The Feynman Matrix of the Harmonic Oscillator

Abstract

The Schrödinger equation allows to determine the state of a physical system, SF, at any instant t, expressed by a wave function \psi_t when the state \psi_{t_0} of the SF at a previous instant t_o < t is known. Feynman invented another way to determine \psi_t from a previous state \psi_{t_0}, resorting to the classical Lagrangian of the SF. This classical Lagrangian allows to construct a matrix K(t, t_o), such that psi_t = K(t,t_o)\psi_{t_0} . The construction of the matrix K(t, t_o); that is, of its elements K(t, x; t_o, x_0) = K(t,t_0)_{xx_0} requires a very special integration process that is only analytically feasible in a few cases, such as that of the free particle and the harmonic oscillator. Here, after introductory considerations, we present the details of the analytical calculation of the Feynman matrix for the harmonic oscillator.