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--- forcebalance:equation_of_moiton [2023/11/21 13:45] – created cjoens
+++ forcebalance:equation_of_moiton [2023/11/21 14:03] (current) – cjoens
@@ Line 1: / Line 1: @@
-======Equation of motion======
+======DeleteMe======
-The equations of motion are derived here with the help of the [[QM representations|Heisenberg EoM]].
-Starting in the [[QM representations|Heisenberg picture]] the time evolution of an operator is
-\begin{equation}
-  \frac{\partial}{\partial t} \hat{O}_\text{H} (x,t)
-  =
-  \left[
-    \hat{O}_\text{H} (x,t)
-  ,
-    \hat{H}_\text{H} (t)
-  \right]
-  +
-  \left(
-    \frac{\partial}{\partial t} \hat{O}_\text{S} (x)
-  \right)_\text{H}
-  \quad .
-\end{equation}
-Under the following conditions, the EoM for the electron current $J(x,t)$ becomes
-\begin{equation}
-  \frac{\partial}{\partial t} J (x,t)
-  =
-  \dots
-  \quat .
-\end{equation}
-This holds for
-\begin{equation}
-  \nu (\vec{r},\vec{r}^\prime)
-  =
-  \nu (\vec{r},\vec{r}^\prime)
-\end{equation}
-and
-\begin{equation}
-  \frac{\partial}{\partial t}
-  \frac{\partial}{\partial k}
-  =
-  \frac{\partial}{\partial k}
-  \frac{\partial}{\partial t}
-  \quad .
-\end{equation}
-The calculation can be found [[forcebalance:calculation:EoM electron current|here]].