This document provides a continuation of the discussion on path integral approaches from a previous document. It begins by defining the matrix elements of the time evolution operator and explains how they give the probability amplitudes for a particle to transition between different states over time. It then shows how the time evolution operator and its matrix elements can be decomposed into multiple steps involving integrals over possible particle paths. The document derives expressions for the matrix elements and probability amplitudes involving position, momentum, and wavefunction terms. It provides equations for the Hamiltonian and time evolution operator in terms of these quantities. Finally, it takes the limit of many short time steps to express the probability amplitude as a continuous path integral over time.