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Summer Research Fellowship Programme of India's Science Academies

Localization in one-dimensional relativistic quantum mechanics

Abhay Mehta

St. Xavier's College, Mumbai, 5, Mahapalika Marg, Mumbai, Maharasthra 400001

Sudhir R. Jain

Bhabha Atomic Research Centre, Trombay, Mumbai, Maharashtra 400085

Abstract

In this project, we consider a relativistic particle incident upon an infinite number of identical potential barriers placed randomly in one dimension. We consider two cases of incident momentum; momentum component only along the direction of motion and momentum component in all three directions. The latter is an interesting case as previous work shows occurrence of spin flip in the reflection and transmission coefficients when a particle is incident upon a potential step. We first calculate the reflection and transmission coefficient for a particle incident upon a single barrier for each case. For the spin flip case, we consider an incident beam polarized in spin-up direction. We put a constraint on the energy of the particle such that it is always greater than that of the potential barrier. This is done so as to avoid the regime of negative-energy solutions and the Klein Paradox. Similarly, we try and calculate the said coefficients for a large number of barriers. We hope to comment upon the localization of the wave function due to the presence of disorder in the positions of the potential barriers.

Keywords: Dirac equation.

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