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diff --git a/static/know/concept/blochs-theorem/index.html b/static/know/concept/blochs-theorem/index.html deleted file mode 100644 index f977739..0000000 --- a/static/know/concept/blochs-theorem/index.html +++ /dev/null @@ -1,103 +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 | Bloch’s theorem</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="blochs-theorem">Bloch’s theorem</h1> -<p>In quantum mechanics, <em>Bloch’s theorem</em> states that, given a potential <span class="math inline">\(V(\vec{r})\)</span> which is periodic on a lattice, i.e. <span class="math inline">\(V(\vec{r}) = V(\vec{r} + \vec{a})\)</span> for a primitive lattice vector <span class="math inline">\(\vec{a}\)</span>, then it follows that the solutions <span class="math inline">\(\psi(\vec{r})\)</span> to the time-independent Schrödinger equation take the following form, where the function <span class="math inline">\(u(\vec{r})\)</span> is periodic on the same lattice, i.e. <span class="math inline">\(u(\vec{r}) = u(\vec{r} + \vec{a})\)</span>:</p> -<p><span class="math display">\[ -\begin{aligned} - \boxed{ - \psi(\vec{r}) = u(\vec{r}) e^{i \vec{k} \cdot \vec{r}} - } -\end{aligned} -\]</span></p> -<p>In other words, in a periodic potential, the solutions are simply plane waves with a periodic modulation, known as <em>Bloch functions</em> or <em>Bloch states</em>.</p> -<p>This is suprisingly easy to prove: if the Hamiltonian <span class="math inline">\(\hat{H}\)</span> is lattice-periodic, then it will commute with the unitary translation operator <span class="math inline">\(\hat{T}(\vec{a})\)</span>, i.e. <span class="math inline">\([\hat{H}, \hat{T}(\vec{a})] = 0\)</span>. Therefore <span class="math inline">\(\hat{H}\)</span> and <span class="math inline">\(\hat{T}(\vec{a})\)</span> must share eigenstates <span class="math inline">\(\psi(\vec{r})\)</span>:</p> -<p><span class="math display">\[ -\begin{aligned} - \hat{H} \:\psi(\vec{r}) = E \:\psi(\vec{r}) - \qquad - \hat{T}(\vec{a}) \:\psi(\vec{r}) = \tau \:\psi(\vec{r}) -\end{aligned} -\]</span></p> -<p>Since <span class="math inline">\(\hat{T}\)</span> is unitary, its eigenvalues <span class="math inline">\(\tau\)</span> must have the form <span class="math inline">\(e^{i \theta}\)</span>, with <span class="math inline">\(\theta\)</span> real. Therefore a translation by <span class="math inline">\(\vec{a}\)</span> causes a phase shift, for some vector <span class="math inline">\(\vec{k}\)</span>:</p> -<p><span class="math display">\[ -\begin{aligned} - \psi(\vec{r} + \vec{a}) - = \hat{T}(\vec{a}) \:\psi(\vec{r}) - = e^{i \theta} \:\psi(\vec{r}) - = e^{i \vec{k} \cdot \vec{a}} \:\psi(\vec{r}) -\end{aligned} -\]</span></p> -<p>Let us now define the following function, keeping our arbitrary choice of <span class="math inline">\(\vec{k}\)</span>:</p> -<p><span class="math display">\[ -\begin{aligned} - u(\vec{r}) - = e^{- i \vec{k} \cdot \vec{r}} \:\psi(\vec{r}) -\end{aligned} -\]</span></p> -<p>As it turns out, this function is guaranteed to be lattice-periodic for any <span class="math inline">\(\vec{k}\)</span>:</p> -<p><span class="math display">\[ -\begin{aligned} - u(\vec{r} + \vec{a}) - &= e^{- i \vec{k} \cdot (\vec{r} + \vec{a})} \:\psi(\vec{r} + \vec{a}) - \\ - &= e^{- i \vec{k} \cdot \vec{r}} e^{- i \vec{k} \cdot \vec{a}} e^{i \vec{k} \cdot \vec{a}} \:\psi(\vec{r}) - \\ - &= e^{- i \vec{k} \cdot \vec{r}} \:\psi(\vec{r}) - \\ - &= u(\vec{r}) -\end{aligned} -\]</span></p> -<p>Then Bloch’s theorem follows from isolating the definition of <span class="math inline">\(u(\vec{r})\)</span> for <span class="math inline">\(\psi(\vec{r})\)</span>.</p> -<hr> -© "Prefetch". 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