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.

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 difficult 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.

Thesis (MSc)--University of Stellenbosch, 2006. / Bibliography. Also available via the Internet.

Thesis (Ocean Engineer and M.S.)--Massachusetts Institute of Technology, 1988. / Supervised by George V. Frisk. "June 1988." Funding provided through MIT by the Office of Naval Research Fellowship Program. Includes bibliographical references (p. 99-103).

No description available.

We give a construction of an eigenstate for a non-critical level of the Hamiltonian function, and investigate the contribution of Morse critical points to the spectral decomposition. We compare the rigorous result with the series obtained by a perturbation theory. As an example the relation to the spectral asymptotics is discussed.

The main idea of this paper is inspired by a paper written together by my advisor Dr. Hang, Dr. Huang of the Industrial Technology Research Institute, and Dr. Chao of Institute of Applied Mechanics of National Taiwan University[quant-ph/0506153 v1,2005]. After some mathematical calculations we can extend the WKB approximation to treat position-dependent effective mass problem (PDEM). Then we did simulation on a model PDEM problem to compare the well-know closed form solution and the extended WKB approximation. We demonstrated that the extended WKB approximation not only can obtain the eigenvalues very accurately, but also is very useful to estimate the distribution of the wave function. We also found the modulation on the oscillations of wave function under PDEM by the extended WKB approximation.

The position-dependent effective mass (PDEM) problem is of enormous importance to the realization of the extended Wentzel-Kramers-Brillouin (WKB) approximation in bound state calculations for semiconductor heterostructures. By studying some model problems, we show that the extended WKB method provides good approximations for the bound states with the high eigenenergies. In addition, the effect of the smoothness of the effective mass distribution functions and potential barrier in the PDEM problems is discussed in our work. We found the precision can be affected by the effective mass and potential barrier in the PDEM.

The thesis is divided into two parts. The first part deals with radiative cor- rections in muonic hydrogen. The effect of vacuum polarization is studied, and the simplified derivation of the Wichmann-Kroll potential is presented. The en- ergy shift caused by vacuum polarization to the Lamb shift in muonic hydrogen is calculated and it agrees with results in literature. Further, the concept of the extended Bethe logarithm is introduced and its advantages are shown and used to calculate the combined self-energy vacuum polarization contribution to the Lamb shift in muonic hydrogen. The results given here are more accurate and somewhat different from others given in literature. In the second part, the ground-state en- ergy splitting due to the tunneling in a two-dimensional double-well potential is calculated. A systematic WKB expansion of the energy splitting is given. An in- terplay between curvature of the classical tunneling path and quantum nature of motion is observed. A series is found that describes systems with strong coupling like the proton transfer in malonaldehyde. The results show a strong sensitivity of the splitting on slight variations of the parameters entering the Hamiltonian linearly. This indicates a presence of quantum chaos in this problem. 1

Thesis (MSc (Physics))--University of Stellenbosch, 2006. / The aim of this thesis is the semi-classical implementation of Wegner’s flow equations and comparison with the well-established Wentzel-Kramers-Brillouin method. We do this by converting operators, in particular the Hamiltonian, into scalar functions, while an isomorphism with the operator product is maintained by the introduction of the Moyal product. A flow equation in terms of these scalar functions is set up and then approximated by expanding it to first order in ~. We apply this method to two potentials, namely the quartic anharmonic oscillator and the symmetric double-well potential. Results obtained via the flow equations are then compared with those obtained from the WKB method.