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Diffstat (limited to 'source/know/concept/lehmann-representation/index.md')
| -rw-r--r-- | source/know/concept/lehmann-representation/index.md | 28 |
1 files changed, 17 insertions, 11 deletions
diff --git a/source/know/concept/lehmann-representation/index.md b/source/know/concept/lehmann-representation/index.md index 0dec718..d96eb66 100644 --- a/source/know/concept/lehmann-representation/index.md +++ b/source/know/concept/lehmann-representation/index.md @@ -78,7 +78,8 @@ $$\begin{aligned} } \end{aligned}$$ -We now go through the same process for the lesser Green's function $$G_{\nu \nu'}^<(t, t')$$: +We now go through the same process +for the lesser Green's function $$G_{\nu \nu'}^<(t, t')$$: $$\begin{aligned} G_{\nu \nu'}^<(t - t') @@ -137,20 +138,22 @@ $$\begin{aligned} \\ &= - \frac{i}{\hbar Z} \Theta(t \!-\! t') \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} - \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) e^{i (E_n - E_{n'}) (t - t') / \hbar} + \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) e^{i (E_n - E_{n'}) (t - t') / \hbar} \end{aligned}$$ We take the Fourier transform, but to ensure convergence, -we must introduce an infinitesimal positive $$\eta \to 0^+$$ to the exponent +we introduce an infinitesimal positive $$\eta \to 0^+$$ to the exponent (and eventually take the limit): $$\begin{aligned} G_{\nu \nu'}^R(\omega) - &= - \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_{-\infty}^\infty \Theta(t) e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega + i \eta) t} \dd{t} + &= - \frac{i}{\hbar Z} \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) + \int_{-\infty}^\infty \Theta(t) e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega + i \eta) t} \dd{t} \\ - &= - \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_0^\infty e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega + i \eta) t} \dd{t} + &= - \frac{i}{\hbar Z} \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) + \int_0^\infty e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega + i \eta) t} \dd{t} \\ - &= - \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) + &= - \frac{i}{\hbar Z} \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) \bigg[ \frac{\hbar e^{i (\hbar \omega + E_n - E_{n'}) t / \hbar} e^{- \eta t}}{i (\hbar \omega + E_n - E_{n'}) - \hbar \eta} \bigg]_0^\infty \end{aligned}$$ @@ -166,7 +169,8 @@ $$\begin{aligned} } \end{aligned}$$ -Finally, we go through the same steps for the advanced Green's function $$G_{\nu \nu'}^A(t, t')$$: +Finally, we go through the same steps +for the advanced Green's function $$G_{\nu \nu'}^A(t, t')$$: $$\begin{aligned} G_{\nu \nu'}^A(t \!-\! t') @@ -174,7 +178,7 @@ $$\begin{aligned} \\ &= \frac{i}{\hbar Z} \Theta(t' \!-\! t) \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} - \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) e^{i (E_n - E_{n'}) (t - t') / \hbar} + \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) e^{i (E_n - E_{n'}) (t - t') / \hbar} \end{aligned}$$ For the Fourier transform, we must again introduce $$\eta \to 0^+$$ @@ -182,11 +186,13 @@ For the Fourier transform, we must again introduce $$\eta \to 0^+$$ $$\begin{aligned} G_{\nu \nu'}^A(\omega) - &= \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_{-\infty}^\infty \Theta(-t) e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega - i \eta) t} \dd{t} + &= \frac{i}{\hbar Z} \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) + \int_{-\infty}^\infty \Theta(-t) e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega - i \eta) t} \dd{t} \\ - &= \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_{-\infty}^0 e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega - i \eta) t} \dd{t} + &= \frac{i}{\hbar Z} \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) + \int_{-\infty}^0 e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega - i \eta) t} \dd{t} \\ - &= \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) + &= \frac{i}{\hbar Z} \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) \bigg[ \frac{\hbar e^{i (\hbar \omega + E_n - E_{n'}) t / \hbar} e^{\eta t}}{i (\hbar \omega + E_n - E_{n'}) + \hbar \eta} \bigg]_{-\infty}^0 \end{aligned}$$ |