Date of Award

8-31-2015

Document Type

Open Access Thesis

Degree Name

Master of Science (MS)

Department

Physics, Applied

First Advisor

Maxim Olchanyi

Second Advisor

Adolfo del Campo

Third Advisor

Bala Sundaram

Abstract

Creation of three-dimensional matter waves, the three-dimensional analog of one-dimensional solitons, has been a goal of experimental physics for some time. A recent proposal has suggested that changing the dispersion law from quadratic to quartic for ultra cold atoms in a shaken lattice should allow for the creation of these objects. In this thesis, we develop the theoretical basis for quantum mechanics with a quartic dispersion law. The probability current functional is constructed from the corresponding time-dependent Schrödinger equation, and used to derive the junction conditions that connect the derivatives of the wavefunction on one side of a potential discontinuity to the ones on the other side. Reflection and transmission amplitudes are determined for scattering problems concerning both step potentials and rectangular barriers/wells. For sufficiently narrow barriers/wells, we show that a -potential constitutes a simple but reliable model for the scatterer. The scattering properties of wide barriers/wells are consistent with the predictions of the classical theory. Finally, we find the eigenstates and eigenenergies of a particle in an infinitely deep well. A simple approximate expression for the high-energy spectrum is obtained; it is found to be fully consistent with Weyl’s law. Our results should aid in the development of experimental systems capable of creating and sustaining self-supporting, mobile, three-dimensional matter waves.

Comments

Free and open access to this Campus Access Thesis is made available to the UMass Boston community by ScholarWorks at UMass Boston. Those not on campus and those without a UMass Boston campus username and password may gain access to this thesis through resources like Proquest Dissertations & Theses Global or through Interlibrary Loan. If you have a UMass Boston campus username and password and would like to download this work from off-campus, click on the "Off-Campus UMass Boston Users" link above.

Download

Share

COinS