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

---
 source/know/concept/dirac-notation/index.md | 16 ++++++++++------
 1 file changed, 10 insertions(+), 6 deletions(-)

(limited to 'source/know/concept/dirac-notation/index.md')

diff --git a/source/know/concept/dirac-notation/index.md b/source/know/concept/dirac-notation/index.md
index 2830a33..bbf31e5 100644
--- a/source/know/concept/dirac-notation/index.md
+++ b/source/know/concept/dirac-notation/index.md
@@ -27,7 +27,8 @@ that maps kets $$\ket{V}$$ to other kets $$\ket{V'}$$.
 Recall that by definition the Hilbert inner product must satisfy:
 
 $$\begin{aligned}
-    \inprod{V}{W} = \inprod{W}{V}^*
+    \inprod{V}{W}
+    = \inprod{W}{V}^*
 \end{aligned}$$
 
 So far, nothing has been said about the actual representation of bras or kets.
@@ -36,12 +37,14 @@ the corresponding bras are given by the kets' adjoints,
 i.e. their transpose conjugates:
 
 $$\begin{aligned}
-    \ket{V} =
+    \ket{V}
+    =
     \begin{bmatrix}
         v_1 \\ \vdots \\ v_N
     \end{bmatrix}
-    \quad \implies \quad
-    \bra{V} =
+    \qquad \implies \qquad
+    \bra{V}
+    =
     \begin{bmatrix}
         v_1^* & \cdots & v_N^*
     \end{bmatrix}
@@ -88,8 +91,9 @@ then the bras are *functionals* $$F[u(x)]$$
 that take an arbitrary function $$u(x)$$ as an argument and return a scalar:
 
 $$\begin{aligned}
-    \ket{f} = f(x)
-    \quad \implies \quad
+    \ket{f}
+    = f(x)
+    \qquad \implies \qquad
     \bra{f}
     = F[u(x)]
     = \int_a^b f^*(x) \: u(x) \dd{x}
-- 
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