Free particle propagator in polymer quantum mechanics

2012 ◽  
Author(s):  
Ernesto Flores-González ◽  
Hugo A. Morales-Técotl ◽  
Juan D. Reyes
Keyword(s):  
Quantum Mechanics ◽  
Free Particle ◽  
Particle Propagator

scholarly journals Investigation of Free Particle Propagator with Generalized Uncertainty Problem

2016 ◽  
Vol 2016 ◽  
pp. 1-4 ◽  
Author(s):  
F. Ghobakhloo ◽  
H. Hassanabadi
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Free Particle ◽  
Generalized Uncertainty Principle ◽  
Minimal Length ◽  
Order Ordinary Differential Equation ◽  
Particle Propagator ◽  
Ordinary Quantum

We consider the Schrödinger equation with a generalized uncertainty principle for a free particle. We then transform the problem into a second-order ordinary differential equation and thereby obtain the corresponding propagator. The result of ordinary quantum mechanics is recovered for vanishing minimal length parameter.

scholarly journals Quantum backflow in solutions to the Dirac equation of the spin-1 2 free particle

2018 ◽  
Vol 33 (32) ◽  
pp. 1850186 ◽  
Author(s):  
Hong-Yi Su ◽  
Jing-Ling Chen
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Relativistic Particle ◽  
Relativistic Quantum ◽  
Free Particle ◽  
Negative Energy ◽  
Dirac Particle ◽  
Relativistic Quantum Mechanics ◽  
Probability Current ◽  
Negative Probability

It was known that a free, non-relativistic particle in a superposition of positive momenta can, in certain cases, bear a negative probability current — hence termed quantum backflow. Here, it is shown that more variations can be brought about for a free Dirac particle, particularly when negative-energy solutions are taken into account. Since any Dirac particle can be understood as an antiparticle that acts oppositely (and vice versa), quantum backflow is found to arise in the superposition (i) of a well-defined momentum but different signs of energies, or more remarkably (ii) of different signs of both momenta and energies. Neither of these cases has a counterpart in non-relativistic quantum mechanics. A generalization by using the field-theoretic formalism is also presented and discussed.

Quantum Mechanics

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Quantum Mechanics ◽  
Ground State ◽  
Variational Principles ◽  
Free Particle ◽  
Uncertainty Relations ◽  
Probabilistic Interpretation ◽  
Historical Account ◽  
Potential Wells ◽  
Harmonic Oscillator Potential ◽  
Ground State Energies

In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.

scholarly journals Quantum mechanics of a free particle on a plane with an extracted point

2002 ◽  
Vol 66 (3) ◽  
Author(s):  
K. Kowalski ◽  
K. Podlaski ◽  
J. Rembieliński
Keyword(s):  
Quantum Mechanics ◽  
Free Particle

scholarly journals SOME ASPECTS OF QUANTUM MECHANICS AND FIELD THEORY IN A LORENTZ INVARIANT NONCOMMUTATIVE SPACE

2013 ◽  
Vol 28 (07) ◽  
pp. 1350017 ◽  
Author(s):  
EVERTON M. C. ABREU ◽  
M. J. NEVES
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Spectral Representation ◽  
Free Particle ◽  
Transition Amplitude ◽  
Noncommutative Space ◽  
Moyal Product ◽  
Massive Scalar Field ◽  
Scalar Action

We obtained the Feynman propagators for a noncommutative (NC) quantum mechanics defined in the recently developed Doplicher–Fredenhagen–Roberts–Amorim (DFRA) NC background that can be considered as an alternative framework for the NC space–time of the early universe. The operators' formalism was revisited and we applied its properties to obtain an NC transition amplitude representation. Two examples of DFRA's systems were discussed, namely, the NC free particle and NC harmonic oscillator. The spectral representation of the propagator gave us the NC wave function and energy spectrum. We calculated the partition function of the NC harmonic oscillator and the distribution function. Besides, the extension to NC DFRA quantum field theory is straightforward and we used it in a massive scalar field. We had written the scalar action with self-interaction ϕ4 using the Weyl–Moyal product to obtain the propagator and vertex of this model needed to perturbation theory. It is important to emphasize from the outset, that the formalism demonstrated here will not be constructed by introducing an NC parameter in the system, as usual. It will be generated naturally from an already existing NC space. In this extra dimensional NC space, we presented also the idea of dimensional reduction to recover commutativity.

scholarly journals On optical transmittance of ultra diluted gas

2020 ◽  
Author(s):  
Jakub Ratajczak
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Quantum Effect ◽  
Free Particle ◽  
Optical Transmittance ◽  
Interpretations Of Quantum Mechanics ◽  
Non Locality ◽  
Astrophysical Phenomena

Abstract The paper proposes a model of optical transmittance of ultra diluted gas taking into account gas particles non-locality, the quantum effect of wave function spreading derived from solving the Schr ̈odinger equation for a free particle. A significant increase in the transmittance of such gas is envisaged as compared to the classical predictions. Some quantitative and qualitative consequences of the model are indicated and falsifying experiments are proposed. The classic Beer-Lambert law equation within range of its applicability is derived from the model. Remarks to some astrophysical phenomena and possible interpretations of Quantum Mechanics are made. An experiment consistent with the predictions of this model is referenced.

scholarly journals Direct Relativistic Extension of the Madelung-de-Broglie-Bohm Reformulations of Quantum Mechanics and Quantum Hydrodynamics

2021 ◽  
Author(s):  
Arquimedes Ruiz-Columbié ◽  
Luis Grave de Peralta
Keyword(s):  
Quantum Mechanics ◽  
Kinetic Energy ◽  
Relativistic Quantum ◽  
Free Particle ◽  
Linear Momentum ◽  
Relativistic Quantum Mechanics ◽  
Quantum Hydrodynamics ◽  
Relativistic Extension ◽  
Basic Equations ◽  
De Broglie Bohm

Abstract Using a Schrödinger-like equation, which describes a particle with mass and spin-0 and with the correct relativistic relation between its linear momentum and kinetic energy, the basic equations of the non-relativistic quantum mechanics with trajectories and quantum hydrodynamics are extended to the relativistic domain. Some simple but instructive free particle examples are discussed.

scholarly journals Times of arrival and gauge invariance

2021 ◽  
Vol 477 (2250) ◽  
pp. 20210101
Author(s):  
Siddhant Das ◽  
Markus Nöth
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Arrival Time ◽  
Free Particle ◽  
Natural Generalization ◽  
Arrival Times ◽  
Quantum Flux ◽  
Freely Moving ◽  
Set Up ◽  
Aharonov Bohm

We revisit the arguments underlying two well-known arrival-time distributions in quantum mechanics, viz., the Aharonov–Bohm–Kijowski (ABK) distribution, applicable for freely moving particles, and the quantum flux (QF) distribution. An inconsistency in the original axiomatic derivation of Kijowski’s result is pointed out, along with an inescapable consequence of the ‘negative arrival times’ inherent to this proposal (and generalizations thereof). The ABK free-particle restriction is lifted in a discussion of an explicit arrival-time set-up featuring a charged particle moving in a constant magnetic field. A natural generalization of the ABK distribution is in this case shown to be critically gauge-dependent. A direct comparison to the QF distribution, which does not exhibit this flaw, is drawn (its acknowledged drawback concerning the quantum backflow effect notwithstanding).

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