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Mathematical aspects of the Feynman path integral

Boutros, Daniel William (2020) Mathematical aspects of the Feynman path integral. Bachelor's Thesis, Mathematics.

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Abstract

Path integration methods are of crucial importance to quantum mechanics and quantum field theory. There are multiple ways the path integral can be constructed, one method uses the link between the Fokker-Planck and the Langevin equations, as is covered in this thesis. This is related to the standard derivation of the path integral in physics, in which the time is discretised. We study the mathematical problems related to the Feynman path integral, in particular the impossibility of a Lebesgue-type measure on the space of paths. It is discussed how oscillatory integrals can be used to have a well-defined `integral' on an infinite dimensional space. This formalism is subsequently applied to both the harmonic and the anharmonic oscillator. We prove an infinite dimensional oscillatory integral exists and obtain a convergent expression for both these cases, under conditions. Examples of these conditions are the initial wavefunctions being Schwartz functions as well as conditions on the endtime, angular frequency and coupling constant. Finally, a possible approach to establish an oscillatory integral for the hydrogen atom is discussed. It is proven that the result is no longer independent of the sequence of projection operators, which is a key step towards a rigorous path integral for this system.

Item Type: Thesis (Bachelor's Thesis)
Supervisor name: Seri, M. and Boer, D.
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 14 Jul 2020 10:58
Last Modified: 14 Jul 2020 10:58
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/22669

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