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Sobre a introdução de um operador de translação generalizado na mecânica quântica não-relativística

Francisco Ancelmo Pinheiro Ferreira

Published Jan 1, 2019• 0 citations
OpenAlex

Abstract

In this work we analyze some aspects related to the introduction, in the non-relativistic quantum mechanics,of a translation operator whose action is a function of the point of application. From the introduction of this operator, we derive a modified canonical commutation relation and a modified representation for the momentum operator and we show that such changes require a restructuring of the Hilbert space. We also see that the “new” Hilbert space can be associated with the case of a particle with position dependent mass and also that of a particle moving in a curved space. Still, we compare our formalism with the formalism of von Roos, we analyze the classical limit of this formalism,we solve the problem of a particle with position-dependent mass in two dimensions and we end up analyzing some quantities of information theory for the case of a one-dimensional harmonic particle.

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In this work we analyze some aspects related to the introduction, in the non-relativistic quantum mechanics,of a translation operator whose action is a function of the point of application. From the introduction of this operator, we derive a modified canonical commutation relation and a modified representation for the momentum operator and we show that such changes require a restructuring of the Hilbert space. We also see that the “new” Hilbert space can be associated with the case of a particle with position dependent mass and also that of a particle moving in a curved space. Still, we compare our formalism with the formalism of von Roos, we analyze the classical limit of this formalism,we solve the problem of a particle with position-dependent mass in two dimensions and we end up analyzing some quantities of information theory for the case of a one-dimensional harmonic particle.
The present work aimed to discuss quantum mechanics in phase space, propose a modification of the Wigner function through the position-dependent translation operator (PDTO) and apply the proposed theory to systems of physical interest. Thus, using a one-dimensional non-Euclidean metric, we construct a modified Wigner function whose form can be expressed in terms of the position-dependent translation operator for the case of finite translations. Through this metric, we define a point transformation that acts as a parameterization that preserves the Euclidean form of the equations. Thus, it was possible to find a modified expression for the Weyl-Wigner quantization map, whose objective is to map functions in phase space into operators in Hilbert space. We show that, when using this modified map to quantize the classical momentum, we obtain the same result as the modified momentum operator found via the position-dependent translation operator. Furthermore, through the inverse operation of the parameterized Weyl-Wigner map, it was possible to define a modified form of the Wigner function, whose mathematical form from the Euclidean case was preserved. We apply the proposed Wigner function to the particle trapped in the potential well and to a quantum oscillator-type system, with the aim of evaluating the influence of the metric on the system dynamics. We plotted the graph of the modified Wigner function for these two systems, where it was possible to verify the appearance of deformations due to the influence of the metric used. Thus, we concluded that the dynamics of the system undergoes a change when the structure of the space is deformed.
In the context of quantum mechanics reformulated in a modified Hilbert space, we can formulate the Feynman’s path integral approach for the quantum systems with position-dependent mass particle using the formulation of position-dependent infinitesimal translation operator. Which is similar a deformed quantum mechanics based on modified commutation relations. An illustration of calculation is given in the case of the harmonic oscillator, the infinite square well and the inverse square plus Coulomb potentials.
We present an algebra of deformed Heisenberg-type [^ x; p ^] = i~f(^ x) arising from a rede nition of the translation operator. We point out that the Hilbert space should be used for the representation of the algebra in question and show the explicit form of the position and momentum operators in this space, in addition, we also write the equations that governs the time evolution of quantum systems (the Schrödinger equation) and use it to solve a traditional problem in the usual quantum mechanics: the harmonic oscillator immersed in a uniform electric eld . It should also point out that we used an unusual method to the solution of the harmonic oscillator, the method of the Laplace transform.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.
Although gravity could not be quantized yet, there are good theoretical evidences that the unification of General Relativity and Quantum Mechanics should lead to the existence of a minimal observable length, a length scale below which the very notion of length looses meaning. Such effect proves to be worth some further analysis, for it acts as a natural regulator parameter, thus avoiding the divergences that plague Quantum Field Theories. In order to investigate some of the physical consequences of the existence of this peculiar effect, we propose to include it in the framework of Quantum Mechanics by modifying the Heisenberg algebra (i.e., the commutation relation of position and momentum operators, Xˆ and Pˆ), so that a minimum non-zero value for the uncertainty ?x emerges, which, being a limitation to the localizability of particles, acts as a minimal length. The Hilbert space of the theory must be modified accordingly. As we will see, the changes are not merely quantitative. On the contrary, our main result is that the familiar concept of “position measurement” must be reformulated, as well as other concepts related to it. We present a proposal for such reformulation.