Liouville-von-Neumann-Gleichung: Difference between revisions

Revision as of 23:21, 6 September 2009

$\dot{ρ} = ℒ ρ = - \frac{i}{ℏ} [H, ρ]$ mit

$ρ$	Dichteoperator
H	Hamiltonoperator
$[_,_]$	Kommutator

^[1]

Herleitung

Schrödingergleichung

$i \partial_{t} Ψ (t) = \hat{H} Ψ (t)$

Dirac Notation

Ket: $\begin{aligned} | i \partial_{t} Ψ (t) ⟩ = | \hat{H} Ψ (t) ⟩ \\ i \partial_{t} | Ψ (t) ⟩ = \hat{H} | Ψ (t) ⟩ \Rightarrow \partial_{t} | Ψ (t) ⟩ = - i \hat{H} | Ψ (t) ⟩ \end{aligned}$

Bra:

$\begin{aligned} ⟨ i \partial_{t} Ψ (t) | = ⟨ \hat{H} Ψ (t) | \\ - i \partial_{t} ⟨ Ψ (t) | = \hat{H} ⟨ Ψ (t) |, \hat{H} = {\hat{H}}^{+} \Rightarrow \partial_{t} ⟨ Ψ (t) | = i \hat{H} ⟨ Ψ (t) | \end{aligned}$

Dichtematrix

$\hat{ρ} = | Ψ (t) ⟩ ⟨ Ψ (t) |$

Einzelnachweise

↑ Schöll, QM 2.5 Teil 1 Seite 77,

Wikipedia-Eintrag

[1] Schöll, QM 2.5 Teil 1 Seite 77,

[1]

@@ Line 13: / Line 13: @@
 {{Quelle|Schöll, QM 2.5 Teil 1 Seite 77}}
 ==Herleitung==
-Schrödingergleichung
+[[Schrödingergleichung]]
@@ Line 21: / Line 21: @@
 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  \\
-<math>\mathfrak{i} \partial_t \left| \Psi \left( t \right) \right\rangle =
+\end{align}</math>
-\hat{H}\left| \Psi (t) \right\rangle </math>
 Bra:
 <math>\begin{align}
-   & \left\langle  \text{-}\mathfrak{i}{{\partial }_{t}}\Psi \left( t \right) \right|=\left\langle  \hat{H}\Psi \left( t \right) \right| \\
+   & \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}}}^{+}} \\
+  & \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{align}</math>
+[[Dichtematrix]]
+<math>\hat{\rho }=\left| \Psi \left( t \right) \right\rangle \left\langle  \Psi \left( t \right) \right|</math>
 == Einzelnachweise ==
 <references />
 [http://de.wikipedia.org/wiki/Von-Neumann-Gleichung Wikipedia-Eintrag]

Liouville-von-Neumann-Gleichung: Difference between revisions

Revision as of 23:21, 6 September 2009

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