Liouville-von-Neumann-Gleichung

$\dot{\rho }=\mathcal{ℒ}\rho =-\frac{i}{\hslash }\left[H,\rho \right]$ mit

$\rho$	Dichteoperator
H	Hamiltonoperator
$[\_,\_]$	Kommutator

Herleitung

Schrödingergleichung

$\mathfrak{i}{\partial }_t\mathrm{\Psi }(t)=\hat{H}\mathrm{\Psi }(t)$

Dirac Notation

Ket: $\begin{array}{rl}& |\mathfrak{i}{\partial }_t\mathrm{\Psi }\left(t\right)⟩=|\hat{H}\mathrm{\Psi }\left(t\right)⟩\\ & \mathfrak{i}{\partial }_t|\mathrm{\Psi }\left(t\right)⟩=\hat{H}|\mathrm{\Psi }\left(t\right)⟩\Rightarrow {\partial }_t|\mathrm{\Psi }\left(t\right)⟩=-\mathfrak{i}\hat{H}|\mathrm{\Psi }\left(t\right)⟩\\ \end{array}$

Bra:

$\begin{array}{rl}& ⟨\mathfrak{i}{\partial }_t\mathrm{\Psi }\left(t\right)|=⟨\hat{H}\mathrm{\Psi }\left(t\right)|\\ & \text{-}\mathfrak{i}{\partial }_t⟨\mathrm{\Psi }\left(t\right)|=⟨\mathrm{\Psi }\left(t\right)|\hat{H},(\hat{H}={\hat{H}}^{+})\Rightarrow {\partial }_t⟨\mathrm{\Psi }\left(t\right)|=\mathfrak{i}⟨\mathrm{\Psi }\left(t\right)|\hat{H}\\ \end{array}$

Dichtematrix

$\rho =|\mathrm{\Psi }\left(t\right)⟩⟨\mathrm{\Psi }\left(t\right)|$

einsetzen:

$\begin{array}{rl}& \dot{\rho }={\partial }_t\left(|\mathrm{\Psi }\left(t\right)⟩⟨\mathrm{\Psi }\left(t\right)|\right)\\ & =\left({\partial }_t|\mathrm{\Psi }\left(t\right)⟩\right)⟨\mathrm{\Psi }\left(t\right)|+|\mathrm{\Psi }\left(t\right)⟩\left({\partial }_t⟨\mathrm{\Psi }\left(t\right)|\right)\\ & =-\mathfrak{i}\hat{H}|\mathrm{\Psi }\left(t\right)⟩⟨\mathrm{\Psi }\left(t\right)|+|\mathrm{\Psi }\left(t\right)⟩⟨\mathrm{\Psi }\left(t\right)|\mathfrak{i}\hat{H}\\ & =-\mathfrak{i}(\hat{H}\rho -\rho \hat{H})\equiv -\mathfrak{i}[\hat{H},\rho ]=\mathfrak{i}[\rho ,\hat{H}]\end{array}$

Einzelnachweise

Wikipedia-Eintrag