Pauli Exclusion Principle

Pauli observed that relativistic quantum field theory requires that particles with half-integer spin (s=1/2, 3/2, ...) must have antisymmetric wave functions and particles with integer spin (s=0, 1, ...) must have symmetric wave functions. Such observation is usually introduced as an additional postulate of quantum mechanics: The wave function of a system of electrons must be antisymmetric with respect to interchange of any two electrons.

As a consequence of such principle is that two electrons with the same spin can not have the same coordinates, since the wavefunction must satisfy the following condition:

$$\psi_{(x_1, x_2)} = - \psi_{(x_2, x_1)},$$

and, therefore, $\psi_{(x_1, x_1)} =0$. For this reason the principle is known as the Pauli Exclusion Principle.

Another consequence of the Pauli Principle is that since the ground state wave function of the He atom must also be anti-symmetric, and since the spatial part of the zeroth order wave function is symmetric, $\Psi = 1S(1) 1S(2)$, then the spin wave function $\chi$ must be anti-symmetric,

$$\chi=\frac{1}{\sqrt{2}} \left \vert\begin{matrix}\alpha(1) & \beta(1) \\ \alpha(2) & \beta(2) \\ \end{matrix} \right \vert,$$

and the overall zeroth-order wave function becomes,

$$\psi = 1S(1) 1S(2) \frac{1}{\sqrt{2}} \left[ \alpha (1) \beta(2) - \beta (1) \alpha(2) \right].$$ (46)

Note that this anti-symmetric spin-atom wave function can be written in the form of the Slater determinant,

$$\psi = \frac{1}{\sqrt{2}} \left \vert\begin{matrix}1S(1) \alpha(1... ...1S(2) \beta(1) \\ 1S(1) \alpha(2) & 1S(2)\beta(2) \\ \end{matrix} \right \vert.$$