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diff --git a/static/know/blochs-theorem/index.html b/static/know/blochs-theorem/index.html new file mode 100644 index 0000000..26e3480 --- /dev/null +++ b/static/know/blochs-theorem/index.html @@ -0,0 +1,103 @@ +<!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> + <style> + body { + background:#ddd; + color:#222; + max-width:72ch; + 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%);} + } + math[display="inline"] { + font-size:110%; + } + math[display="block"] { + font-size:130%; + } </style> + </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 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">V(\vec{r})</annotation></semantics></math> which is periodic on a lattice, i.e. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo>+</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">V(\vec{r}) = V(\vec{r} + \vec{a})</annotation></semantics></math> for a primitive lattice vector <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>a</mi><mo accent="true">⃗</mo></mover><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math>, then it follows that the solutions <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\psi(\vec{r})</annotation></semantics></math> to the time-independent <a href="/know/page/schroedinger-equation">Schrödinger equation</a> take the following form, where the function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">u(\vec{r})</annotation></semantics></math> is periodic on the same lattice, i.e. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo>+</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">u(\vec{r}) = u(\vec{r} + \vec{a})</annotation></semantics></math>:</p> +<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable><mtr><mtd columnalign="right"><menclose notation="box"><mrow><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><msup><mi>e</mi><mrow><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover></mrow></msup></mrow></menclose></mtd></mtr></mtable><annotation encoding="application/x-tex"> +\begin{aligned} + \boxed{ + \psi(\vec{r}) = u(\vec{r}) e^{i \vec{k} \cdot \vec{r}} + } +\end{aligned} +</annotation></semantics></math></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 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>H</mi><mo accent="true">̂</mo></mover><annotation encoding="application/x-tex">\hat{H}</annotation></semantics></math> is lattice-periodic, then it will commute with the unitary translation operator <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mi>T</mi><mo accent="true">̂</mo></mover><mo stretchy="false" form="prefix">(</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\hat{T}(\vec{a})</annotation></semantics></math>, i.e. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">[</mo><mover><mi>H</mi><mo accent="true">̂</mo></mover><mo>,</mo><mover><mi>T</mi><mo accent="true">̂</mo></mover><mo stretchy="false" form="prefix">(</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo stretchy="false" form="postfix">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">[\hat{H}, \hat{T}(\vec{a})] = 0</annotation></semantics></math>. Therefore <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>H</mi><mo accent="true">̂</mo></mover><annotation encoding="application/x-tex">\hat{H}</annotation></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mi>T</mi><mo accent="true">̂</mo></mover><mo stretchy="false" form="prefix">(</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\hat{T}(\vec{a})</annotation></semantics></math> must share eigenstates <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\psi(\vec{r})</annotation></semantics></math>:</p> +<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable><mtr><mtd columnalign="right"><mover><mi>H</mi><mo accent="true">̂</mo></mover><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mi>E</mi><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mspace width="2.0em"></mspace><mover><mi>T</mi><mo accent="true">̂</mo></mover><mo stretchy="false" form="prefix">(</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mi>τ</mi><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr></mtable><annotation encoding="application/x-tex"> +\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} +</annotation></semantics></math></p> +<p>Since <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>T</mi><mo accent="true">̂</mo></mover><annotation encoding="application/x-tex">\hat{T}</annotation></semantics></math> is unitary, its eigenvalues <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>τ</mi><annotation encoding="application/x-tex">\tau</annotation></semantics></math> must have the form <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>e</mi><mrow><mi>i</mi><mi>θ</mi></mrow></msup><annotation encoding="application/x-tex">e^{i \theta}</annotation></semantics></math>, with <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>θ</mi><annotation encoding="application/x-tex">\theta</annotation></semantics></math> real. Therefore a translation by <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>a</mi><mo accent="true">⃗</mo></mover><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math> causes a phase shift, for some vector <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>k</mi><mo accent="true">⃗</mo></mover><annotation encoding="application/x-tex">\vec{k}</annotation></semantics></math>:</p> +<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable><mtr><mtd columnalign="right"><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo>+</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mover><mi>T</mi><mo accent="true">̂</mo></mover><mo stretchy="false" form="prefix">(</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><msup><mi>e</mi><mrow><mi>i</mi><mi>θ</mi></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><msup><mi>e</mi><mrow><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr></mtable><annotation encoding="application/x-tex"> +\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} +</annotation></semantics></math></p> +<p>Let us now define the following function, keeping our arbitrary choice of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>k</mi><mo accent="true">⃗</mo></mover><annotation encoding="application/x-tex">\vec{k}</annotation></semantics></math>:</p> +<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable><mtr><mtd columnalign="right"><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr></mtable><annotation encoding="application/x-tex"> +\begin{aligned} + u(\vec{r}) + = e^{- i \vec{k} \cdot \vec{r}} \:\psi(\vec{r}) +\end{aligned} +</annotation></semantics></math></p> +<p>As it turns out, this function is guaranteed to be lattice-periodic for any <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>k</mi><mo accent="true">⃗</mo></mover><annotation encoding="application/x-tex">\vec{k}</annotation></semantics></math>:</p> +<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable><mtr><mtd columnalign="right"><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo>+</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd><mtd columnalign="left"><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo>+</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo>+</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover></mrow></msup><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover></mrow></msup><msup><mi>e</mi><mrow><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>a</mi><mo accent="true">⃗</mo></mover></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr></mtable><annotation encoding="application/x-tex"> +\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} +</annotation></semantics></math></p> +<p>Then Bloch’s theorem follows from isolating the definition of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">u(\vec{r})</annotation></semantics></math> for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\psi(\vec{r})</annotation></semantics></math>.</p> + +<hr> +© "Prefetch". Licensed under <a href="https://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA 4.0</a>. +</body> +</html> |