Finite Size Scaling in Quantum Mechanics

Abstract

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. I. Introduction In statistical mechanics, the singularities in thermodynamic functions associated with a critical point occur only in the thermodynamic limit, when all the dimensions of the system under consideration tend to infinity. Strictly speaking, there are no phase transitions in a finite system at nonzero temper- ature, and yet, experiments as well as numerical calculations all use finite systems. 1 To address this problem, the finite size scaling method was formulated by Fisher 2 and others 3 to extrapolate information obtained from a finite system to the thermodynamic limit. In quantum mechanics, when using variation methods, one encounters the same finite size problem in studying the critical behavior of a quantum Hamiltonian H(I1,...,Ik) as a function of its set of parameters {Ii}. In this context, critical means the values of {Ii} for which a bound-state energy is nonanalytic. In many cases, as in this study, this critical point is the point where a bound-state energy becomes absorbed or degenerate with a continuum. In this case, 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 wave function of a given Hamiltonian. Recently, we used the finite size scaling and phenomenologi- cal renormalization equations for calculations of the critical charges for two- 4,5 and three-electron systems. 6 This approach is based on taking the lowest eigenvalues of a quantum Hamiltonian as the leading eigenvalues of a transfer matrix of a classical pseudosystem. In this paper we will assume that there exists a scaling function for the truncated mean value of a given operator, and with the help of the Hellmann-Feynman theorem we can obtain a direct finite size scaling approach to the Schrodinger equation. 7 This approach is general and can be used to study critical behavior of a quantum Hamiltonian as a function of its parameters. To illustrate this approach, we include detailed calculations for the critical parameters for two cases with qualitatively different behavior: one with short-range interaction, the Yukawa potential, and one with a long-range interaction, the inverse power law potential.