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Simple example and demonstration

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).