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A Física das extensões autoadjuntas: partícula na caixa

Matheus Curado Ferreira

Published Jan 1, 2019• 0 citations
OpenAlex

Abstract

In this paper we will use techniques developed in Mathematical Physics, to try to clarify some questions about one of the most elementary problems addressed in undergraduate Physics courses, usually in the discipline of Quantum Mechanics. When we subject a one dimensional confined particle to the most general boundary conditions possible, negative energy states arise. What are these states? Since there is no interaction with any potential wells. We show a study about the physics of self-adjoint extensions of symmetric operators, and in consequence develop a proposal to explain such phenomenon.

Related papers

Extensoes auto-adjuntas de operadores em mecanica quantica (Self-adjoint extensions of operators in Quantum Mechanics)

Cleverson Filgueiras, Fernando Moraes

Recebido em 1/6/2006; Aceito em 26/9/2006O problema das extens˜oes auto-adjuntas de operadores em mecˆanica quˆantica, um assunto normalmente n˜aoapresentado em livros-texto, ´e brevemente discutido neste trabalho.Palavras-chave: mecˆanica quˆantica, condi¸c˜oes de contorno, extens˜ao auto-adjunta.In this work we discuss briefly the problem of self-adjoint extensions in quantum mechanics, a subject usuallynot presented in textbooks.Keywords: quantum mechanics, boundary conditions, self-adjoint extension.

•Jan 1, 2007•0 citations

Introduction

Fabio Bagarello, J.P. Gazeau, Franciszek Hugon Szafraniec +1 more

Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects presents various mathematical constructions influenced by quantum mechanics and emphasizes the spectral theory of non-adjoint operators. Featuring coverage of functional analysis and algebraic methods in contemporary quantum physics, the book discusses the recent emergence of unboundedness of metric operators, which is a serious issue in the study of parity-time-symmetric quantum mechanics. The book also answers mathematical questions that are currently the subject of rigorous analysis with potentially significant physical consequences. In addition to prompting a discussion on the role of mathematical methods in the contemporary development of quantum physics

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Self-Adjointness and the Renormalization of Singular Potentials

Sarang Gopalakrishnan, William Loinaz

Schrodinger operators with very singular potentials fail to produce reasonable spectra, because the usual boundary conditions are insufficient to make them self-adjoint. If one regularizes a singular potential at a length , the bound state energies diverge as → 0. A meaningful spectrum can be restored by picking a self-adjoint extension of the operator, by renormalizing the theory, or by allowing nonunitary time evolution. We show that renormalizations of the 1/r potential fall into two classes: those that are asymptotically equivalent to self-adjoint extensions, and those that reduce to treatments of delta functions. We also apply the apparatus of self-adjoint extensions and renormalization to clarify aspects of anomalous symmetry breaking, supersymmetric quantum mechanics, and one-dimensional quantum mechanics. Lines composed upon reading a thesis abstract by S. Gopalakrishnan Think a moment, dear reader, of that long-legged fly, The sprung meniscus underfoot, And what asymptotes invisible to well-trained eyes Still break the surface at its root. How a singular potential picks tight-bounded energy Apart till it no longer recognizes Its own mirrored face, its extensions, or symmetry; Unless one cautiously renormalizes. You have tried in these pensive pages to show the stuff Such theory’s made of, its quiddity When logic is applied to the diamond rough, And quantum notes ring purely. May Apollo’s clarity pierce through your reasoning And self-adjointed truth and fact Elevate your thought to that sense so pleasing When truth recovers nature’s pact.

•Jan 1, 2006•12 citations

Non‐Selfadjoint Operators in Quantum Physics

Fabio Bagarello, Jean‐Pierre Gazeau, Franciszek Hugon Szafraniec +1 more

A unique discussion of mathematical methods with applications to quantum mechanics Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects presents various mathematical constructions influenced by quantum mechanics and emphasizes the spectral theory of non-adjoint operators. Featuring coverage of functional analysis and algebraic methods in contemporary quantum physics, the book discusses recent emergence of the unboundedness of metric operators, which is a serious issue in the study of parity-time-symmetric quantum mechanics. The book also answers mathematical questions that are currently the subject of rigorous analysis, with potentially significant physical consequences. In addition to prompting a discussion of the role of mathematical methods in the contemporary development of quantum physics, the book features: * Chapter contributions written by well-known mathematical physicists who clarify numerous misunderstandings and misnomers while shedding light on new approaches in this growing area * An overview of recent inventions and advances in understanding functional analytic and algebraic methods for non-selfadjoint operators as well as the use of Krein space theory and pertubation theory * Rigorous support of the progress in theoretical physics of non-Hermitian systems in addition to mathematically justified applications in various domains of physics such as nuclear and particle physics, condensed matter physics An ideal reference, Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects is useful for researchers, professionals, and academics in applied mathematics and theoretical and/or applied physics who would like to expand their knowledge of classical applications of quantum tools to address problems in their research. Also a useful resource for recent and related trends, the book is appropriate as a graduate-level and/or PhD-level text for courses on quantum mechanics and mathematical models in physics.

Wiley eBooks•Jul 7, 2015•176 citations

On Distributions of Self-Adjoint Extensions of Symmetric Operators

Franco Fagnola, Zheng Li

In quantum probability a self-adjoint operator on a Hilbert space determines a real random variable and one can define a probability distribution with respect to a given state.In this paper we consider self-adjoint extensions of certain symmetric operators, such as momentum and Hamiltonian operators, with various boundary conditions, explicitly compute their probability distributions in some state and study dependence of these probability distributions on boundary conditions.

