Categories: Physics, Quantum mechanics.

# Pauli exclusion principle

In quantum mechanics, the Pauli exclusion principle is a theorem with 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 can occupy 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 $$\lambda = -1$$:

\begin{aligned} \hat{P} \ket{a}\ket{b} = \lambda \ket{a}\ket{b} \end{aligned}

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 type of particle that $$x_1$$ and $$x_2$$ are. 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 is 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}

And 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 atoms only fit a limited number of electrons (which are fermions), since each must have a different quantum number.