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--- forcebalance:equation_of_motion [2023/11/22 12:33] – Create Chapter: Particle Number Operator cjoens
+++ forcebalance:equation_of_motion [2023/11/22 12:42] (current) – [Current Density Operator] cjoens
@@ Line 1: / Line 1: @@
+======Equation of motion======
+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}
+=====Fermionic Field Operator=====
+The EoM for the electronic creation and annihilation operator is
+\begin{equation}
+  \frac{\partial}{\partial t} \hat{\Psi}_\text{H} (x,t)
+  =
+  \dots
+\end{equation}
+according to the calculation [[forcebalance:calculation:EoM fermionic field operator|here]].
+Condition:
+\begin{equation}
+  \nu (\vec{r},\vec{r}^\prime)
+  =
+  \nu (\vec{r}^\prime,\vec{r})
+\end{equation}
+=====Particle Number Operator=====
+The EoM of the particle number operator
+\begin{equation}
+  \frac{\partial}{\partial t} \hat{n}_\text{H}=
+  \dots
+\end{equation}
+is of the form of a [[forcebalance:continuity equation]].
+The calculation is done [[forcebalance:calculation:EoM particle number operator|here]].
+=====Current Density Operator=====
+Wtih the help of the EoM of the fermionic field operator, the EoM for the current density operator $J(x,t)$ becomes
+\begin{equation}
+  \frac{\partial}{\partial t} \hat{J}_\text{H} (x,t)
+  =
+  \frac{\partial}{\partial t} \hat{j}_\text{H} (x,t)
+  +
+  \frac{\partial}{\partial t} \hat{j}_\text{d,H} (x,t)
+  =
+  \dots
+  +
+  \dots
+  \quad .
+\end{equation}
+FIXME Insert Link to para and diramagnetic definitions
+This holds for symmetric correlation potentials
+\begin{equation}
+  \nu (\vec{r},\vec{r}^\prime)
+  =
+  \nu (\vec{r}^\prime,\vec{r})
+\end{equation}
+and (Do a page where I discuss where this occurs)
+\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 current density operator|here]].