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<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>
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