.

The Hartree-Fock equations in the position representation are

$$\left[ \hat{H_0} + {\sum_i \int d{\bf r'} \vert\psi_{\nu_{i}}({\bf r'})\vert^2 U(\vert{r-r'}\vert)} \right] \psi_{\nu_{j}}({\bf r})$$

$$- {\sum_i \int d{\bf r'} \psi_{\nu_{i}}^*({\bf r'}) U(\vert{r-r'}... ...t) \psi_{\nu_{j}}({\bf r'}) \psi_{\nu_{i}}({\bf r}) \delta_{\sigma_i \sigma_j}} = \varepsilon_j \psi_{\nu_{j}}({\bf r}) .$$ (4.2)

In this problem, we consider the case where the interaction is constant, $U$=const.

a) Simplify the Hartree-Fock equations by using the orthonormality of the $\psi_{\nu_{i}}$.

b) Thus solve the Hartree-Fock equations for the ground state wave function $\vert\Psi\rangle_{\rm HF}$ explicitly.

c) Using b), calculate the Hartree-Fock ground state energy

$$E_{\Psi}=\frac{1}{2}\sum_{i=1}^{N} \left[ \varepsilon_i+ \langle\nu_{i}\vert\hat{H}_0 \vert \nu_{i}\rangle\right].$$ (4.3)

d) Compare the Hartree-Fock ground state energy $E_{\Psi}$ with the exact ground state energy $E_0$ from the exact solution of the previous problem and briefly discuss your result.