Slater Determinant

\[\Psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_N) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \dots & \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \dots & \chi_N(\mathbf{x}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \dots & \chi_N(\mathbf{x}_N) \end{vmatrix}\]

Eigenvector and Eigenvalues

Given a matrix \(\mathbf{A}\), the eigenvector equation is:

\[\mathbf{A} \mathbf{v} = \lambda \mathbf{v}\]

Finding Eigenvalues:

To find the eigenvalues \(\lambda\), solve the characteristic equation:

\[\det(\mathbf{A} - \lambda \mathbf{I}) = 0\]

Finding Eigenvectors:

Once the eigenvalues are determined, substitute each \(\lambda\) into the equation:

\[(\mathbf{A} - \lambda \mathbf{I}) \mathbf{v} = 0\]

Solve for the eigenvector \(\mathbf{v}\).