diff options
Diffstat (limited to 'source/know/concept/matsubara-greens-function/index.md')
| -rw-r--r-- | source/know/concept/matsubara-greens-function/index.md | 10 |
1 files changed, 5 insertions, 5 deletions
diff --git a/source/know/concept/matsubara-greens-function/index.md b/source/know/concept/matsubara-greens-function/index.md index 5e753db..6f60edf 100644 --- a/source/know/concept/matsubara-greens-function/index.md +++ b/source/know/concept/matsubara-greens-function/index.md @@ -64,7 +64,7 @@ $$\begin{aligned} With $$-$$ for bosons, and $$+$$ for fermions, due to the time-ordered product for $$\tau > \tau'$$. -On this domain $$[-\hbar \beta, \hbar \beta]$$, +On this domain $$]\!-\!\hbar \beta, \hbar \beta[$$, the Matsubara Green's function $$C_{AB}$$ obeys a useful shift relation: it is $$\hbar \beta$$-periodic for bosons, @@ -133,7 +133,7 @@ $$\begin{aligned} {% include proof/end.html id="proof-period" %} -Due to this limited domain $$\tau \in [-\hbar \beta, \hbar \beta]$$, +Due to this limited domain $$\tau \in \:]\!-\!\hbar \beta, \hbar \beta[$$, the [Fourier transform](/know/concept/fourier-transform/) of $$C_{AB}(\tau)$$ consists of discrete frequencies $$k_n \equiv n \pi / (\hbar \beta)$$. @@ -288,7 +288,7 @@ $$\begin{aligned} \matrixel{n'}{\hat{B}}{n} e^{(E_n - E_{n'})(\tau - \tau') / \hbar} \end{aligned}$$ -We take the Fourier transform by integrating over $$[0, \hbar \beta]$$: +We take the Fourier transform by integrating over $$]0, \hbar \beta[$$: $$\begin{aligned} C_{AB}(i \omega_m) @@ -324,7 +324,7 @@ $$\begin{aligned} \end{aligned}$$ Since $$\tau \!-\! \tau' < 0$$ this time, -we take the Fourier transform over $$[-\hbar \beta, 0]$$: +we take the Fourier transform over $$]\!-\!\hbar \beta, 0[$$: $$\begin{aligned} C_{AB}(i \omega_m) @@ -341,7 +341,7 @@ $$\begin{aligned} \Big( e^{-\beta E_n} - e^{-i \hbar \omega_m \beta} e^{-\beta E_{n'}} \Big) \\ &= \mp \frac{1}{Z} \sum_{n n'} \frac{\matrixel{n}{\hat{B}}{n'} \matrixel{n'}{\hat{A}}{n}}{i \hbar \omega_m - E_n + E_{n'}} - \Big( e^{- \beta E_n} \pm e^{-\beta E_{n'}} \Big) + \Big( e^{- \beta E_n} \mp e^{-\beta E_{n'}} \Big) \\ &= \frac{1}{Z} \sum_{n n'} \frac{\matrixel{n}{\hat{B}}{n'} \matrixel{n'}{\hat{A}}{n}}{i \hbar \omega_m - E_n + E_{n'}} \Big( e^{- \beta E_{n'}} \mp e^{-\beta E_n} \Big) |