On two-component Bose-Einstein condensates in a ring
(2019) In Bachelor’s Theses in Mathematical Sciences MATK01 20181Mathematics (Faculty of Sciences)
- Abstract
- A Bose-Einstein condensate is a type of gas consisting of one or more types of particles called bosons which are cooled to a temperature very close the absolute zero. Under these conditions the particles all start to occupy their lowest quantum state. Once in these states it is necessary to use quantum physics to describe the behaviour of the particles by the means of a wave function which describes the probability of finding the particles in different locations. In this thesis we will study a gas consisting of two components. The wave function satisfies the Schrödinger equation, which is a linear partial differential equation. The gas will be considered to be confined within a thin ring with a cross section small enough to treat it as one... (More)
- A Bose-Einstein condensate is a type of gas consisting of one or more types of particles called bosons which are cooled to a temperature very close the absolute zero. Under these conditions the particles all start to occupy their lowest quantum state. Once in these states it is necessary to use quantum physics to describe the behaviour of the particles by the means of a wave function which describes the probability of finding the particles in different locations. In this thesis we will study a gas consisting of two components. The wave function satisfies the Schrödinger equation, which is a linear partial differential equation. The gas will be considered to be confined within a thin ring with a cross section small enough to treat it as one dimensional. By using the mean field approximation we're able to consider a much simpler model than the individual particle interaction and reduce the many-body problem to a one-body problem. However, the linear Schrödinger equation is replaced by the non-linear Schrödinger equation. We will also investigate the mean-field yrast spectrum, where these states are the ones with minimum energy for a given angular momentum. The existence of a minimum gives the possibility of having persistent currents as argued in previous research. In order to identify the yrast states, we first look for critical points of the energy under the constraints of total probability mass and angular momentum using a special Ansatz. We then try to determine if they are minimizers using analytic and numerical methods. (Less)
- Popular Abstract (Swedish)
- Vi har studerat en matematisk modell för ett Bose-Einstein kondensat bestående av två komponenter. Gasen är uppbyggd av bosoner, t.ex. fotoner eller atomer som har en jämn summa av protoner, neutroner och elektroner. Denna typ av gas uppvisar en del intressanta egenskaper vid väldigt låga temperaturer nära den absoluta nollpunkten, främst att den är superflytande, vilket upptäcktes redan 1938 i flytande helium. Detta innebär att gasen påverkas väldigt lite av friktion, och skulle kunna ge ett alternativ till att transportera information liknande ett superledande material. Allt eftersom temperaturen sänks så antar atomerna sina lägsta tillåtna kvantmekaniska tillstånd. Genom att placera gasen i ett magnetfält kan man fånga atomerna i en... (More)
- Vi har studerat en matematisk modell för ett Bose-Einstein kondensat bestående av två komponenter. Gasen är uppbyggd av bosoner, t.ex. fotoner eller atomer som har en jämn summa av protoner, neutroner och elektroner. Denna typ av gas uppvisar en del intressanta egenskaper vid väldigt låga temperaturer nära den absoluta nollpunkten, främst att den är superflytande, vilket upptäcktes redan 1938 i flytande helium. Detta innebär att gasen påverkas väldigt lite av friktion, och skulle kunna ge ett alternativ till att transportera information liknande ett superledande material. Allt eftersom temperaturen sänks så antar atomerna sina lägsta tillåtna kvantmekaniska tillstånd. Genom att placera gasen i ett magnetfält kan man fånga atomerna i en ring-potential där man sedan kan försätta kondensatet i rotation och studera superfluiditeten. I kvantmekaniken beskrivs gasen med hjälp av en vågfunktion som anger sannolikheten för att hitta partiklarna i olika positioner. Vi använder medelfältsapproximationen där flerkroppsproblemet ersätts med ett enkroppsproblem. Vågfunktionen uppfyller då en icke-linjär differential ekvation. Vi löser denna och undersöker vilka av lösningarna som ger lägst energi. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/8969891
- author
- Sköld, Marcus LU
- supervisor
-
- Erik Wahlén LU
- organization
- course
- MATK01 20181
- year
- 2019
- type
- M2 - Bachelor Degree
- subject
- publication/series
- Bachelor’s Theses in Mathematical Sciences
- report number
- LUNFMA-4081-2018
- ISSN
- 1654-6229
- other publication id
- 2018:K28
- language
- English
- id
- 8969891
- date added to LUP
- 2019-03-18 14:17:07
- date last changed
- 2019-03-18 14:17:07
@misc{8969891,
abstract = {{A Bose-Einstein condensate is a type of gas consisting of one or more types of particles called bosons which are cooled to a temperature very close the absolute zero. Under these conditions the particles all start to occupy their lowest quantum state. Once in these states it is necessary to use quantum physics to describe the behaviour of the particles by the means of a wave function which describes the probability of finding the particles in different locations. In this thesis we will study a gas consisting of two components. The wave function satisfies the Schrödinger equation, which is a linear partial differential equation. The gas will be considered to be confined within a thin ring with a cross section small enough to treat it as one dimensional. By using the mean field approximation we're able to consider a much simpler model than the individual particle interaction and reduce the many-body problem to a one-body problem. However, the linear Schrödinger equation is replaced by the non-linear Schrödinger equation. We will also investigate the mean-field yrast spectrum, where these states are the ones with minimum energy for a given angular momentum. The existence of a minimum gives the possibility of having persistent currents as argued in previous research. In order to identify the yrast states, we first look for critical points of the energy under the constraints of total probability mass and angular momentum using a special Ansatz. We then try to determine if they are minimizers using analytic and numerical methods.}},
author = {{Sköld, Marcus}},
issn = {{1654-6229}},
language = {{eng}},
note = {{Student Paper}},
series = {{Bachelor’s Theses in Mathematical Sciences}},
title = {{On two-component Bose-Einstein condensates in a ring}},
year = {{2019}},
}