Betrachtung eines mikr. Hamiltonoperators
bestehend aus
Die Umgebung setzt sich aus einem Reservoir
Wechselwirkung besteht aus 4 Teilen
erzeugt ein Electron im System mit Energieniveau i.
vernichtet ...
Transformation ins WW-Bild[edit | edit source]
Operator ins WWBild
![{\displaystyle {\tilde {A}}\left(t\right):=U_{0}^{\dagger }A{{U}_{0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e26725cbb17a0250324b4e0a5b1a31accbb9c5d8)
mit
und
Starte von Liouville-von-Neumann-Gleichung
![{\displaystyle {\dot {\rho }}=-{\mathfrak {i}}\left[{H,\rho }\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a110105817768be16429d4c18e915aa4242f3b8d)
mit der Lösung
![{\displaystyle \rho \left(t\right)={{U}^{\dagger }}{{\rho }_{0}}U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1038dc3281443573b56e2fb1ea4361766197557f)
mit
Beweis
sowie ![{\displaystyle {{\partial }_{t}}{{U}^{\dagger }}={\mathfrak {i}}HU}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e480215e0e7cdae278743c684a15bd1e64d05b09)
Dann ist
![{\displaystyle {{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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fa8f27e571b02b0296437369dc56e0fbecee5a2)
beweis ende
lösung ende
Die LVN-Gln wird zu
![{\displaystyle {\begin{aligned}&{{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}}\left[{{H}_{0}},{\tilde {\rho }}\right]-{\mathfrak {i}}U_{0}^{\dagger }\left[H,\rho \right]{{U}_{0}}\\&={\mathfrak {i}}\left[{{H}_{0}},{\tilde {\rho }}\right]-{\mathfrak {i}}U_{0}^{\dagger }\left[{{H}_{0}}+{{H}_{I}},\rho \right]{{U}_{0}}\\&={\mathfrak {i}}\left[{{H}_{0}},{\tilde {\rho }}\right]-{\mathfrak {i}}\left[{{H}_{0}},{\tilde {\rho }}\right]-{\mathfrak {i}}U_{0}^{\dagger }\left[{{H}_{I}},\rho \right]{{U}_{0}}\\&=-{\mathfrak {i}}\left[{{\tilde {H}}_{I}},{\tilde {\rho }}\right]\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dcdaef93a6167f2d75edd3485ec4210b599e4c1)
Integrieren
![{\displaystyle {\tilde {\rho }}=\rho _{0}-{\mathfrak {i}}\int _{0}^{t}[{\tilde {H_{I}}},{\tilde {\rho }}]\,dt'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13cc0ac8a3bf5da19758fc1c2a84588cd4ba3bb7)
auf rechter Seite einsetzen
![{\displaystyle {\begin{aligned}&{{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]-\int _{0}^{t}{\left[{{\tilde {H}}_{I}},\left[{{\tilde {H}}_{I}},\,{\tilde {\rho }}\right]\right]}d{t}'\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c7612f51db4063ae9c38dbd937cd04d282cf925)
Der Dichteoperator des Systems ist die Spur über das Bad
![{\displaystyle {{\rho }_{S}}={{\operatorname {Tr} }_{B}}\left[\rho \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f60c091fc36f7c3b82e43ee2f8c55caca0188bdf)
![{\displaystyle {{\tilde {\rho }}_{S}}=U_{S}^{\dagger }{{\rho }_{S}}{{U}_{S}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0709b1db1bc4be404e51d8ee64d977f984d2bb94)
![{\displaystyle {{U}_{S}}=\exp \left(-{\mathsf {\mathfrak {i}}}{{H}_{S}}t\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b17445923477c4214b67b8a764f290fdd46005f3)
damit folgt für
![{\displaystyle {\begin{aligned}&{{d}_{t}}{\tilde {\rho _{S}}}=-{\mathfrak {i}}\operatorname {Tr} _{B}\left[{{\tilde {H}}_{I}},{{\rho }_{0}}\right]-\int _{0}^{t}{\operatorname {Tr} _{B}\left[{{\tilde {H}}_{I}},\left[{{\tilde {H}}_{I}},\,{\tilde {\rho }}\right]\right]}d{t}'\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbde1d2e2f77e65e6d2cf53061c0406beb4e2cf)
- WW zur Zeit t=0 eingeschaltet
- no korrelation beteween System and Bath at t=0
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![{\displaystyle {{\tilde {\rho }}_{0}}={{\rho }_{0}}={{\rho }_{S,0}}{{R}_{B,0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a766c95abb866d3bb2ea292316c6395ee015cb4)
- Kopplung Reservoiroperatoren ans System in Zustand R_0 liefern keinen Beitrag.
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![{\displaystyle {{\operatorname {Tr} }_{S}}\left[{{\tilde {H}}_{I}}{{R}_{B,0}}\right]=0\Rightarrow \left[{{\tilde {H}}_{I}},{{\rho }_{0}}\right]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92490ec42e2965c25ac15b62a6fe75f3801477dd)
- Dichtematrix zu t=0 Sperabel
- Schwache Kopplung zwischen System und Bad H_I
- Systemgröße von B größer als S daher B nicht beeinflusst
![{\displaystyle {\tilde {\rho }}={{\tilde {\rho }}_{S,0}}{{R}_{B,0}}+O\left({{H}_{I}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90c204b7ac4b97c14d5f4cb016c5a9c7e1c6c883)
- Jetzt vernachlässigen von Termen mit Ordnung von H_I>2
![{\displaystyle {{d}_{t}}{{\tilde {\rho }}_{S}}=-\int _{0}^{t}{{{\operatorname {Tr} }_{B}}\left[{{\tilde {H}}_{I}},\left[{\tilde {H}}{{'}_{I}},\,{\tilde {\rho }}{{'}_{S}}{{R}_{B,0}}\right]\right]}d{t}'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b6441494765a8b43edcb45c8d632bef23f3efa3)
- Zukunft hängt nur von aktuellem Zustand ab
![{\displaystyle {{\rho }_{S}}=\rho {{'}_{S}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99601c59182fe934e725cd6b7488e7084747f7b6)
Kategorie:Thermodynamik