Supercriticality and Transmission Resonances in the Dirac Equation (original) (raw)
Phase Shifts and Resonances in the Dirac Equation
International Journal of Modern Physics A, 2004
We review the analytic results for the phase shifts δl(k) in nonrelativistic scattering from a spherical well. The conditions for the existence of resonances are established in terms of time-delays. Resonances are shown to exist for p-waves (and higher angular momenta) but not for s-waves. These resonances occur when the potential is not quite strong enough to support a bound p-wave of zero energy. We then examine relativistic scattering by spherical wells and barriers in the Dirac equation. In contrast to the nonrelativistic situation, s-waves are now seen to possess resonances in scattering from both wells and barriers. When s-wave resonances occur for scattering from a well, the potential is not quite strong enough to support a zero momentum s-wave solution at E=m. Resonances resulting from scattering from a barrier can be explained in terms of the "crossing" theorem linking s-wave scattering from barriers to p-wave scattering from wells. A numerical procedure to extrac...
2020
In a recent work we have proven the existence of degenerate solutions to the Dirac equation, corresponding to an infinite number of different electromagnetic fields, providing also some examples regarding massless particles. In the present article our results are extended significantly, providing degenerate solutions to the Dirac equation for particles with arbitrary mass, which, under certain conditions, could be interpreted as pairs of particles (or antiparticles) moving in a potential barrier with energy equal to the height of the barrier and spin opposite to each other. We calculate the electromagnetic fields corresponding to these solutions, providing also some examples regarding both spatially constant electromagnetic fields and electromagnetic waves. Further, we discuss some potential applications of our work, mainly regarding the control of the particles outside the potential barrier, without affecting their state inside the barrier. Finally, we study the effect of small per...
Overcritical -symmetric square well potential in the Dirac equation
Physics Letters A, 2008
We study scattering properties of a PT -symmetric square well potential with real depth larger than the threshold of particle-antiparticle pair production as the time component of a vector potential in the Dirac equation. Spontaneous pair production inside the well becomes tiny beyond the strength at which discrete bound states with real energies disappear, consistently with a spontaneous breakdown of PT symmetry.
On absence of threshold resonances for Schrödinger and Dirac operators
Discrete & Continuous Dynamical Systems - S, 2018
Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrödinger operators with sufficiently short-range interactions in general space dimensions. More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions n 3, the absence of resonances at ±m for massive Dirac operators (with mass m > 0) in dimensions n 5, and recall the well-known case of absence of zero-energy resonances for Schrödinger operators in dimension n 5.
Inverse resonance scattering for massless Dirac operators on the real line
arXiv: Mathematical Physics, 2020
We consider massless Dirac operators on the real line with compactly supported potentials. We solve two inverse problems (including characterization): in terms of zeros of reflection coefficient and in terms of poles of reflection coefficients (i.e. resonances). We prove that a potential is uniquely determined by zeros of reflection coefficients and there exist distinct potentials with the same resonances. We describe the set of "isoresonance potentials". Moreover, we prove the following: 1) a zero of the reflection coefficient can be arbitrarily shifted, such that we obtain the sequence of zeros of the reflection coefficient for an other compactly supported potential, 2) the forbidden domain for resonances is estimated, 3) asymptotics of resonances counting function is determined, 4) these results are applied to canonical systems.
Superluminal Tunneling of a Relativistic Half-Integer Spin Particle Through a Potential Barrier
This paper investigates the problem of a relativistic Dirac half-integer spin free particle tunneling through a rectangular quantum-mechanical barrier. If the energy difference between the barrier and the particle is positive, and the barrier width is large enough, there is proof that the tunneling is always superluminal. For antiparticle states, the tunneling may be either subluminal or superluminal instead, depending on the barrier width. These results derive from studying the tunneling time in terms of phase time. For particle states these are always negatives while for antiparticle states they are always positives, whatever the height and width of the barrier. The scattering also leads to an anomalous distortion of the Dirac spinor that tends to disappear as the particle velocity approaches the speed of light. Moreover, the phase time tends to zero, increasing the potential barrier both for particle and antiparticle states. This agrees with the interpretation of quantum tunneling that the Heisenberg uncertainty principle provides. This study's results are innovative with respect to those available in the literature. Moreover, they show that the superluminal behaviour of particles occurs in those processes with high-energy confinement. 1 Introduction Several theoretical and experimental studies in the past decades have examined phenomena involving superluminal waves and objects because of their implication in quantum and cosmological physics [1-6]. Among them, the study of the tunneling time problem is one of the topics that has most attracted the interest of quantum physicists [7-12]. Researchers have approached this issue both from the perspective of non-relativistic [13] and relativistic [14] quantum theory. In both cases the tunneling time does not depend on the barrier width (at least for large enough barriers), thus proving superluminal behaviour of the quantum object (wave or particle). However, the tunneling time problem remains a controversial one in quantum physics. A comprehensive and clear theory to explain how long does it take a particle to tunnel through a barrier still does not exist [15]. As is well known, classical quantum mechanics does not treat time as an Hermitian operator but rather as a parameter [16]. Time does not appear in the commutation relationships typical of the Hermitian operators, even if it appears in one of the forms of the Heisenberg uncertainty principle, being a physical variable conjugated to the energy. For this reason we have to give up directly knowing the tunneling time. We may bypass the obstacle by assuming that the wave packet inside the barrier is stationary, with an imaginary wave vector. We can then interpret the tunneling time as the phase variation of the evanescent stationary wave that crossing
The cylindrical δ-potential and the Dirac equation
Journal of Physics A: Mathematical and Theoretical, 2012
In this article we discuss the Dirac equation in the presence of an attractive cylindrical δ-shell potential V (ρ) = −aδ(ρ − ρ 0), where ρ is the radial coordinate and a > 0. We present a detailed discussion on the boundary conditions the wave function has to satisfy when crossing the support of the potential, proceeding then to explore the dependence of the ground state on the parameter a, analyzing the occurrence of supercritical effects. We also apply the Foldy-Wouthuysen transformation, discussing the non-relativistic limit of this problem.
Overcritical PT-symmetric square well potential in the Dirac equation
Physics Letters A, 2008
We study scattering properties of a PT -symmetric square well potential with real depth larger than the threshold of particle-antiparticle pair production as the time component of a vector potential in the Dirac equation. Spontaneous pair production inside the well becomes tiny beyond the strength at which discrete bound states with real energies disappear, consistently with a spontaneous breakdown of PT symmetry.
The European Physical Journal C, 2009
In the present article we show that the energy spectrum of the one-dimensional Dirac equation, in the presence of an attractive vectorial delta potential, exhibits a resonant behavior when one includes an asymptotically spatially vanishing weak electric field associated with a hyperbolic tangent potential. We solve the Dirac equation in terms of Gauss hyper-geometric functions and show explicitly how the resonant behavior depends on the strength of the electric field evaluated at the support of the point interaction. We derive an approximate expression for the value of the resonances and compare the results calculated for the hyperbolic potential with those obtained for a linear perturbative potential. Finally, we characterize the resonances with the help of the phase shift and the Wigner delay time.