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Generalized Lagrangian-Path Representation of Non-Relativistic Quantum Mechanics

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Abstract

In this paper a new trajectory-based representation to non-relativistic quantum mechanics is formulated. This is ahieved by generalizing the notion of Lagrangian path (LP) which lies at the heart of the deBroglie-Bohm “ pilot-wave” interpretation. In particular, it is shown that each LP can be replaced with a statistical ensemble formed by an infinite family of stochastic curves, referred to as generalized Lagrangian paths (GLP). This permits the introduction of a new parametric representation of the Schrödinger equation, denoted as GLP-parametrization, and of the associated quantum hydrodynamic equations. The remarkable aspect of the GLP approach presented here is that it realizes at the same time also a new solution method for the N-body Schrödinger equation. As an application, Gaussian-like particular solutions for the quantum probability density function (PDF) are considered, which are proved to be dynamically consistent. For them, the Schrödinger equation is reduced to a single Hamilton–Jacobi evolution equation. Particular solutions of this type are explicitly constructed, which include the case of free particles occurring in 1- or N-body quantum systems as well as the dynamics in the presence of suitable potential forces. In all these cases the initial Gaussian PDFs are shown to be free of the spreading behavior usually ascribed to quantum wave-packets, in that they exhibit the characteristic feature of remaining at all times spatially-localized.

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Notes

  1. Which was formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927.

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Acknowledgments

Work developed within the research Projects of the Czech Science Foundation GAČR Grant No. 14-07753P (C.C.) and Albert Einstein Center for Gravitation and Astrophysics, Czech Science Foundation No. 14-37086G (M.T.).

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Authors and Affiliations

  1. Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127, Trieste, Italy

    Massimo Tessarotto

  2. Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám.13, 74601, Opava, Czech Republic

    Massimo Tessarotto & Claudio Cremaschini

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  1. Massimo Tessarotto
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  2. Claudio Cremaschini
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Correspondence to Claudio Cremaschini.

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Tessarotto, M., Cremaschini, C. Generalized Lagrangian-Path Representation of Non-Relativistic Quantum Mechanics. Found Phys 46, 1022–1061 (2016). https://doi.org/10.1007/s10701-016-9989-7

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