Continuum limit Tonks-Girardeau matrix elements. Part I. The ground state and the uniform density state

Abstract

The Tonks-Girardeau model is a quantum mechanical model of N impenetrable bosons in 1+1 dimensions. A Vandermonde determinant provides the exact N -particle wave function of the ground state, or equivalently the matrix elements with respect to position eigenstates. We consider the large N limit of these matrix elements. We present a binning prescription which calculates the leading terms of the matrix elements in a time which is independent of N, and so is suitable for this limit. In this sense, it allows one to solve for the ground state of a strongly coupled continuum quantum field theory in the field eigenstate basis. As examples, we calculate the matrix elements with respect to states with uniform density and also states consisting of two regions with distinct densities.