used vector potentials (i.e., \(\vec{A} (\vec{x}) \sim (-x_{2}, x_{1}, 0)\)) is found eligible and is considered for our Illustrative examples.

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Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsEastern Mediterranean University, G. MagusaMersin, North CyprusTurkey

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