forcebalance:equation_of_motion
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| forcebalance:equation_of_motion [2023/11/21 14:05] – typo cjoens | forcebalance:equation_of_motion [2023/11/22 12:42] (current) – [Current Density Operator] cjoens | ||
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| Line 16: | Line 16: | ||
| \quad . | \quad . | ||
| \end{equation} | \end{equation} | ||
| - | Under the following conditions, the EoM for the electron current $J(x,t)$ becomes | + | |
| + | =====Fermionic Field Operator===== | ||
| + | The EoM for the electronic creation and annihilation operator is | ||
| \begin{equation} | \begin{equation} | ||
| - | \frac{\partial}{\partial t} J (x,t) | + | \frac{\partial}{\partial t} \hat{\Psi}_\text{H} |
| = | = | ||
| + | \dots | ||
| + | \end{equation} | ||
| + | according to the calculation [[forcebalance: | ||
| + | |||
| + | Condition: | ||
| + | \begin{equation} | ||
| + | \nu (\vec{r}, | ||
| + | = | ||
| + | \nu (\vec{r}^\prime, | ||
| + | \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: | ||
| + | The calculation is done [[forcebalance: | ||
| + | |||
| + | =====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, | ||
| + | = | ||
| + | \dots | ||
| + | + | ||
| \dots | \dots | ||
| \quad . | \quad . | ||
| \end{equation} | \end{equation} | ||
| + | FIXME Insert Link to para and diramagnetic definitions | ||
| This holds for symmetric correlation potentials | This holds for symmetric correlation potentials | ||
| \begin{equation} | \begin{equation} | ||
| Line 29: | Line 63: | ||
| \nu (\vec{r}^\prime, | \nu (\vec{r}^\prime, | ||
| \end{equation} | \end{equation} | ||
| - | and | + | and (Do a page where I discuss where this occurs) |
| \begin{equation} | \begin{equation} | ||
| \frac{\partial}{\partial t} | \frac{\partial}{\partial t} | ||
| Line 38: | Line 72: | ||
| \quad . | \quad . | ||
| \end{equation} | \end{equation} | ||
| - | The calculation can be found [[forcebalance: | + | The calculation can be found [[forcebalance: |
forcebalance/equation_of_motion.1700571923.txt.gz · Last modified: 2023/11/21 14:05 by cjoens