# 2.1: System of identical fermions and Slater Determinants

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In this section we construct a set of basis vectors for a system of n identical fermions.

### Basis vectors for the one-fermion system

$| \phi_r \rangle_{\alpha} \label{3.1}$

where $$r =1, 2, \ldots ,\infty$$.be a complete orthonormal set of vectors spanning the Hilbert space '¡.l for fermion $$\alpha$$. That is

$\sum_{r=1}^{\infty} | \phi_r \rangle \langle \phi _r | = 1_{\alpha} \label{3.2}$

$_{\alpha} \langle \phi_r | \phi_{s} \rangle_{\alpha} = \delta_{rs} \label{3.3}$

where $$1_{\alpha}$$ is the unit operator in the one-particle space for fermion number $$\alpha$$