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Role of nonlinearity in non-Hermitian quantum mechanics: Description of linear quantum electrodynamics from the nonlinear Schrödinger-Poisson equation

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Abstract.

In this paper we consider a basic two-level nonlinear quantum model consisting in a two-particle interacting bound-state system. It is described by means of two different approaches: i) the mean-field stationary nonlinear Schrödinger-Poisson equation with classical Coulomb interaction and harmonic potential; ii) the linear quantum electrodynamics Hamiltonian of a quantized field coupled to two fixed charges. Computing numerically the ground state and the first excited state about the maximum eigenstate overlap (which is not zero because of eigenstate non-orthogonality), we numerically demonstrate that these two descriptions coincide at first order. As a consequence, a specific definition of the fine-structure constant \( \alpha\) is provided within 99.95% accuracy by the present first-order non-relativistic and nonlinear quantum description. This result also means that the internal Coulomb interaction commutes with external particle confinement for the calculation of the ground state. Consequently peculiar nonlinear quantum properties become observable (an experiment with GaAs quantum-dot helium is suggested).

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Authors and Affiliations

  1. Laboratoire Lagrange, UMR7293, Université de Nice Sophia-Antipolis, CNRS, Observatoire de la Côte d’Azur, Bd. de Observatoire, 06304, Nice, France

    Gilbert C. Reinisch

  2. University of Toronto, Department of Mathematics, 40 St. George St., M5S 2E4, Toronto, Canada

    Maxime Gazeau

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  1. Gilbert C. Reinisch
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  2. Maxime Gazeau
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Correspondence to Gilbert C. Reinisch.

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Reinisch, G., Gazeau, M. Role of nonlinearity in non-Hermitian quantum mechanics: Description of linear quantum electrodynamics from the nonlinear Schrödinger-Poisson equation. Eur. Phys. J. Plus 131, 220 (2016). https://doi.org/10.1140/epjp/i2016-16220-6

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  • DOI: https://doi.org/10.1140/epjp/i2016-16220-6

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