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

TL;DR

This paper extends Feynman's sum-over-paths method to wave optics and quantum probability currents, providing a new integration approach for phase vectors that simplifies diffraction calculations and offers insights into quantum flux behavior.

Contribution

It introduces a novel phase vector integration method based on Feynman's approach, applicable to wave optics and quantum probability currents, including Babinet's principle for phases.

Findings

Accurate diffraction patterns matching Fresnel integrals.

First presentation of Babinet's principle for phases.

Quantum probability current distribution aligns with probability density.

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 Schr\"odinger- 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.