From 9d9693af6fb94ef4404a3c2399cb38842e5ca822 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Fri, 12 May 2023 21:19:19 +0200
Subject: Improve knowledge base

---
 source/know/concept/dirac-notation/index.md | 76 +++++++++++++++--------------
 1 file changed, 40 insertions(+), 36 deletions(-)

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

diff --git a/source/know/concept/dirac-notation/index.md b/source/know/concept/dirac-notation/index.md
index 46cc325..2830a33 100644
--- a/source/know/concept/dirac-notation/index.md
+++ b/source/know/concept/dirac-notation/index.md
@@ -8,47 +8,49 @@ categories:
 layout: "concept"
 ---
 
-**Dirac notation** is a notation to do calculations in a [Hilbert space](/know/concept/hilbert-space/)
-without needing to worry about the space's representation. It is
-basically the *lingua franca* of quantum mechanics.
-
-In Dirac notation there are **kets** $$\Ket{V}$$ from the Hilbert space
-$$\mathbb{H}$$ and **bras** $$\Bra{V}$$ from a dual $$\mathbb{H}'$$ of the
-former. Crucially, the bras and kets are from different Hilbert spaces
-and therefore cannot be added, but every bra has a corresponding ket and
-vice versa.
+**Dirac notation** enables us to do calculations
+in a general [Hilbert space](/know/concept/hilbert-space/)
+without needing to worry about the space's representation.
+It is the *lingua franca* of quantum mechanics.
+
+In Dirac notation there are
+**kets** $$\ket{V}$$ from the Hilbert space $$\mathbb{H}$$
+and **bras** $$\bra{V}$$ from its dual space $$\mathbb{H}'$$.
+Crucially, the bras and kets are from different Hilbert spaces
+and therefore cannot be added,
+but every bra has a corresponding ket and vice versa.
 
 Bras and kets can be combined in two ways: the **inner product**
-$$\Inprod{V}{W}$$, which returns a scalar, and the **outer product**
-$$\Ket{V} \Bra{W}$$, which returns a mapping $$\hat{L}$$ from kets $$\Ket{V}$$
-to other kets $$\Ket{V'}$$, i.e. a linear operator. Recall that the
-Hilbert inner product must satisfy:
+$$\inprod{V}{W}$$, which returns a scalar, and the **outer product**
+$$\ket{V} \bra{W}$$, which returns a linear operator
+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. If we represent kets as $$N$$-dimensional columns vectors, the
-corresponding bras are given by the kets' adjoints, i.e. their transpose
-conjugates:
+So far, nothing has been said about the actual representation of bras or kets.
+If we represent kets as $$N$$-dimensional columns vectors,
+the corresponding bras are given by the kets' adjoints,
+i.e. their transpose conjugates:
 
 $$\begin{aligned}
-    \Ket{V} =
+    \ket{V} =
     \begin{bmatrix}
         v_1 \\ \vdots \\ v_N
     \end{bmatrix}
     \quad \implies \quad
-    \Bra{V} =
+    \bra{V} =
     \begin{bmatrix}
         v_1^* & \cdots & v_N^*
     \end{bmatrix}
 \end{aligned}$$
 
-The inner product $$\Inprod{V}{W}$$ is then just the familiar dot product $$V \cdot W$$:
+The inner product $$\inprod{V}{W}$$ is then just the familiar dot product $$V \cdot W$$:
 
 $$\begin{gathered}
-    \Inprod{V}{W}
+    \inprod{V}{W}
     =
     \begin{bmatrix}
         v_1^* & \cdots & v_N^*
@@ -60,10 +62,11 @@ $$\begin{gathered}
     = v_1^* w_1 + ... + v_N^* w_N
 \end{gathered}$$
 
-Meanwhile, the outer product $$\Ket{V} \Bra{W}$$ creates an $$N \cross N$$ matrix:
+Meanwhile, the outer product $$\ket{V} \bra{W}$$ creates an $$N \cross N$$ matrix,
+which can be thought of as applying an operation to any vector it multiplies:
 
 $$\begin{gathered}
-    \Ket{V} \Bra{W}
+    \ket{V} \bra{W}
     =
     \begin{bmatrix}
         v_1 \\ \vdots \\ v_N
@@ -80,15 +83,14 @@ $$\begin{gathered}
     \end{bmatrix}
 \end{gathered}$$
 
-If the kets are instead represented by functions $$f(x)$$ of
-$$x \in [a, b]$$, then the bras represent *functionals* $$F[u(x)]$$ which
-take an unknown function $$u(x)$$ as an argument and turn it into a scalar
-using integration:
+If the kets are instead represented by continuous functions $$f(x)$$ of $$x \in [a, b]$$,
+then the bras are *functionals* $$F[u(x)]$$
+that take an arbitrary function $$u(x)$$ as an argument and return a scalar:
 
 $$\begin{aligned}
-    \Ket{f} = f(x)
+    \ket{f} = f(x)
     \quad \implies \quad
-    \Bra{f}
+    \bra{f}
     = F[u(x)]
     = \int_a^b f^*(x) \: u(x) \dd{x}
 \end{aligned}$$
@@ -96,23 +98,25 @@ $$\begin{aligned}
 Consequently, the inner product is simply the following familiar integral:
 
 $$\begin{gathered}
-    \Inprod{f}{g}
+    \inprod{f}{g}
     = F[g(x)]
     = \int_a^b f^*(x) \: g(x) \dd{x}
 \end{gathered}$$
 
-However, the outer product becomes something rather abstract:
+However, the outer product is then rather abstract:
+a continuous analogue of a matrix:
 
 $$\begin{gathered}
-    \Ket{f} \Bra{g}
+    \ket{f} \bra{g}
     = f(x) \: G[u(x)]
     = f(x) \int_a^b g^*(\xi) \: u(\xi) \dd{\xi}
 \end{gathered}$$
 
-This result makes more sense if we surround it by a bra and a ket:
+This maybe makes more sense if we surround it
+by a bra $$\bra{u}$$ and a ket $$\ket{w}$$ and rearrange:
 
 $$\begin{aligned}
-    \Bra{u} \!\Big(\!\Ket{f} \Bra{g}\!\Big)\! \Ket{w}
+    \bra{u} \!\Big(\!\ket{f} \bra{g}\!\Big)\! \ket{w}
     &= U\big[f(x) \: G[w(x)]\big]
     = U\Big[ f(x) \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big]
     \\
@@ -120,7 +124,7 @@ $$\begin{aligned}
     \\
     &= \Big( \int_a^b u^*(x) \: f(x) \dd{x} \Big) \Big( \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big)
     \\
-    &= \Inprod{u}{f} \Inprod{g}{w}
+    &= \inprod{u}{f} \inprod{g}{w}
 \end{aligned}$$
 
 
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