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arXiv:quant-ph/0104124 · American Journal of Physics

n+1 dimensional Dirac equation and the Klein paradox

Antonio S. de Castro

Published Oct 1, 2001• 4 citations
arXivHTMLPDF

Abstract

The purpose of this comment is to clarify two points related to the Dirac equation. First, the Lorentz structure of the potential and its connection with the Klein paradox. Second, the connection between the number of space dimensions and the number of spinor components.
DOI
10.1119/1.1376376

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The Klein paradox is one of the cornerstones in the development of quantum mechanics, and its consequences were used in various branches of physics, ranging from elementary particles to solid state. Yet its mathematical derivation is questionable in a number of steps, resulting in the wrong solution of Dirac equation. In this paper, the paradox is analysed in more detail, and its mathematical content is emphasized in order to show that in the study of extreme conditions of matter simple arguments may result in erroneous predictions.
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In this chapter, we investigate the Dirac equation, which is named after P. A. M. Dirac, who is one of the fathers of quantum field theory. The Dirac equation is a relativistic quantum mechanical wave equation for spin-1/2 particles (e.g. electrons), which was derived by Dirac in 1928. The difficulties in finding a consistent single-particle theory from the Klein–Gordon equation led Dirac to search for an equation that had a positive-definite conserved probability density and was first order both in time and space. One can show that these two conditions imply that a matrix equation is required. The reason why the Klein–Gordon equation did not yield a positive-definite probability density is connected with the second-order time derivative in this equation, which arises because the Klein–Gordon equation is related to the relativistic energy–momentum relation E 2 = m 2 + p 2 via the correspondence principle that includes a term E 2 . Thus, a ‘better’ Lorentz covariant wave equation with a positive-definite probability density should have a first-order time derivative only. However, the equivalence of time and space coordinates in Minkowski space requires that such an equation also have only first-order space derivatives.