Exact solution of the Schrödinger equation for the movement of a particle in a parametric magnetic field
| Autor(a) principal: | |
|---|---|
| Data de Publicação: | 2021 |
| Outros Autores: | , , , |
| Tipo de documento: | Artigo |
| Idioma: | por |
| Título da fonte: | Research, Society and Development |
| Texto Completo: | https://rsdjournal.org/index.php/rsd/article/view/16401 |
Resumo: | We solved the Schrödinger equation exactly for a time-dependent quadratic system for a particle that moves under the influence of a magnetic field with parametric oscillation. We apply the decoupling method, which adopts a transformation of Ray-Reid's spatio-temporal coordinates (Nassar, 1990). The fundamental idea of the problem is to obtain a Schrödinger free particle equation. In this way, it was possible to determine the wave function and the probability density of the particle in the form of a parametric vibration function. We show that the regions of stability and instability are determined by the phase space defined by the equation's control parameters. We determined, as an unprecedented result, the discrete values that the magnetic field can assume in terms of Mathieu functions. |
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Exact solution of the Schrödinger equation for the movement of a particle in a parametric magnetic field Solución exacta de la ecuación de Schrödinger para el movimiento de una partícula en un campo magnético paramétricoSolução exata da equação de Schrödinger para movimento de uma partícula em um campo magnético paramétricoEquação de SchrödingerSistema quadrático dependente do tempoTransformação espaço-temporalEquação e funções de MathieuRessonância e oscilações paramétricas.Ecuación de SchrödingerSistema cuadrático dependiente del tiempoTransformación espacio-temporalEcuación y funciones de MathieuResonancia y oscilaciones paramétricas.Schrödinger's EquationTime-dependent quadratic systemSpatio-temporal transformationMathieu equation and functionsResonance and parametric oscillations.We solved the Schrödinger equation exactly for a time-dependent quadratic system for a particle that moves under the influence of a magnetic field with parametric oscillation. We apply the decoupling method, which adopts a transformation of Ray-Reid's spatio-temporal coordinates (Nassar, 1990). The fundamental idea of the problem is to obtain a Schrödinger free particle equation. In this way, it was possible to determine the wave function and the probability density of the particle in the form of a parametric vibration function. We show that the regions of stability and instability are determined by the phase space defined by the equation's control parameters. We determined, as an unprecedented result, the discrete values that the magnetic field can assume in terms of Mathieu functions.Resolvimos la ecuación de Schrödinger exactamente para un sistema cuadrático dependiente del tiempo para una partícula que se mueve bajo la influencia de un campo magnético con oscilación paramétrica. Aplicamos el método de desacoplamiento, que adopta una transformación de las coordenadas espacio-temporales de Ray-Reid (Nassar, 1990). La idea fundamental del problema es obtener una ecuación de partículas libres de Schrödinger. De esta forma, fue posible determinar la función de onda y la densidad de probabilidad de la partícula en forma de función de vibración paramétrica. Mostramos que las regiones de estabilidad e inestabilidad están determinadas por el espacio de fase definido por los parámetros de control de la ecuación. Determinamos, como resultado sin precedentes, los valores discretos que el campo magnético puede asumir en términos de funciones de Mathieu.Resolvemos de modo exato a equação de Schrödinger para um sistema quadrático dependente do tempo para uma partícula que se movimenta sob a influência de um campo magnético com oscilação paramétrica. Aplicamos o método de desacoplamento, o qual adota uma transformação de coordenadas espaço-temporal de Ray-Reid (Nassar, 1990). A ideia fundamental do problema é obter uma equação tipo partícula livre de Schrödinger. Desse modo, foi possível determinar a função de onda e a densidade de probabilidade da partícula na forma de uma função de vibração paramétrica. Mostramos que as regiões de estabilidades e instabilidades são determinadas pelo espaço de fase definidos pelos parâmetros de controle da equação. Determinamos, como resultado inédito, os valores discretos que o campo magnético pode assumir em termos das funções de Mathieu.Research, Society and Development2021-06-17info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://rsdjournal.org/index.php/rsd/article/view/1640110.33448/rsd-v10i7.16401Research, Society and Development; Vol. 10 No. 7; e16310716401Research, Society and Development; Vol. 10 Núm. 7; e16310716401Research, Society and Development; v. 10 n. 7; e163107164012525-3409reponame:Research, Society and Developmentinstname:Universidade Federal de Itajubá (UNIFEI)instacron:UNIFEIporhttps://rsdjournal.org/index.php/rsd/article/view/16401/14655Copyright (c) 2021 João Bosco Soares Pampolha Junior; Charles da Rocha Silva; João Paulo da Silva Alves; Renato Germano; Damião Pedro Meira Filhohttps://creativecommons.org/licenses/by/4.0info:eu-repo/semantics/openAccessPampolha Junior, João Bosco Soares Silva, Charles da Rocha Alves, João Paulo da Silva Germano, RenatoMeira Filho, Damião Pedro 2021-07-18T21:07:03Zoai:ojs.pkp.sfu.ca:article/16401Revistahttps://rsdjournal.org/index.php/rsd/indexPUBhttps://rsdjournal.org/index.php/rsd/oairsd.articles@gmail.com2525-34092525-3409opendoar:2024-01-17T09:36:57.487017Research, Society and Development - Universidade Federal de Itajubá (UNIFEI)false |
| dc.title.none.fl_str_mv |
Exact solution of the Schrödinger equation for the movement of a particle in a parametric magnetic field Solución exacta de la ecuación de Schrödinger para el movimiento de una partícula en un campo magnético paramétrico Solução exata da equação de Schrödinger para movimento de uma partícula em um campo magnético paramétrico |
