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Journal of Mathematical Physics

Lower bounds for quantum energy levels through statistical mechanics methods

Luigi Nitti, M. Pellicoro, M. Villani

Published Apr 1, 1980• 5 citations
Publisher

Abstract

A perturbative variational method for the calculation of lower bounds to energy eigenvalues is given for a wide class of quantum mechanical systems. The method makes use of the quantum statistical partition function and exploits the inequalities satisfied by the generalized Padè approximants. The mathematical and physical content of our approach is illustrated through a standard model.

DOI
10.1063/1.524463

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International Journal of Modern Physics E•Jun 1, 2008•0 citations
A perturbative variational method for the calculation of lower bounds to energy eigenvalues is given for a wide class of quantum mechanical systems. The method makes use of the quantum statistical partition function and exploits the inequalities satisfied by the generalized Padè approximants. The mathematical and physical content of our approach is illustrated through a standard model.
A class of quasiclassical adiabatic approximations is described that enables upper bounds to be determined for the partition function of a system and, hence, lower bounds for its Helmholtz function. Based upon a neglect of the lack of commutativity of selected observables with others, these adiabatic approximations become exact in classical statistical mechanics. The effects of carrying out sequences of such adiabatic approximations and of relaxing intrinsic exchange-symmetry restrictions is to produce additional upper-bound expressions for the partition function. The lower-bound property of the adiabatically approximated Helmholtz function provides an alternative proof of the inequality that relates the lowest-energy eigenvalue of a system and that given by the Born-Oppenheimer approximation, as well as improvements of it.
The well known and analytically solvable Lipkin–Meshkov–Glick model was devised to test several many-body approximation approaches used in nuclear physics. Here, we use this model to illustrate a method that sets lower bounds for the ground state energy of a quantum Hamiltonian. The idea of the method consists of dividing the Hamiltonian into identical groups of terms that represent subdivisions in the particle population of the system, and then to make an appropriate use of the ground energy of one of these groups. The bounds clearly improve as the size of these groups increases.