From cc391ce3b9867d88d124e147931d33be34e756fc Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Mon, 14 Sep 2026 18:11:38 +0200
Subject: Improve knowledge base

---
 source/know/concept/rabi-oscillation/index.md | 37 ++++++++++++++-------------
 1 file changed, 19 insertions(+), 18 deletions(-)

(limited to 'source/know/concept/rabi-oscillation/index.md')

diff --git a/source/know/concept/rabi-oscillation/index.md b/source/know/concept/rabi-oscillation/index.md
index 9497ad4..a2cea23 100644
--- a/source/know/concept/rabi-oscillation/index.md
+++ b/source/know/concept/rabi-oscillation/index.md
@@ -14,12 +14,12 @@ In quantum mechanics, we know from the
 [amplitude rate equations](/know/concept/amplitude-rate-equations/)
 that a time-dependent term $$\hat{H}_1$$ in the Hamiltonian
 affects the state as follows,
-where $$c_n(t)$$ are the coefficients of the linear combination
+where $$c_n(t)$$ are coefficients of a linear combination
 of unperturbed basis states $$\ket{n} e^{-i E_n t / \hbar}$$:
 
 $$\begin{aligned}
     i \hbar \dv{c_m}{t}
-    = \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1}{n} e^{i \omega_{mn} t}
+    = \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1(t)}{n} e^{i \omega_{mn} t}
 \end{aligned}$$
 
 Where $$\omega_{mn} \equiv (E_m \!-\! E_n) / \hbar$$
@@ -36,7 +36,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Where $$\omega_0 \equiv \omega_{ba}$$ is positive.
-We assume that $$\hat{H}_1$$ has odd spatial parity,
+It is realistic to assume that $$\hat{H}_1$$ has odd spatial parity,
 in which case [Laporte's selection rule](/know/concept/selection-rules/)
 states that the diagonal matrix elements vanish, leaving:
 
@@ -49,16 +49,16 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We now choose $$\hat{H}_1$$ to be as follows,
-sinusoidally oscillating with a spatially odd $$V(\vec{r})$$:
+sinusoidally oscillating with a spatially odd $$\hat{V}(\vec{r})$$:
 
 $$\begin{aligned}
     \hat{H}_1(t)
-    = V \cos(\omega t)
-    = \frac{V}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big)
+    = \hat{V} \cos(\omega t)
+    = \frac{\hat{V}}{2} \Big( e^{i \omega t} + e^{-i \omega t} \Big)
 \end{aligned}$$
 
 We insert this into the equations for $$c_a$$ and $$c_b$$,
-and define $$V_{ab} \equiv \matrixel{a}{V}{b}$$, leading us to:
+and define $$V_{ab} \equiv \matrixel{a}{\hat{V}}{b}$$, leading us to:
 
 $$\begin{aligned}
     \dv{c_a}{t}
@@ -124,7 +124,7 @@ which are found to be:
 $$\begin{aligned}
     \lambda_1
     = i \frac{\omega - \omega_0 + \tilde{\Omega}}{2}
-    \qquad \quad
+    \qquad \qquad
     \lambda_2
     = i \frac{\omega - \omega_0 - \tilde{\Omega}}{2}
 \end{aligned}$$
@@ -140,7 +140,7 @@ $$\begin{aligned}
 
 So that the general solution $$c_a(t)$$ is as follows,
 where $$A$$ and $$B$$ are arbitrary constants,
-to be determined from initial conditions (and normalization):
+determined by the initial conditions and normalization:
 
 $$\begin{aligned}
     \boxed{
@@ -154,7 +154,7 @@ from the coupled equation we started at,
 or, if we only care about the probability density $$|c_a|^2$$,
 we can use $$|c_b|^2 = 1 - |c_a|^2$$.
 For example, if $$A = 0$$ and $$B = 1$$,
-we get the following probabilities
+we get the following probabilities:
 
 $$\begin{aligned}
     |c_a(t)|^2
@@ -168,9 +168,8 @@ $$\begin{aligned}
 
 Note that the period was halved by squaring.
 This periodic "flopping" of the particle between $$\ket{a}$$ and $$\ket{b}$$
-is known as **Rabi oscillation**, **Rabi flopping** or the **Rabi cycle**.
-This is a more accurate treatment
-of the flopping found from first-order
+is called **Rabi oscillation**, **Rabi flopping** or the **Rabi cycle**.
+This is a more accurate treatment of the flopping found from first-order
 [time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/).
 
 The name **generalized Rabi frequency** suggests
@@ -183,11 +182,13 @@ $$\begin{aligned}
     \equiv \frac{V_{ba}}{\hbar}
 \end{aligned}$$
 
-Some authors use $$|V_{ba}|$$ instead,
-but not doing that lets us use $$\Omega$$ as a nice abbreviation.
-As an example, Rabi oscillation arises
-in the [electric dipole approximation](/know/concept/electric-dipole-approximation/),
-where $$\hat{H}_1$$ is:
+Some authors write $$|V_{ba}|$$ instead,
+but not doing so lets us use $$\Omega$$ as a nice abbreviation.
+For example, Rabi oscillation arises for electrons
+in an oscillating electric field (e.g. a light wave),
+in which case $$\hat{H}_1$$ is as follows according to
+the [electric dipole approximation](/know/concept/electric-dipole-approximation/),
+with charge $$q < 0$$:
 
 $$\begin{aligned}
     \hat{H}_1(t)
-- 
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