Application Of Self Adjoint Extensions To The Relativistic And Non Relativistic Coulomb Problem
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Application of Self-adjoint Extensions to the Relativistic and Non-relativistic Coulomb Problem
The Coulomb problem was one of the first successful applications of quantum theory and is a staple topic in textbooks. However there is an ambiguity in the solution to the problem that is seldom discussed in either textbooks or the literature. The ambiguity arises in the boundary conditions that must be applied at the origin where the Coulomb potential is singular. The textbook boundary condition is generally not the only one that is permissible or the one that is most appropriate. Here we revisit the question of boundary conditions using the mathematical method of self-adjoint extensions in context of modern realizations of the Coulomb problem in electrons on helium, Rydberg atoms and graphene. We determine the family of allowed boundary conditions for the non-relativistic Schr\"{o}dinger equation in one and three dimensions and the relativistic Dirac equation in two dimensions. The boundary conditions are found to break the classical SO$(4)$ Runge-Lenz symmetry of the non-relativistic Coulomb problem in three dimensions and to break scale invariance for the two dimensional Dirac problem. The symmetry breaking is analogous to the anomaly phenomenon in quantum field theory. Electrons on helium have been extensively studied for their potential use in quantum computing and as a laboratory for condensed matter physics. The trapped electrons provide a realization of the one dimensional non-relativistic Coulomb problem. Using the method of self-adjoint extensions we are able to reproduce the observed energy levels of electrons on helium which are known to deviate from the textbook Balmer formula. We also study the connection between the method of self-adjoint extensions and an older theoretical model introduced by Cole. Rydberg atoms have potential applications to atomic clocks and precision atomic experiments. They are hydrogen-like in that they have a single highly excited electron that orbits a small positively charged core. We compare the observed spectrum of several species of Rydberg atoms to the predictions of the Coulomb model with self-adjoint extension and to the predictions of the more elaborate quantum defect model which is generally found to be more accurate. The motion of electrons on atomically flat sheets of graphene is governed by the massless Dirac equation. The effect of charged impurities on the electronic states of graphene has been studied using scanning probe microscopy. Here we use the method of self-adjoint extensions to analyze the scattering of electrons from the charged impurities; our results generalize prior theoretical work which considered only one of the family of possible boundary conditions.
Self-Adjoint Extension Schemes and Modern Applications to Quantum Hamiltonians
This book introduces and discusses the self-adjoint extension problem for symmetric operators on Hilbert space. It presents the classical von Neumann and Krein–Vishik–Birman extension schemes both in their modern form and from a historical perspective, and provides a detailed analysis of a range of applications beyond the standard pedagogical examples (the latter are indexed in a final appendix for the reader’s convenience). Self-adjointness of operators on Hilbert space representing quantum observables, in particular quantum Hamiltonians, is required to ensure real-valued energy levels, unitary evolution and, more generally, a self-consistent theory. Physical heuristics often produce candidate Hamiltonians that are only symmetric: their extension to suitably larger domains of self-adjointness, when possible, amounts to declaring additional physical states the operator must act on in order to have a consistent physics, and distinct self-adjoint extensions describe different physics. Realising observables self-adjointly is the first fundamental problem of quantum-mechanical modelling. The discussed applications concern models of topical relevance in modern mathematical physics currently receiving new or renewed interest, in particular from the point of view of classifying self-adjoint realisations of certain Hamiltonians and studying their spectral and scattering properties. The analysis also addresses intermediate technical questions such as characterising the corresponding operator closures and adjoints. Applications include hydrogenoid Hamiltonians, Dirac–Coulomb Hamiltonians, models of geometric quantum confinement and transmission on degenerate Riemannian manifolds of Grushin type, and models of few-body quantum particles with zero-range interaction. Graduate students and non-expert readers will benefit from a preliminary mathematical chapter collecting all the necessary pre-requisites on symmetric and self-adjoint operators on Hilbert space (including the spectral theorem), and from a further appendix presenting the emergence from physical principles of the requirement of self-adjointness for observables in quantum mechanics.
Self-adjoint Extensions in Quantum Mechanics
Author: D.M. Gitman
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-04-27
This exposition is devoted to a consistent treatment of quantization problems, based on appealing to some nontrivial items of functional analysis concerning the theory of linear operators in Hilbert spaces. The authors begin by considering quantization problems in general, emphasizing the nontriviality of consistent operator construction by presenting paradoxes to the naive treatment. It then builds the necessary mathematical background following it by the theory of self-adjoint extensions. By considering several problems such as the one-dimensional Calogero problem, the Aharonov-Bohm problem, the problem of delta-like potentials and relativistic Coulomb problemIt then shows how quantization problems associated with correct definition of observables can be treated consistently for comparatively simple quantum-mechanical systems. In the end, related problems in quantum field theory are briefly introduced. This well-organized text is most suitable for students and post graduates interested in deepening their understanding of mathematical problems in quantum mechanics. However, scientists in mathematical and theoretical physics and mathematicians will also find it useful.