Basic Quantum Monte Carlo, repetition from last week

We start with the variational principle. Given a hamiltonian $H$ and a trial wave function $\Psi_T(\boldsymbol{R};\boldsymbol{\alpha})$, the variational principle states that the expectation value of $\cal{E}[H]$, defined through

$$\cal {E}[H] = \frac{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R};\boldsymbol{\alpha})H(\boldsymbol{R})\Psi_T(\boldsymbol{R};\boldsymbol{\alpha})} {\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R};\boldsymbol{\alpha})\Psi_T(\boldsymbol{R};\boldsymbol{\alpha})},$$

is an upper bound to the ground state energy $E_0$ of the hamiltonian $H$, that is

$$E_0 \le {\cal E}[H].$$

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