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: $\left|{{{\mathfrak {i}}{\partial }_{t}}\Psi (t)}\right\rangle =\left|{\hat {H}}\Psi (t)\right\rangle$

${\mathfrak {i}}\partial _{t}\left|\Psi \left(t\right)\right\rangle ={\hat {H}}\left|\Psi (t)\right\rangle$

Bra:

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

Einzelnachweise

Wikipedia-Eintrag