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Journal of Longyan University

Comparing Path Integrals with Riemann Integrals

Lai Guo-zhon

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

From double slit diffraction experiment and the image of interference choice in quantum mechanics,the idea of is introduced.Using the sum of probability breadth and comparison with Riemann Integrals,the idea and method of path integrals are vividly explained.

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Abstract Basic explanations of the double slit diffraction phenomenon include a description of waves that emanate from two slits and interfere. The locations of the interference minima and maxima are determined by the phase difference of the waves. An optical wave, which has a wavelength and propagates a distance accumulates a phase of A matter wave, also having wavelength that propagates the same distance accumulates a phase of which is a factor of two different from the optical case. Nevertheless, in most situations, the phase difference, for interfering matter waves that propagate distances that differ by is approximately which is the same value computed in the optical case. The difference between the matter and optical case hinders conceptual explanations of diffraction from two slits based on the matter–optics analogy. In the following article we provide a path integral description for matter waves with a focus on conceptual explanation. A thought experiment is provided to illustrate the validity range of the approximation .

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

Josef Joerg

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.

arXiv:2202.09232•Feb 18, 2022•0 citations
Abstract Basic explanations of the double slit diffraction phenomenon include a description of waves that emanate from two slits and interfere. The locations of the interference minima and maxima are determined by the phase difference of the waves. An optical wave, which has a wavelength and propagates a distance accumulates a phase of A matter wave, also having wavelength that propagates the same distance accumulates a phase of which is a factor of two different from the optical case. Nevertheless, in most situations, the phase difference, for interfering matter waves that propagate distances that differ by is approximately which is the same value computed in the optical case. The difference between the matter and optical case hinders conceptual explanations of diffraction from two slits based on the matter–optics analogy. In the following article we provide a path integral description for matter waves with a focus on conceptual explanation. A thought experiment is provided to illustrate the validity range of the approximation .
One of the important classic experiments that led to the fundamental difference between quantum and classical mechanics is the double slit experiment, which is interesting with respect to the path integral formalism since it leads to a conceptual motivation for introducing it. From Quantum Mechanics, the probability amplitude can also be determined from the Schrodinger equation as it describes the wave properties of the particles. From Quantum Mechanics, the probability amplitude (square of the probability function or wave function) can also be determined from the Schrodinger equation as it describes the wave properties of the particles. Here, the role of the trajectory of a point-like particle, formally is revived in a way compatible with the indetermination principle and is indeed the path integral approach to quantum mechanics.
In this article we present an analytic solution of the famous problem of diffraction and interference of electrons through one and two slits (for simplicity, only the one-dimensional case is considered). In addition to exact formulas, we exhibit various approximations of the electron distribution which facilitate the interpretation of the results. Our derivation is based on the Feynman path integral formula and this work could therefore also serve as an interesting pedagogical introduction to Feynman's formulation of quantum mechanics for university students dealing with the foundations of quantum mechanics.
The probabilistic interference fringes observed in the double slit experiment vividly demonstrate the quantum superposition principle, yet they also highlight a fundamental conceptual challenge: the relationship between a system before and after the measurement. According to Copenhagen interpretation, an unobserved quantum system evolves continuously based on the Schrodinger equation, whereas observation induces an instantaneous collapse of the wave function to an eigenstate. This contrast between continuous evolution and sudden collapse renders the single particle behavior particularly enigmatic, especially given that quantum mechanics itself is constructed upon the statistical behavior of ensembles rather than individual entities. In this study, we introduce a Double Slit Diffraction Surrogate Model DSM based on deep learning, designed to capture the mapping between wave functions and probability distributions. The DSM explores multiple potential propagation paths and adaptively selects optimal transmission channels using gradient descent, forming a backbone for the information through the network. By comparing the interpretability of paths and interference, we propose an intuitive physical analogy: the particle behaves like a stretchable dough, extending across both slits, reconnecting after transmission, allowing detachment before the barrier. Monte Carlo simulations confirm that this framework can naturally reproduce the characteristic interference and diffraction probability patterns. Our approach offers a novel, physically interpretable perspective on quantum superposition and measurement induced collapse. The dough analogy is expected to extend to other quantum phenomena. Finally, we provide a dough based picture, attempting to unify interference, entanglement, and tunneling as manifestations of the same underlying phenomenon.
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.