From 075683cdf4588fe16f41d9f7b46b9720b42b2553 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Wed, 17 Jul 2024 10:01:43 +0200
Subject: Improve knowledge base

---
 .../concept/rotating-wave-approximation/index.md   | 32 +++++++++-------------
 1 file changed, 13 insertions(+), 19 deletions(-)

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

diff --git a/source/know/concept/rotating-wave-approximation/index.md b/source/know/concept/rotating-wave-approximation/index.md
index edb13e9..54e0675 100644
--- a/source/know/concept/rotating-wave-approximation/index.md
+++ b/source/know/concept/rotating-wave-approximation/index.md
@@ -16,7 +16,7 @@ in the [electric dipole approximation](/know/concept/electric-dipole-approximati
 
 $$\begin{aligned}
     \hat{H}_1(t)
-    = \hat{V} \cos(\omega t)
+    \equiv \hat{V} \cos(\omega t)
     = \frac{\hat{V}}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big)
 \end{aligned}$$
 
@@ -26,17 +26,17 @@ 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 $$\omega_0 = (E_e \!-\! E_g) / \hbar$$.
+with a resonance frequency $$\omega_0 \equiv (E_e \!-\! E_g) / \hbar$$.
 From the [amplitude rate equations](/know/concept/amplitude-rate-equations/),
 we know that the general superposition state
 $$\ket{\Psi} = c_g \ket{g} + c_e \ket{e}$$ evolves as:
 
 $$\begin{aligned}
     i \hbar \dv{c_g}{t}
-    &= \matrixel{g}{\hat{H}_1(t)}{g} \: c_g(t) + \matrixel{g}{\hat{H}_1(t)}{e} \: c_e(t) \: e^{- i \omega_0 t}
+    &= \matrixel{g}{\hat{H}_1(t)}{g} c_g(t) + \matrixel{g}{\hat{H}_1(t)}{e} c_e(t) \: e^{- i \omega_0 t}
     \\
     i \hbar \dv{c_e}{t}
-    &= \matrixel{e}{\hat{H}_1(t)}{g} \: c_g(t) \: e^{i \omega_0 t} + \matrixel{e}{\hat{H}_1(t)}{e} \: c_e(t)
+    &= \matrixel{e}{\hat{H}_1(t)}{g} c_g(t) \: e^{i \omega_0 t} + \matrixel{e}{\hat{H}_1(t)}{e} c_e(t)
 \end{aligned}$$
 
 Typically, $$\hat{V}$$ has odd spatial parity, in which case
@@ -66,15 +66,10 @@ $$\begin{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 $$\omega \!-\! \omega_0$$
-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.
-Furthermore, since they average out to zero over most realistic time intervals,
-the fast terms can be dropped, leaving:
+we argue that $$\omega \!+\! \omega_0$$ is much larger than $$\omega \!-\! \omega_0$$,
+so that those oscillations average out to zero
+when the system is observed over a realistic time interval.
+Hence we drop those terms:
 
 $$\begin{aligned}
     \boxed{
@@ -103,13 +98,12 @@ $$\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 $$e^{- i \omega_0 t}$$
+involves removing the exponentials of $$\omega_0$$ from the above equations,
+i.e. they are multiplied 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 $$\cos(\omega t)$$
-into two exponentials, one co-rotates, and the other counter-rotates.
-We keep only the co-rotating waves, hence the name.
+When we split the wave $$\cos(\omega t)$$ into two exponentials,
+one co-rotates relative to this rotation, and the other counter-rotates.
+We keep only the co-rotating terms, hence the name.
 
 The rotating wave approximation is usually used in the context
 of the two-level quantum system for light-matter interactions,
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