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Graduate texts in contemporary physics

The Klein Paradox: An Example from the History of Negative Energy State Difficulties: The Positron Interpretation

K. T. Hecht

Published Jan 1, 2000• 0 citations
Publisher
DOI
10.1007/978-1-4612-1272-0_75

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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.

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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.

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The de Broglie-Bohm pilot wave interpretation of quantum mechanics is shown to provide a consistent explanation for a single relativistic particle (more accurately, a single particle process since pair production is addressed). This is accomplished by incorporating the Feynman-Stueckelberg interpretation of anti-matter. The lack of conserved probability and existence of negative energy solutions is studied through this interpretation. We discuss the resolution of Klein's original paradox through the pilot wave interpretation, and similarly analyze the potential barrier problem. Simulations are presented for a wave packet incident on potential steps & barriers of various heights.

arXiv:1312.1377•Dec 4, 2013•0 citations
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.
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 de Broglie-Bohm pilot wave interpretation of quantum mechanics is shown to provide a consistent explanation for a single relativistic particle (more accurately, a single particle process since pair production is addressed). This is accomplished by incorporating the Feynman-Stueckelberg interpretation of anti-matter. The lack of conserved probability and existence of negative energy solutions is studied through this interpretation. We discuss the resolution of Klein's original paradox through the pilot wave interpretation, and similarly analyze the potential barrier problem. Simulations are presented for a wave packet incident on potential steps & barriers of various heights.