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--- forcebalance:start [2023/11/14 14:35] – created cjoens
+++ forcebalance:start [2023/11/22 10:57] (current) – [Force-Balance Approach To Functional Approximations] cjoens
@@ Line 2: / Line 2: @@
 The purpose of this wiki is to create a more or less consistent nomenclature within this project for all participants. Additionally, this is a way to create a collection of relevant sources:
   * Framework of the project
-  * Definitions
+  * [[forcebalance:definitions|Definitions]]
-  * Analytical calculations
+  * [[forcebalance:calculation|Analytical calculations]]
-  * Theorems
+  * [[forcebalance:theorems|Theorems]]
   * Citations
   * Additional information
   * Numerical calculations with code or prompt
+  * Open Questions
   * etc.
 ===== Overview =====
+Starting with the Hamiltonian
+\begin{equation}
+  \hat{H} (t)
+  =
+  \hat{H}_0 (t) + H_{\text{int}} (t)
+\end{equation}
+describing electorns in an external Potential $V(r,t)$ and an external vectorpotential \vector{A}(\vector{r},t).
+Here the singele particle part of the Hamiltonian is
+\begin{equation}
+  H_0 (t)
+  =
+  \sum_{\sigma^\prime, \sigma^{\prime\prime}}
+  \int \!\mathrm{d}r^\prime
+  \hat{\Psi}^{\dagger}(\vec{r}^\prime,\sigma^\prime)
+  h_{\sigma^\prime, \sigma^{\prime\prime}} (\vec{r},-i\nabla,\vec{s},t)
+  \hat{\Psi}(\vec{r}^\prime,\sigma^{\prime\prime})
+  \quad ,
+\end{equation}
+with
+\begin{equation}
+  h_{\sigma^\prime, \sigma^{\prime\prime}} (\vec{r},-i\nabla,\vec{s},t)
+  =
+  \dots
+\end{equation}
+and the interaction part of the Hamiltonian is
+\begin{equation}
+  H_{\text{int}} (t)
+  =
+  \frac{1}{2}
+  \int \!\mathrm{d}x^{\prime}
+  \int \!\mathrm{d}x^{\prime\prime}
+  \nu (\vec{r}^\prime,\vec{r}^{\prime\prime})
+  \hat{\Psi}^{\dagger}(x^{\prime})
+  \hat{\Psi}^{\dagger}(x^{\prime\prime})
+  \hat{\Psi} (x^{\prime\prime})
+  \hat{\Psi} (x^\prime)
+  \quad .
+\end{equation}
+From this the [[equation of motion|equation of motion(EoM)]] of the electron current $\frac{\partial}{\partial t}J(\vec{r},t)$ can be derived.
+((in there describe: starting from the Heisenberg EoM and the rest is transforming the equation))
 TODO: Main ideas, formulas (-> with additional "calculations nodes"), variables (-> "definition node"), Hamiltonian [(:cite:forcebalance:tchenkoue_force_2019)] and linkts to "assumption nodes"
@@ Line 26: / Line 69: @@
 TODO: Short overview on the main formulas variables and abbreviations. Further abbreviations can be found here and important variables can be found here.
 "complementary nodes"
+=====Abandoned Links=====
+  *equation of moiton