Evaluation Of Path Integrals Research Articles

Path-integral evaluation of the space–time propagator for quadratic Hamiltonian systems

Path-integral methods are used to derive an exact expression for the space–time propagator for systems with quadratic Hamiltonians. For a certain subclass of such systems, the result is reduced to a simplified closed form. The propagators for several illustrative elementary cases are exhibited in detail.

Direct evaluation of statistical mechanical path integrals

The discrete step Wiener integral of Trotter formula approximation to the density operator of equilibrium statistical mechanics is evaluated for the li for the harmonic oscillator potential. Explicit results show the speed of convergence of the approximation.

Summation over Feynman Histories in Polar Coordinates

Use of polar coordinates is examined in performing summation over all Feynman histories. Several relationships for the Lagrangian path integral and the Hamiltonian path integral are derived in the central-force problem. Applications are made for a harmonic oscillator, a charged particle in a uniform magnetic field, a particle in an inverse-square potential, and a rigid rotator. Transformations from Cartesian to polar coordinates in path integrals are rather different from those in ordinary calculus and this complicates evaluation of path integrals in polars. However, it is observed that for systems of central symmetry use of polars is often advantageous over Cartesians.