Studies in one dimension of the quantum crystal problem

Abstract

Various methods proposed for the quantum crystal problem are studied in one dimension, using a simple linear lattice whose particle mass, spacing, and Lennard-Jones potential parameters are chosen as the values for a normal rare gas crystal (Kr) or as those of a quantum crystal (He). It is found that the Hartree approximation to the cohesive energy of such a lattice fails for the model of solid Helium just as in three dimensions. The cluster expansion method of L. H. Nosanow is applied, using a parameterized correlation function, and the cohesive energy is considerably better than the Hartree result for the model Helium, though the system remains unbound. An integro-differential equation is derived for the single particle function from the first two terms of the Brueckner-Frohberg expansion, but the resulting equation gives no improvement over the Nosanow approximation at greatly increased cost in computing time. Frohberg's pair function equation is examined without his approximations and is found to be quite intractable, even in one dimension. An alternative to the cluster methods is developed for a special version of the one dimensional model. In this case, a positive integral operator can be found whose highest eigenvalue and eigenfunction characterize the system completely. This method provides a rigorous upper bound on the cohesive energy and we find that Nosanow's method actually results in an estimate of the cohesive energy which is lower than the rigorous upper bound. A brief discussion of the convergence of the cluster expansion, based on the behavior of the positive integral operator, is also included.