Liouville-von-Neumann-Gleichung: Difference between revisions
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<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 | Dirac Notation | ||
Ket: | Ket: | ||
<math>\begin{align} | :<math>\begin{align} | ||
& \left| \mathfrak{i}{{\partial }_{t}}\Psi \left( t \right) \right\rangle =\left| \hat{H}\Psi \left( t \right) \right\rangle \\ | & \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 \\ | & \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 \\ | ||
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Bra: | Bra: | ||
<math>\begin{align} | :<math>\begin{align} | ||
& \left\langle \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|=\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} \\ | & \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> | \end{align}</math> | ||
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[[Dichtematrix]] | [[Dichtematrix]] | ||
<math>\rho =\left| \Psi \left( t \right) \right\rangle \left\langle \Psi \left( t \right) \right|</math> | :<math>\rho =\left| \Psi \left( t \right) \right\rangle \left\langle \Psi \left( t \right) \right|</math> | ||
einsetzen: | einsetzen: | ||
<math>\begin{align} | |||
:<math>\begin{align} | |||
& \dot{\rho }={{\partial }_{t}}\left( \left| \Psi \left( t \right) \right\rangle \left\langle \Psi \left( t \right) \right| \right) \\ | & \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) \\ | & =\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}\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 -\rho \hat{H} \right)\equiv -\mathfrak{i}\left[ \hat{H},\rho \right]=\mathfrak{i}\left[ \rho ,\hat{H} \right] | ||
\end{align}</math> | \end{align}</math> | ||
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<references /> | <references /> | ||
[http://de.wikipedia.org/wiki/Von-Neumann-Gleichung Wikipedia-Eintrag] | [http://de.wikipedia.org/wiki/Von-Neumann-Gleichung Wikipedia-Eintrag] | ||
[[Kategorie:Thermodynamik]] | |||
Latest revision as of 23:13, 16 September 2010
| Dichteoperator | |
| H | Hamiltonoperator |
| Kommutator |
Herleitung[edit | edit source]
Dirac Notation
Ket:
Bra:
einsetzen:
Einzelnachweise[edit | edit source]
- ↑ Schöll, QM 2.5 Teil 1 Seite 77,