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arXiv:1902.06232

Localized states of Dirac equation

Syed Badiuzzaman Faruque, S.D. Shuvo, Pradip Das

Published Feb 17, 2019• 0 citations
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Abstract

In this paper, we introduce an extension of the Dirac equation, very similar to Dirac oscillator, that gives stationary localized wave packets as eigenstates of the equation. The extension to the Dirac equation is achieved through the replacement of the momentum operator by a PT-symmetric generalized momentum operator. In the 1D case, the solutions represent bound particles carrying spin having continuous energy spectrum, where the envelope parameter defines the width of the packet without affecting the dispersion relation of the original Dirac equation. In the 2D case, the solutions are localized wave packets and are eigenstates of the third component of total angular momentum and involve Bessel functions of integral order. In the 3D case, the solutions are localized spherical wave packets with definite total angular momentum.

DOI
10.48550/arxiv.1902.06232

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In this paper, we introduce an extension of the Dirac equation, very similar to Dirac oscillator, that gives stationary localized wave packets as eigenstates of the equation. The extension to the Dirac equation is achieved through the replacement of the momentum operator by a PT-symmetric generalized momentum operator. In the 1D case, the solutions represent bound particles carrying spin having continuous energy spectrum, where the envelope parameter defines the width of the packet without affecting the dispersion relation of the original Dirac equation. In the 2D case, the solutions are localized wave packets and are eigenstates of the third component of total angular momentum and involve Bessel functions of integral order. In the 3D case, the solutions are localized spherical wave packets with definite total angular momentum.
In this paper, we introduce an extension of the Dirac equation, very similar to Dirac oscillator, that gives stationary localized wave packets as eigenstates of the equation. The extension to the Dirac equation is achieved through the replacement of the momentum operator by a PT-symmetric generalized momentum operator. In the 1D case, the solutions represent bound particles carrying spin having continuous energy spectrum, where the envelope parameter defines the width of the packet without affecting the dispersion relation of the original Dirac equation. In the 2D case, the solutions are localized wave packets and are eigenstates of the third component of total angular momentum and involve Bessel functions of integral order. In the 3D case, the solutions are localized spherical wave packets with definite total angular momentum.
The free particle solutions of the relativistic Dirac equation are characterized by plane waves with infinite uncertainty in position. However, many practical implementations of the solutions require a wave packet configuration, which can be utilized to represent a localized Dirac particle. Unlike the traditional wave packet generation method by superposing multiple plane waves, this study of ours presents an alternative approach towards obtaining a wave packet solution of a free particle relativistic Dirac equation. In this paper, we present Dirac’s free particle equation with a modification in the generalized momentum. The modification is achieved by coupling the momentum with a spatially varying logarithmic function, and this alteration does not affect the relativistic dispersion relation of the particle. Moreover, a solution of this modified Dirac equation is provided as well, which has been calculated using a trial wave function. The wave function solution is carried out in one dimension, where it behaves as a wave packet for a given ratio of the envelope parameter to the reduced Planck's constant greater than unity, where the envelope parameter regulates the width of the wave packet. The solution, being subject to this constraint, represents a bound particle with spin and a continuous energy spectrum.
Quantum-mechanical spin is often thought of in terms of classical angular momentum. In fact spin is defined by its commutation relations and the spin and orbital angular momentum operators are very different. Here we solve the Dirac equation in a rotating frame of reference and create a localized wave packet from the resulting wave functions to examine the consequences of the rotational motion. We highlight some unexpected effects in the properties of the wave packet and show that the spin operator and orbital angular momentum operator describe different aspects of the rotational properties of the wave packet. It is also observed that neither quantum-mechanical spin nor orbital angular momentum can be fully understood within an inertial frame.
Abstract Localized spinor solutions of the zero-mass free-space Dirac equation, which are eigenstates of the angular momentum component along the direction of propagation, satisfy the three-dimensional wave equation. General expressions for and special compact forms of space-time localized spinors are given. An exact evaluation of the norm, energy, momentum, and angular momentum of such wavepackets is made. These are all finite. The integrals of the densities over all space, at fixed time, are shown to be equivalent to integrals over the wavenumber weight function. The localized wavepackets converge onto or diverge from a focal region, and their net momentum is smaller than their energy divided by the speed of light. They can be transformed to a (net) zero-momentum frame.
We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second-order separated operators. Such operators, defined by complex dilation, give the resonances of the problem. Their unique extensions as closed operators with a purely point spectrum are PT -symmetric with positive eigenvalues converging to the Schrödinger ones as c → ∞ . Precise numerical computations show that these eigenvalues coincide with the positions of the resonances up to the order of the width. The corresponding eigenfunctions are a definite choice of metastable states of the problem. Similar results are found for the Klein–Gordon oscillator: here also we have two closed, isospectral and complex conjugate extensions of the formal operator with PT -symmetry, but an infinite number of self-adjoint extensions and physical dynamics. The infinitely many pairs of eigenvectors of the two closed PT -symmetric operators give metastable states for any choice of the dynamics. The eigenvalues of the operator defined by complex dilation are resonances, although not according to the standard definition, for any dynamics.
Solving the Dirac equation is a formidable task due to the high frequency and the degrees of freedom involved. However, this high frequency allows one to obtain an approximation to the equation. Here, we directly solve the Dirac equation using an envelope method and derive analytical solutions of Dirac wave packets to first order for the small momentum spread. We apply the insight gained from this solution to the Zitterbewegung behavior in a Dirac-like system, where we show that it is crucial to include the first-order term in our solution to correctly describe a Dirac packet.