Finite size scaling in quantum mechanics

The finite size scaling ansatz is combined with the variational method to e xtract information about critical behavior of quantum Hamiltonians. This ap proach 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 scali ng can then be used directly in the Schrodinger equation. This approach is general and gives very accurate results for the critical parameters, for wh ich the bound-state energy becomes absorbed or degenerate with a continuum. To illustrate the applications in quantum calculations, we present detaile d calculations for both short- and long-range potentials.