On the mean value of the force operator for 1D particles in the step potential
Autor(a) principal: | |
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Data de Publicação: | 2021 |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Revista Brasileira de Ensino de Física (Online) |
Texto Completo: | http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172021000100423 |
Resumo: | In the one-dimensional Klein-Fock-Gordon theory, the probability density is a discontinuous function at the point where the step potential is discontinuous. Thus, the mean value of the external classical force operator cannot be calculated from the corresponding formula of the mean value. To resolve this issue, we obtain this quantity directly from the Klein-Fock-Gordon equation in Hamiltonian form, or the Feshbach-Villars wave equation. Not without surprise, the result obtained is not proportional to the average of the discontinuity of the probability density but to the size of the discontinuity. In contrast, in the one-dimensional Schrödinger and Dirac theories this quantity is proportional to the value that the respective probability density takes at the point where the step potential is discontinuous. We examine these issues in detail in this paper. The presentation is suitable for the advanced undergraduate level. |
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On the mean value of the force operator for 1D particles in the step potentialSchrödinger wave equationKlein-Fock-Gordon wave equationDirac wave equationthe external classical force operatorboundary conditionsIn the one-dimensional Klein-Fock-Gordon theory, the probability density is a discontinuous function at the point where the step potential is discontinuous. Thus, the mean value of the external classical force operator cannot be calculated from the corresponding formula of the mean value. To resolve this issue, we obtain this quantity directly from the Klein-Fock-Gordon equation in Hamiltonian form, or the Feshbach-Villars wave equation. Not without surprise, the result obtained is not proportional to the average of the discontinuity of the probability density but to the size of the discontinuity. In contrast, in the one-dimensional Schrödinger and Dirac theories this quantity is proportional to the value that the respective probability density takes at the point where the step potential is discontinuous. We examine these issues in detail in this paper. The presentation is suitable for the advanced undergraduate level.Sociedade Brasileira de Física2021-01-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersiontext/htmlhttp://old.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172021000100423Revista Brasileira de Ensino de Física v.43 2021reponame:Revista Brasileira de Ensino de Física (Online)instname:Sociedade Brasileira de Física (SBF)instacron:SBF10.1590/1806-9126-rbef-2020-0422info:eu-repo/semantics/openAccessDe Vincenzo,Salvatoreeng2021-02-18T00:00:00Zoai:scielo:S1806-11172021000100423Revistahttp://www.sbfisica.org.br/rbef/https://old.scielo.br/oai/scielo-oai.php||marcio@sbfisica.org.br1806-91261806-1117opendoar:2021-02-18T00:00Revista Brasileira de Ensino de Física (Online) - Sociedade Brasileira de Física (SBF)false |
dc.title.none.fl_str_mv |
On the mean value of the force operator for 1D particles in the step potential |
title |
On the mean value of the force operator for 1D particles in the step potential |
spellingShingle |
On the mean value of the force operator for 1D particles in the step potential De Vincenzo,Salvatore Schrödinger wave equation Klein-Fock-Gordon wave equation Dirac wave equation the external classical force operator boundary conditions |
title_short |
On the mean value of the force operator for 1D particles in the step potential |
title_full |
On the mean value of the force operator for 1D particles in the step potential |
title_fullStr |
On the mean value of the force operator for 1D particles in the step potential |
title_full_unstemmed |
On the mean value of the force operator for 1D particles in the step potential |
title_sort |
On the mean value of the force operator for 1D particles in the step potential |
author |
De Vincenzo,Salvatore |
author_facet |
De Vincenzo,Salvatore |
author_role |
author |
dc.contributor.author.fl_str_mv |
De Vincenzo,Salvatore |
dc.subject.por.fl_str_mv |
Schrödinger wave equation Klein-Fock-Gordon wave equation Dirac wave equation the external classical force operator boundary conditions |
topic |
Schrödinger wave equation Klein-Fock-Gordon wave equation Dirac wave equation the external classical force operator boundary conditions |
description |
In the one-dimensional Klein-Fock-Gordon theory, the probability density is a discontinuous function at the point where the step potential is discontinuous. Thus, the mean value of the external classical force operator cannot be calculated from the corresponding formula of the mean value. To resolve this issue, we obtain this quantity directly from the Klein-Fock-Gordon equation in Hamiltonian form, or the Feshbach-Villars wave equation. Not without surprise, the result obtained is not proportional to the average of the discontinuity of the probability density but to the size of the discontinuity. In contrast, in the one-dimensional Schrödinger and Dirac theories this quantity is proportional to the value that the respective probability density takes at the point where the step potential is discontinuous. We examine these issues in detail in this paper. The presentation is suitable for the advanced undergraduate level. |
publishDate |
2021 |
dc.date.none.fl_str_mv |
2021-01-01 |
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=S1806-11172021000100423 |
url |
http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172021000100423 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
10.1590/1806-9126-rbef-2020-0422 |
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 |
Revista Brasileira de Ensino de Física v.43 2021 reponame:Revista Brasileira de Ensino de Física (Online) instname:Sociedade Brasileira de Física (SBF) instacron:SBF |
instname_str |
Sociedade Brasileira de Física (SBF) |
instacron_str |
SBF |
institution |
SBF |
reponame_str |
Revista Brasileira de Ensino de Física (Online) |
collection |
Revista Brasileira de Ensino de Física (Online) |
repository.name.fl_str_mv |
Revista Brasileira de Ensino de Física (Online) - Sociedade Brasileira de Física (SBF) |
repository.mail.fl_str_mv |
||marcio@sbfisica.org.br |
_version_ |
1752122425206636544 |