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| author | Prefetch | 2021-02-20 20:21:32 +0100 |
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| committer | Prefetch | 2021-02-20 20:21:32 +0100 |
| commit | 88d42030530abeba4f3ceaf63da71e6cdfa71267 (patch) | |
| tree | f16d7f173d8cc4c0ba9eaaf8467099a2bd90555e /static/know/concept/pauli-exclusion-principle | |
| parent | 92a86d01c4837901fa433158294a9ce23cbfcefa (diff) | |
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diff --git a/static/know/concept/pauli-exclusion-principle/index.html b/static/know/concept/pauli-exclusion-principle/index.html deleted file mode 100644 index 5389c02..0000000 --- a/static/know/concept/pauli-exclusion-principle/index.html +++ /dev/null @@ -1,107 +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 | Pauli exclusion principle</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="pauli-exclusion-principle">Pauli exclusion principle</h1> -<p>In quantum mechanics, the <em>Pauli exclusion principle</em> is a theorem that has profound consequences for how the world works.</p> -<p>Suppose we have a composite state <span class="math inline">\(\ket*{x_1}\ket*{x_2} = \ket*{x_1} \otimes \ket*{x_2}\)</span>, where the two identical particles <span class="math inline">\(x_1\)</span> and <span class="math inline">\(x_2\)</span> each can occupy the same two allowed states <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span>. We then define the permutation operator <span class="math inline">\(\hat{P}\)</span> as follows:</p> -<p><span class="math display">\[\begin{aligned} - \hat{P} \ket{a}\ket{b} = \ket{b}\ket{a} -\end{aligned}\]</span></p> -<p>That is, it swaps the states of the particles. Obviously, swapping the states twice simply gives the original configuration again, so:</p> -<p><span class="math display">\[\begin{aligned} - \hat{P}^2 \ket{a}\ket{b} = \ket{a}\ket{b} -\end{aligned}\]</span></p> -<p>Therefore, <span class="math inline">\(\ket{a}\ket{b}\)</span> is an eigenvector of <span class="math inline">\(\hat{P}^2\)</span> with eigenvalue <span class="math inline">\(1\)</span>. Since <span class="math inline">\([\hat{P}, \hat{P}^2] = 0\)</span>, <span class="math inline">\(\ket{a}\ket{b}\)</span> must also be an eigenket of <span class="math inline">\(\hat{P}\)</span> with eigenvalue <span class="math inline">\(\lambda\)</span>, satisfying <span class="math inline">\(\lambda^2 = 1\)</span>, so we know that <span class="math inline">\(\lambda = 1\)</span> or <span class="math inline">\(\lambda = -1\)</span>.</p> -<p>As it turns out, in nature, each class of particle has a single associated permutation eigenvalue <span class="math inline">\(\lambda\)</span>, or in other words: whether <span class="math inline">\(\lambda\)</span> is <span class="math inline">\(-1\)</span> or <span class="math inline">\(1\)</span> depends on the species of particle that <span class="math inline">\(x_1\)</span> and <span class="math inline">\(x_2\)</span> represent. Particles with <span class="math inline">\(\lambda = -1\)</span> are called <em>fermions</em>, and those with <span class="math inline">\(\lambda = 1\)</span> are known as <em>bosons</em>. We define <span class="math inline">\(\hat{P}_f\)</span> with <span class="math inline">\(\lambda = -1\)</span> and <span class="math inline">\(\hat{P}_b\)</span> with <span class="math inline">\(\lambda = 1\)</span>, such that:</p> -<p><span class="math display">\[\begin{aligned} - \hat{P}_f \ket{a}\ket{b} = \ket{b}\ket{a} = - \ket{a}\ket{b} - \qquad - \hat{P}_b \ket{a}\ket{b} = \ket{b}\ket{a} = \ket{a}\ket{b} -\end{aligned}\]</span></p> -<p>Another fundamental fact of nature is that identical particles cannot be distinguished by any observation. Therefore it is impossible to tell apart <span class="math inline">\(\ket{a}\ket{b}\)</span> and the permuted state <span class="math inline">\(\ket{b}\ket{a}\)</span>, regardless of the eigenvalue <span class="math inline">\(\lambda\)</span>. There is no physical difference!