From 073a0fb3b666bcb7ee49757ac8724867a9efdce1 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 21 Feb 2021 15:34:10 +0100
Subject: Add "Dirac delta function"

---
 content/know/category/index.md                    |  6 ++
 content/know/category/mathematics.md              | 10 +++
 content/know/category/physics.md                  | 27 +++++++
 content/know/concept/index.md                     |  1 +
 latex/know/concept/dirac-delta-function/source.md | 99 +++++++++++++++++++++++
 5 files changed, 143 insertions(+)
 create mode 100644 content/know/category/mathematics.md
 create mode 100644 content/know/category/physics.md
 create mode 100644 latex/know/concept/dirac-delta-function/source.md

diff --git a/content/know/category/index.md b/content/know/category/index.md
index b8f0456..ba4936c 100644
--- a/content/know/category/index.md
+++ b/content/know/category/index.md
@@ -9,5 +9,11 @@ Alphabetical list of categories in this knowledge base.
 Note that these categories might overlap,
 i.e. share certain concepts.
 
+## M
+* [Mathematics](/know/category/mathematics/)
+
+## P
+* [Physics](/know/category/physics/)
+
 ## Q
 * [Quantum mechanics](/know/category/quantum-mechanics/)
diff --git a/content/know/category/mathematics.md b/content/know/category/mathematics.md
new file mode 100644
index 0000000..e873d90
--- /dev/null
+++ b/content/know/category/mathematics.md
@@ -0,0 +1,10 @@
++++
+title = "Category: mathematics"
++++
+
+#
+
+Alphabetical list of concepts in this category.
+
+## D
+* [Dirac delta function](/know/concept/dirac-delta-function/)
diff --git a/content/know/category/physics.md b/content/know/category/physics.md
new file mode 100644
index 0000000..8bcba4d
--- /dev/null
+++ b/content/know/category/physics.md
@@ -0,0 +1,27 @@
++++
+title = "Category: physics"
++++
+
+#
+
+Alphabetical list of concepts in this category.
+
+## B
+* [Bloch's theorem](/know/concept/blochs-theorem/)
+
+## D
+* [Dirac delta function](/know/concept/dirac-delta-function/)
+* [Dirac notation](/know/concept/dirac-notation/)
+
+## P
+* [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/)
+* [Probability current](/know/concept/probability-current/)
+
+## S
+* [Slater determinant](/know/concept/slater-determinant/)
+
+## T
+* [Time-independent perturbation theory](/know/concept/time-independent-perturbation-theory/)
+
+## W
+* [Wentzel-Kramers-Brillouin approximation](/know/concept/wentzel-kramers-brillouin-approximation/)
diff --git a/content/know/concept/index.md b/content/know/concept/index.md
index 7d9c183..697c622 100644
--- a/content/know/concept/index.md
+++ b/content/know/concept/index.md
@@ -10,6 +10,7 @@ Alphabetical list of concepts in this knowledge base.
 * [Bloch's theorem](/know/concept/blochs-theorem/)
 
 ## D
+* [Dirac delta function](/know/concept/dirac-delta-function/)
 * [Dirac notation](/know/concept/dirac-notation/)
 
 ## P
diff --git a/latex/know/concept/dirac-delta-function/source.md b/latex/know/concept/dirac-delta-function/source.md
new file mode 100644
index 0000000..478efb4
--- /dev/null
+++ b/latex/know/concept/dirac-delta-function/source.md
@@ -0,0 +1,99 @@
+% Dirac delta function
+
+
+# Dirac delta function
+
+The **Dirac delta function** $\delta(x)$, often just called the **delta function**,
+is an infinitely narrow discontinuous "spike" at $x = 0$ whose area is
+defined to be 1:
+
+$$\begin{aligned}
+    \boxed{
+        \delta(x) =
+        \begin{cases}
+            +\infty & \mathrm{if}\: x = 0 \\
+            0 & \mathrm{if}\: x \neq 0
+        \end{cases}
+        \quad \mathrm{and} \quad
+        \int_{-\varepsilon}^\varepsilon \delta(x) \dd{x} = 1
+    }
+\end{aligned}$$
+
+It is sometimes also called the **sampling function**, due to its most
+important property: the so-called **sampling property**:
+
+$$\begin{aligned}
+    \boxed{
+        \int f(x) \: \delta(x - x_0) \: dx = \int f(x) \: \delta(x_0 - x) \: dx = f(x_0)
+    }
+\end{aligned}$$
+
+$\delta(x)$ is thus an effective weapon against integrals. This may not seem very
+useful due to its "unnatural" definition, but in fact it appears as the
+limit of several reasonable functions:
+
+$$\begin{aligned}
+    \delta(x)
+    = \lim_{n \to +\infty} \!\Big\{ \frac{n}{\sqrt{\pi}} \exp(- n^2 x^2) \Big\}
+    = \lim_{n \to +\infty} \!\Big\{ \frac{n}{\pi} \frac{1}{1 + n^2 x^2} \Big\}
+    = \lim_{n \to +\infty} \!\Big\{ \frac{\sin(n x)}{\pi x} \Big\}
+\end{aligned}$$
+
+The last one is especially important, since it is equivalent to the
+following integral, which appears very often in the context of
+Fourier transforms:
+
+$$\begin{aligned}
+    \boxed{
+        \delta(x)
+        %= \lim_{n \to +\infty} \!\Big\{\frac{\sin(n x)}{\pi x}\Big\}
+        = \frac{1}{2\pi} \int_{-\infty}^\infty \exp(i k x) \dd{k}
+        \propto \hat{\mathcal{F}}\{1\}
+    }
+\end{aligned}$$
+
+When the argument of $\delta(x)$ is scaled, the delta function is itself scaled:
+
+$$\begin{aligned}
+    \boxed{
+        \delta(s x) = \frac{1}{|s|} \delta(x)
+    }
+\end{aligned}$$
+
+*__Proof.__ Because it is symmetric, $\delta(s x) = \delta(|s| x)$. Then by
+substituting $\sigma = |s| x$:*
+
+$$\begin{aligned}
+    \int \delta(|s| x) \dd{x}
+    &= \frac{1}{|s|} \int \delta(\sigma) \dd{\sigma} = \frac{1}{|s|}
+\end{aligned}$$
+
+*__Q.E.D.__*
+
+An even more impressive property is the behaviour of the derivative of
+$\delta(x)$:
+
+$$\begin{aligned}
+    \boxed{
+        \int f(\xi) \: \delta'(x - \xi) \dd{\xi} = f'(x)
+    }
+\end{aligned}$$
+
+*__Proof.__ Be careful about which variable is used for the
+differentiation:*
+
+$$\begin{aligned}
+    \int f(\xi) \: \frac{d\delta(x - \xi)}{dx} \dd{\xi}
+    &= \frac{d}{dx} \int f(\xi) \: \delta(x - \xi) \dd{x}
+    = f'(x)
+\end{aligned}$$
+
+*__Q.E.D.__*
+
+This property also generalizes nicely for the higher-order derivatives:
+
+$$\begin{aligned}
+    \boxed{
+        \int f(\xi) \: \frac{d^n \delta(x - \xi)}{dx^n} \dd{\xi} = \dv[n]{f(x)}{x}
+    }
+\end{aligned}$$
-- 
cgit v1.2.3