Liouville-von-Neumann-Gleichung

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\dot \rho = \mathcal{L} \rho = - \frac{i}{\color{Gray}\hbar }\left[ {H,\rho } \right] mit

ρ Dichteoperator
H Hamiltonoperator
\left[\_ \,,\_ \right] Kommutator

[1]

Herleitung

Schrödingergleichung


{\mathfrak{i}{\partial }_{t}}\Psi (t) =\hat{H}\Psi (t)

Dirac Notation

Ket:

\begin{align} & \left| \mathfrak{i}{{\partial }_{t}}\Psi \left( t \right) \right\rangle =\left| \hat{H}\Psi \left( t \right) \right\rangle \\ & \mathfrak{i}{{\partial }_{t}}\left| \Psi \left( t \right) \right\rangle =\hat{H}\left| \Psi \left( t \right) \right\rangle \Rightarrow {{\partial }_{t}}\left| \Psi \left( t \right) \right\rangle =-\mathfrak{i}\hat{H}\left| \Psi \left( t \right) \right\rangle \\ \end{align}

Bra:

\begin{align} & \left\langle \mathfrak{i}{{\partial }_{t}}\Psi \left( t \right) \right|=\left\langle \hat{H}\Psi \left( t \right) \right| \\ & \text{-}\mathfrak{i}{{\partial }_{t}}\left\langle \Psi \left( t \right) \right|=\left\langle \Psi \left( t \right) \right|\hat{H},\,\left( \hat{H}={{{\hat{H}}}^{+}} \right)\Rightarrow {{\partial }_{t}}\left\langle \Psi \left( t \right) \right|=\mathfrak{i}\left\langle \Psi \left( t \right) \right|\hat{H} \\ \end{align}


Dichtematrix

\rho =\left| \Psi \left( t \right) \right\rangle \left\langle \Psi \left( t \right) \right|

einsetzen:


\begin{align} & \dot{\rho }={{\partial }_{t}}\left( \left| \Psi \left( t \right) \right\rangle \left\langle \Psi \left( t \right) \right| \right) \\ & =\left( {{\partial }_{t}}\left| \Psi \left( t \right) \right\rangle \right)\left\langle \Psi \left( t \right) \right|+\left| \Psi \left( t \right) \right\rangle \left( {{\partial }_{t}}\left\langle \Psi \left( t \right) \right| \right) \\ & =-\mathfrak{i}\hat{H}\left| \Psi \left( t \right) \right\rangle \left\langle \Psi \left( t \right) \right|+\left| \Psi \left( t \right) \right\rangle \left\langle \Psi \left( t \right) \right|\mathfrak{i}\hat{H} \\ & =-\mathfrak{i}\left( \hat{H}\rho -\rho \hat{H} \right)\equiv -\mathfrak{i}\left[ \hat{H},\rho \right]=\mathfrak{i}\left[ \rho ,\hat{H} \right] \end{align}

Einzelnachweise

  1. ↑ Schöll, QM 2.5 Teil 1 Seite 77,

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