The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint
Autor(a) principal: | |
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Data de Publicação: | 2008 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Brazilian Journal of Physics |
Texto Completo: | http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332008000100030 |
Resumo: | We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems - the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone\'s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the time-dependent Schrödinger equation. |
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Brazilian Journal of Physics |
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The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjointOperator domainsSelf-adjointnessStone theoremQuantum MechanicsOperator exponentialWe present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems - the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone\'s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the time-dependent Schrödinger equation.Sociedade Brasileira de Física2008-03-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersiontext/htmlhttp://old.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332008000100030Brazilian Journal of Physics v.38 n.1 2008reponame:Brazilian Journal of Physicsinstname:Sociedade Brasileira de Física (SBF)instacron:SBF10.1590/S0103-97332008000100030info:eu-repo/semantics/openAccessAraujo,Vanilse S.Coutinho,F. A. B.Toyama,F. M.eng2008-03-27T00:00:00Zoai:scielo:S0103-97332008000100030Revistahttp://www.sbfisica.org.br/v1/home/index.php/pt/ONGhttps://old.scielo.br/oai/scielo-oai.phpsbfisica@sbfisica.org.br||sbfisica@sbfisica.org.br1678-44480103-9733opendoar:2008-03-27T00:00Brazilian Journal of Physics - Sociedade Brasileira de Física (SBF)false |
dc.title.none.fl_str_mv |
The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint |
title |
The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint |
spellingShingle |
The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint Araujo,Vanilse S. Operator domains Self-adjointness Stone theorem Quantum Mechanics Operator exponential |
title_short |
The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint |
title_full |
The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint |
title_fullStr |
The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint |
title_full_unstemmed |
The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint |
title_sort |
The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint |
author |
Araujo,Vanilse S. |
author_facet |
Araujo,Vanilse S. Coutinho,F. A. B. Toyama,F. M. |
author_role |
author |
author2 |
Coutinho,F. A. B. Toyama,F. M. |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
Araujo,Vanilse S. Coutinho,F. A. B. Toyama,F. M. |
dc.subject.por.fl_str_mv |
Operator domains Self-adjointness Stone theorem Quantum Mechanics Operator exponential |
topic |
Operator domains Self-adjointness Stone theorem Quantum Mechanics Operator exponential |
description |
We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems - the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone\'s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the time-dependent Schrödinger equation. |
publishDate |
2008 |
dc.date.none.fl_str_mv |
2008-03-01 |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332008000100030 |
url |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332008000100030 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
10.1590/S0103-97332008000100030 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
text/html |
dc.publisher.none.fl_str_mv |
Sociedade Brasileira de Física |
publisher.none.fl_str_mv |
Sociedade Brasileira de Física |
dc.source.none.fl_str_mv |
Brazilian Journal of Physics v.38 n.1 2008 reponame:Brazilian Journal of Physics instname:Sociedade Brasileira de Física (SBF) instacron:SBF |
instname_str |
Sociedade Brasileira de Física (SBF) |
instacron_str |
SBF |
institution |
SBF |
reponame_str |
Brazilian Journal of Physics |
collection |
Brazilian Journal of Physics |
repository.name.fl_str_mv |
Brazilian Journal of Physics - Sociedade Brasileira de Física (SBF) |
repository.mail.fl_str_mv |
sbfisica@sbfisica.org.br||sbfisica@sbfisica.org.br |
_version_ |
1754734864451502080 |