Liouville-von-Neumann-Gleichung: Unterschied zwischen den Versionen
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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 \\ | ||
\end{align}</math> | \end{align}</math> | ||
| Zeile 30: | Zeile 30: | ||
<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|= | & \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> | ||
[[Dichtematrix]] | [[Dichtematrix]] | ||
<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)=-\mathfrak{i}\left[ \hat{H},\rho \right]=\mathfrak{i}\left[ \rho ,\hat{H} \right] | |||
\end{align}</math> | |||
== Einzelnachweise == | == Einzelnachweise == | ||
<references /> | <references /> | ||
[http://de.wikipedia.org/wiki/Von-Neumann-Gleichung Wikipedia-Eintrag] | [http://de.wikipedia.org/wiki/Von-Neumann-Gleichung Wikipedia-Eintrag] | ||
Version vom 7. September 2009, 00:23 Uhr
mit
![{\displaystyle {\dot {\rho }}={\mathcal {L}}\rho =-{\frac {i}{\color {Gray}\hbar }}\left[{H,\rho }\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fae42ea1de04b81ad7659db21e691e245f55a1e)
| Dichteoperator | |
| H | Hamiltonoperator |
| Kommutator |
![{\displaystyle \left[\_\,,\_\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc3c634206ee5c8982a5eef8128d7f11f3f6c734)
Herleitung
Dirac Notation
Ket:
Bra:
einsetzen:
![{\displaystyle {\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)=-{\mathfrak {i}}\left[{\hat {H}},\rho \right]={\mathfrak {i}}\left[\rho ,{\hat {H}}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91b0f43754f15bb8e057f36eff71d75b4e6c65a3)
Einzelnachweise
- ↑ Schöll, QM 2.5 Teil 1 Seite 77,