Quantum mechanics of a free particle beyond differential equations

Abstract

With Feynman's path- integral method we can obtain the quantum mechanics of a quantum system like a free particle outside Schroedinger's method of differential equations and Heisenberg's method of algebra. The work involves obtaining the quantum propagator Kf, of the system which leads to summation over infinite number of paths. With Van Vleck's formula in one dimension, the classical propagator Kcl for a free particle is computed as the analytical result. This then serves as a yardstick for justifying the theoretical method used to compute the quantum propagator, kf, by direct path summation. The graphical display of the results shows that the Feynman – Schulman's checkerboard model used to enumerate the paths is reliable. Furthermore, this work shows that by windowing off a large number of paths and weighting the rest non uniformly we can compute the required propagator. The weights used in this case are the random and exponential window functions, wr and we respectively, which yield kwr and kwe to compare with kf. Key words: Propagator, action, path - integral, window function, free particle. (Global Journal of Pure and Applied Sciences: 2003 9(4): 561-566)