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arXiv:2001.10239 · Journal of Physics Conference Series

Generalized Uncertainty Principle and Momentum-Dependent Effective Mass Schrödinger Equation

B Bagchi, R Ghosh, P Goswami

Published Apr 1, 2020• 6 citations
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Abstract

Abstract We show in this paper that the basic representations of position and momentum in a quantum mechanical system, that are guided by a generalized uncertainty principle and lead to a corresponding one-parameter eigenvalue problem, can be interpreted in terms of an extended Schrödinger equation embodying momentum-dependent mass. Some simple consequences are pointed out.

DOI
10.1088/1742-6596/1540/1/012004

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Abstract We show in this paper that the basic representations of position and momentum in a quantum mechanical system, that are guided by a generalized uncertainty principle and lead to a corresponding one-parameter eigenvalue problem, can be interpreted in terms of an extended Schrödinger equation embodying momentum-dependent mass. Some simple consequences are pointed out.
Abstract We study and generalize the momentum operator satisfying the extended uncertainty principle relation (EUP). This generalized extended momentum operator (GEMO) consists of an arbitrary auxiliary function of position operator, μ x , in such a combination that not only GEMO satisfies the EUP relation but also it is Hermitian. Next, we apply the GEMO to construct the generalized one-dimensional Schrödinger equation. Upon using the so called point canonical transformation (PCT), we transform the generalized Schrödinger equation from x -space to z -space where in terms of the transformed coordinate, z , it is of the standard form of the Schrödinger equation. In continuation, we study two illustrative examples and solve the corresponding equations analytically to find the energy spectrum.
A systematic procedure to study one-dimensional Schrodinger equation with a position-dependent effective mass (PDEM) in the kinetic energy operator is explored. The conventional free-particle problem reveals an interesting situation in that, in the presence of a mass background, formation of bound states is signalled. We also discuss coordinate-transformed, constant-mass Schrodinger equation, its matching with the PDEM form and the consequent decoupling of the ambiguity parameters. Many exact results are derived in this way.
In this work, we propose an alternative formulation for the description of quantum dynamics based on a displacement interpretation of the Schrödinger equation. Startingfrom a generalized wave propagation framework, we show that the Schrödinger equation can be recovered in a form that admits a displacement-type interpretation ofthe wave function. Within this formulation, we derive a momentum-related relation, referred to as the Schrödinger momentum equation, which provides an effective description of quantumpropagation. This relation allows for a unified discussion of several foundational aspects of quantum mechanics, including interference phenomena, coherence anddecoherence, and quantum correlations. The proposed framework is deterministic and explicitly nonlocal, and remains consistent with established experimental observations. In addition, we examine theimplications of this formulation for the interpretation of the Heisenberg Uncertainty Principle, showing that uncertainty-type relations can be naturally obtained within thisapproach while preserving measurement-dependent limitations. Overall, the present work offers a complementary perspective on quantum mechanics that may contribute to ongoing discussions on its conceptual foundations.