Master of Science
B. N. Narahari Achar
John W. Hanneken
Donald R. Franceschetti
The Schrödinger equation which is fractional in space only has been previously derived by Laskin in terms of the Riesz fractional derivative, and the familiar Schrödinger equation is recovered when the fractional order equals 2. The objective of the present thesis is to derive a Schrödinger equation which is fractional in time, such that the standard Schrödinger equation is recovered when the fractional order equals unity, using the path integral method of Feynman. This time-fractional Schrödinger equation will be solved for a free particle, and the fractional wave packet and Green's function solutions will be obtained. Other topics such as the uncertainty product of a Gaussian under fractionalized time will be discussed.It will be shown that the action integral itself must be fractionalized to the same order as the Lagrangian used for the Feynman path integral kernel, in order to maintain the correct order of the fractional derivative in the resulting Schrödinger equation. This suggests that all fractional classical mechanics problems involving Hamilton's principle must be treated in this way as well.In order to maintain correct units and the normalization condition for all fractional orders, it is suggested that space and time be fractionalized as a pair, with a related fractal index, suggesting a fundamental relationship between fractal space and fractal time similar to standard spacetime.
Dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.
Yale, Bradley Thomas, "Path Integral Approach to Time-Fractional Quantum Mechanics" (2012). Electronic Theses and Dissertations. 587.