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.

Towards a Relativistic Quantum Mechanics

2017 ◽  
Author(s):  
Laurent Baulieu ◽  
John Iliopoulos ◽  
Roland Sénéor
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Degrees Of Freedom ◽  
Relativistic Quantum ◽  
Negative Energy ◽  
Relativistic Quantum Mechanics ◽  
Relativistic Limit ◽  
Probability Current ◽  
Klein Gordon ◽  
Relativistic Equations

Towards a relativistic quantum mechanics. Klein–Gordon and the problems of the probability current and the negative energy solutions. The Dirac equation and negative energies. P, C, and T symmetries. Positrons. The Schrödinger equation as the non-relativistic limit of relativistic equations. Majorana and Weyl equations. Relativistic corrections in hydrogen-like atoms. The Dirac equation as a quantum system with an infinite number of degrees of freedom.

Nuclear Physics Methods for Problems in Relativistic Quantum Mechanics

1998 ◽  
Vol 07 (05) ◽  
pp. 559-571
Author(s):  
Marcos Moshinsky ◽  
Verónica Riquer
Keyword(s):  
Quantum Mechanics ◽  
Nuclear Physics ◽  
Bound States ◽  
Relativistic Quantum ◽  
Negative Energy ◽  
Relativistic Quantum Mechanics ◽  
Variational Parameter ◽  
Approximate Methods ◽  
Rest Energy ◽  
Quark Models

Atomic and molecular physicists have developed extensive and detailed approximate methods for dealing with the relativistic versions of the Hamiltonians appearing in their fields. Nuclear physicists were originally more concerned with non-relativistic problems as the energies they were dealing with were normally small compared with the rest energy of the nucleon. This situation has changed with the appearance of the quark models of nucleons and thus the objective of this paper is to use the standard variational procedures of nuclear physics for problems in relativistic quantum mechanics. The 4 × 4α and β matrices in the Dirac equation are replaced by 2 × 2 matrices, one associated with ordinary spin and the other, which we call sign spin, is mathematically identical to the isospin of nuclear physics. The states on which our Hamiltonians will act will be the usual harmonic oscillator ones with ordinary and sign spin and the frequency ω of the oscillator will be our only variational parameter. The example discussed as an illustration will still be the Coulomb problem as the exact energies of the relativistic bound states are available for comparison. A gap of the order of 2mc2 is observed between states of positive and negative energy, that permits the former to be compared with the exact results.

scholarly journals An extended Dirac equation in noncommutative spacetime

2016 ◽  
Vol 31 (15) ◽  
pp. 1650089 ◽  
Author(s):  
R. Vilela Mendes
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Relativistic Quantum ◽  
Large Mass ◽  
Exterior Algebra ◽  
Relativistic Quantum Mechanics ◽  
Spacetime Geometry ◽  
Noncommutative Spacetime ◽  
Extended Dirac Equation

Stabilizing, by deformation, the algebra of relativistic quantum mechanics a noncommutative spacetime geometry is obtained. The exterior algebra of this geometry leads to an extended massless Dirac equation which has both a massless and a large mass solution. The nature of the solutions is discussed as well as the effects of coupling the two solutions.

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.

The Dirac Equation

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Relativistic Quantum ◽  
Type Equation ◽  
Nonrelativistic Limit ◽  
Relativistic Quantum Mechanics ◽  
Electronic Systems ◽  
Nonrelativistic Case ◽  
Relativistic Description ◽  
Basic Features

The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions In terms of the momentum four-vector introduced earlier, this yields In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrödinger-type equation containing spin.

A time operator in the simulations of the Dirac equation

2019 ◽  
Vol 34 (22) ◽  
pp. 1950114 ◽  
Author(s):  
M. Bauer
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Relativistic Quantum ◽  
Einstein Condensate ◽  
Time Operator ◽  
Relativistic Quantum Mechanics ◽  
Electron Motion ◽  
Klein Paradox ◽  
Bose Einstein Condensate ◽  
Bose Einstein

Simulations of the Dirac equation have allowed to mimic measurably the predicted unusual characteristics of the electron motion, e.g. Zitterbewegung and Klein paradox that are beyond current technical capabilities. In this paper, it is shown that a Bose–Einstein condensate experiment carried out corroborates these results, but in addition exhibits a particular feature of an observable represented by a Dirac self-adjoint time operator introduced in relativistic quantum mechanics.

scholarly journals QUANTIZATION OF POINTLIKE PARTICLES AND CONSISTENT RELATIVISTIC QUANTUM MECHANICS

2000 ◽  
Vol 15 (28) ◽  
pp. 4499-4538 ◽  
Author(s):  
S. P. GAVRILOV ◽  
D. M. GITMAN
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Relativistic Particle ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Quantum Field ◽  
Positive Spectrum ◽  
The One ◽  
Relativistic Particles

We revise the problem of the quantization of relativistic particle models (spinless and spinning), presenting a modified consistent canonical scheme. One of the main point of the modification is related to a principally new realization of the Hilbert space. It allows one not only to include arbitrary backgrounds in the consideration but to get in the course of the quantization consistent relativistic quantum mechanics, which reproduces literally the behavior of the one-particle sector of the corresponding quantum field. In particular, in a physical sector of the Hilbert space, a complete positive spectrum of energies of relativistic particles and antiparticles is reproduced, and all state vectors have only positive norms.

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