About the quantum Talbot effect on the sphere

Autores
Chamizo, Fernando; Santillán, Osvaldo Pablo
Año de publicación
2023
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
The Schrödinger equation on a circle with an initially localized profile of the wave function is known to give rise to revivals or replications, where the probability density of the particle is partially reproduced at rational times. As a consequence of the convolutional form of the general solution it is deduced that a piecewise constant initial wave function remains piecewise constant at rational times as well. For a sphere instead, it is known that this piecewise revival does not necessarily occur, indeed the wave function becomes singular at some specific locations at rational times. It may be desirable to study the same problem, but with an initial condition being a localized Dirac delta instead of a piecewise constant function, and this is the purpose of the present work. By use of certain summation formulas for the Legendre polynomials together with properties of Gaussian sums, it is found that revivals on the sphere occur at rational times for some specific locations, and the structure of singularities of the resulting wave function is characterized in detail. In addition, a partial study of the regions where the density vanishes, named before valley of shadows in the context of the circle, is initiated here. It is suggested that, differently from the circle case, these regions are not lines but instead some specific set of points along the sphere. A conjecture about the precise form of this set is stated and the intuition behind it is clarified.
Fil: Chamizo, Fernando. Universidad Autónoma de Madrid; España
Fil: Santillán, Osvaldo Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Materia
TALBOT EFFECT
QUANTUM REVIVALS
GAUSSIAN SUMS
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/231443

id CONICETDig_717e5a78bf871be9f529034bc30bd25c
oai_identifier_str oai:ri.conicet.gov.ar:11336/231443
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling About the quantum Talbot effect on the sphereChamizo, FernandoSantillán, Osvaldo PabloTALBOT EFFECTQUANTUM REVIVALSGAUSSIAN SUMShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1The Schrödinger equation on a circle with an initially localized profile of the wave function is known to give rise to revivals or replications, where the probability density of the particle is partially reproduced at rational times. As a consequence of the convolutional form of the general solution it is deduced that a piecewise constant initial wave function remains piecewise constant at rational times as well. For a sphere instead, it is known that this piecewise revival does not necessarily occur, indeed the wave function becomes singular at some specific locations at rational times. It may be desirable to study the same problem, but with an initial condition being a localized Dirac delta instead of a piecewise constant function, and this is the purpose of the present work. By use of certain summation formulas for the Legendre polynomials together with properties of Gaussian sums, it is found that revivals on the sphere occur at rational times for some specific locations, and the structure of singularities of the resulting wave function is characterized in detail. In addition, a partial study of the regions where the density vanishes, named before valley of shadows in the context of the circle, is initiated here. It is suggested that, differently from the circle case, these regions are not lines but instead some specific set of points along the sphere. A conjecture about the precise form of this set is stated and the intuition behind it is clarified.Fil: Chamizo, Fernando. Universidad Autónoma de Madrid; EspañaFil: Santillán, Osvaldo Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaIOP Publishing2023-05info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/231443Chamizo, Fernando; Santillán, Osvaldo Pablo; About the quantum Talbot effect on the sphere; IOP Publishing; Journal of Physics A: Mathematical and Theoretical; 56; 25; 5-2023; 1-221751-8121CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1088/1751-8121/acd489info:eu-repo/semantics/altIdentifier/url/https://iopscience.iop.org/article/10.1088/1751-8121/acd489info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:22:16Zoai:ri.conicet.gov.ar:11336/231443instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:22:16.54CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv About the quantum Talbot effect on the sphere
title About the quantum Talbot effect on the sphere
spellingShingle About the quantum Talbot effect on the sphere
Chamizo, Fernando
TALBOT EFFECT
QUANTUM REVIVALS
GAUSSIAN SUMS
title_short About the quantum Talbot effect on the sphere
title_full About the quantum Talbot effect on the sphere
title_fullStr About the quantum Talbot effect on the sphere
title_full_unstemmed About the quantum Talbot effect on the sphere
title_sort About the quantum Talbot effect on the sphere
dc.creator.none.fl_str_mv Chamizo, Fernando
Santillán, Osvaldo Pablo
author Chamizo, Fernando
author_facet Chamizo, Fernando
Santillán, Osvaldo Pablo
author_role author
author2 Santillán, Osvaldo Pablo
author2_role author
dc.subject.none.fl_str_mv TALBOT EFFECT
QUANTUM REVIVALS
GAUSSIAN SUMS
topic TALBOT EFFECT
QUANTUM REVIVALS
GAUSSIAN SUMS
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv The Schrödinger equation on a circle with an initially localized profile of the wave function is known to give rise to revivals or replications, where the probability density of the particle is partially reproduced at rational times. As a consequence of the convolutional form of the general solution it is deduced that a piecewise constant initial wave function remains piecewise constant at rational times as well. For a sphere instead, it is known that this piecewise revival does not necessarily occur, indeed the wave function becomes singular at some specific locations at rational times. It may be desirable to study the same problem, but with an initial condition being a localized Dirac delta instead of a piecewise constant function, and this is the purpose of the present work. By use of certain summation formulas for the Legendre polynomials together with properties of Gaussian sums, it is found that revivals on the sphere occur at rational times for some specific locations, and the structure of singularities of the resulting wave function is characterized in detail. In addition, a partial study of the regions where the density vanishes, named before valley of shadows in the context of the circle, is initiated here. It is suggested that, differently from the circle case, these regions are not lines but instead some specific set of points along the sphere. A conjecture about the precise form of this set is stated and the intuition behind it is clarified.
Fil: Chamizo, Fernando. Universidad Autónoma de Madrid; España
Fil: Santillán, Osvaldo Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
description The Schrödinger equation on a circle with an initially localized profile of the wave function is known to give rise to revivals or replications, where the probability density of the particle is partially reproduced at rational times. As a consequence of the convolutional form of the general solution it is deduced that a piecewise constant initial wave function remains piecewise constant at rational times as well. For a sphere instead, it is known that this piecewise revival does not necessarily occur, indeed the wave function becomes singular at some specific locations at rational times. It may be desirable to study the same problem, but with an initial condition being a localized Dirac delta instead of a piecewise constant function, and this is the purpose of the present work. By use of certain summation formulas for the Legendre polynomials together with properties of Gaussian sums, it is found that revivals on the sphere occur at rational times for some specific locations, and the structure of singularities of the resulting wave function is characterized in detail. In addition, a partial study of the regions where the density vanishes, named before valley of shadows in the context of the circle, is initiated here. It is suggested that, differently from the circle case, these regions are not lines but instead some specific set of points along the sphere. A conjecture about the precise form of this set is stated and the intuition behind it is clarified.
publishDate 2023
dc.date.none.fl_str_mv 2023-05
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/231443
Chamizo, Fernando; Santillán, Osvaldo Pablo; About the quantum Talbot effect on the sphere; IOP Publishing; Journal of Physics A: Mathematical and Theoretical; 56; 25; 5-2023; 1-22
1751-8121
CONICET Digital
CONICET
url http://hdl.handle.net/11336/231443
identifier_str_mv Chamizo, Fernando; Santillán, Osvaldo Pablo; About the quantum Talbot effect on the sphere; IOP Publishing; Journal of Physics A: Mathematical and Theoretical; 56; 25; 5-2023; 1-22
1751-8121
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1088/1751-8121/acd489
info:eu-repo/semantics/altIdentifier/url/https://iopscience.iop.org/article/10.1088/1751-8121/acd489
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv IOP Publishing
publisher.none.fl_str_mv IOP Publishing
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
_version_ 1844614213656379392
score 13.070432