Pauli exclusion principle

The Pauli exclusion principle states that no two identical fermions (particles with spin $s=\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots$ ) can occupy the same state.

The principle is a consequence of the postulate that the wavefunction of a system of fermions is antisymmetric with respect to label exchange, i.e. the wavefunction changes sign when the labels are exchanged.

The identical property of particles is fulfilled by the linear combinations:

$\psi_{\pm}(r_1,r_2)=c\left [ \psi_{r_1}(1)\psi_{r_2}(2)\pm\psi_{r_1}(2)\psi_{r_2}(1)\right ]$

where the labels 1 and 2 denote the first particle and second particle respectively, $\psi_{r_j}(i)$ denotes particle $i$ in the state $\psi_{r_j}$ and there is equal probability $\vert c\vert^{2}$ of the wavefunction $\psi_{\pm}$ collapsing to $\psi_{r_1}(1)\psi_{r_2}(2)$ and $\psi_{r_1}(2)\psi_{r_2}(1)$ upon measurement.

Question

Show that $\psi_+$ is symmetric with respect to label exchange, while $\psi_-$ is antisymmetric with respect to label exchange.

Answer

Exchanging the labels of $\psi_+=c\left [ \psi_{r_1}(1)\psi_{r_2}(2)+\psi_{r_1}(2)\psi_{r_2}(1)\right ]$ and $\psi_-=c\left [ \psi_{r_1}(1)\psi_{r_2}(2)-\psi_{r_1}(2)\psi_{r_2}(1)\right ]$ ,

$c\left [ \psi_{r_1}(2)\psi_{r_2}(1)+\psi_{r_1}(1)\psi_{r_2}(2)\right ]=\psi_+$

$c\left [ \psi_{r_1}(2)\psi_{r_2}(1)-\psi_{r_1}(1)\psi_{r_2}(2)\right ]=-\psi_-$

Since the wavefunction of a system of fermions is antisymmetric with respect to label exchange, it must be expressed by

$\psi_-(r_1,r_2)=c\left [ \psi_{r_1}(1)\psi_{r_2}(2)-\psi_{r_1}(2)\psi_{r_2}(1)\right ]$

When $\psi_{r_1}=\psi_{r_2}$ , the wavefunction $\psi_-=0$ , which contradicts the proposition that an eigenfunction is a non-zero function. Therefore, no two identical fermions can occupy the same state.

If the system is an atom, the total wavefunction (spatial + spin) describing two identical electrons ( $s=\frac{1}{2}$ ) is

$\Psi(n,l,m_l,m_s)=\psi(n,l,m_l)\sigma(m_s)$

$\psi(n,l,m_l)=\psi_{\pm}=\phi_{n_1,l_1,m_{l1}}(1)\phi_{n_2,l_2,m_{l2}}(2)\pm\phi_{n_1,l_1,m_{l1}}(2)\phi_{n_2,l_2,m_{l2}}(1)$ is the composite spatial wavefunction of the system, where $\phi$ is a one-electron spatial wavefunction (also known as hydrogenic wavefunction). $\sigma(m_s)$ is the composite spin wavefunction of the system and is explicitly given by eq222 through eq225. Since $\Psi$ must be antisymmetric with respect to label exchange

$\Psi=\left\{\begin{matrix} \psi_+\sigma_d\\\psi_-\sigma_{a\: or\: b\: or\: c} \end{matrix}\right.$

When $n_1=n_2$ , $l_1=l_2$ and $m_{l1}=m_{l2}$ only $\psi_+\sigma_d$ is non-zero. Since $\sigma_d$ refers to the singlet state (i.e. a state with anti-parallel spins, where $m_{s1}\neq m_{s2}$ ), no two identical electrons in an atom can have the same set of quantum numbers. This is consistent with the general statement that no two identical fermions can occupy the same state because a distinct quantum state of an atom is characterised by a particular set of quantum numbers, e.g. $n=1$ , $l=1$ , $m_l=0$ and $m_s=\frac{1}{2}$ .