Liouville-von-Neumann-Gleichung: Unterschied zwischen den Versionen

Aktuelle Version vom 17. September 2010, 00:13 Uhr

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

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

^[1]

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|=\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{aligned}}

Dichtematrix

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

einsetzen:

{\begin{aligned}&{\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{aligned}}

Einzelnachweise

↑ Schöll, QM 2.5 Teil 1 Seite 77,

Wikipedia-Eintrag

[1] Schöll, QM 2.5 Teil 1 Seite 77,

[1]

@@ Zeile 13: / Zeile 13: @@
 {{Quelle|Schöll, QM 2.5 Teil 1 Seite 77}}
 ==Herleitung==
-Schrödingergleichung
+[[Schrödingergleichung]]
-<math>{\mathfrak{i}{\partial }_{t}}\Psi (t) =\hat{H}\Psi (t)</math>
+:<math>{\mathfrak{i}{\partial }_{t}}\Psi (t) =\hat{H}\Psi (t)</math>
 Dirac Notation
 Ket:
-<math>\left| {{\mathfrak{i}{\partial }_{t}}\Psi (t)} \right\rangle =
+:<math>\begin{align}
-\left| \hat{H}\Psi (t) \right\rangle </math>
+  & \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}</math>
-<math>\mathfrak{i} \partial_t \left| \Psi \left( t \right) \right\rangle =
+Bra:
-\hat{H}\left| \Psi (t) \right\rangle </math>
-Bra:
+:<math>\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}</math>
-<math>\begin{align}
+[[Dichtematrix]]
-  & \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{align}</math>
+:<math>\rho =\left| \Psi \left( t \right) \right\rangle \left\langle  \Psi \left( t \right) \right|</math>
+einsetzen:
+:<math>\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}</math>
 == Einzelnachweise ==
 <references />
 [http://de.wikipedia.org/wiki/Von-Neumann-Gleichung Wikipedia-Eintrag]
+[[Kategorie:Thermodynamik]]

Liouville-von-Neumann-Gleichung: Unterschied zwischen den Versionen

Aktuelle Version vom 17. September 2010, 00:13 Uhr

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