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authorPrefetch2021-02-20 13:20:32 +0100
committerPrefetch2021-02-20 13:20:32 +0100
commit8688d88c5d56541c8d06bcff4eeabe9ebf8a2fb8 (patch)
treeff74485b6d697dfa88eeaa2eb15d038250d324c5 /static/know/concept
parentfd5f0b15129417aa965d84ea46e5a9fc92a3c196 (diff)
Trial MathJax for knowledge base + Update CSS
Diffstat (limited to 'static/know/concept')
-rw-r--r--static/know/concept/blochs-theorem/index.html41
1 files changed, 18 insertions, 23 deletions
diff --git a/static/know/concept/blochs-theorem/index.html b/static/know/concept/blochs-theorem/index.html
index 58a934e..9710710 100644
--- a/static/know/concept/blochs-theorem/index.html
+++ b/static/know/concept/blochs-theorem/index.html
@@ -9,7 +9,7 @@
body {
background:#ddd;
color:#222;
- max-width:72ch;
+ max-width:80ch;
text-align:justify;
margin:auto;
padding:1em 0;
@@ -29,13 +29,8 @@
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>
+ <script src="/mathjax/tex-svg.js" type="text/javascript"></script>
</head>
<body>
<div class="nav">
@@ -49,41 +44,41 @@
<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 Schrödinger equation 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">
+<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">\[
\begin{aligned}
\boxed{
\psi(\vec{r}) = u(\vec{r}) e^{i \vec{k} \cdot \vec{r}}
}
\end{aligned}
-</annotation></semantics></math></p>
+\]</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 <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">
+<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}
-</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">
+\]</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">\[
\begin{aligned}
\psi(\vec{r} + \vec{a})
= \hat{T}(\vec{a}) \:\psi(\vec{r})
= e^{i \theta} \:\psi(\vec{r})
= e^{i \vec{k} \cdot \vec{a}} \:\psi(\vec{r})
\end{aligned}
-</annotation></semantics></math></p>
-<p>Let us now define the following function, keeping our arbitrary choice of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover><mi>k</mi><mo accent="true">⃗</mo></mover><annotation encoding="application/x-tex">\vec{k}</annotation></semantics></math>:</p>
-<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable><mtr><mtd columnalign="right"><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mover><mi>k</mi><mo accent="true">⃗</mo></mover><mo>⋅</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover></mrow></msup><mspace width="0.222em"></mspace><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mtd></mtr></mtable><annotation encoding="application/x-tex">
+\]</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}
-</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">
+\]</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">\[
\begin{aligned}
u(\vec{r} + \vec{a})
&amp;= e^{- i \vec{k} \cdot (\vec{r} + \vec{a})} \:\psi(\vec{r} + \vec{a})
@@ -94,8 +89,8 @@
\\
&amp;= u(\vec{r})
\end{aligned}
-</annotation></semantics></math></p>
-<p>Then Bloch’s theorem follows from isolating the definition of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">u(\vec{r})</annotation></semantics></math> for <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ψ</mi><mo stretchy="false" form="prefix">(</mo><mover><mi>r</mi><mo accent="true">⃗</mo></mover><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\psi(\vec{r})</annotation></semantics></math>.</p>
+\]</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>
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