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Path Integral Solution for some Time-Dependient Potential

S. N. Storchak

Published Jan 1, 1989• 0 citations
OpenAlex

Abstract

<meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-1252"> <meta name=3DGenerator content=3D"Microsoft Word 11"> <meta name=3DOriginator content=3D"Microsoft Word 11"> <link rel=3DFile-List href=3D"IC_89_367yatodo_archivos/filelist.xml"> RESUMEN <!--[if gte mso 9]> Un usuario de Microsoft Office satisfecho. Normal Un usuario de Microsoft Office satisfecho. 2 2 2004-06-23T15:40:00Z 2004-06-23T15:40:00Z 1 44 243 UCLA 2 1 286 11.6568 <![endif]--><!--[if gte mso 9]> 0 Clean Clean 21 false false false MicrosoftInternetExplorer4 <![endif]--><!--[if gte mso 9]> <![endif]--> <!-- /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} span.SpellE {mso-style-name:""; mso-spl-e:yes;} @page Section1 {size:612.0pt 792.0pt; margin:70.85pt 3.0cm 70.85pt 3.0cm; mso-header-margin:35.4pt; mso-footer-margin:35.4pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> <!--[if gte mso 10]> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} <![endif]--> RESUMEN The quantum-mechanical problem with a time-dependent potencial is solved by the path integral method. The solution is obtained by the application of the previously derived general formula for rheonomic homogeneous point transformation and reparametrization in the path integral.

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<meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-1252"> <meta name=3DGenerator content=3D"Microsoft Word 11"> <meta name=3DOriginator content=3D"Microsoft Word 11"> <link rel=3DFile-List href=3D"IC_89_10_archivos/filelist.xml"> RESUMEN <!--[if gte mso 9]> Un usuario de Microsoft Office satisfecho. Normal Biblioteca de Ciencias y Tecnología 3 3 2006-02-20T13:06:00Z 2006-02-20T13:19:00Z 1 56 309 UCLA 2 1 364 11.6568 <![endif]--><!--[if gte mso 9]> 0 Clean Clean 21 false false false MicrosoftInternetExplorer4 <![endif]--><!--[if gte mso 9]> <![endif]--> <!-- /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"; mso-ansi-language:ES-VE; mso-fareast-language:ES-VE;} span.SpellE {mso-style-name:""; mso-spl-e:yes;} @page Section1 {size:612.0pt 792.0pt; margin:70.85pt 3.0cm 70.85pt 3.0cm; mso-header-margin:35.4pt; mso-footer-margin:35.4pt; mso-paper-source:0;} div.Section1 {page:Section1;} /* List Definitions */ @list l0 {mso-list-id:1399013862; mso-list-type:hybrid; mso-list-template-ids:-1000807380 70317156 201981955 201981957 201981953 201981955 201981957 201981953 201981955 201981957;} @list l0:level1 {mso-level-number-format:bullet; mso-level-text:\F0B7; mso-level-tab-stop:36.0pt; mso-level-number-position:left; text-indent:-18.0pt; font-family:Symbol; color:windowtext;} @list l0:level2 {mso-level-tab-stop:72.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level3 {mso-level-tab-stop:108.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level4 {mso-level-tab-stop:144.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level5 {mso-level-tab-stop:180.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level6 {mso-level-tab-stop:216.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level7 {mso-level-tab-stop:252.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level8 {mso-level-tab-stop:288.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level9 {mso-level-tab-stop:324.0pt; mso-level-number-position:left; text-indent:-18.0pt;} ol {margin-bottom:0cm;} ul {margin-bottom:0cm;} --> <!--[if gte mso 10]> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} <![endif]--> RESUMEN The classical limit of the Schrodinger equation is analyzed in the frame of its hydrodynamical formulation. In that limit the probability density develops shock waves, clearly contradicting the exact result. By an approximated analysis in the point on which these discontinuities appear, it is shown that their formulation is inhibited by quantum effects.

