Scaling-variational treatment of anharmonic oscillators

The scaling-variational method is applied to anharmonic-oscillator models with the Hamiltonian $H=p^2+h{x}^2+g{x}^{2k}$ to enable the discussion of two important aspects not previously analyzed. First, it is shown that the introduction of a scaling factor which is variationally optimized assures us of the correct dependence of the approximate eigenvalue with $g$. Second, it is shown that quantities ${\overline{E}}_n=〈{\phi }_n|H{\phi }_n〉$ are very good approximations to the exact eigenvalues whenever the trial function ${\phi }_n$ satisfies the quantum virial theorem.