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Diffstat (limited to 'content/know/concept/lehmann-representation/index.pdc')
| -rw-r--r-- | content/know/concept/lehmann-representation/index.pdc | 11 |
1 files changed, 5 insertions, 6 deletions
diff --git a/content/know/concept/lehmann-representation/index.pdc b/content/know/concept/lehmann-representation/index.pdc index f38f803..5808934 100644 --- a/content/know/concept/lehmann-representation/index.pdc +++ b/content/know/concept/lehmann-representation/index.pdc @@ -18,9 +18,8 @@ is an alternative way to write the [Green's functions](/know/concept/greens-func obtained by expanding in the many-particle eigenstates under the assumption of a time-independent Hamiltonian $\hat{H}$. -We start by writing out the -greater Green's function $G_{\nu \nu'}(t, t')$, -and then expanding its thermal expectation value $\expval{}$ +First, we write out the greater Green's function $G_{\nu \nu'}(t, t')$, +and then expand its expected value $\expval{}$ (at thermodynamic equilibrium) into a sum of many-particle basis states $\ket{n}$: $$\begin{aligned} @@ -29,9 +28,9 @@ $$\begin{aligned} &= - \frac{i}{\hbar Z} \sum_{n} \matrixel**{n}{\hat{c}_\nu(t) \hat{c}_{\nu'}^\dagger(t') e^{-\beta \hat{H}}}{n} \end{aligned}$$ -Where $\beta = 1 / (k_B T)$, and $Z$ is the partition function -(see [canonical ensemble](/know/concept/canonical-ensemble/)); -the operator $e^{\beta \hat{H}}$ gives the weight of each term at thermal equilibrium. +Where $\beta = 1 / (k_B T)$, and $Z$ is the grand partition function +(see [grand canonical ensemble](/know/concept/grand-canonical-ensemble/)); +the operator $e^{\beta \hat{H}}$ gives the weight of each term at equilibrium. Since $\ket{n}$ is an eigenstate of $\hat{H}$ with energy $E_n$, this gives us a factor of $e^{\beta E_n}$. Furthermore, we are in the [Heisenberg picture](/know/concept/heisenberg-picture/), |