The WKB approximation for a linear potential and ceiling
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The physical problem this thesis deals with is a quantum system with linear potential driving a particle away from a ceiling (impenetrable barrier). This thesis will construct the WKB approximation of the quantum mechanical propagator. The application of the approximation will be for propagators corresponding to both initial momentum data and initial position data. Although 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 is less problematic because it is representable in exact form, and integration schemes (both numerical and analytic) to obtain probability densities are straightforward to implement. 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, impenetrable barrier. Research pertaining to this thesis includes determining all classical paths and constraints for the one-dimensional linear potential with ceiling, and using these equations to construct the classical action, and hence the WKB approximation. Also, evaluation of final quantum wave functions using numerical integration to check and better understand the approximation is part of the research. The results indicate that the validity of the WKB approximation depends on the type 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 to become vanishingly small in one representation of initial classical data, while not effecting the packets in another. The conclusion of this phenomenon is that the representation where the packets are not annihilated is the correct representation.
Zapata, Todd Austin (2007). The WKB approximation for a linear potential and ceiling. Master's thesis, Texas A&M University. Available electronically from