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diff --git a/content/know/category/quantum-mechanics.md b/content/know/category/quantum-mechanics.md index 944e491..d61f03f 100644 --- a/content/know/category/quantum-mechanics.md +++ b/content/know/category/quantum-mechanics.md @@ -13,6 +13,7 @@ Alphabetical list of concepts in this category. * [Dirac notation](/know/concept/dirac-notation/) ## P +* [Pauli exclusion principle](/known/concept/pauli-exclusion-principle/) * [Probability current](/know/concept/probability-current/) ## T diff --git a/content/know/concept/index.md b/content/know/concept/index.md index db1c81c..43f5928 100644 --- a/content/know/concept/index.md +++ b/content/know/concept/index.md @@ -13,6 +13,7 @@ Alphabetical list of concepts in this knowledge base. * [Dirac notation](/know/concept/dirac-notation/) ## P +* [Pauli exclusion principle](/known/concept/pauli-exclusion-principle/) * [Probability current](/know/concept/probability-current/) ## T diff --git a/latex/know/concept/pauli-exclusion-principle/source.md b/latex/know/concept/pauli-exclusion-principle/source.md new file mode 100644 index 0000000..0a35869 --- /dev/null +++ b/latex/know/concept/pauli-exclusion-principle/source.md @@ -0,0 +1,112 @@ +% Pauli exclusion principle + + +# Pauli exclusion principle + +In quantum mechanics, the *Pauli exclusion principle* is a theorem that +has profound consequences for how the world works. + +Suppose we have a composite state +$\ket*{x_1}\!\ket*{x_2} = \ket*{x_1} \otimes \ket*{x_2}$, where the two +identical particles $x_1$ and $x_2$ each have the same two allowed +states $a$ and $b$. We then define the permutation operator $\hat{P}$ as +follows: + +$$\begin{aligned} + \hat{P} \ket{a}\!\ket{b} = \ket{b}\!\ket{a} +\end{aligned}$$ + +That is, it swaps the states of the particles. Obviously, swapping the +states twice simply gives the original configuration again, so: + +$$\begin{aligned} + \hat{P}^2 \ket{a}\!\ket{b} = \ket{a}\!\ket{b} +\end{aligned}$$ + +Therefore, $\ket{a}\!\ket{b}$ is an eigenvector of $\hat{P}^2$ with +eigenvalue $1$. Since $[\hat{P}, \hat{P}^2] = 0$, $\ket{a}\!\ket{b}$ +must also be an eigenket of $\hat{P}$ with eigenvalue $\lambda$, +satisfying $\lambda^2 = 1$, so we know that $\lambda = 1$ or +$\lambda = -1$. + +As it turns out, in nature, each class of particle has a single +associated permutation eigenvalue $\lambda$, or in other words: whether +$\lambda$ is $-1$ or $1$ depends on the species of particle that $x_1$ +and $x_2$ represent. Particles with $\lambda = -1$ are called +*fermions*, and those with $\lambda = 1$ are known as *bosons*. We +define $\hat{P}_f$ with $\lambda = -1$ and $\hat{P}_b$ with +$\lambda = 1$, such that: + +$$\begin{aligned} + \hat{P}_f \ket{a}\!\ket{b} = \ket{b}\!\ket{a} = - \ket{a}\!\ket{b} + \qquad + \hat{P}_b \ket{a}\!\ket{b} = \ket{b}\!\ket{a} = \ket{a}\!\ket{b} +\end{aligned}$$ + +Another fundamental fact of nature is that identical particles cannot be +distinguished by any observation. Therefore it is impossible to tell +apart $\ket{a}\!\ket{b}$ and the permuted state $\ket{b}\!\ket{a}$, +regardless of the eigenvalue $\lambda$. There is no physical difference! + +But this does not mean that $\hat{P}$ is useless: despite not having any +observable effect, the resulting difference between fermions and bosons +is absolutely fundamental. Consider the following superposition state, +where $\alpha$ and $\beta$ are unknown: + +$$\begin{aligned} + \ket{\Psi(a, b)} + = \alpha \ket{a}\!\ket{b} + \beta \ket{b}\!\ket{a} +\end{aligned}$$ + +When we apply $\hat{P}$, 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: + +$$\begin{aligned} + \hat{P} \ket{\Psi(a, b)} + &= \lambda \alpha \ket{a}\!\ket{b} + \lambda \beta \ket{b}\!\ket{a} + \\ + \hat{P} \ket{\Psi(a, b)} + = \alpha \ket{b}\!\ket{a} + \beta \ket{a}\!\ket{b} +\end{aligned}$$ + +This gives us the equations $\lambda \alpha = \beta$ and +$\lambda \beta = \alpha$. In fact, just from this we could have deduced +that $\lambda$ can be either $-1$ or $1$. In any case, for bosons +($\lambda = 1$), we thus find that $\alpha = \beta$: + +$$\begin{aligned} + \ket{\Psi(a, b)}_b = C \big( \ket{a}\!\ket{b} + \ket{b}\!\ket{a} \!\big) +\end{aligned}$$ + +Where $C$ is a normalization constant. As expected, this state is +*symmetric*: switching $a$ and $b$ gives the same result. Meanwhile, for +fermions ($\lambda = -1$), we find that $\alpha = -\beta$: + +$$\begin{aligned} + \ket{\Psi(a, b)}_f = C \big( \ket{a}\!\ket{b} - \ket{b}\!\ket{a} \!\big) +\end{aligned}$$ + +This state called *antisymmetric* under exchange: switching $a$ and $b$ +causes a sign change, as we would expect for fermions. + +Now, what if the particles $x_1$ and $x_2$ are in the same state $a$? +For bosons, we just need to update the normalization constant $C$: + +$$\begin{aligned} + \ket{\Psi(a, a)}_b + = C \ket{a}\!\ket{a} +\end{aligned}$$ + +However, for fermions, the state is unnormalizable and thus unphysical: + +$$\begin{aligned} + \ket{\Psi(a, a)}_f + = C \big( \ket{a}\!\ket{a} - \ket{a}\!\ket{a} \!\big) + = 0 +\end{aligned}$$ + +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. diff --git a/static/know/concept/pauli-exclusion-principle/index.html b/static/know/concept/pauli-exclusion-principle/index.html new file mode 100644 index 0000000..74a0954 --- /dev/null +++ b/static/know/concept/pauli-exclusion-principle/index.html @@ -0,0 +1,107 @@ +<!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 have 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> |