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/rotating-wave-approximation/index.md   | 34 +++++++++++-----------
 1 file changed, 17 insertions(+), 17 deletions(-)

(limited to 'source/know/concept/rotating-wave-approximation')

diff --git a/source/know/concept/rotating-wave-approximation/index.md b/source/know/concept/rotating-wave-approximation/index.md
index 7066f37..63efc9c 100644
--- a/source/know/concept/rotating-wave-approximation/index.md
+++ b/source/know/concept/rotating-wave-approximation/index.md
@@ -10,7 +10,7 @@ categories:
 layout: "concept"
 ---
 
-Consider the following periodic perturbation $\hat{H}_1$ to a quantum system,
+Consider the following periodic perturbation $$\hat{H}_1$$ to a quantum system,
 which represents e.g. an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
 in the [electric dipole approximation](/know/concept/electric-dipole-approximation/):
 
@@ -20,16 +20,16 @@ $$\begin{aligned}
     = \frac{\hat{V}}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big)
 \end{aligned}$$
 
-Where $\hat{V}$ is some operator, and we assume that $\omega$
-is fairly close to a resonance frequency $\omega_0$
-of the system that is getting perturbed by $\hat{H}_1$.
+Where $$\hat{V}$$ is some operator, and we assume that $$\omega$$
+is fairly close to a resonance frequency $$\omega_0$$
+of the system that is getting perturbed by $$\hat{H}_1$$.
 
 As an example, consider a two-level system
-consisting of states $\Ket{g}$ and $\Ket{e}$,
-with a resonance frequency $\omega_0 = (E_e \!-\! E_g) / \hbar$.
+consisting of states $$\Ket{g}$$ and $$\Ket{e}$$,
+with a resonance frequency $$\omega_0 = (E_e \!-\! E_g) / \hbar$$.
 From the derivation of
 [time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/),
-we know that the state $\Ket{\Psi} = c_g \Ket{g} + c_e \Ket{e}$ evolves as:
+we know that the state $$\Ket{\Psi} = c_g \Ket{g} + c_e \Ket{e}$$ evolves as:
 
 $$\begin{aligned}
     i \hbar \dv{c_g}{t}
@@ -39,7 +39,7 @@ $$\begin{aligned}
     &= \matrixel{e}{\hat{H}_1(t)}{g} \: c_g(t) \: e^{i \omega_0 t} + \matrixel{e}{\hat{H}_1(t)}{e} \: c_e(t)
 \end{aligned}$$
 
-Typically, $\hat{V}$ has odd spatial parity, in which case
+Typically, $$\hat{V}$$ has odd spatial parity, in which case
 [Laporte's selection rule](/know/concept/selection-rules/)
 reduces this to:
 
@@ -51,8 +51,8 @@ $$\begin{aligned}
     &= \frac{1}{i \hbar} \matrixel{e}{\hat{H}_1}{g} \: c_g \: e^{i \omega_0 t}
 \end{aligned}$$
 
-We now insert the general $\hat{H}_1$ defined above,
-and define $V_{eg} \equiv \matrixel{e}{\hat{V}}{g}$ to get:
+We now insert the general $$\hat{H}_1$$ defined above,
+and define $$V_{eg} \equiv \matrixel{e}{\hat{V}}{g}$$ to get:
 
 $$\begin{aligned}
     \dv{c_g}{t}
@@ -65,14 +65,14 @@ $$\begin{aligned}
 \end{aligned}$$
 
 At last, here we make the **rotating wave approximation**:
-since $\omega$ is assumed to be close to $\omega_0$,
-we argue that $\omega \!+\! \omega_0$ is so much larger than $\omega \!-\! \omega_0$
+since $$\omega$$ is assumed to be close to $$\omega_0$$,
+we argue that $$\omega \!+\! \omega_0$$ is so much larger than $$\omega \!-\! \omega_0$$
 that those oscillations turn out negligible
 if the system is observed over a reasonable time interval.
 
 Specifically, since both exponentials have the same weight,
-the fast ($\omega \!+\! \omega_0$) oscillations
-have a tiny amplitude compared to the slow ($\omega \!-\! \omega_0$) ones.
+the fast ($$\omega \!+\! \omega_0$$) oscillations
+have a tiny amplitude compared to the slow ($$\omega \!-\! \omega_0$$) ones.
 Furthermore, since they average out to zero over most realistic time intervals,
 the fast terms can be dropped, leaving:
 
@@ -103,11 +103,11 @@ $$\begin{aligned}
 This approximation's name is a bit confusing:
 the idea is that going from the Schrödinger to
 the [interaction picture](/know/concept/interaction-picture/)
-has the effect of removing the exponentials of $\omega_0$ from the above equations,
-i.e. multiplying them by $e^{i \omega_0 t}$ and $e^{- i \omega_0 t}$
+has the effect of removing the exponentials of $$\omega_0$$ from the above equations,
+i.e. multiplying them by $$e^{i \omega_0 t}$$ and $$e^{- i \omega_0 t}$$
 respectively, which can be regarded as a rotation.
 
-Relative to this rotation, when we split the wave $\cos(\omega t)$
+Relative to this rotation, when we split the wave $$\cos(\omega t)$$
 into two exponentials, one co-rotates, and the other counter-rotates.
 We keep only the co-rotating waves, hence the name.
 
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