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<meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-1252"> <meta name=3DGenerator content=3D"Microsoft Word 11"> <meta name=3DOriginator content=3D"Microsoft Word 11"> <link rel=3DFile-List href=3D"IC_89_367yatodo_archivos/filelist.xml"> RESUMEN <!--[if gte mso 9]> Un usuario de Microsoft Office satisfecho. Normal Un usuario de Microsoft Office satisfecho. 2 2 2004-06-23T15:40:00Z 2004-06-23T15:40:00Z 1 44 243 UCLA 2 1 286 11.6568 <![endif]--><!--[if gte mso 9]> 0 Clean Clean 21 false false false MicrosoftInternetExplorer4 <![endif]--><!--[if gte mso 9]> <![endif]--> <!-- /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} span.SpellE {mso-style-name:""; mso-spl-e:yes;} @page Section1 {size:612.0pt 792.0pt; margin:70.85pt 3.0cm 70.85pt 3.0cm; mso-header-margin:35.4pt; mso-footer-margin:35.4pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> <!--[if gte mso 10]> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} <![endif]--> RESUMEN The quantum-mechanical problem with a time-dependent potencial is solved by the path integral method. The solution is obtained by the application of the previously derived general formula for rheonomic homogeneous point transformation and reparametrization in the path integral.
<meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-1252"> <meta name=3DGenerator content=3D"Microsoft Word 11"> <meta name=3DOriginator content=3D"Microsoft Word 11"> <link rel=3DFile-List href=3D"IC_89_10_archivos/filelist.xml"> RESUMEN <!--[if gte mso 9]> Un usuario de Microsoft Office satisfecho. Normal Biblioteca de Ciencias y Tecnología 3 3 2006-02-20T13:06:00Z 2006-02-20T13:19:00Z 1 56 309 UCLA 2 1 364 11.6568 <![endif]--><!--[if gte mso 9]> 0 Clean Clean 21 false false false MicrosoftInternetExplorer4 <![endif]--><!--[if gte mso 9]> <![endif]--> <!-- /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"; mso-ansi-language:ES-VE; mso-fareast-language:ES-VE;} span.SpellE {mso-style-name:""; mso-spl-e:yes;} @page Section1 {size:612.0pt 792.0pt; margin:70.85pt 3.0cm 70.85pt 3.0cm; mso-header-margin:35.4pt; mso-footer-margin:35.4pt; mso-paper-source:0;} div.Section1 {page:Section1;} /* List Definitions */ @list l0 {mso-list-id:1399013862; mso-list-type:hybrid; mso-list-template-ids:-1000807380 70317156 201981955 201981957 201981953 201981955 201981957 201981953 201981955 201981957;} @list l0:level1 {mso-level-number-format:bullet; mso-level-text:\F0B7; mso-level-tab-stop:36.0pt; mso-level-number-position:left; text-indent:-18.0pt; font-family:Symbol; color:windowtext;} @list l0:level2 {mso-level-tab-stop:72.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level3 {mso-level-tab-stop:108.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level4 {mso-level-tab-stop:144.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level5 {mso-level-tab-stop:180.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level6 {mso-level-tab-stop:216.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level7 {mso-level-tab-stop:252.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level8 {mso-level-tab-stop:288.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l0:level9 {mso-level-tab-stop:324.0pt; mso-level-number-position:left; text-indent:-18.0pt;} ol {margin-bottom:0cm;} ul {margin-bottom:0cm;} --> <!--[if gte mso 10]> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} <![endif]--> RESUMEN The classical limit of the Schrodinger equation is analyzed in the frame of its hydrodynamical formulation. In that limit the probability density develops shock waves, clearly contradicting the exact result. By an approximated analysis in the point on which these discontinuities appear, it is shown that their formulation is inhibited by quantum effects.
<meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-1252"> <meta name=3DGenerator content=3D"Microsoft Word 11"> <meta name=3DOriginator content=3D"Microsoft Word 11"> <link rel=3DFile-List href=3D"IC_89_139_archivos/filelist.xml"> RESUMEN <!--[if gte mso 9]> Un usuario de Microsoft Office satisfecho. Normal Biblioteca de Ciencias y Tecnología 2 56 2006-03-13T20:51:00Z 2006-03-13T20:51:00Z 1 53 296 UCLA 2 1 348 11.6568 <![endif]--><!--[if gte mso 9]> 75 Clean Clean 21 false false false MicrosoftInternetExplorer4 <![endif]--><!--[if gte mso 9]> <![endif]--> <!-- /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} span.SpellE {mso-style-name:""; mso-spl-e:yes;} @page Section1 {size:612.0pt 792.0pt; margin:70.85pt 3.0cm 70.85pt 3.0cm; mso-header-margin:35.4pt; mso-footer-margin:35.4pt; mso-paper-source:0;} div.Section1 {page:Section1;} /* List Definitions */ @list l0 {mso-list-id:665935979; mso-list-template-ids:1870185836;} @list l0:level1 {mso-level-number-format:bullet; mso-level-text:\F0B7; mso-level-tab-stop:36.0pt; mso-level-number-position:left; text-indent:-18.0pt; mso-ansi-font-size:10.0pt; font-family:Symbol;} @list l1 {mso-list-id:1584024425; mso-list-type:hybrid; mso-list-template-ids:-1703523192 657982034 201981955 201981957 201981953 201981955 201981957 201981953 201981955 201981957;} @list l1:level1 {mso-level-number-format:bullet; mso-level-text:\F0B7; mso-level-tab-stop:36.0pt; mso-level-number-position:left; text-indent:-18.0pt; font-family:Symbol; color:windowtext;} @list l1:level2 {mso-level-tab-stop:72.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level3 {mso-level-tab-stop:108.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level4 {mso-level-tab-stop:144.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level5 {mso-level-tab-stop:180.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level6 {mso-level-tab-stop:216.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level7 {mso-level-tab-stop:252.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level8 {mso-level-tab-stop:288.0pt; mso-level-number-position:left; text-indent:-18.0pt;} @list l1:level9 {mso-level-tab-stop:324.0pt; mso-level-number-position:left; text-indent:-18.0pt;} ol {margin-bottom:0cm;} ul {margin-bottom:0cm;} --> <!--[if gte mso 10]> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} <![endif]--> RESUMEN The inverse scattering method is applied to the integrable nonlinear system describing temperature correlators of the impenetrable bosons in one space dimension. The corresponding matrix Riemann problems are constructed for two-point as well as for multi-point correlators. Long-distance asymptotics of two-point correlators is calculated.
Normal 0 21 false false false MicrosoftInternetExplorer4 /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabela normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} In this work we intend to develop the Mathematical Formalism of Non-Relativistic Quantum Mechanics in terms of two Basics Operators: Evolution Operator, and Density Operator and some of its properties. This article has didactic purposes.