From 7cb1bd307e6d3f1279731bebadbc6f994ed1105a Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Fri, 3 Jul 2026 17:18:50 +0200
Subject: Improve knowledge base

---
 source/know/concept/larmor-precession/index.md | 24 ++++++++++++------------
 1 file changed, 12 insertions(+), 12 deletions(-)

(limited to 'source/know/concept/larmor-precession')

diff --git a/source/know/concept/larmor-precession/index.md b/source/know/concept/larmor-precession/index.md
index 774af7b..601dae7 100644
--- a/source/know/concept/larmor-precession/index.md
+++ b/source/know/concept/larmor-precession/index.md
@@ -36,8 +36,8 @@ and the exponentials are "twiddle factors":
 
 $$\begin{aligned}
     \Ket{\chi(t)}
-    = a \exp(- i E_{\downarrow} t / \hbar) \: \Ket{\downarrow}
-    \:+\: b \exp(- i E_{\uparrow} t / \hbar) \: \Ket{\uparrow}
+    = a e^{- i E_{\downarrow} t / \hbar} \Ket{\downarrow}
+    \:+\: b e^{- i E_{\uparrow} t / \hbar} \Ket{\uparrow}
 \end{aligned}$$
 
 For our purposes, we can safely assume that $$a$$ and $$b$$ are real,
@@ -45,8 +45,8 @@ and then say that there exists an angle $$\theta$$
 satisfying $$a = \sin(\theta / 2)$$ and $$b = \cos(\theta / 2)$$, such that:
 
 $$\begin{aligned}
-    \Ket{\chi(t)} = \sin(\theta / 2) \exp(- i E_{\downarrow} t / \hbar) \: \Ket{\downarrow}
-    \:+\: \cos(\theta / 2) \exp(- i E_{\uparrow} t / \hbar) \: \Ket{\uparrow}
+    \Ket{\chi(t)} = \sin(\theta / 2) \: e^{- i E_{\downarrow} t / \hbar} \Ket{\downarrow}
+    \:+\: \cos(\theta / 2) \: e^{- i E_{\uparrow} t / \hbar} \Ket{\uparrow}
 \end{aligned}$$
 
 Now, we find the expectation values of the spin operators
@@ -56,23 +56,23 @@ The first is:
 $$\begin{aligned}
     \matrixel{\chi}{\hat{S}_x}{\chi}
     &= \frac{\hbar}{2}
-    \begin{bmatrix} a \exp(i E_{\downarrow} t / \hbar) \\ b \exp(i E_{\uparrow} t / \hbar) \end{bmatrix}^{\mathrm{T}}
+    \begin{bmatrix} a e^{i E_{\downarrow} t / \hbar} \\ b e^{i E_{\uparrow} t / \hbar} \end{bmatrix}^{\mathrm{T}}
     \cdot
     \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
     \cdot
-    \begin{bmatrix} a \exp(- i E_{\downarrow} t / \hbar) \\ b \exp(- i E_{\uparrow} t / \hbar) \end{bmatrix}
+    \begin{bmatrix} a e^{- i E_{\downarrow} t / \hbar} \\ b e^{- i E_{\uparrow} t / \hbar} \end{bmatrix}
     \\
     &= \frac{\hbar}{2}
-    \begin{bmatrix} a \exp(i E_{\downarrow} t / \hbar) \\ b \exp(i E_{\uparrow} t / \hbar) \end{bmatrix}^{\mathrm{T}}
+    \begin{bmatrix} a e^{i E_{\downarrow} t / \hbar} \\ b e^{i E_{\uparrow} t / \hbar} \end{bmatrix}^{\mathrm{T}}
     \cdot
-    \begin{bmatrix} b \exp(- i E_{\uparrow} t / \hbar) \\ a \exp(- i E_{\downarrow} t / \hbar) \end{bmatrix}
+    \begin{bmatrix} b e^{- i E_{\uparrow} t / \hbar} \\ a e^{- i E_{\downarrow} t / \hbar} \end{bmatrix}
     \\
-    &= \frac{\hbar}{2} \Big( a b \exp(i (E_{\downarrow} \!-\! E_{\uparrow}) t / \hbar)
-    + b a \exp(i (E_{\uparrow} \!-\! E_{\downarrow}) t / \hbar) \Big)
+    &= \frac{\hbar}{2} \Big( a b e^{i (E_{\downarrow} \!-\! E_{\uparrow}) t / \hbar}
+    + b a e^{i (E_{\uparrow} \!-\! E_{\downarrow}) t / \hbar} \Big)
     \\
-    &= \frac{\hbar}{2} \cos(\theta/2) \sin(\theta/2) \Big( \exp(i \gamma B t) + \exp(- i \gamma B t) \Big)
+    &= \frac{\hbar}{2} \cos(\theta/2) \sin(\theta/2) \Big( e^{i \gamma B t} + e^{- i \gamma B t} \Big)
     \\
-    &= \frac{\hbar}{2} \cos(\gamma B t) \Big( \cos(\theta/2) \sin(\theta/2) + \cos(\theta/2) \sin(\theta/2) \Big)
+    &= \frac{\hbar}{2} \cos(\gamma B t) \cdot 2 \cos(\theta/2) \sin(\theta/2)
     \\
     &= \frac{\hbar}{2} \sin(\theta) \cos(\gamma B t)
 \end{aligned}$$
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