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Title: Beam spread functions calculated using Feynman path integrals

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Journal Article: Publisher's Accepted Manuscript
Journal Name:
Journal of Quantitative Spectroscopy and Radiative Transfer
Additional Journal Information:
Journal Volume: 196; Journal Issue: C; Related Information: CHORUS Timestamp: 2017-12-18 18:58:19; Journal ID: ISSN 0022-4073
Country of Publication:
United Kingdom

Citation Formats

Kilgo, Paul, and Tessendorf, Jerry. Beam spread functions calculated using Feynman path integrals. United Kingdom: N. p., 2017. Web. doi:10.1016/j.jqsrt.2017.03.030.
Kilgo, Paul, & Tessendorf, Jerry. Beam spread functions calculated using Feynman path integrals. United Kingdom. doi:10.1016/j.jqsrt.2017.03.030.
Kilgo, Paul, and Tessendorf, Jerry. 2017. "Beam spread functions calculated using Feynman path integrals". United Kingdom. doi:10.1016/j.jqsrt.2017.03.030.
title = {Beam spread functions calculated using Feynman path integrals},
author = {Kilgo, Paul and Tessendorf, Jerry},
abstractNote = {},
doi = {10.1016/j.jqsrt.2017.03.030},
journal = {Journal of Quantitative Spectroscopy and Radiative Transfer},
number = C,
volume = 196,
place = {United Kingdom},
year = 2017,
month = 7

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on April 14, 2018
Publisher's Accepted Manuscript

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  • Evolution semigroups generated by pseudo-differential operators are considered. These operators are obtained by different (parameterized by a number τ) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains Hamilton functions of particles with variable mass in magnetic and potential fields and more general symbols given by the Lévy-Khintchine formula. The considered semigroups are represented as limits of n-fold iterated integrals when n tends to infinity. Such representations are called Feynman formulae. Some of these representations are constructed with the help of another pseudo-differential operator, obtained by the same procedure ofmore » quantization; such representations are called Hamiltonian Feynman formulae. Some representations are based on integral operators with elementary kernels; these are called Lagrangian Feynman formulae. Langrangian Feynman formulae provide approximations of evolution semigroups, suitable for direct computations and numerical modeling of the corresponding dynamics. Hamiltonian Feynman formulae allow to represent the considered semigroups by means of Feynman path integrals. In the article, a family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented as phase space Feynman path integrals with respect to these Feynman pseudomeasures, i.e., different quantizations correspond to Feynman path integrals with the same integrand but with respect to different pseudomeasures. This answers Berezin’s problem of distinguishing a procedure of quantization on the language of Feynman path integrals. Moreover, the obtained Lagrangian Feynman formulae allow also to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures.« less
  • In this paper the authors report the initial steps in the development of a Monte Carlo method for evaluation of real-time Feynman path integrals for many-particle dynamics. The approach leads to Gaussian factors. These Gaussian factors result from the use of a generalization of their new discrete distributed approximating functions (DDAFs) to continuous distributed approximating functions (CDAFs) so as to replace the exact coordinate representation free-particle propagator by a {open_quotes}CDAF-class, free-particle propagator{close_quotes} which is highly banded. The envelope of the CDAF-class free propagator is the product of a {open_quotes}bare Gaussian{close_quotes}, exp[{minus}(x{prime} {minus} x){sup 2}{sigma}{sup 2}(0)/(2{sigma}{sup 4}(0) + {h_bar}{sup 2}{tau}{sup 2}/m{supmore » 2})], with a {open_quotes}shape polynomial{close_quotes} in (x{prime}{minus}x){sup 2}, where {sigma}(0) is a width parameter at zero time (associated with the description of the wavepacket in terms of Hermite functions), {tau} is the time step ({tau} = t/N, where t is the total propagation time), and x and x{prime} are any two configurations of the system. The bare Gaussians are used for Monte Carlo integration of a path integral for the survival probability of a Gaussian wavepacket in a Morse potential. The approach appears promising for real-time quantum Monte Carlo studies based on the time-dependent Schroedinger equation, the time-dependent von Neumann equation, and related equations. 38 refs., 3 figs., 3 tabs.« less
  • We show that it is possible to do numerical calculations in elementary quantum mechanics using Feynman path integrals. Our method involves discretizing both time and space, and summing paths through matrix multiplication. We give numerical results for various one-dimensional potentials. The calculations of energy levels and wavefunctions take approximately 100 times longer than with standard methods, but there are other problems for which such an approach should be more efficient.
  • Thermal ionization of hydrogen at temperatures on the order of 10{sup 4}-10{sup 5} K and densities within 10{sup 24}-10{sup 28} m{sup -3} has been simulated using Feynman path integrals. This method has been realized for the first time under conditions of a statistical ensemble with fluctuating volume. Multidimensional integrals have been calculated using the Monte Carlo simulation method that was preliminarily tested numerically on a problem of the quantum ground state of a confined hydrogen atom, which admits analytical solution. The position of isolines of the degree of ionization has been determined on the p-T plane of plasma states. Themore » spatial correlation functions for electrons and nuclei are calculated, and the quantum effects in behavior of the electron component are evaluated. It is shown that, owing to the presence of strong Coulomb interactions, plasma retains a substantially quantum character in a broad domain of thermodynamic states, where a formal use of the degeneracy criterion predicts a classical regime. A basically exact stochastic method is developed for calculating the equilibrium kinetic energy of a spatially bounded system of quantum particles free of the dispersion divergence.« less