From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Fri, 14 Oct 2022 23:25:28 +0200
Subject: Initial commit after migration from Hugo
---
.../concept/equation-of-motion-theory/index.md | 197 +++++++++++++++++++++
1 file changed, 197 insertions(+)
create mode 100644 source/know/concept/equation-of-motion-theory/index.md
(limited to 'source/know/concept/equation-of-motion-theory/index.md')
diff --git a/source/know/concept/equation-of-motion-theory/index.md b/source/know/concept/equation-of-motion-theory/index.md
new file mode 100644
index 0000000..81d4d46
--- /dev/null
+++ b/source/know/concept/equation-of-motion-theory/index.md
@@ -0,0 +1,197 @@
+---
+title: "Equation-of-motion theory"
+date: 2021-11-08
+categories:
+- Physics
+- Quantum mechanics
+layout: "concept"
+---
+
+In many-body quantum theory, **equation-of-motion theory**
+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 $G_{\nu \nu'}^A$):
+
+$$\begin{aligned}
+ i \hbar \pdv{G^R_{\nu \nu'}(t, t')}{t}
+ &= \pdv{\Theta(t \!-\! t')}{t} \Expval{\comm{\hat{c}_\nu(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}}
+ + \Theta(t \!-\! t') \pdv{}{t}\Expval{\comm{\hat{c}_\nu(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}}
+ \\
+ &= \delta(t \!-\! t') \Expval{\comm{\hat{c}_\nu(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}}
+ + \Theta(t \!-\! t') \Expval{\Comm{\dv{\hat{c}_\nu(t)}{t}}{\hat{c}_{\nu'}^\dagger(t)}_{\mp}}
+\end{aligned}$$
+
+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/) $\delta$.
+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 $\nu \neq \nu'$.
+
+Since we are in the [Heisenberg picture](/know/concept/heisenberg-picture/),
+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 $\hat{H}_\mathrm{int}$
+are assumed to be time-independent in the Schrödinger picture.
+We thus get:
+
+$$\begin{aligned}
+ i \hbar \pdv{G^R_{\nu \nu'}}{t}
+ &= \delta_{\nu \nu'} \delta(t \!-\! t')+ \frac{i}{\hbar} \Theta(t \!-\! t')
+ \Expval{\Comm{\comm{\hat{H}_0}{\hat{c}_\nu} + \comm{\hat{H}_\mathrm{int}}{\hat{c}_\nu}}{\hat{c}_{\nu'}^\dagger}_{\mp}}
+\end{aligned}$$
+
+The most general form of $\hat{H}_0$, for any basis,
+is as follows, where $u_{\nu' \nu''}$ are constants:
+
+$$\begin{aligned}
+ \hat{H}_0
+ = \sum_{\nu' \nu''} u_{\nu' \nu''} \hat{c}_{\nu'}^\dagger \hat{c}_{\nu''}
+ \quad \implies \quad
+ \comm{\hat{H}_0}{\hat{c}_\nu}
+ = - \sum_{\nu''} u_{\nu \nu''} \hat{c}_{\nu''}
+\end{aligned}$$
+
+