Journal of Stochastic Analysis•Jun 3, 2021•0 citations

One-Dimensional Tunnel-Junction Formula for the Schrödinger Particle

Masao Hirokawa, Takuya Kosaka

We handle all the self-adjoint extensions of the minimal Schrödinger operator for the nonrelativistic electron living in the one-dimensional configuration space with a junction. We are interested in every boundary condition corresponding to the individual self-adjoint extension. Thus, we clarify all the types of those boundary conditions of the wave functions of the nonrelativistic electron. We find concrete tunnel-junction formulae for the nonrelativistic electron passing through the junction, which reveals the phase factor caused by the tunneling through the junction. Using this tunnel-junction formula, we propose a mathematical possibility of a tunnel-junction device for qubit.

SIAM Journal on Applied Mathematics•Jan 1, 2013•3 citations
In this paper we will use techniques developed in Mathematical Physics, to try to clarify some questions about one of the most elementary problems addressed in undergraduate Physics courses, usually in the discipline of Quantum Mechanics. When we subject a one dimensional confined particle to the most general boundary conditions possible, negative energy states arise. What are these states? Since
there is no interaction with any potential wells. We show a study about the physics of self-adjoint extensions of symmetric operators, and in consequence develop a proposal to explain such phenomenon.
Recebido em 1/6/2006; Aceito em 26/9/2006O problema das extens˜oes auto-adjuntas de operadores em mecˆanica quˆantica, um assunto normalmente n˜aoapresentado em livros-texto, ´e brevemente discutido neste trabalho.Palavras-chave: mecˆanica quˆantica, condi¸c˜oes de contorno, extens˜ao auto-adjunta.In this work we discuss briefly the problem of self-adjoint extensions in quantum mechanics, a subject usuallynot presented in textbooks.Keywords: quantum mechanics, boundary conditions, self-adjoint extension.
Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects presents various mathematical constructions influenced by quantum mechanics and emphasizes the spectral theory of non-adjoint operators. Featuring coverage of functional analysis and algebraic methods in contemporary quantum physics, the book discusses the recent emergence of unboundedness of metric operators, which is a serious issue in the study of parity-time-symmetric quantum mechanics. The book also answers mathematical questions that are currently the subject of rigorous analysis with potentially significant physical consequences. In addition to prompting a discussion on the role of mathematical methods in the contemporary development of quantum physics
Schrodinger operators with very singular potentials fail to produce reasonable spectra, because the usual boundary conditions are insufficient to make them self-adjoint. If one regularizes a singular potential at a length , the bound state energies diverge as → 0. A meaningful spectrum can be restored by picking a self-adjoint extension of the operator, by renormalizing the theory, or by allowing nonunitary time evolution. We show that renormalizations of the 1/r potential fall into two classes: those that are asymptotically equivalent to self-adjoint extensions, and those that reduce to treatments of delta functions. We also apply the apparatus of self-adjoint extensions and renormalization to clarify aspects of anomalous symmetry breaking, supersymmetric quantum mechanics, and one-dimensional quantum mechanics. Lines composed upon reading a thesis abstract by S. Gopalakrishnan Think a moment, dear reader, of that long-legged fly, The sprung meniscus underfoot, And what asymptotes invisible to well-trained eyes Still break the surface at its root. How a singular potential picks tight-bounded energy Apart till it no longer recognizes Its own mirrored face, its extensions, or symmetry; Unless one cautiously renormalizes. You have tried in these pensive pages to show the stuff Such theory’s made of, its quiddity When logic is applied to the diamond rough, And quantum notes ring purely. May Apollo’s clarity pierce through your reasoning And self-adjointed truth and fact Elevate your thought to that sense so pleasing When truth recovers nature’s pact.
A unique discussion of mathematical methods with applications to quantum mechanics Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects presents various mathematical constructions influenced by quantum mechanics and emphasizes the spectral theory of non-adjoint operators. Featuring coverage of functional analysis and algebraic methods in contemporary quantum physics, the book discusses recent emergence of the unboundedness of metric operators, which is a serious issue in the study of parity-time-symmetric quantum mechanics. The book also answers mathematical questions that are currently the subject of rigorous analysis, with potentially significant physical consequences. In addition to prompting a discussion of the role of mathematical methods in the contemporary development of quantum physics, the book features: * Chapter contributions written by well-known mathematical physicists who clarify numerous misunderstandings and misnomers while shedding light on new approaches in this growing area * An overview of recent inventions and advances in understanding functional analytic and algebraic methods for non-selfadjoint operators as well as the use of Krein space theory and pertubation theory * Rigorous support of the progress in theoretical physics of non-Hermitian systems in addition to mathematically justified applications in various domains of physics such as nuclear and particle physics, condensed matter physics An ideal reference, Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects is useful for researchers, professionals, and academics in applied mathematics and theoretical and/or applied physics who would like to expand their knowledge of classical applications of quantum tools to address problems in their research. Also a useful resource for recent and related trends, the book is appropriate as a graduate-level and/or PhD-level text for courses on quantum mechanics and mathematical models in physics.
We handle all the self-adjoint extensions of the minimal Schrödinger operator for the nonrelativistic electron living in the one-dimensional configuration space with a junction. We are interested in every boundary condition corresponding to the individual self-adjoint extension. Thus, we clarify all the types of those boundary conditions of the wave functions of the nonrelativistic electron. We find concrete tunnel-junction formulae for the nonrelativistic electron passing through the junction, which reveals the phase factor caused by the tunneling through the junction. Using this tunnel-junction formula, we propose a mathematical possibility of a tunnel-junction device for qubit.