Finite Size Scaling for Criticality of the Schrödinger Equation

By solving the Schrödinger equation one obtains the whole energy spectrum, both the bound and the continuum states. If the Hamiltonian depends on a set of parameters, these could be tuned to a transition from bound to continuum states. The behavior of systems near the threshold, which separates bound-states from continuum states, is important in the study of such phenomenon as: ionization of atoms and molecules, molecule dissociation, scattering collisions and stability of matter. In general, the energy is non-analytic as a function of the Hamiltonian parameters or a bound-state does not exist at the threshold energy. The overall goal of this chapter is to show how one can predict, generate and identify new class of stable quantum systems using large-dimensional models and the finite size scaling approach. Within this approach, the finite size corresponds not to the spatial dimension but to the number of elements in a complete basis set used to expand the exact eigenfunction of a given Hamiltonian. This method is efficient and very accurate for estimating the critical parameters, {λi}, for stability of a given Hamiltonian, H(λi). We present two methods of obtaining critical parameters using finite size scaling for a given quantum Hamiltonian: The finite element method and the basis set expansion method. The long term goal of developing finite size scaling is treating criticality from first principles at quantum phase transitions. In the last decade considerable attention has concentrated on a new class of phase transitions, transitions which occur at the absolute zero of temperature. These are quantum phase transitions which are driven by quantum fluctuations as a consequence of Heisenberg’s uncertainty principle. These new transitions are tuned by parameters in the Hamiltonian. Finite size scaling might be useful in predicting the quantum critical parameters for systems going through quantum phase transitions.