$$ \Psi(x_1, x_2, \ldots, x_A) = \frac{1}{\sqrt{N!}} \det \left | \begin{array}{cccc} \phi_1(x_1) & \phi_1(x_2) & \ldots & \phi_1(x_A) \\ \phi_2(x_1) & \phi_2(x_2) & \ldots & \phi_2(x_A) \\ \vdots & & & \\ \phi_A(x_1) & \phi_A(x_2) & \ldots & \phi_A(x_A) \end{array} \right | $$ Properties of the determinant (interchange of any two rows or any two columns yields a change in sign; thus no two rows and no two columns can be the same) lead to the Pauli principle:

• No two particles can be at the same place (two columns the same); and
• No two particles can be in the same state (two rows the same).