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
Abstract The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.