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AIP conference proceedings

Polymer quantum mechanics some examples using path integrals

Lorena Parra‐Rodríguez, J. David Vergara

Published Jan 1, 2014• 6 citations
Publisher

Abstract

In this work we analyze several physical systems in the context of polymer quantum mechanics using path integrals. First we introduce the group averaging method to quantize constrained systems with path integrals and later we use this procedure to compute the effective actions for the polymer non-relativistic particle and the polymer harmonic oscillator. We analyze the measure of the path integral and we describe the semiclassical dynamics of the systems.

DOI
10.1063/1.4861963

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In this work we analyze several physical systems in the context of polymer quantum mechanics using path integrals. First we introduce the group averaging method to quantize constrained systems with path integrals and later we use this procedure to compute the effective actions for the polymer non-relativistic particle and the polymer harmonic oscillator. We analyze the measure of the path integral and we describe the semiclassical dynamics of the systems.
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Fundamentals Path Integrals - Elementary Properties and Simple Solutions External Sources, Correlations, and Perturbation Theory Semiclassical Time Evolution Amplitude Variational Perturbation Theory Path Integrals with Topological Constrainsts Many Particle Orbits - Statistics and Second Quantization Path Integrals in Spherical Coordinates Fixed-Energy Amplitude and Wave Functions Spaces with Curvature and Torsion Schrodinger Equation in General Metric-Affine Spaces New Path Integral Formula for Singular Potentials Path Integral of Coulomb System Solution of Further Path Integrals by the Duru-Kleinert Method Path Integrals in Polymer Physics Polymers and Particle Orbits in Multiply Connected Spaces Tunneling Nonequilibrium Quantum Statistics Relativistic Particle Orbits Path Integrals and Financial Markets.
This paper investigated the basic formalism for treating the dissipative Hamiltonian systems within the framework of the canonical method using the path integral quantization. The Hamiltonian treatment of the dissipative systems leads to obtain the equations of motion as a total differential equation in many variables which require the investigation of integrability conditions on the action and the equations of motion. In this formalism, the action integral for dissipative systems is obtained from equations of motion and written in phase space. Besides, the quantization of these systems is investigated using the action to construct the wave function in terms of the canonical phase space coordinates. Two examples are examined, the free particle and the simple harmonic oscillator.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
In this paper, we provide the representation of the symplectic group Sp(2n,R)Sp(2n,\mathbb{R})Sp(2n,R) in polymer quantum mechanics. We derive the propagator of the polymer free particle and the polymer harmonic oscillator without considering a polymer scale. The polymer scale is then introduced to reconcile our results with those expressions for the polymer free particle. The propagator for the polymer harmonic oscillator implies nonunitary evolution.