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Section 6.4 Fermions and the Pauli Exclusion Principle

A very interesting thing happens to fermions in the antisymmetric two-particle state when the single-particle states \(|\alpha\rangle\) and \(|\beta\rangle\) are taken to be the same state. Then we have

\begin{equation} |\psi_A\rangle = \frac{1}{\sqrt 2}|\alpha\, \alpha\rangle - \frac{1}{\sqrt 2}|\alpha\, \alpha\rangle = 0\text{.}\tag{6.6} \end{equation}

What does this mean? It means that there is no such state! This is known as the Pauli exclusion principle, which can be stated as follows:

Pauli exclusion principle: it is not possible to put two identical fermions into the same single-particle state.

Let's see what it does in practice. Electrons are fermions. Consider an electron in an infinite square-well potential (particle-in-a-box) in the lowest energy level \(E_1\text{.}\) There are two possible states: \(|E_1\hspace{-1mm}\uparrow\rangle\) and \(|E_1\hspace{-1mm}\downarrow\rangle\text{,}\) meaning “ground state, spin up” and “ground state, spin down.”

In the ground state, there are two electrons in this lowest energy level. If we attempt to put these two electrons in both with spin up we get

\begin{equation} |\psi_A\rangle = \frac{1}{\sqrt 2}|E_1\hspace{-1mm}\uparrow\hspace{2mm} E_1\hspace{-1mm}\uparrow\rangle - \frac{1}{\sqrt 2}|E_1\hspace{-1mm}\uparrow\hspace{2mm} E_1\hspace{-1mm}\uparrow\rangle = 0\text{.}\tag{6.7} \end{equation}

We find that the state doesn't exist! The Pauli exclusion principle tells us we cannot put the two electrons into the ground state with the spins both up.

But we can put the two electrons into the ground state when their spins are opposite. That is, the real ground state is

\begin{equation} |\text{ He g.s. } \rangle = \frac{1}{\sqrt 2}|E_1\hspace{-1mm}\uparrow\hspace{2mm} E_1\hspace{-1mm}\downarrow\rangle - \frac{1}{\sqrt 2}|E_1\hspace{-1mm}\downarrow\hspace{2mm} E_1\hspace{-1mm}\uparrow\rangle\text{.}\label{eq_helium_ground_state}\tag{6.8} \end{equation}

This expression has the proper anti-symmetry and is not zero.

This raises the question of whether we could put three electrons into the ground state. As you might guess, there is no three-electron antisymmetric state \(|\psi_A\rangle\text{,}\) so the upper limit on the number of electrons we can put into the ground state is two.

Five identical particles with spin \(s=1/2\) are put into a one-dimensional infinite square well with energy levels \(E_n=n^2 E_1\) and \(E_1=1\Xunits{eV}\text{.}\) What is the lowest combined energy that these particles can have?

Solution.

Two particles can fit into the lowest energy level, in an antisymmetric combination like (6.8). Two more particles can fit into the second lowest energy level, and then one particle must be in the third energy level. This is illustrated schematically on the right. Adding up these energies:

\begin{equation*} E_\text{ combined } = 1 + 1 + 4 + 4 + 9 = 19\Xunits{eV}\text{.} \end{equation*}

Pauli's Exclusion Principle has tremendously important implications for the behavior of atoms and for the entire field of chemistry (and, therefore, biology as well). We'll discuss these implications in the next chapter.