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Journal of Wuhan Institute of Technology

The new scheme for solving relativistic quantum mechanics equations

Ming Li

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The new concept of the particle-antiparticle index is introduced into relativistic quantum mechanics.A new scheme for solving relativistic quantum mechanics equations is presented.The Dirac equation is resolved under the new scheme: and the difficulty about negative energy is cleared up.The wave-functions of free spinor particle and antiparticle coincided with quantum field Theory is obtained directly.At last the energy and momentum operators of antiparticle are redefined.

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arXiv:1711.05891•Modern Physics Letters A•Sep 25, 2018•15 citations
The new concept of the particle-antiparticle index is introduced into relativistic quantum mechanics.A new scheme for solving relativistic quantum mechanics equations is presented.The Dirac equation is resolved under the new scheme: and the difficulty about negative energy is cleared up.The wave-functions of free spinor particle and antiparticle coincided with quantum field Theory is obtained directly.At last the energy and momentum operators of antiparticle are redefined.
As in non-relativistic quantum theory, the simplest problem to solve in relativistic quantum theory is that of describing a free particle. Much can be learned from this case which will be of use in interpreting the topics covered in later chapters. Furthermore, some of the most profound features of relativistic quantum theory are well illustrated by the free particle, so it is a very instructive problem to consider in detail. Another advantage of the free-particle problem is that the mathematics involved in solving it is not nearly as involved as that necessary for solving problems involving particles under the influence of potentials. Firstly we shall look briefly at the formulae for the current and probability density, then we shall go on to examine the solutions of the Dirac equation and investigate their behaviour. This leads us to a discussion of spin, the Pauli limit, and the relativistic spin operator. Next we consider the negative energy solutions and show how relativistic quantum theory predicts the existence of antiparticles. Some of the dilemmas this concept introduces and their resolution are discussed. At the end of the chapter we go back to the Klein paradox, and examine it for an incident spin-1/2 particle. We find that the Klein paradox exists for Dirac particles in exactly the same way as it existed for Klein—Gordon particles and has the same resolution and interpretation.
Spin is one of the most important characteristics of particles, which results in a fine and hyperfine energy structure and Zeeman energy shift. Spin is a property of particles, independent of the density distribution given by the potential field. The spin eigenfunction should be described using a vector, and the spin operator is given by a matrix. The property of spin is not derived from the Schroedinger equation, which treats wave functions as scalars. The Dirac equation was developed to obtain wave functions from the relationship between frequency (energy) and wavenumber (momentum) given by the theory of special relativity, for which the equation is given by the 4 x 4 matrix and the wave functions are given by four-dimensional vectors. There are four solutions for one equation, which correspond to ± 1/2 spin states and the positive and negative rest energies. The existence of negative rest energy was confirmed by the discovery of positrons (antiparticles). The Zeeman shift induced by spin is derived from the exchange of matrix production.
We underscore the need of relativistic quantum mechanics in dealing with particles. Various notations used in relativity are defined and explained. We derive the Klein–Gordon equation for free particle as well as for particle in electromagnetic field. We obtain a solution and interpret it. We derive the Dirac equation for free particle as well as for particle in electromagnetic field. Plane wave solution to the Dirac equation is obtained. We show how spin of electron, idea of antiparticle , phenomena of pair production and pair annihilation follow from the Dirac equation.
Abstract The Pryce (e) spin and position operators of the quantum theory of Dirac's free field were re-defined and studied recently with the help of a new spin symmetry and suitable spectral representations [Eur. Phys. J. C 82, 1073 (2022)]. This approach is generalized here, associating a pair of integral operators acting directly on particle and antiparticle wave spinors in momentum representation to any integral operator in configuration representation, acting on mode spinors. This framework allows an effective quantization procedure, giving a large set of one-particle operators with physical meaning as the spin and orbital parts of the isometry generators, the Pauli-Lubanski and position operators, or other spin-type operators proposed to date. Special attention is paid to the operators that mix the particle and antiparticle sectors whose off-diagonal associated operators have oscillating terms producing Zitterbevegung. The principal operators of this type, including the usual coordinate operator, are derived here for the first time. As an application, it is shown that an apparatus measuring these new observables may prepare and detect one-particle wave packets moving uniformly without Zitterbewegung or spin dynamics, spreading in time normally as any other relativistic or even non-relativistic wave packet.
Based on complex quantum Hamilton-Jacobi theory and its natural complex spacetime configuration, we realized new states for both positive and negative values of energy and momentum, and we discuss this complex configuration in a relativistic entangled "space-time" state and the resultant 12 extra wave functions than the four solutions of Dirac equation for a free particle. Accordingly, we calculate the quantum forces between particles and antiparticles in a relativistic entangled "space-time" state.
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.