Key equations from quantum statistical tools

Equilibrium

Expectation values

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A = \mathrm{Tr}[\hat{\rho}_\mathrm{eq} \hat{A}] }

Density matrix of fermions in equilibrium

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho}_\mathrm{eq}=\sum_\alpha f(E_\alpha-\mu)|E_\alpha\rangle \langle E_\alpha| = f(\hat{H} - \mu\hat{I})}

• Fermi-Dirac distribution function: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(x) = 1/[\exp(x/k_BT)+1] }
• Hamiltonian and its spectral decomposition: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{H} = \sum_\alpha E_\alpha |E_\alpha \rangle \langle E_\alpha| }
• function of Hamiltonian: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(\hat{H}) = \sum_\alpha F(E_\alpha) |E_\alpha \rangle \langle E_\alpha| }

Charge density

• charge density operator: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{n}(\mathbf{r}) = |\mathbf{r} \rangle \langle \mathbf{r}| }

• expectation value: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 } (in some discrete representation these is just diagonal matrix element)

Density of states

• local density of states: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n(\mathbf{r}) = \int dE\, g(r,E) f(E) }

Nonequilibrium

• Expectation values:

• Current operator:

• Spin torque operator: