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American Journal of Physics

A sum-over-paths approach to one-dimensional time-independent quantum systems

Massimiliano Malgieri, Pasquale Onorato, Anna De Ambrosis

Published Aug 20, 2016• 6 citations
Publisher

Abstract

We present an alternative treatment for simple time-independent quantum systems in one dimension, which can be used in the context of an elementary introduction to quantum physics using the Feynman approach. The method is based on representation of the energy-dependent propagator (or Green function) as a sum of complex amplitudes over all possible paths, classical and non-classical, at fixed energy. We treat both confined and open systems with piecewise-constant potentials, obtaining exact results. We introduce an approximation scheme to extend the method to smooth potentials, recovering the Van Vleck-Gutzwiller propagator. Finally, we discuss the educational application of the method.

DOI
10.1119/1.4953344

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We present an alternative treatment for simple time-independent quantum systems in one dimension, which can be used in the context of an elementary introduction to quantum physics using the Feynman approach. The method is based on representation of the energy-dependent propagator (or Green function) as a sum of complex amplitudes over all possible paths, classical and non-classical, at fixed energy. We treat both confined and open systems with piecewise-constant potentials, obtaining exact results. We introduce an approximation scheme to extend the method to smooth potentials, recovering the Van Vleck-Gutzwiller propagator. Finally, we discuss the educational application of the method.
Abstract Moving from the research tradition on the use of Feynman’s sum over paths approach in education, we have developed an educational reconstruction of elementary quantum physics which uses Feynman paths at fixed energy and is capable of a simple conceptual explanation of discrete energy levels in bound systems. The reconstruction has been found to offer several educational advantages, may help students think of quantum physics in a way more akin to how the subject is used by scientists, and allows to shed light on the relationship between energy and time.
A new approach to finding the exact Green's function to the time-dependent Schr\"odinger equation is described. The approach is conceptually simple: one seeks an operator which when commuted through the Schr\"odinger operator transforms it into the Schr\"odinger operator of a soluble problem. This approach provides an alternative to both the eigenfunction expansion of the propagator and the path integral. The first example of an exact propagator represented as a ``sum over classical paths'' which is not equal to the Gaussian approximation of the path integral is found.
A direct procedure for determining the propagator associated with a quantum mechanical problem was given by the Path Integration Procedure of Feynman.The Green function, which is the Fourier Transform with respect to the time variable of the propagator, can be derived later.In our approach, with the help of a Laplace transform, a direct way to get the energy dependent Green function is presented, and the propagator can be obtained later with an inverse Laplace transform.The method is illustrated through simple one dimensional examples and for time independent potentials, though it can be generalized to the derivation of more complicated propagators.
Abstract The one-sided bouncer (1SB) and the symmetric bouncer (SB) involve a one-dimensional particle in a piecewise linear potential. For such problems, the time-dependent quantum mechanical propagator cannot be found in closed form. The semiclassical Feynman path integral (PI) is a very appealing approach, as it approximates the propagator by a closed-form expression (a sum over a finite number of classical paths). In this paper we solve the classical path enumeration problem. We obtain closed-form expressions for the initial velocity, bounce times, focal times, action, van Vleck determinant, and Morse index for each classical path. We calculate the propagator within the semiclassical approximation. The numerical results agree with eigenfunction expansion results away from caustics. We derive mappings between the 1SB and SB, which explains why each bounce of the 1SB increases the Morse index by 2 and results in a phase change of π . We interpret the semiclassical Feynman PI to obtain visualizations of matter wave propagation based on interference between classical paths, in analogy with the traditional visualization of light wave propagation as interference between classical ray paths.
In this chapter we will derive the expression for the sum over paths from the usual formulation of non-relativistic quantum mechanics, considering the particularly simple case of a quantum system with only one degree of freedom. After deriving the expression for the transition amplitudes by means of the sum over paths, we will show how this method also allows calculations of the Green’s functions in quantum mechanics. We will conclude the chapter by showing how these results can be extended to the case of systems with more discrete degrees of freedom and to field theory.
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.