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diff --git a/static/know/concept/dirac-notation/index.html b/static/know/concept/dirac-notation/index.html deleted file mode 100644 index 5da8c1b..0000000 --- a/static/know/concept/dirac-notation/index.html +++ /dev/null @@ -1,137 +0,0 @@ -<!DOCTYPE html> -<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang=""> -<head> - <meta charset="utf-8" /> - <meta name="generator" content="pandoc" /> - <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> - <title>Prefetch | Dirac notation</title> - <link rel="icon" href="data:,"> - <style> - body { - background:#ddd; - color:#222; - max-width:80ch; - text-align:justify; - margin:auto; - padding:1em 0; - font-family:sans-serif; - line-height:1.3; - } - a {text-decoration:none;color:#00f;} - h1,h2,h3 {text-align:center} - h1 {font-size:200%;} - h2 {font-size:160%;} - h3 {font-size:120%;} - .nav {height:3rem;font-size:250%;} - .nav a:link,a:visited {color:#222;} - .nav a:hover,a:focus,a:active {color:#00f;} - .navl {width:30%;float:left;text-align:left;} - .navr {width:70%;float:left;text-align:right;} - pre {filter:invert(100%);} - @media (prefers-color-scheme: dark) { - body {background:#222;filter:invert(100%);} - } </style> - <script> - MathJax = { - loader: {load: ["[tex]/physics"]}, - tex: {packages: {"[+]": ["physics"]}} - }; - </script> - <script src="/mathjax/tex-svg.js" type="text/javascript"></script> - </head> -<body> -<div class="nav"> -<div class="navl"><a href="/">PREFETCH</a></div> -<div class="navr"> -<a href="/blog/">blog</a>  -<a href="/code/">code</a>  -<a href="/know/">know</a> -</div> -</div> -<hr> -<h1 id="dirac-notation">Dirac notation</h1> -<p><em>Dirac notation</em> is a notation to do calculations in a Hilbert space without needing to worry about the space’s representation. It is basically the <em>lingua franca</em> of quantum mechanics.</p> -<p>In Dirac notation there are <em>kets</em> <span class="math inline">\(\ket{V}\)</span> from the Hilbert space <span class="math inline">\(\mathbb{H}\)</span> and <em>bras</em> <span class="math inline">\(\bra{V}\)</span> from a dual <span class="math inline">\(\mathbb{H}'\)</span> 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.</p> -<p>Bras and kets can only be combined in two ways: the <em>inner product</em> <span class="math inline">\(\braket{V}{W}\)</span>, which returns a scalar, and the <em>outer product</em> <span class="math inline">\(\ket{V} \bra{W}\)</span>, which returns a mapping <span class="math inline">\(\hat{L}\)</span> from kets <span class="math inline">\(\ket{V}\)</span> to other kets <span class="math inline">\(\ket{V'}\)</span>, i.e. a linear operator. Recall that the Hilbert inner product must satisfy:</p> -<p><span class="math display">\[\begin{aligned} - \braket{V}{W} = \braket{W}{V}^* -\end{aligned}\]</span></p> -<p>So far, nothing has been said about the actual representation of bras or kets. If we represent kets as <span class="math inline">\(N\)</span>-dimensional columns vectors, the corresponding bras are given by the kets’ adjoints, i.e. their transpose conjugates:</p> -<p><span class="math display">\[\begin{aligned} - \ket{V} = - \begin{bmatrix} - v_1 \\ \vdots \\ v_N - \end{bmatrix} - \quad \implies \quad - \bra{V} = - \begin{bmatrix} - v_1^* & \cdots & v_N^* - \end{bmatrix} -\end{aligned}\]</span></p> -<p>The inner product <span class="math inline">\(\braket{V}{W}\)</span> is then just the familiar dot product <span class="math inline">\(V \cdot W\)</span>:</p> -<p><span class="math display">\[\begin{gathered} - \braket{V}{W} - = - \begin{bmatrix} - v_1^* & \cdots & v_N^* - \end{bmatrix} - \cdot - \begin{bmatrix} - w_1 \\ \vdots \\ w_N - \end{bmatrix} - = v_1^* w_1 + ... + v_N^* w_N -\end{gathered}\]</span></p> -<p>Meanwhile, the outer product <span class="math inline">\(\ket{V} \bra{W}\)</span> creates an <span class="math inline">\(N \cross N\)</span> matrix:</p> -<p><span class="math display">\[\begin{gathered} - \ket{V} \bra{W} - = - \begin{bmatrix} - v_1 \\ \vdots \\ v_N - \end{bmatrix} - \cdot - \begin{bmatrix} - w_1^* & \cdots & w_N^* - \end{bmatrix} - = - \begin{bmatrix} - v_1 w_1^* & \cdots & v_1 w_N^* \\ - \vdots & \ddots & \vdots \\ - v_N w_1^* & \cdots & v_N w_N^* - \end{bmatrix} -\end{gathered}\]</span></p> -<p>If the kets are instead represented by functions <span class="math inline">\(f(x)\)</span> of <span class="math inline">\(x \in [a, b]\)</span>, then the bras represent <em>functionals</em> <span class="math inline">\(F[u(x)]\)</span> which take an unknown function <span class="math inline">\(u(x)\)</span> as an argument and turn it into a scalar using integration:</p> -<p><span class="math display">\[\begin{aligned} - \ket{f} = f(x) - \quad \implies \quad - \bra{f} - = F[u(x)] - = \int_a^b f^*(x) \: u(x) \dd{x} -\end{aligned}\]</span></p> -<p>Consequently, the inner product is simply the following familiar integral:</p> -<p><span class="math display">\[\begin{gathered} - \braket{f}{g} - = F[g(x)] - = \int_a^b f^*(x) \: g(x) \dd{x} -\end{gathered}\]</span></p> -<p>However, the outer product becomes something rather abstract:</p> -<p><span class="math display">\[\begin{gathered} - \ket{f} \bra{g} - = f(x) \: G[u(x)] - = f(x) \int_a^b g^*(\xi) \: u(\xi) \dd{\xi} -\end{gathered}\]</span></p> -<p>This result makes more sense if we surround it by a bra and a ket:</p> -<p><span class="math display">\[\begin{aligned} - \bra{u} \!\Big(\!\ket{f} \bra{g}\!\Big)\! \ket{w} - &= U\big[f(x) \: G[w(x)]\big] - = U\Big[ f(x) \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big] - \\ - &= \int_a^b u^*(x) \: f(x) \: \Big(\int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big) \dd{x} - \\ - &= \Big( \int_a^b u^*(x) \: f(x) \dd{x} \Big) \Big( \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big) - \\ - &= \braket{u}{f} \braket{g}{w} -\end{aligned}\]</span></p> -<hr> -© "Prefetch". 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