Monte Carlo evaluation of Feynman Path Integrals in imaginary time and spherical polar coordinates

The Feynman Path Goes Monte Carlo

“…They investigated the ground state wave functions of simple one-dimensional problems (harmonic oscillator, square well, and Morse potential) and, theoretically, also addressed the problem of extracting information about excited energies and of simulating many particle problems. In a follow-up paper [11] they presented investigations of the Coulomb problem using Monte Carlo simulations of the path integral in polar coordinates. Not surprisingly, the singularity at the origin had to be avoided by artificial constraints and the authors admitted that a more rigorous justification of their procedure was called for.…”

Application of Monte Carlo simulation methods to quantum cosmology

“…The Gaussian integrations over momenta can be per- [15,351 formed to yield the Lagrangian form [15] Expression (2.11) is easy to simulate on the computer work directly with the Euclideanized Schrodinger equa- [13,14,20]. One constructs a discrete path with points tion regarded to be a diffusion equation [22,36].)…”

“…Most schemes take advantage of some version of the Metropolis importance sampling algorithm [41] (MA) to preferentially select paths which contribute significantly to the PI. In its simplest version [13], one moves a single point on the original path to define a new path. The Euclidean actions of the two paths are compared.…”

“…A fruitful methodology used throughout physics to study complicated many D F systems is quantum Monte Carlo (MC) simulation [12]. For example, in MC path-integral (PI) simulations [13,14], the Feynman PI formulation of quantum mechanics [15] is used to construct the quantum propagator (schematically) as K = I: exp(iS) , paths where S is the (Lorentzian) action for the system. Boundary conditions are implemented by the choice of paths (in the dynamical configuration space of the system) over which the (formal) sum is to be taken.…”