Classical paths and the WKB approximation

Tapan Kumar Das

K. T. Hecht

Barry R. Holstein, Arthur R. Swift

The WKB approximation is demonstrated to be formally identical to the semiclassical limit of the Feynman path integral propagator. In the limit that the path integral propagator is dominated by the classical particle trajectory, it is identical to the propagator calculated directly from WKB wave functions in the stationary phase approximation.

N. G. van Kampen

Todd Zapata

The physical problem this thesis deals with is a quantum system with linear\npotential driving a particle away from a ceiling (impenetrable barrier). This thesis\nwill construct the WKB approximation of the quantum mechanical propagator. The\napplication of the approximation will be for propagators corresponding to both initial\nmomentum data and initial position data.\nAlthough the analytic solution for the propagator exists, it is an indefinite integral of Airy functions and di±cult to use in obtaining probability densities by numerical integration or other schemes considered by the author. The WKB construction\nis less problematic because it is representable in exact form, and integration schemes\n(both numerical and analytic) to obtain probability densities are straightforward to\nimplement. Another purpose of this thesis is to be a starting point for the construction of WKB propagators with general potentials but the same type of boundary,\nimpenetrable barrier.\nResearch pertaining to this thesis includes determining all classical paths and\nconstraints for the one-dimensional linear potential with ceiling, and using these\nequations to construct the classical action, and hence the WKB approximation. Also,\nevaluation of final quantum wave functions using numerical integration to check and\nbetter understand the approximation is part of the research.\nThe results indicate that the validity of the WKB approximation depends on the\ntype of classical paths (i.e. the initial data of the path) used in the construction. Specifically, the presence of the ceiling may cause the semi-classical wave packets\nto become vanishingly small in one representation of initial classical data, while not\neffecting the packets in another. The conclusion of this phenomenon is that the\nrepresentation where the packets are not annihilated is the correct representation.

D. A. Goodings, T. Szeredi

The path integral formulation of quantum mechanics in the semiclassical or WKB approximation provides a physically intuitive way of relating a classical system to its quantum analog. A fruitful way of studying quantum chaos is based upon applying the Gutzwiller periodic orbit sum rule, a result derived by the path integral method in the WKB approximation. This provides some motivation for learning about path integral techniques. In this paper a pedagogical example of the path integral formalism is presented in the hope of conveying the basic physical and mathematical concepts. The ‘‘quantum bouncer’’ is studied—the quantum version of a particle moving in one dimension above a perfectly reflecting surface while subject to a constant force directed toward the surface. The classical counterpart of this system is a ball bouncing on a floor in a constant gravitational field, collisions with the floor being assumed to be elastic. Path integration is used to derive the energy eigenvalues and eigenfunctions of the quantum bouncer in the WKB or semiclassical approximation. The results are shown to be the same as those obtained by solving the Schrödinger equation in the same approximation.