Hamiltonian for $N$ Fermions

This is a preparation for the new method (Hartree-Fock) we learn in the next section where we deal with interactions between a large number of Fermions.

The Hamiltonian for $N$ Fermions is given by the generalization of the $N=2$ case, Eq. (III.2.17), and reads

$$\hat{H} = \hat{\mathcal H}_0+\hat{U}\equiv \sum_{i=1}^N \hat{H}_0^{(i)}+ \frac{1}{2}\sum_{i\ne j}^N U(\xi_i,\xi_j)$$

$$\hat{H}_0^{(i)} = -\frac{\hbar^2}{2m}\Delta_i+ V({\bf r}_i).$$ (2.1)