From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- .../concept/equation-of-motion-theory/index.md | 55 +++++++++++----------- 1 file changed, 28 insertions(+), 27 deletions(-) (limited to 'source/know/concept/equation-of-motion-theory') diff --git a/source/know/concept/equation-of-motion-theory/index.md b/source/know/concept/equation-of-motion-theory/index.md index f62fb56..02ed856 100644 --- a/source/know/concept/equation-of-motion-theory/index.md +++ b/source/know/concept/equation-of-motion-theory/index.md @@ -13,9 +13,9 @@ is a method to calculate the time evolution of a system's properties using [Green's functions](/know/concept/greens-functions/). Starting from the definition of -the retarded single-particle Green's function $G_{\nu \nu'}^R(t, t')$, -we simply take the $t$-derivative -(we could do the same with the advanced function $G_{\nu \nu'}^A$): +the retarded single-particle Green's function $$G_{\nu \nu'}^R(t, t')$$, +we simply take the $$t$$-derivative +(we could do the same with the advanced function $$G_{\nu \nu'}^A$$): $$\begin{aligned} i \hbar \pdv{G^R_{\nu \nu'}(t, t')}{t} @@ -27,22 +27,22 @@ $$\begin{aligned} \end{aligned}$$ Where we have used that the derivative -of a [Heaviside step function](/know/concept/heaviside-step-function/) $\Theta$ -is a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$. +of a [Heaviside step function](/know/concept/heaviside-step-function/) $$\Theta$$ +is a [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta$$. Also, from the [second quantization](/know/concept/second-quantization/), -$\expval{\comm{\hat{c}_\nu(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}}$ -for $t = t'$ is zero when $\nu \neq \nu'$. +$$\expval{\comm{\hat{c}_\nu(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}}$$ +for $$t = t'$$ is zero when $$\nu \neq \nu'$$. Since we are in the [Heisenberg picture](/know/concept/heisenberg-picture/), -we know the equation of motion of $\hat{c}_\nu(t)$: +we know the equation of motion of $$\hat{c}_\nu(t)$$: $$\begin{aligned} \dv{\hat{c}_\nu(t)}{t} = \frac{i}{\hbar} \comm{\hat{H}_0(t)}{\hat{c}_\nu(t)} + \frac{i}{\hbar} \comm{\hat{H}_\mathrm{int}(t)}{\hat{c}_\nu(t)} \end{aligned}$$ -Where the single-particle part of the Hamiltonian $\hat{H}_0$ -and the interaction part $\hat{H}_\mathrm{int}$ +Where the single-particle part of the Hamiltonian $$\hat{H}_0$$ +and the interaction part $$\hat{H}_\mathrm{int}$$ are assumed to be time-independent in the Schrödinger picture. We thus get: @@ -52,8 +52,8 @@ $$\begin{aligned} \Expval{\Comm{\comm{\hat{H}_0}{\hat{c}_\nu} + \comm{\hat{H}_\mathrm{int}}{\hat{c}_\nu}}{\hat{c}_{\nu'}^\dagger}_{\mp}} \end{aligned}$$ -The most general form of $\hat{H}_0$, for any basis, -is as follows, where $u_{\nu' \nu''}$ are constants: +The most general form of $$\hat{H}_0$$, for any basis, +is as follows, where $$u_{\nu' \nu''}$$ are constants: $$\begin{aligned} \hat{H}_0 @@ -68,7 +68,7 @@ $$\begin{aligned} -Substituting this into $G_{\nu \nu'}^R$'s equation of motion, -we recognize another Green's function $G_{\nu'' \nu'}^R$: +Substituting this into $$G_{\nu \nu'}^R$$'s equation of motion, +we recognize another Green's function $$G_{\nu'' \nu'}^R$$: $$\begin{aligned} i \hbar \pdv{G^R_{\nu \nu'}}{t} @@ -132,9 +133,9 @@ $$\begin{aligned} } \end{aligned}$$ -Where $D_{\nu \nu'}^R$ represents a correction due to interactions $\hat{H}_\mathrm{int}$, +Where $$D_{\nu \nu'}^R$$ represents a correction due to interactions $$\hat{H}_\mathrm{int}$$, and also has the form of a retarded Green's function, -but with $\hat{c}_{\nu}$ replaced by $\comm{-\hat{H}_\mathrm{int}}{\hat{c}_\nu}$: +but with $$\hat{c}_{\nu}$$ replaced by $$\comm{-\hat{H}_\mathrm{int}}{\hat{c}_\nu}$$: $$\begin{aligned} \boxed{ @@ -143,19 +144,19 @@ $$\begin{aligned} } \end{aligned}$$ -Unfortunately, calculating $D_{\nu \nu'}^R$ -might still not be doable due to $\hat{H}_\mathrm{int}$. -The key idea of equation-of-motion theory is to either approximate $D_{\nu \nu'}^R$ now, -or to differentiate it again $i \hbar \idv{D_{\nu \nu'}^R}{t}$, +Unfortunately, calculating $$D_{\nu \nu'}^R$$ +might still not be doable due to $$\hat{H}_\mathrm{int}$$. +The key idea of equation-of-motion theory is to either approximate $$D_{\nu \nu'}^R$$ now, +or to differentiate it again $$i \hbar \idv{D_{\nu \nu'}^R}{t}$$, and try again for the resulting corrections, until a solvable equation is found. There is no guarantee that that will ever happen; if not, one of the corrections needs to be approximated. -For non-interacting particles $\hat{H}_\mathrm{int} = 0$, -so clearly $D_{\nu \nu'}^R$ trivially vanishes then. -Let us assume that $\hat{H}_0$ is also time-independent, -such that $G_{\nu'' \nu'}^R$ only depends on the difference $t - t'$: +For non-interacting particles $$\hat{H}_\mathrm{int} = 0$$, +so clearly $$D_{\nu \nu'}^R$$ trivially vanishes then. +Let us assume that $$\hat{H}_0$$ is also time-independent, +such that $$G_{\nu'' \nu'}^R$$ only depends on the difference $$t - t'$$: $$\begin{aligned} \sum_{\nu''} \Big( i \hbar \delta_{\nu \nu''} \pdv{}{t} - u_{\nu \nu''} \Big) G^R_{\nu'' \nu'}(t - t') @@ -163,14 +164,14 @@ $$\begin{aligned} \end{aligned}$$ We take the [Fourier transform](/know/concept/fourier-transform/) -$(t \!-\! t') \to (\omega + i \eta)$, where $\eta \to 0^+$ ensures convergence: +$$(t \!-\! t') \to (\omega + i \eta)$$, where $$\eta \to 0^+$$ ensures convergence: $$\begin{aligned} \sum_{\nu''} \Big( \hbar \delta_{\nu \nu''} (\omega + i \eta) - u_{\nu \nu''} \Big) G^R_{\nu'' \nu'}(\omega) = \delta_{\nu \nu'} \end{aligned}$$ -If we assume a diagonal basis $u_{\nu \nu''} = \varepsilon_\nu \delta_{\nu \nu''}$, +If we assume a diagonal basis $$u_{\nu \nu''} = \varepsilon_\nu \delta_{\nu \nu''}$$, this reduces to the following: $$\begin{aligned} -- cgit v1.2.3