Self-consistent Hartree-Fock approach to Phase Transitions

Hartree-Fock approach to Phase Transitions

Hartree-Fock in second quantized form

The interaction energy operator for a two body system with pairwise interactions is given by

$\nu ={\frac {1}{2}}\sum _{\sigma \sigma ^{\prime }}\int d^{3}rd^{3}r^{\prime }v({\vec {r}}-{\vec {r}}^{\prime })\psi _{\sigma }^{\dagger }({\vec {r}})\psi _{\sigma ^{\prime }}^{\dagger }({\vec {r}}^{\prime })\psi _{\sigma ^{\prime }}({\vec {r}}^{\prime })\psi _{\sigma }({\vec {r}})$

In the above operator, it is important to note the order of the operators. Using

(19.58)

we can write the pairwise interaction Hamiltonian for particles of mass m in second quantized formalism as

Failed to parse (unknown function "\grad"): {\displaystyle H = \sum_{\sigma \sigma^{\prime}} \int d^{3}r \frac{1}{2m} \grad \psi_{\sigma}^{\dagger}(\vec{r}) \dot \grad \psi_{\sigma}(\vec{r}) + \frac{1}{2} \sum_{\sigma \sigma^{\prime}} \int d^{3}r d^{3}r^{\prime} v(\vec{r}-\vec{r}^{ \prime}) \psi_{\sigma}^{\dagger}(\vec{r}) \psi_{\sigma^{\prime}}^{\dagger}(\vec{r}^{ \prime}) \psi_{\sigma^{\prime}}(\vec{r}^{ \prime}) \psi_{\sigma}(\vec{r}) }