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arXiv:2402.15045

Effective description of a quantum particle constrained to a catenoid

Guillermo Chacón‐Acosta, Héctor H. Hernández, J. Ruvalcaba-Rascón

Published Feb 23, 2024• 0 citations
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Abstract

We describe a quantum particle constrained on a catenoid, employing an effective description of quantum mechanics based on expected values of observables and quantum dispersions. We obtain semiclassical trajectories for particles, displaying general features of the quantum behaviour; most interestingly, particles present tunneling through the throat of the catenoid, a characteristic having important physical applications

DOI
10.48550/arxiv.2402.15045

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In this work, we study the quantum tunnel effect through a potential barrier within a semiclassical formulation of quantum mechanics based on expectation values of configuration variables and quantum dispersions as dynamical variables. The evolution of the system is given in terms of a dynamical system for which we are able to determine numerical effective trajectories for individual particles, similar to the Bohmian description of quantum mechanics. We obtain a complete description of the possible trajectories of the system, finding semiclassical reflected, tunneled and confined paths due to the appearance of an effective time-dependent potential.
We study the Hydrogen atom as a quantum mechanical system with a Coulomb like potential, with a semiclassical approach based on an effective description of quantum mechanics. This treatment allows us to describe the quantum state of the system as a system of infinite many classical equations for expectation values of configuration variables, their moments and quantum dispersions. It also provides a semiclassical description of the orbits and the evolution of observables and spreadings and their back-reaction on the evolution.
In this work we analyze the effective evolution of a one dimensional Bose-Einstein Condensate (BEC) within a semiclassical description of quantum systems based on expectation values of quantum dispersions and physical observables, known as momentous quantum mechanics. We show that the most prominent features and physical parameters of the system can be determined from the dynamics of the corresponding semiclassical system, consisting of an extended phase space including original classical observables and quantum dispersions, and we also show that particle trajectories for expectation values of observables are a particular characteristic in this framework. We also demonstrate that interactions with several potentials can be implemented in an intuitive way.
Abstract We provide a semiclassical description of the double-slit experiment given by the momentous quantum mechanics. This formulation allows the study of quantum systems by an effective augmented Hamiltonian. The use of canonical variables enables us to comprise all the quantum information into a finite system of equations. We show the evolution of individual particles and their semiclassical trajectories, and how their collective behavior seems to reproduce the well-known quantum interference pattern. We are able to follow the individual evolution of each particle and its interaction with the effective quantum potential, showing that, contrary to the non-crossing rule present in Bohmian mechanics, particle trajectories actually cross each other in our description. By discussing the quantum-effective potential obtained, we mention possible extensions and applications to other areas.
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