Liouville-von-Neumann-Gleichung

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

$\rho$	Dichteoperator
H	Hamiltonoperator
$[\mathrm{\_},\mathrm{\_}]$	Kommutator

Herleitung

Schrödingergleichung

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

Dirac Notation

Ket:

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

Bra:

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

Dichtematrix

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

einsetzen:

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

Einzelnachweise

Wikipedia-Eintrag Kategorie:Thermodynamik