Distributional approach to point interactions in one-dimensional quantum mechanics

Abstract

We consider the one-dimensional quantum mechanical problem of defining interactions concentrated at a single point in the framework of the theory of distributions. The often ill-defined product which describes the interaction term in the Schrödinger and Dirac equations is replaced by a well-defined distribution satisfying some simple mathematical conditions and, in addition, the physical requirement of probability current conservation is imposed. A four-parameter family of interactions thus emerges as the most general point interaction both in the non-relativistic and in the relativistic theories (in agreement with results obtained by self-adjoint extensions). Since the interaction is given explicitly, the distributional method allows one to carry out symmetry investigations in a simple way, and it proves to be useful to clarify some ambiguities related to the so-called ${\delta }^{\prime }$ interaction.

Open Access link: http://journal.frontiersin.org/Journal/10.3389/fphy.2014.00023/abstract

References in corpus (2)

• Self-adjoint extensions of Coulomb systems in 1,2 and 3 dimensions
• Products of random matrices and generalised quantum point scatterers

Cited by in corpus (1)

• Point interactions with bias potentials