Liouville-von-Neumann-Gleichung

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

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

Herleitung

Schrödingergleichung

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

Dirac Notation

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

Bra:

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

Dichtematrix

${\hat {\rho }}=\left|\Psi \left(t\right)\right\rangle \left\langle \Psi \left(t\right)\right|$

Einzelnachweise

Wikipedia-Eintrag