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arXiv:2202.09232

Feynman's Sum-over-Paths method applied in wave optics and for calculating the quantum probability current

Josef Joerg

Published Feb 18, 2022• 0 citations
arXivHTMLPDF

Abstract

Based on the Sum-over-Paths approach of Richard Feynman, an integration method for calculating wave phase vectors is derived. The diffraction and interference patterns of various slit masks can be calculated from such phase vectors. The results obtained match with those computed using the more complex Fresnel integrals. Babinet's principle for phases is presented for the first time. As an example, the diffraction pattern of a double slit can also be calculated by subtracting the phase vectors of a smaller slit from those of a larger one. This method is also demonstrated by using a diffraction calculation tool when an obstacle is positioned behind a double slit. The approach of phase vector integration is expanded to calculate the quantum probability current. Based on the Hamilton-Jacobi- and the stationary Schrödinger- equation, the quantum probability current is calculated and its continuity is shown. The time-dependent differential equation of the probabilty current velocity is numerically solved. For a large number of integrations it is statistically shown that the distribution of the flux per area is consistent with the probability density profile. The quantum probability current is also simulated for photons traversing an asymmetrical double slit and the special results are discussed.

DOI
10.48550/arxiv.2202.09232

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Based on the Sum-over-Paths approach of Richard Feynman, an integration method for calculating wave phase vectors is derived. The diffraction and interference patterns of various slit masks can be calculated from such phase vectors. The results obtained match with those computed using the more complex Fresnel integrals. Babinet's principle for phases is presented for the first time. As an example, the diffraction pattern of a double slit can also be calculated by subtracting the phase vectors of a smaller slit from those of a larger one. This method is also demonstrated by using a diffraction calculation tool when an obstacle is positioned behind a double slit. The approach of phase vector integration is expanded to calculate the quantum probability current. Based on the Hamilton-Jacobi- and the stationary Schrödinger- equation, the quantum probability current is calculated and its continuity is shown. The time-dependent differential equation of the probabilty current velocity is numerically solved. For a large number of integrations it is statistically shown that the distribution of the flux per area is consistent with the probability density profile. The quantum probability current is also simulated for photons traversing an asymmetrical double slit and the special results are discussed.
This paper provides a complete physical model associated with mathematical analysis and formulation that is based on Schrödinger equation and Feynman path integral. This model answers to single Photon (electron) double slit experiment with high degree of accuracy. It also gives a new method of analysis and formulation for Fresnel and Fraunhofer diffractions that are completely according to the Schrödinger equation and Feynman path concept. The results of these methods are precisely in agreement with the near field and far field diffraction experiments. In other words the new formulation in all above mentioned phenomena i.e. single photon (electron) double slit experiment, Fresnel and Fraunhofer diffractions are completely confirmed by lab experiments(measurements). Lab experiments mean common intensity and pattern experiments resulting from Fresnel (near field) diffraction, Fraunhofer (far field) diffraction and patterns resulting in single photon (electron) lab experiment. The resultant formulas are sketched and calculated by matlab program, the resultant graphs and values are compared with lab experiments in each section. This analysis substitutes wave function and probability density function concepts instead of interference of light beams emitted from infinitesimal pinholes or classical interpretation of near field diffraction and far field diffraction phenomena. The ultimate goal of this paper is to give a very accurate answer to the single particle double slit experiment and also to unify diffraction concept of classical optics in to quantum mechanics, deleting the traditional concept of phases in classical optics and substituting the concept of deflection of Feynman paths. The above mentioned analysis may be a glad tiding to a reliable method to unify classical optics in to quantum mechanics.
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