View source for Key equations from quantum statistical tools

==Equilibrium==

===Expectation values===

<math> A = \mathrm{Tr}[\hat{\rho}_\mathrm{eq} \hat{A}] </math>

===Density matrix of fermions in equilibrium===

*using spectral decomposition: 

<math> \hat{\rho}_\mathrm{eq}=\sum_\alpha f(E_\alpha)|E_\alpha\rangle \langle E_\alpha| = f(\hat{H} - \mu\hat{I})</math>

*using Green functions:

<math> \hat{\rho}_\mathrm{eq} = - \frac{1}{\pi} \int dE\, \mathrm{Im} G^r f(E) </math>

* Fermi-Dirac distribution function: <math> f(E) = 1/[\exp((E-\mu)/k_BT)+1] </math>

* Hamiltonian and its spectral decomposition: <math> \hat{H} = \sum_\alpha E_\alpha |E_\alpha \rangle \langle E_\alpha| </math>

* function of Hamiltonian: <math> F(\hat{H}) =  \sum_\alpha F(E_\alpha) |E_\alpha \rangle \langle E_\alpha| </math>

* Green operators: 

<math> \hat{G}^{r,a}(E) = [E\hat{I}-\hat{H} \pm i\eta]^{-1} </math>

<math> \mathrm{Im} \hat{G}^r = (\hat{G}^{r} - \hat{G}^a)/2i </math>

===Charge density===

*charge density operator: <math> \hat{n}(\mathbf{r}) = |\mathbf{r} \rangle \langle \mathbf{r}| </math>

*expectation value: <math> n(\mathbf{r})  = \mathrm{Tr}[\hat{\rho}_\mathrm{eq}|\mathbf{r} \rangle \langle \mathbf{r}|] = \langle \mathbf{r} | \hat{\rho}_\mathrm{eq}|\mathbf{r} \rangle </math> (in some discrete representation these is just diagonal matrix element)

===Density of states===

* definition of total DOS: <math> g(E) = \sum_\alpha \delta(E-E_\alpha) </math> (with possible normalization factors like <math> 2_s/V </math>)

* definition of LDOS: <math> g(E) = \int d^3 \mathbf{r} g(\mathbf{r},E) </math>

* LDOS using wavefunctions: <math> n(\mathbf{r}) = \mathrm{Tr}[\hat{\rho}_\mathrm{eq}|\mathbf{r} \rangle \langle \mathbf{r}|] = \sum_\alpha |\Psi_\alpha(\mathbf{r})|^2 f(E_\alpha) = \int dE \left[\sum_\alpha |\Psi_\alpha(\mathbf{r})|^2 \delta(E-E_\alpha)\right]f(E) = \int dE\,  g(\mathbf{r},E) f(E) </math>

* LDOS using Green functions:

<math> g(\mathbf{r},E) = -\frac{1}{\pi} \langle \mathbf{r} |\mathrm{Im} \hat{G}^r(E) | \mathbf{r} \rangle </math>

* total DOS using Green functions:

<math> g(E) = -\frac{1}{\pi} \mathrm{Tr}[ \hat{G}^r(E)] = -\frac{1}{\pi} \int d^3 \mathbf{r} \, \langle \mathbf{r} |\mathrm{Im} \hat{G}^r(E) | \mathbf{r} \rangle  </math>

==Nonequilibrium==

===Expectation values===

<math> A = \mathrm{Tr}[\hat{\rho}_\mathrm{neq} \hat{A}] </math>

*Current operators: