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Path Integral for the Hydrogen Atom: Solutions in two and three dimensions
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.
2016 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesisAlternative title
Vägintegral för Väteatomen : Lösningar i två och tre dimensioner (Swedish)
Abstract [en]

The path integral formulation of quantum mechanics generalizes the action principle of classical mechanics. The Feynman path integral is, roughly speaking, a sum over all possible paths that a particle can take between fixed endpoints, where each path contributes to the sum by a phase factor involving the action for the path. The resulting sum gives the probability amplitude of propagation between the two endpoints, a quantity called the propagator. Solutions of the Feynman path integral formula exist, however, only for a small number of simple systems, and modifications need to be made when dealing with more complicated systems involving singular potentials, including the Coulomb potential. We derive a generalized path integral formula, that can be used in these cases, for a quantity called the pseudo-propagator from which we obtain the fixed-energy amplitude, related to the propagator by a Fourier transform. The new path integral formula is then successfully solved for the Hydrogen atom in two and three dimensions, and we obtain integral representations for the fixed-energy amplitude.

Abstract [sv]

Vägintegral-formuleringen av kvantmekanik generaliserar minsta-verkanprincipen från klassisk mekanik. Feynmans vägintegral kan ses som en summa över alla möjliga vägar en partikel kan ta mellan två givna ändpunkter A och B, där varje väg bidrar till summan med en fasfaktor innehållande den klassiska verkan för vägen. Den resulterande summan ger propagatorn, sannolikhetsamplituden att partikeln går från A till B. Feynmans vägintegral är dock bara lösbar för ett fåtal simpla system, och modifikationer behöver göras när det gäller mer komplexa system vars potentialer innehåller singulariteter, såsom Coulomb--potentialen. Vi härleder en generaliserad vägintegral-formel som kan användas i dessa fall, för en pseudo-propagator, från vilken vi erhåller fix-energi-amplituden som är relaterad till propagatorn via en Fourier-transform. Den nya vägintegral-formeln löses sedan med framgång för väteatomen i två och tre dimensioner, och vi erhåller integral-representationer för fix-energi-amplituden.

Place, publisher, year, edition, pages
2016. , 55 p.
Keyword [en]
physics, quantum mechanics, path integral, feynman, hydrogen atom, propagator
Keyword [sv]
fysik, kvantmekanik, vägintegral, feynman, väteatom, propagator
National Category
Physical Sciences
Identifiers
URN: urn:nbn:se:kau:diva-44425OAI: oai:DiVA.org:kau-44425DiVA: diva2:947133
Subject / course
Physics
Presentation
2016-06-07, 21D304, Karlstads universitet, Kronoparken, Karlstad, 13:00 (English)
Supervisors
Examiners
Available from: 2016-08-09 Created: 2016-07-07 Last updated: 2016-08-09Bibliographically approved

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