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American Journal of Physics

Motion of a wave packet in the Klein paradox

H. Nitta, Tomoshige Kudo, Hiroaki Minowa

Published Nov 1, 1999• 53 citations
Publisher

Abstract

Computer simulations for the scattering of a relativistic wave packet by a step potential are performed. Under the condition of the Klein paradox, where the height of the potential barrier becomes supercritical, the scattering induces the creation of electron–positron pairs. The “pair creation” is demonstrated in the present simulation by making use of the hole theory.

DOI
10.1119/1.19174

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Computer simulations for the scattering of a relativistic wave packet by a step potential are performed. Under the condition of the Klein paradox, where the height of the potential barrier becomes supercritical, the scattering induces the creation of electron–positron pairs. The “pair creation” is demonstrated in the present simulation by making use of the hole theory.
The early papers by Klein, Sauter and Hund which investigate scattering off a high step potential in the context of the Dirac equation are discussed to derive the 'paradox' first obtained by Klein. The explanation of this effect in terms of electron-positron production is re-assessed. It is shown that a potential well or barrier in the Dirac equation can become supercritical and emit positrons or electrons spontaneously if the potential is strong enough. If the well or barrier is wide enough, a seemingly constant current is emitted. This phenomenon is transient whereas the tunnelling first calculated by Klein is time-independent. It is shown that tunnelling without exponential suppression occurs when an electron is incident on a high barrier, even when the barrier is not high enough to radiate. Klein tunnelling is therefore a property of relativistic wave equations and is not necessarily connected to particle emission. The Coulomb potential is investigated and it is shown that a heavy nucleus of sufficiently large Z will bind positrons. Correspondingly, as Z increases the Coulomb barrier should become increasingly transparent to positrons. This is an example of Klein tunnelling. Phenomena akin to supercritical positron emission may be studied experimentally in superfluid 3 He.
The well-known Klein paradox for the relativistic Dirac wave equation consists in the computation of possible "negative probabilities" induced by certain potentials in some regimes of energy. The paradox may be resolved by employing the notion of electron–positron pair production in which the number of electrons present in a process can increase. The Klein paradox also exists in the Maxwell's equations viewed as the wave equation for photons. In a medium containing "inverted energy populations" of excited atoms, e.g. in a LASER medium, one may again compute possible "negative probabilities." The resolution of the electromagnetic Klein paradox is that when the atoms decay, the final state may contain more photons then were contained the initial state. The optical theorem total cross-section for scattering photons from excited state atoms may then be computed as negative within a frequency band with matter induced amplification.
In this paper, the electron-positron creation process in a double well scheme is investigated. A series of simulations is conducted by solving the quantized Dirac equation numerically. Here the split operator scheme is used to solve the Dirac equation, and the Fourier analysis is adopted to study the evolution of the wave function. The evolution starts from the state that all the negative energy eigenstates are occupied. By projecting the time dependent wave function to the positive energy eigenstates, the distributions of electrons and positrons in coordinate space and momentum space would be calculated. The total number of the electrons and positrons can be obtained by integrating the momentum distributions, and the number of the positrons in different parts of coordinate space can be achieved by integrating the space distributions. At first the electron-positron is created at the double-well edge, and positrons are emitted from the edges of double-well potential and propagate out while the electrons are bounded by the barriers. It is found that when the positron waves from different double-well edges encounter in the double-well for the first time, there occurs no positron wave interference phenomenon. The wave interference emerges after the positron no indent wave is reflected by the barriers. At the same time, because of Klein tunneling the number of positrons outside the double well begin to surpass the positrons inside the double well. After a piece of time, the amplitude of interference wave would reach its peak, and then collapses since Klein tunneling. If the double-well potential meets the standing-wave conditions, a stationary wave would be found before the interference wave reaches its peak if the distance between the double wells is short, and a stationary wave would be found after the interference wave has reached its peak if the distance between the double wells is long. And the stationary wave occurs when the positron wave is reflected by the barriers for the second time. The occurring of the stationary wave would affect the pairs producing process by making the number of pairs fluctuate. Because of Klein tunneling, the wave packages close to the double-well would disappear first, and the others can last for a longer time when the standing-wave condition is fulfilled, but all of the stationary wave packages disappear in the double well finally. And there is barely no positrons left inside the double well to the end since Klein tunneling.
The “particle-in-a-box” problem is investigated for a relativistic particle obeying the Klein–Gordon equation. To find the bound states, the standard methods known from elementary non-relativistic quantum mechanics can only be employed for “shallow” wells. For deeper wells, when the confining potentials become supercritical, we show that a method based on a scattering expansion accounts for Klein tunneling (undamped propagation outside the well) and the Klein paradox (charge density increase inside the well). We will see that in the infinite well limit, the wave function outside the well vanishes, and Klein tunneling is suppressed: Quantization is, thus, recovered, similar to the non-relativistic particle in a box. In addition, we show how wave packets can be constructed semi-analytically from the scattering expansion, accounting for the dynamics of Klein tunneling in a physically intuitive way.
We investigate wave-packet dynamics across supercritical barriers for the Klein-Gordon and Dirac equations. Our treatment is based on a multiple scattering expansion (MSE). For spin-0 particles, the MSE diverges, rendering invalid the use of the usual connection formulas for the scattering basis functions. In a time-dependent formulation, the divergent character of the MSE naturally accounts for charge creation at the barrier boundaries. In the Dirac case, the MSE converges and no charge is created. We show that this time-dependent charge behavior dynamics can adequately explain the Klein paradox in a first-quantized setting. We further compare our semianalytical wave-packet approach to exact finite-difference solutions of the relativistic wave equations.