Relativistic spin-0 particle in a box: bound states, wavepackets, and the disappearance of the Klein paradox

Abstract

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 wavefunction outside the well vanishes and Klein tunneling is suppressed: quantization is thus recovered, similarly to the non-relativistic particle in a box. In addition, we show how wavepackets can be constructed semi-analytically from the scattering expansion, accounting for the dynamics of Klein tunneling in a physically intuitive way

arXiv commentSlightly extended with additional explanations. Typos corrected

References in corpus (10)

• Dirac and Klein-Gordon particles in one-dimensional periodic potentials
• Visualizing the collapse and revival of wavepackets in the infinite square well using expectation values
• Quantization of charged fields in the presence of critical potential steps
• Classical antiparticles
• Relativistic time-dependent quantum dynamics across supercritical barriers for Klein-Gordon and Dirac particles
• Relativistic particle in a box: Klein-Gordon vs Dirac Equations
• Klein paradox for bosons, wave packets and negative tunnelling times
• Solutions of the Klein-Gordon equation in an infinite square-well potential with a moving wall
• Non-locality and time-dependent boundary conditions: a Klein-Gordon perspective
• Seed and vacuum pair production in strong laser field

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• Effects of chiral MIT boundary conditions for a Dirac particle in a box
• Hydrodynamic Analog of the Klein Paradox: Vacuum Instability and Pair Production in a Linear Elastic Medium
• Revivals and quantum carpets for the relativistic Schrödinger equation