Theses and Dissertations

Issuing Body

Mississippi State University

Advisor

Novotny, Mark A.

Committee Member

Arnoldus, Henk F.

Committee Member

Rupak, Gautam

Committee Member

Clay, R. Torsten

Committee Member

Koshka, Yaroslav

Date of Degree

5-12-2023

Document Type

Dissertation - Open Access

Major

Physics

Degree Name

Doctor of Philosophy (Ph.D)

College

College of Arts and Sciences

Department

Department of Physics and Astronomy

Abstract

A prescription is given to obtain some exact results for certain external potentials �� (r) of the time-independent Gross-Pitaevskii and Schrodinger equations. The study motivation is the ability to program �� (r) experimentally in cold atom Bose-Einstein condensates. Rather than derive wavefunctions that are solutions for a given �� (r), we ask which �� (r) will have a given pdf (probability density function) �� (r). Several examples in 1 dimension (1D), 2 dimensions (2D), and 3 dimensions (3D) are presented for well-known pdfs in the position space. Exact potentials with zero, one and two walls are obtained and explained in detail. Apart from position space, the method is also applicable to obtain exact solutions for the Time-independent Schr¨odinger equation (TISE) and Gross-Pitaevskii equation (GPeq) for pdfs in momentum space. For this, we derived the potentials which are generated from the pdfs of the hydrogen atom in the real space as well as in the momentum space.

However, the method was also extended for the time-dependent case. The prescription is also applicable to solve time-dependent pdfs. The aim is to find the ��(r, ��) which generates the pdf ��(r, ��). As a special case, we tested our method by studying the well known case for the Gaussian wave packet in 1D with zero potential ��(��, ��) = 0.

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