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Parichart Journal

ทฤษฎการอนทเกรตตามเสนทางเบองตน

นิคม ชูศิริ

Published Oct 26, 2016• 0 citations
OpenAlex

Abstract

The method of Feynman in evaluationg the particle propagator is reviewed. According to Feynman's ideas, apart from classical path, there are many possible paths of a partlcle to go from polnt a to point b in a given time interval. The propagator is the summation of the probability amplitude of all possible paths. By replacing the summation with integration the calculation can be done. Furthermore, lhe obrained propagator also leads ro lhe wave functions and energy of the system.

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•Jan 1, 2016•0 citations
The method of Feynman in evaluationg the particle propagator is reviewed. According to Feynman's ideas, apart from classical path, there are many possible paths of a partlcle to go from polnt a to point b in a given time interval. The propagator is the summation of the probability amplitude of all possible paths. By replacing the summation with integration the calculation can be done. Furthermore, lhe obrained propagator also leads ro lhe wave functions and energy of the system.
A path , in general, is defined by a determinate trajectory in time, from an initial to a final point. The classical trajectory is only one of the possible trajectories, and in quantum mechanics all the possible paths between the initial and final point come into play. Recall that the probability amplitude is a complex number that is assigned to each determinate path. Indeterminate paths are defined as a collection of determinate paths that are experimentally indistinguishable . In the Dirac–Feynman approach, the inherent indeterminacy of the quantum entity is realized by the degree of freedom – in undergoing time evolution – “taking” indeterminate paths [Baaquie (2013e)]. For a quantum degree of freedom evolving from an observed initial state to the observed final state – and with no other observations made – the Feynman path integral is a mathematical construction that computes the probability amplitudes by summing over all the allowed determinate paths of the degree of freedom – discussed in Feynman and Hibbs (1965), Zinn-Justin (1993), Zinn-Justin (2005) and Baaquie (2013e). Probability amplitude and time evolution Recall that the description of a quantum system, at a particular instant,isgiven by its state vector, namely |ψ>. To avoid confusion with the concept of a state vector, the term probability amplitude is used for describing a quantum entity undergoing transitions in time.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.
As an alternative formulation of quantum mechanics, path integral is based on the notion of transition amplitude which gives the wave function of a quantum system at a time t f by acting on the wave function at an earlier time ti.We show that for a general quadratic form for the Lagrangian of the system, transition amplitude has the form f (t f -ti)e i h S class., where S class. is the classical action.We then present an algebraic method to evaluate the function f (t f -ti) without refereing to the path integral calculations.We examine the presented method to the cases of free particle and harmonic oscillator and obtain their propagators.
With Feynman's path- integral method we can obtain the quantum mechanics of a quantum system like a free particle outside Schroedinger's method of differential equations and Heisenberg's method of algebra. The work involves obtaining the quantum propagator Kf, of the system which leads to summation over infinite number of paths. With Van Vleck's formula in one dimension, the classical propagator Kcl for a free particle is computed as the analytical result. This then serves as a yardstick for justifying the theoretical method used to compute the quantum propagator, kf, by direct path summation. The graphical display of the results shows that the Feynman – Schulman's checkerboard model used to enumerate the paths is reliable. Furthermore, this work shows that by windowing off a large number of paths and weighting the rest non uniformly we can compute the required propagator. The weights used in this case are the random and exponential window functions, wr and we respectively, which yield kwr and kwe to compare with kf. Key words: Propagator, action, path - integral, window function, free particle. (Global Journal of Pure and Applied Sciences: 2003 9(4): 561-566)
The path integral formulation of quantum mechanics generalizes the action principle of classical mechanics. The Feynman path integral is, roughly speaking, a sum over all possible paths that a particle can take between fixed endpoints, where each path contributes to the sum by a phase factor involving the action for the path. The resulting sum gives the probability amplitude of propagation between the two endpoints, a quantity called the propagator. Solutions of the Feynman path integral formula exist, however, only for a small number of simple systems, and modifications need to be made when dealing with more complicated systems involving singular potentials, including the Coulomb potential. We derive a generalized path integral formula, that can be used in these cases, for a quantity called the pseudo-propagator from which we obtain the fixed-energy amplitude, related to the propagator by a Fourier transform. The new path integral formula is then successfully solved for the Hydrogen atom in two and three dimensions, and we obtain integral representations for the fixed-energy amplitude.