| title |
Exact solution of the Schrödinger equation for the movement of a particle in a parametric magnetic field |
| spellingShingle |
Exact solution of the Schrödinger equation for the movement of a particle in a parametric magnetic field Pampolha Junior, João Bosco Soares Equação de Schrödinger Sistema quadrático dependente do tempo Transformação espaço-temporal Equação e funções de Mathieu Ressonância e oscilações paramétricas. Ecuación de Schrödinger Sistema cuadrático dependiente del tiempo Transformación espacio-temporal Ecuación y funciones de Mathieu Resonancia y oscilaciones paramétricas. Schrödinger's Equation Time-dependent quadratic system Spatio-temporal transformation Mathieu equation and functions Resonance and parametric oscillations. |
| title_short |
Exact solution of the Schrödinger equation for the movement of a particle in a parametric magnetic field |
| title_full |
Exact solution of the Schrödinger equation for the movement of a particle in a parametric magnetic field |
| title_fullStr |
Exact solution of the Schrödinger equation for the movement of a particle in a parametric magnetic field |
| title_full_unstemmed |
Exact solution of the Schrödinger equation for the movement of a particle in a parametric magnetic field |
| title_sort |
Exact solution of the Schrödinger equation for the movement of a particle in a parametric magnetic field |
| author |
Pampolha Junior, João Bosco Soares |
| author_facet |
Pampolha Junior, João Bosco Soares Silva, Charles da Rocha Alves, João Paulo da Silva Germano, Renato Meira Filho, Damião Pedro |
| author_role |
author |
| author2 |
Silva, Charles da Rocha Alves, João Paulo da Silva Germano, Renato Meira Filho, Damião Pedro |
| author2_role |
author author author author |
| dc.contributor.author.fl_str_mv |
Pampolha Junior, João Bosco Soares Silva, Charles da Rocha Alves, João Paulo da Silva Germano, Renato Meira Filho, Damião Pedro |
| dc.subject.por.fl_str_mv |
Equação de Schrödinger Sistema quadrático dependente do tempo Transformação espaço-temporal Equação e funções de Mathieu Ressonância e oscilações paramétricas. Ecuación de Schrödinger Sistema cuadrático dependiente del tiempo Transformación espacio-temporal Ecuación y funciones de Mathieu Resonancia y oscilaciones paramétricas. Schrödinger's Equation Time-dependent quadratic system Spatio-temporal transformation Mathieu equation and functions Resonance and parametric oscillations. |
| topic |
Equação de Schrödinger Sistema quadrático dependente do tempo Transformação espaço-temporal Equação e funções de Mathieu Ressonância e oscilações paramétricas. Ecuación de Schrödinger Sistema cuadrático dependiente del tiempo Transformación espacio-temporal Ecuación y funciones de Mathieu Resonancia y oscilaciones paramétricas. Schrödinger's Equation Time-dependent quadratic system Spatio-temporal transformation Mathieu equation and functions Resonance and parametric oscillations. |
| description |
We solved the Schrödinger equation exactly for a time-dependent quadratic system for a particle that moves under the influence of a magnetic field with parametric oscillation. We apply the decoupling method, which adopts a transformation of Ray-Reid's spatio-temporal coordinates (Nassar, 1990). The fundamental idea of the problem is to obtain a Schrödinger free particle equation. In this way, it was possible to determine the wave function and the probability density of the particle in the form of a parametric vibration function. We show that the regions of stability and instability are determined by the phase space defined by the equation's control parameters. We determined, as an unprecedented result, the discrete values that the magnetic field can assume in terms of Mathieu functions. |
| publishDate |
2021 |
| dc.date.none.fl_str_mv |
2021-06-17 |
| dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion |
| format |
article |
| status_str |
publishedVersion |
| dc.identifier.uri.fl_str_mv |
https://rsdjournal.org/index.php/rsd/article/view/16401 10.33448/rsd-v10i7.16401 |
| url |
https://rsdjournal.org/index.php/rsd/article/view/16401 |
| identifier_str_mv |
10.33448/rsd-v10i7.16401 |
| dc.language.iso.fl_str_mv |
por |
| language |
por |
| dc.relation.none.fl_str_mv |
https://rsdjournal.org/index.php/rsd/article/view/16401/14655 |
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https://creativecommons.org/licenses/by/4.0 info:eu-repo/semantics/openAccess |
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https://creativecommons.org/licenses/by/4.0 |
| eu_rights_str_mv |
openAccess |
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application/pdf |
| dc.publisher.none.fl_str_mv |
Research, Society and Development |
| publisher.none.fl_str_mv |
Research, Society and Development |
| dc.source.none.fl_str_mv |
Research, Society and Development; Vol. 10 No. 7; e16310716401 Research, Society and Development; Vol. 10 Núm. 7; e16310716401 Research, Society and Development; v. 10 n. 7; e16310716401 2525-3409 reponame:Research, Society and Development instname:Universidade Federal de Itajubá (UNIFEI) instacron:UNIFEI |
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Universidade Federal de Itajubá (UNIFEI) |
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UNIFEI |
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UNIFEI |
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Research, Society and Development |
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Research, Society and Development |
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Research, Society and Development - Universidade Federal de Itajubá (UNIFEI) |
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rsd.articles@gmail.com |
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1797052786225971200 |