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.
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April 2022
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April 01 2022
Relativistic spin-0 particle in a box: Bound states, wave packets, and the disappearance of the Klein paradox
M. Alkhateeb;
M. Alkhateeb
a)
Laboratoire de Physique Théorique et Modélisation, CNRS Unité 8089, CY Cergy Paris Université
, 95302 Cergy-Pontoise Cedex, France
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A. Matzkin
A. Matzkin
b)
Laboratoire de Physique Théorique et Modélisation, CNRS Unité 8089, CY Cergy Paris Université
, 95302 Cergy-Pontoise Cedex, France
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a)
Electronic mail: mohammed.alkhateeb@cyu.fr
b)
Electronic mail: alexandre.matzkin@cnrs.fr
Am. J. Phys. 90, 297–304 (2022)
Article history
Received:
March 19 2021
Accepted:
January 11 2022
Citation
M. Alkhateeb, A. Matzkin; Relativistic spin-0 particle in a box: Bound states, wave packets, and the disappearance of the Klein paradox. Am. J. Phys. 1 April 2022; 90 (4): 297–304. https://doi.org/10.1119/10.0009408
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