Liouville-von-Neumann-Gleichung: Unterschied zwischen den Versionen
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| Zeile 16: | Zeile 16: | ||
<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 \\ | ||
| Zeile 28: | Zeile 28: | ||
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} \\ | ||
| Zeile 37: | Zeile 37: | ||
[[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) \\ | ||
Version vom 12. September 2010, 18:35 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)\equiv -{\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/2e8661d27e86e10052366ceb0c0e62455a2a6a94)
Einzelnachweise
- ↑ Schöll, QM 2.5 Teil 1 Seite 77,