testwiki - User contributions [en]

Liouville-von-Neumann-Gleichung

2009-09-06T22:24:24Z

85.178.233.148: /* Herleitung */

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

{| class="wikitable"
|-
| <math>\rho</math> || Dichteoperator
|-
| H || Hamiltonoperator
|-
| <math>\left[\_ \,,\_ \right]</math>|| Kommutator
|}
{{Quelle|Schöll, QM 2.5 Teil 1 Seite 77}}
==Herleitung==
[[Schrödingergleichung]]

<math>{\mathfrak{i}{\partial }_{t}}\Psi (t) =\hat{H}\Psi (t)</math>

Dirac Notation

Ket:
<math>\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}</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>

[[Dichtematrix]]

<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]

Liouville-von-Neumann-Gleichung

2009-09-06T22:23:52Z

85.178.233.148: /* Herleitung */

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

{| class="wikitable"
|-
| <math>\rho</math> || Dichteoperator
|-
| H || Hamiltonoperator
|-
| <math>\left[\_ \,,\_ \right]</math>|| Kommutator
|}
{{Quelle|Schöll, QM 2.5 Teil 1 Seite 77}}
==Herleitung==
[[Schrödingergleichung]]

<math>{\mathfrak{i}{\partial }_{t}}\Psi (t) =\hat{H}\Psi (t)</math>

Dirac Notation

Ket:
<math>\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}</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>

[[Dichtematrix]]

<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)=-\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]

Liouville-von-Neumann-Gleichung

2009-09-06T22:21:13Z

85.178.233.148: /* Herleitung */

Liouville-von-Neumann-Gleichung

2009-09-06T21:57:02Z

85.178.233.148:

Schrödinger-Gleichung

2009-09-06T20:56:05Z

85.178.233.148:

<math>{\mathfrak{i}{\partial }_{t}}\Psi =\hat{H}\Psi ,\hat{H}=-\frac{\Delta }{2m}</math>

<math>\mathfrak{i} {\color{Gray} \hbar} \frac{\partial }{\partial t}|\,\psi (t)\rangle =\hat{H}|\,\psi (t)\rangle </math>

[[Kategorie:Gleichung]]