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diff --git a/static/know/concept/blochs-theorem/index.html b/static/know/concept/blochs-theorem/index.html index 9710710..d900aba 100644 --- a/static/know/concept/blochs-theorem/index.html +++ b/static/know/concept/blochs-theorem/index.html @@ -5,6 +5,7 @@ <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; @@ -30,7 +31,13 @@ @media (prefers-color-scheme: dark) { body {background:#222;filter:invert(100%);} } </style> - <script src="/mathjax/tex-svg.js" type="text/javascript"></script> + <script> + MathJax = { + loader: {load: ["[tex]/physics"]}, + tex: {packages: {"[+]": ["physics"]}} + }; + </script> + <script src="/mathjax/tex-chtml.js" type="text/javascript"></script> </head> <body> <div class="nav"> @@ -42,7 +49,6 @@ </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">\[ @@ -53,7 +59,7 @@ \end{aligned} \]</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>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">\(\comm{\hat{H}}{\hat{T}(\vec{a})} = 0\)</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">\[ \begin{aligned} \hat{H} \:\psi(\vec{r}) = E \:\psi(\vec{r}) @@ -91,7 +97,6 @@ \end{aligned} \]</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". Licensed under <a href="https://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA 4.0</a>. </body> |