</p> -<p>But this does not mean that <span class="math inline">\(\hat{P}\)</span> is useless: despite not having any observable effect, the resulting difference between fermions and bosons is absolutely fundamental. Consider the following superposition state, where <span class="math inline">\(\alpha\)</span> and <span class="math inline">\(\beta\)</span> are unknown:</p> -<p><span class="math display">\[\begin{aligned} - \ket{\Psi(a, b)} - = \alpha \ket{a}\ket{b} + \beta \ket{b}\ket{a} -\end{aligned}\]</span></p> -<p>When we apply <span class="math inline">\(\hat{P}\)</span>, we can “choose” between two “intepretations” of its action, both shown below. Obviously, since the left-hand sides are equal, the right-hand sides must be equal too:</p> -<p><span class="math display">\[\begin{aligned} - \hat{P} \ket{\Psi(a, b)} - &= \lambda \alpha \ket{a}\ket{b} + \lambda \beta \ket{b}\ket{a} - \\ - \hat{P} \ket{\Psi(a, b)} - &= \alpha \ket{b}\ket{a} + \beta \ket{a}\ket{b} -\end{aligned}\]</span></p> -<p>This gives us the equations <span class="math inline">\(\lambda \alpha = \beta\)</span> and <span class="math inline">\(\lambda \beta = \alpha\)</span>. In fact, just from this we could have deduced that <span class="math inline">\(\lambda\)</span> can be either <span class="math inline">\(-1\)</span> or <span class="math inline">\(1\)</span>. In any case, for bosons (<span class="math inline">\(\lambda = 1\)</span>), we thus find that <span class="math inline">\(\alpha = \beta\)</span>:</p> -<p><span class="math display">\[\begin{aligned} - \ket{\Psi(a, b)}_b = C \big( \ket{a}\ket{b} + \ket{b}\ket{a} \big) -\end{aligned}\]</span></p> -<p>Where <span class="math inline">\(C\)</span> is a normalization constant. As expected, this state is <em>symmetric</em>: switching <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span> gives the same result. Meanwhile, for fermions (<span class="math inline">\(\lambda = -1\)</span>), we find that <span class="math inline">\(\alpha = -\beta\)</span>:</p> -<p><span class="math display">\[\begin{aligned} - \ket{\Psi(a, b)}_f = C \big( \ket{a}\ket{b} - \ket{b}\ket{a} \big) -\end{aligned}\]</span></p> -<p>This state called <em>antisymmetric</em> under exchange: switching <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span> causes a sign change, as we would expect for fermions.</p> -<p>Now, what if the particles <span class="math inline">\(x_1\)</span> and <span class="math inline">\(x_2\)</span> are in the same state <span class="math inline">\(a\)</span>? For bosons, we just need to update the normalization constant <span class="math inline">\(C\)</span>:</p> -<p><span class="math display">\[\begin{aligned} - \ket{\Psi(a, a)}_b - = C \ket{a}\ket{a} -\end{aligned}\]</span></p> -<p>However, for fermions, the state is unnormalizable and thus unphysical:</p> -<p><span class="math display">\[\begin{aligned} - \ket{\Psi(a, a)}_f - = C \big( \ket{a}\ket{a} - \ket{a}\ket{a} \big) - = 0 -\end{aligned}\]</span></p> -<p>At last, this is the Pauli exclusion principle: fermions may never occupy the same quantum state. One of the many notable consequences of this is that the shells of an atom only fit a limited number of electrons, since each must have a different quantum number.</p> -<hr> -© "Prefetch". Licensed under <a href="https://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA 4.0</a>. -</body> -</html> |