We present a simple numerical approach to compute dynamic correlation functions in quantum many-body systems at zero temperature. The method is based on a projective technique for the memory-function formalism, and requires as a starting point an independent evaluation of the ground-state energy and wave function. We illustrate the method with a one-dimensional model of spinless fermions and calculate the density-density, current-current, and single-particle correlation functions. The obtained spectral functions show different types of elementary excitations: particle-hole, soliton-antisoliton, exciton, etc. We compare our results with analytical results, and with those obtained through different approaches available in the literature.
In this paper we have solved analytically the Schrodinger equation for a microscopic particle in a one-dimensional box with two infinite walls, which the potential function inside it, has a linear form. Based on the solutions of this special quantum mechanical system, we have shown that as the quantum number approaches infinity the expectation values of microscopic particle position and square of the linear momentum are equal to the classical time average of particle position and its square of the linear momentum, respectively.