The propagator for the one dimensional Lagrangian is calculated in the frame work of path integrals, via Matrix algebra, and by the polygonal paths approach. When m = 0, the propagator is reduced to the distribution function for a Brownian particle.
Concepts of Lagrangian and Hamiltonian path integrals Wiener integrals and Brownian motion representation of the statistical mechanical density matrix as a path integral propagators for local and non-local quadratic actions path integrals in polar and curvilinear coordinates and group manifolds local and global time transformation techniques approximation and numerical methods of evaluating path integrals and their applications to polaron bi polaron spectrum of disordered systems path integrals with topological constraints and their applications
As an alternative formulation of quantum mechanics, path integral is based on the notion of transition amplitude which gives the wave function of a quantum system at a time t f by acting on the wave function at an earlier time ti.We show that for a general quadratic form for the Lagrangian of the system, transition amplitude has the form f (t f -ti)e i h S class., where S class. is the classical action.We then present an algebraic method to evaluate the function f (t f -ti) without refereing to the path integral calculations.We examine the presented method to the cases of free particle and harmonic oscillator and obtain their propagators.