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Schrödingergleichung mit äußeren Potenzialen
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====Eichtransformation:==== :<math>\begin{align} & \bar{A}\acute{\ }(\bar{r},t)=\bar{A}(\bar{r},t)+\nabla G(\bar{r},t) \\ & \Phi \acute{\ }(\bar{r},t)=\Phi (\bar{r},t)-\dot{G}(\bar{r},t) \\ \end{align}</math> Dies ist eine zulässige Umeichung mit einer beliebigen, zweifach stetig diffbaren Funktion <math>G(\bar{r},t)</math> Durch Einsetzen in :<math>\bar{E}=-\nabla \Phi (\bar{q},t)-\dot{\bar{A}}(\bar{q},t)</math> :<math>\bar{B}=\nabla x\bar{A}(\bar{q},t)</math> zeigt sich :<math>\bar{E}=\bar{E}\acute{\ }</math> :<math>\bar{B}=\bar{B}\acute{\ }</math> Jedoch muss die Wellenfunktion auch umgeeicht werden: :<math>\Psi \acute{\ }(\bar{r},t)=\Psi (\bar{r},t){{e}^{i\frac{e}{\hbar }G(\bar{r},t)}}</math> Die Beschränkung der Eichung auf Phasenfaktoren geschieht wegen der Eichinvarianz der Wahrscheinlichkeitsdichte: :<math>{{\left| \Psi \acute{\ }(\bar{r},t) \right|}^{2}}={{\left| \Psi (\bar{r},t) \right|}^{2}}</math> '''Beweis:''' '''Zeige: '''<math>i\hbar \frac{\partial }{\partial t}\Psi \acute{\ }=\hat{H}\acute{\ }\Psi \acute{\ }\to i\hbar \frac{\partial }{\partial t}\Psi =\hat{H}\Psi </math> # <math>\frac{\hbar }{i}\nabla \Psi \acute{\ }(\bar{r},t)=\frac{\hbar }{i}\nabla \left\{ \Psi (\bar{r},t){{e}^{i\frac{e}{\hbar }G(\bar{r},t)}} \right\}={{e}^{i\frac{e}{\hbar }G(\bar{r},t)}}\left\{ \frac{\hbar }{i}\nabla \Psi (\bar{r},t)+e(\nabla G(\bar{r},t))\Psi (\bar{r},t) \right\}</math> # <math>\left( \frac{\hbar }{i}\nabla -e\bar{A}\acute{\ } \right)\Psi \acute{\ }(\bar{r},t)={{e}^{i\frac{e}{\hbar }G(\bar{r},t)}}\left\{ \frac{\hbar }{i}\nabla -e(\bar{A}\acute{\ }-\nabla G(\bar{r},t)) \right\}\Psi (\bar{r},t)={{e}^{i\frac{e}{\hbar }G(\bar{r},t)}}\left\{ \frac{\hbar }{i}\nabla -e\bar{A} \right\}\Psi (\bar{r},t)</math> # <math>{{\left( \frac{\hbar }{i}\nabla -e\bar{A}\acute{\ } \right)}^{2}}\Psi \acute{\ }(\bar{r},t)={{e}^{i\frac{e}{\hbar }G(\bar{r},t)}}{{\left\{ \frac{\hbar }{i}\nabla -e\bar{A} \right\}}^{2}}\Psi (\bar{r},t)</math> # <math>\frac{1}{2m}{{\left( \frac{\hbar }{i}\nabla -e\bar{A}\acute{\ } \right)}^{2}}\Psi \acute{\ }(\bar{r},t)+e\Phi \acute{\ }\Psi \acute{\ }(\bar{r},t)={{e}^{i\frac{e}{\hbar }G(\bar{r},t)}}{{\left\{ \frac{1}{2m}\left( \frac{\hbar }{i}\nabla -e\bar{A} \right)+e\Phi -e\dot{G}(\bar{r},t) \right\}}^{2}}\Psi (\bar{r},t)</math> dabei: :<math>\frac{1}{2m}{{\left( \frac{\hbar }{i}\nabla -e\bar{A}\acute{\ } \right)}^{2}}\Psi \acute{\ }(\bar{r},t)+e\Phi \acute{\ }\Psi \acute{\ }(\bar{r},t)=\hat{H}\acute{\ }\Psi \acute{\ }(\bar{r},t)</math> :<math>\left( \frac{1}{2m}\left( \frac{\hbar }{i}\nabla -e\bar{A} \right)+e\Phi \right)\Psi (\bar{r},t)=\hat{H}\Psi (\bar{r},t)</math> Schritt 4 repräsentiert die linke Seite der Schrödingergleichung. Gleichzeitig: # <math>i\hbar \frac{\partial }{\partial t}\Psi \acute{\ }(\bar{r},t)=i\hbar \frac{\partial }{\partial t}\left\{ \Psi (\bar{r},t){{e}^{i\frac{e}{\hbar }G(\bar{r},t)}} \right\}={{e}^{i\frac{e}{\hbar }G(\bar{r},t)}}\left\{ i\hbar \dot{\Psi }(\bar{r},t)+e\dot{G}\Psi (\bar{r},t) \right\}</math> Da Gleichung4) und 5) gleich sein müssen folgt als Bedingung :<math>\hat{H}\Psi (\bar{r},t)=i\hbar \dot{\Psi }(\bar{r},t)</math> Was ja gerade die nicht umgeeichte Schrödingergleichung ist. Fazit: Die Schrödingergleichung ist eichinvariant, falls die Wellenfunktion gemäß <math>\Psi \acute{\ }(\bar{r},t)=\Psi (\bar{r},t){{e}^{i\frac{e}{\hbar }G(\bar{r},t)}}</math>umgeeicht wird.
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