A real space split operator method for the Klein–Gordon equation

We study the creation of electron-positron pairs from the vacuum induced by two spatially displaced static electric fields. The strength and spatial width of each localized field is less than required for pair-creation. If, however, the separation between the fields is less than the quantum mechanical tunneling length associated with the corresponding quantum scattering system, the system produces a steady flux of electron-positron pairs. We compute the time-dependence of the pair-creation probability by solving the Dirac equation numerically for various external field sequences. For the special case of two very narrow fields we provide an analytical expression for the pair-creation rate in the long-time limit.2

Section: Summary and Discussionmentioning

confidence: 99%

“…The operator's space and time dependence is obtained by solving the Dirac equation numerically [18][19][20][21]. This computational approach to quantum field theory has been introduced recently to study the pair-creation process with full space-time resolution.…”

Section: Introductionmentioning

confidence: 99%

Electron-positron pair creation induced by quantum-mechanical tunneling

Jiang

et al. 2011

“…(B5), the total number of particleantiparticle pairs produced as a function of time by a potential V (x, t) can be found easily as long as the initial states of the system are given. We have therefore solved the Dirac equation for fermions and Klein-Gordon equation with a split-operator algorithm [35][36][37] with a numerical box of length L with periodic boundary conditions, and we have used up to N x = 2014 and N t = 10 000 spaceand time-grid points. For our parameters this discretization scheme gave converged results with a numerical error of less than 1%.…”

Section: Appendix A: Numerical Solutions Of Qedmentioning

confidence: 99%

Instantaneous rest-frame transformation method for temporally induced pair creation

Vikartofsky

Norris

et al. 2014

We introduce a computational method to determine the rate of the electron-positron pair creation triggered by a time-dependent subcritical external field. It is based on constructing a Lorentz transformation to an instantaneous rest frame, for which the pair-creation rate can be determined by standard techniques. We will discuss the accuracy and efficiency of this method by comparing its predictions with exact time-dependent quantum field theoretical solutions to the Dirac and Klein-Gordon equations for various space-time dependent external fields.

“…The evolution of this operator [4,22,24] can be obtained from either the Schrödinger- [25][26][27][28][29]. A similar but more efficient approach has been developed by Ruf et al [30] to study the quantum mechanical version of the Klein paradox based on the Klein-Gordon equation in two spatial dimensions.…”

mentioning

confidence: 99%

Bosonic analog of the Klein paradox

Wagner

Ware

et al. 2010

The standard Klein paradox describes how an incoming electron scatters off a supercritical electrostatic barrier that is so strong that it can generate electron-positron pairs. This fermionic system has been widely discussed in textbooks to illustrate some of the discrepancies between quantum mechanical and quantum field theoretical descriptions for the pair creation process. We compare the fermionic dynamics with that of the corresponding bosonic system. We point out that the direct counterpart of the Pauli exclusion principle (the central mechanism to resolve the fermionic Klein paradox) is stimulated emission, which leads to the resolution of the analogous bosonic paradox.In 1927 the Swedish scientist Oskar Klein [1] analyzed the quantum-mechanical solution of the Dirac equation for an incoming electron that scatters off a one-dimensional electrostatic barrier of height V . He observed a nonvanishing transmission if this height exceeds twice the rest mass energy of the electron (V > 2 mc 2 ), even for incoming energies E that are far below the potential height, E < V . This classically forbidden nonvanishing transmission has been dubbed the "Klein paradox." Two types of approaches have been used to investigate this paradox [2][3][4][5][6][7][8][9][10][11][12]. The first group analyzed the dynamics in purely quantum-mechanical terms, which cannot predict any of the pair creation that is unavoidably associated with the supercritical barrier by itself. The second approach used quantum field theory, which correctly describes the pair-creation process, but the effect of the incoming electron on the pair-creation process was not included. A complete description has to be quantum field theoretical and must include the incoming particle as well as the pair-creation process at the barrier.Numerical solutions to quantum field theory were used to study the interaction of the pair-creation process at the barrier with an incoming electron with full space-time resolution [10,11]. In agreement with earlier predictions, it was shown that the incoming electron suppresses the pair-creation process during the time it is being (completely) reflected at the barrier. The incoming electron occupies states that are therefore no longer available for the pair-creation process. As a result of this Pauli blocking, the reflected electron gets entangled with the pair-creation process such that it cannot be described by a single-particle wave function. The suppression due to Pauli blocking is in contradiction to statements that the incoming electron can "knock-out" a positron under the barrier [4] or that the incoming electron "stimulates" [7] the pair-creation process.While a wide number of articles have examined the Klein paradox in its historical fermionic context, an analogous analysis within the context of a bosonic system with spacetime resolution has found less attention [5,13]. There are several reasons for this. First, the threshold with regard to the electric field strength to break down the fermionic vacuum is easier to reach tha...