Let us illustrate what is needed in our calculations using a simple example, the harmonic oscillator in one dimension. For the harmonic oscillator in one-dimension we have a trial wave function and probability
$$ \psi_T(x;\alpha) = \exp{-(\frac{1}{2}\alpha^2x^2)}, $$which results in a local energy
$$ \frac{1}{2}\left(\alpha^2+x^2(1-\alpha^4)\right). $$We can compare our numerically calculated energies with the exact energy as function of \( \alpha \)
$$ \overline{E}[\alpha] = \frac{1}{4}\left(\alpha^2+\frac{1}{\alpha^2}\right). $$