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Path Integrals and Quantum Mechanics
Umeå University, Faculty of Science and Technology, Department of Physics.
2015 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesisAlternative title
Banintegraler och Kvantmekanik (Swedish)
Abstract [en]

In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism. It is a generalization of the classical least action principle. The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state. Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral. We look in to the subject of two kinds of slit-experiments: The square slit, the single- and the double-Gaussian slit. Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics. It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral. Further applications of the path integral are discussed.

Abstract [sv]

I detta fördjupningsarbete undersöker vi en annan formalism av icke-relativistisk kvantmekanik kallad banintegral formalismen. Det är en generalisering av den klassiska verkansprincipen. Introduktionen till detta ämne börjar med konstruktionen av banintegralen i termer av sannolikhetsamplituder vars absolutbelopp i kvadrat ger sannolikheten av att finna ett system i ett särskilt tillstånd. Sedan visar vi att om Lagrangianen är av kvadratisk form så krävs endast en beräkning av den klassiska verkan förutom en tidsberoende normaliseringskonstant för att finna ett uttryck för banintegralen. Vi ser på två olika typer av spaltproblem: Den kantinga spalten, enkel- och dubbel Gaussisk spalt. Vi beräknar dessutom propagatorn för banor med restriktioner och applicerar detta på Aharonov-Bohm effekten, som visar att den klassiska vektorpotentialen som definierad i klassisk elektrodynamik har en fysikalisk mening i kvantmekaniken. Vi visar också ekvivalensen av banintegralformalismen med Schrödingerekvationen genom att härleda Schrödingerekvationen från banintegralen. Andra applikationer av banintegralen diskuteras.

Place, publisher, year, edition, pages
2015. , 44 p.
Keyword [en]
quantum, path integral, aharonov-bohm
National Category
Other Physics Topics
URN: urn:nbn:se:umu:diva-110026OAI: diva2:860492
Subject / course
Fysik C - Examensarbete
Educational program
Bachelor of Science in Physics and Applied Mathematics
Available from: 2015-10-12 Created: 2015-10-12 Last updated: 2015-10-12Bibliographically approved

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