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8.Offene Quantensysteme: Interaktion eines Biosystems mit seiner Umgebung

Wie bereits betont wurde, jedes Biosystem Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} ist grundsätzlich offen. Daher muss die Dynamik seines Zustands über eine Wechselwirkung mit der Umgebung modelliert werden $\varepsilon$ . Die Staaten von Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} und Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} sind in den Hilberträumen vertreten Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} und Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} . Das zusammengesetzte System Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S+\varepsilon} wird im Tensorprodukt Hilberträume dargestellt. Dieses System wird als isoliertes System behandelt und gemäß der Quantentheorie kann die Dynamik seines reinen Zustands durch die Schrödinger-Gleichung beschrieben werden:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\tfrac{d}{dt}\Psi(t)=\widehat{H}\Psi(t)(t), \Psi(0)=\Psi_0}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (21)}

wo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(t)} ist der reine Zustand des Systems Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S+\varepsilon} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathcal{H}}} ist sein Hamiltonoperator. Diese Gleichung impliziert, dass der reine Zustand Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(t)} entwickelt sich einheitlich: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(t)=\hat{U}(t)\psi_0} . Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{U}(t)=e^{-it\hat{\mathcal{H}}}} . Hamiltonian (Evolutionsgenerator), der Informationsinteraktionen beschreibt, hat die Form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathcal{H}}=\hat{\mathcal{H}}_s+\hat{\mathcal{H}}_\varepsilon+{\mathcal{\hat H_{S,\varepsilon}}}} , wo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathcal{H}}_s} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathcal{H}}_\varepsilon} sind Hamiltonoperatoren der Systeme und Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal{\hat H_{S,\varepsilon}}}} ist der Wechselwirkungs-Hamiltonoperator.12 Diese Gleichung impliziert, dass die Evolution des Dichteoperators Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathcal{R}}(t)} of the system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S+\varepsilon} wird durch die von Neumann-Gleichung beschrieben:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{d\widehat{R}}{dt}(t)=-i[\widehat{H},\widehat{R},(t)], \widehat{R}(0)=\widehat{R}_0}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (22)}

Allerdings der Staat Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathcal{R}}(t)} ist für jede mathematische Analyse zu komplex: Die Umgebung enthält zu viele Freiheitsgrade. Daher interessiert uns nur der Zustand Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} ; seine Dynamik wird über die Verfolgung des Zustands von erhalten Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S+\varepsilon} w.r.t. die Freiheitsgrade vonFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\rho}(t)=Tr_\mathcal{H}\widehat{R}(t)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (23)}

Im Allgemeinen ist diese Gleichung, die Quanten-Master-Gleichung, mathematisch sehr kompliziert. In Anwendungen wird eine Vielzahl von Näherungen verwendet.

8.1. Quanten-Markov-Modell: Gorini-Kossakowski-Sudarshan-Lindbladequation

Die einfachste Annäherung der Quanten-Master-Gleichung (23) ist die Quanten-Markov-Dynamik, die durch die Gorini-Kossakowski-Sudarshan-Lindblad (GKSL)-Gleichung (Ingarden et al., 1997) gegeben ist (in der Physik wird sie allgemein einfach als Lindblad-Gleichung bezeichnet; Dies ist die einfachste Quanten-Master-Gleichung):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{d\widehat{\rho}}{dt}(t)=-i[\widehat{H},\widehat{\rho},(t)]+ \widehat{L}[\widehat{\rho}(t),\widehat{\rho}(0)=\widehat{\rho}_0}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (24)}

wobei Hermitescher Operator (Hamiltonian) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\mathcal{H}}} beschreibt die innere Dynamik von Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} und der Superoperator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{{L}}} , im Raum der Dichteoperatoren agierend, beschreibt eine Interaktion mit der Umgebung Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} .Dieser Superoperator wird oft Lindbladian genannt. Die GKSL-Gleichung ist eine Quantenmastergleichung für die Markovsche Dynamik. In diesem Beitrag haben wir keine Möglichkeit, den Begriff der Quanten-Markovianität näher zu erläutern. Die Quantenmastergleichung (23) beschreibt im Allgemeinen nicht-Markovsche Dynamik.