Key equations from quantum statistical tools

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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

  • using 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{\rho}_\mathrm{eq}=\sum_\alpha f(E_\alpha)|E_\alpha\rangle \langle E_\alpha| = f(\hat{H} - \mu\hat{I})}

  • using Green functions:

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} = - \frac{1}{\pi} \int dE\, \mathrm{Im} G^r f(E) }

  • 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(E) = 1/[\exp((E-\mu)/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| }
  • Green operators: 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{G}^{r,a} = [E\hat{I}-\hat{H} \pm i\eta]^{-1} }

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

  • definition: 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 g(E) = \sum_\alpha \delta(E-E_\alpha) } (with possible normalization factors like 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 2_s/V } )
  • 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}) = \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(r,E) f(E) }

Nonequilibrium

  • Expectation values:
  • Current operator:
  • Spin torque operator:
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