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The Henstock-Kurzweil-Feynman-Pardy Integral in Quantum Physics

Miroslav Pardy

Published May 1, 2019• 0 citations
OpenAlex

Abstract

The Feynman integral is generalised so as to involve the random fluctuations of vacuum, from this integral the generalized Schroedinger equation is derived and the energy spectrum for the Coulomb potential determined.

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The Schrödinger equation is fundamental in quantum mechanics as it makes it possible to determine the wave function from energies and to use this function in the mean calculation of variables, for example, as the most likely position of a group of one or more massive particles. In this paper, we present a survey on some theories involving the Schrödinger equation and the Feynman path integral. We also consider a Feynman–Kac-type formula, as introduced by Patrick Muldowney, with the Henstock integral in the description of the expectation of random walks of a particle. It is well known that the non-absolute integral defined by R. Henstock “fixes” the defects of the Feynman integral. Possible applications where the potential in the Schrödinger equation can be highly oscillating, discontinuous or delayed are mentioned in the end of the paper.
In the setting of Cameron and Storvick's recent theory we show that the solution of an integral equation formally equivalent to the Schroedinger equation is expressible as the analytic Feynman integral of a function on ^-dimensional Wiener space of the form F(X) = exp{/ o '0(ίs, X(s) + I) ds}ψ(X(t\+ ?).Here Xis an Revalued continuous function on [0, /] such that Z(0) = 0, ?G R\ and ψ and θ(s, ) are Fourier-Stieltjes transforms.REMARK 3.2.Corollary 3.2 and Lemma 3.2 will yield the measurability properties that we will need as we continue.For example, let θ be in % and for λ > 0 and fin W let G: [0, ί 0 ] X C[0, t 0 ] -> C be defined by The Pacific
The path integral is a powerful tool for studying quantum mechanics because it has the merit of establishing the connection between classical mechanics and quantum mechanics. This formalism quickly gained prominence in various fields of theoretical physics, including its generalization to quantum field theory, quantum mechanics, and statistical physics. Using the Feynman propagator, we can calculate the partition function, the free energy, wave functions, and the energy spectrum of the considered physical system. Moreover, the Feynman formalism finds broad applications in geophysics and in the field of financial sciences.