Editing Master Gleichung
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Latest revision | Your text | ||
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* Vor Rechts ins System <math>{}_{S}^{R}{{H}_{I}}</math> | * Vor Rechts ins System <math>{}_{S}^{R}{{H}_{I}}</math> | ||
* Vom System nach Links <math>{}_{L}^{S}{{H}_{I}}</math> | * Vom System nach Links <math>{}_{L}^{S}{{H}_{I}}</math> | ||
* Vom System nach Rechts <math>{}_{R}^{S}{{H}_{I}}</math> mit <math>{}_{S}^{X}{{H}_{I}}=\sum\limits_{k,i}{{}^{X}{{V}_{k}}^{X}a_{k}^{\dagger }{{e}_{i}}}</math> und <math>{}_{X}^{S}{{H}_{I}}=\sum\limits_{k,i}{{}^{X}{{V}_{k}}^{X}{{a}_{k}}e_{i}^{\dagger }}</math> | * Vom System nach Rechts <math>{}_{R}^{S}{{H}_{I}}</math> | ||
mit | |||
<math>{}_{S}^{X}{{H}_{I}}=\sum\limits_{k,i}{{}^{X}{{V}_{k}}^{X}a_{k}^{\dagger }{{e}_{i}}}</math> | |||
und | |||
<math>{}_{X}^{S}{{H}_{I}}=\sum\limits_{k,i}{{}^{X}{{V}_{k}}^{X}{{a}_{k}}e_{i}^{\dagger }}</math> | |||
<math>{{e}_{i}}</math> erzeugt ein Electron im System mit Energieniveau i. | |||
<math>e_{i}^{\dagger }</math> vernichtet ... | |||
==Transformation ins WW-Bild== | ==Transformation ins WW-Bild== | ||
Operator ins WWBild | Operator ins WWBild | ||
<math>\tilde{A}\left( t \right):=U_{0}^{\dagger }A{{U}_{0}}</math> | |||
mit <math>{{U}_{0}}=\exp \left( -\mathfrak{i}{{H}_{0}}t \right)</math> | mit <math>{{U}_{0}}=\exp \left( -\mathfrak{i}{{H}_{0}}t \right)</math> | ||
und <math>{{H}_{0}}={{H}_{S}}+{{H}_{B}}</math> | und <math>{{H}_{0}}={{H}_{S}}+{{H}_{B}}</math> | ||
Starte von [[Liouville-von-Neumann-Gleichung]] | Starte von [[Liouville-von-Neumann-Gleichung]] | ||
<math> | |||
\dot \rho = - \mathfrak{i} \left[ {H,\rho } \right]</math> | \dot \rho = - \mathfrak{i} \left[ {H,\rho } \right]</math> | ||
mit der Lösung | mit der Lösung | ||
<math>\rho \left( t \right)={{U}^{\dagger }}{{\rho }_{0}}U</math> | |||
mit <math>U=\exp \left( -\mathfrak{i}Ht \right)</math> | mit <math>U=\exp \left( -\mathfrak{i}Ht \right)</math> | ||
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Beweis | Beweis | ||
<math>{{\partial }_{t}}U=-\mathfrak{i}HU</math> | |||
sowie | |||
<math>{{\partial }_{t}}{{U}^{\dagger }}=\mathfrak{i}HU</math> | |||
Dann ist | Dann ist | ||
<math>{{d}_{t}}\rho =\underbrace{-\mathfrak{i}HU{{\rho }_{0}}{{U}^{\dagger }}+U{{\rho }_{0}}\mathfrak{i}H{{U}^{\dagger }}}_{-\mathfrak{i}\left[ H,\rho \right]}+\underbrace{U\left( {{\partial }_{t}}{{\rho }_{0}} \right){{U}^{\dagger }}}_{0}</math> | |||
beweis ende | beweis ende | ||
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<math>\begin{align} | |||
& {{d}_{t}}\tilde{\rho }={{d}_{t}}\left( U_{0}^{\dagger }\rho {{U}_{0}} \right) \\ | & {{d}_{t}}\tilde{\rho }={{d}_{t}}\left( U_{0}^{\dagger }\rho {{U}_{0}} \right) \\ | ||
& =\mathfrak{i}{{H}_{0}}U_{0}^{\dagger }\rho {{U}_{0}}-iU_{0}^{\dagger }\rho {{H}_{0}}{{U}_{0}}+U_{0}^{\dagger }{{d}_{t}}\left( \rho \right){{U}_{0}} \\ | & =\mathfrak{i}{{H}_{0}}U_{0}^{\dagger }\rho {{U}_{0}}-iU_{0}^{\dagger }\rho {{H}_{0}}{{U}_{0}}+U_{0}^{\dagger }{{d}_{t}}\left( \rho \right){{U}_{0}} \\ | ||
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===Lösung=== | ===Lösung=== | ||
Integrieren | Integrieren | ||
<math>\tilde{\rho}=\rho_0 - \mathfrak{i} \int_0^t [\tilde{H_I},\tilde{\rho}]\,dt'</math> | |||
auf rechter Seite einsetzen | auf rechter Seite einsetzen | ||
<math>\begin{align} | |||
& {{d}_{t}}\tilde{\rho }=-\mathfrak{i}\left[ {{{\tilde{H}}}_{I}},{{\rho }_{0}}-\mathfrak{i}\int_{0}^{t}{[{{{\tilde{H}}}_{I}},\tilde{\rho }]}\,d{t}' \right] \\ | & {{d}_{t}}\tilde{\rho }=-\mathfrak{i}\left[ {{{\tilde{H}}}_{I}},{{\rho }_{0}}-\mathfrak{i}\int_{0}^{t}{[{{{\tilde{H}}}_{I}},\tilde{\rho }]}\,d{t}' \right] \\ | ||
& =-\mathfrak{i}\left[ {{{\tilde{H}}}_{I}},{{\rho }_{0}} \right]-\left[ {{{\tilde{H}}}_{I}},\int_{0}^{t}{[{{{\tilde{H}}}_{I}},\tilde{\rho }]}\,d{t}' \right] \\ | & =-\mathfrak{i}\left[ {{{\tilde{H}}}_{I}},{{\rho }_{0}} \right]-\left[ {{{\tilde{H}}}_{I}},\int_{0}^{t}{[{{{\tilde{H}}}_{I}},\tilde{\rho }]}\,d{t}' \right] \\ | ||
& =-\mathfrak{i}\left[ {{{\tilde{H}}}_{I}},{{\rho }_{0}} \right]-\int_{0}^{t}{\left[ {{{\tilde{H}}}_{I}},\left[ {{{\tilde{H}}}_{I}},\,\tilde{\rho } \right] \right]}d{t}' | & =-\mathfrak{i}\left[ {{{\tilde{H}}}_{I}},{{\rho }_{0}} \right]-\int_{0}^{t}{\left[ {{{\tilde{H}}}_{I}},\left[ {{{\tilde{H}}}_{I}},\,\tilde{\rho } \right] \right]}d{t}' | ||
\end{align}</math> | \end{align}</math> | ||