The finite size scaling ansatz is combined with the variational method to extract information about critical behavior of quantum Hamiltonians. This approach is based on taking the number of elements in a complete basis set as the size of the system. As in statistical mechanics, the finite size scaling can then be used directly in the Schrodinger equation. This approach is general and gives very accurate results for the critical parameters, for which the bound-state energy becomes absorbed or degenerate with a continuum. To illustrate the applications in quantum calculations, we present detailed calculations for both short- and long-range potentials.