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Physica Scripta

Computing Feynman path integrals in real time

E. Onofri, Giampietro Tecchiolli

Published Mar 1, 1988• 7 citations
Publisher

Abstract

We review the algorithm of Feit, Fleck and Steiger [1] as a method to calculate Feynman Path integrals in real time under very general circumstances ranging from the simple potential problems for the Schroedinger equation to cases involving time dependent perturbations, resonances and coupling to a white noise source. Dirac equation can be studied with minor modifications. We test the algorithm in the one-dimensional Dirac equation using the potential tanh (λ x ) which allows for an exact solution.

DOI
10.1088/0031-8949/37/3/002

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Scattering using real-time path integrals

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Background: Path integrals are a powerful tool for solving problems in quantum theory that are not amenable to a treatment by perturbation theory. Most path integral computations require an analytic continuation to imaginary time. While imaginary time treatments of scattering are possible, imaginary time is not a natural framework for treating scattering problems. More importantly, quantum algorithms for calculating path integrals require real-time evolution.Purpose: We test a recently introduced method for performing direct calculations of scattering observables using real-time path integrals in order to understand the challenges facing real-time path integral calculations of scattering observables.Method: The computations are based on a new interpretation of the path integral as the expectation value of a potential functional on cylinder sets of continuous paths with respect to a complex probability distribution. The method can in principle be applied to arbitrary short-range potentials.Results: The method is applied to compute matrix elements of M\o{}ller wave operators applied to narrow wave packets. These are used to calculate half-shell sharp-momentum transition matrix elements for one-dimensional potential scattering. The calculations for half-shell transition operator matrix elements converge to the numerical solution of the Lippmann-Schwinger equation.Conclusions: This work presents a proof in principle that scattering observables can be computed using real-time Feynman path integrals. While the computational method is not efficient, it can be improved. It provides a laboratory for studying quantum computational algorithms that are applicable to scattering problems.

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Revista Mexicana de Física•Dec 1, 2008•4 citations
We review the algorithm of Feit, Fleck and Steiger [1] as a method to calculate Feynman Path integrals in real time under very general circumstances ranging from the simple potential problems for the Schroedinger equation to cases involving time dependent perturbations, resonances and coupling to a white noise source. Dirac equation can be studied with minor modifications. We test the algorithm in the one-dimensional Dirac equation using the potential tanh (λ x ) which allows for an exact solution.
Background: Path integrals are a powerful tool for solving problems in quantum theory that are not amenable to a treatment by perturbation theory. Most path integral computations require an analytic continuation to imaginary time. While imaginary time treatments of scattering are possible, imaginary time is not a natural framework for treating scattering problems. More importantly, quantum algorithms for calculating path integrals require real-time evolution.Purpose: We test a recently introduced method for performing direct calculations of scattering observables using real-time path integrals in order to understand the challenges facing real-time path integral calculations of scattering observables.Method: The computations are based on a new interpretation of the path integral as the expectation value of a potential functional on cylinder sets of continuous paths with respect to a complex probability distribution. The method can in principle be applied to arbitrary short-range potentials.Results: The method is applied to compute matrix elements of M\o{}ller wave operators applied to narrow wave packets. These are used to calculate half-shell sharp-momentum transition matrix elements for one-dimensional potential scattering. The calculations for half-shell transition operator matrix elements converge to the numerical solution of the Lippmann-Schwinger equation.Conclusions: This work presents a proof in principle that scattering observables can be computed using real-time Feynman path integrals. While the computational method is not efficient, it can be improved. It provides a laboratory for studying quantum computational algorithms that are applicable to scattering problems.
(and its generalisations that include magnetic fields) can be roughly divided into two classes. In the aproaches of the first class that we shall not discuss here in detail the Feynman integral is not supposed to be a genuine integral, but is specified as some generalised functional on an appropriate space of functions, which can be defined, for example, as the limit of certain discrete approximations (see e.g. [16], [44] and more recent papers [24], [25], [34]), by means of analytical continuation (see e.g. [26], [23] and references therein), by extensions of Parceval’s identity and by related axiomatic definitions (see [2], [5], [30], [11], [42] and references therein) or by means of the white noise analysis (see [22]), see also [46] for the discussion of path integral applied to the Dirac equation. These approaches still cover only a very restrictive class of potentials, for example, singular potentials were considered only by white noise analysis approach but only in one-dimensional case (see e.g. [6] and references therein). In the approaches of the second class, one tries to define the infinite-dimensional Feynmann integral as a genuine integral over a bona fide σ-additive measure on an appropriate space of trajectories. The first attempt made in [19] to construct such a measure was erronious and led to understanding that there is no direct generalisation of Wiener measure that can give an analog of FeynmanKac formula for the case of Schrodinger operators. A correct construction of the Feynman integral in terms of the Wiener measure was proposed in [15] and was based on the idea of the rotation of the classical trajectories in complex
Abstract This review presents time-independent perturbative methods for solving the one-dimensional Schrödinger equation, highlighting representative cases that reveal key aspects of the theory. The focus is on their relevance to quantum computing applications, particularly in systems with finite-dimensional state spaces. The main link between these methods and quantum processing emerges when we consider systems with only a finite number of well-defined energy levels, e.g., two levels corresponding to the ‘zero’ and ‘one’ states of a quantum bit (qubit), for which the Schrödinger equation describes a quantum algorithm corresponding to the unitary evolution of the qubits. Perturbative techniques can then be applied to this evolution to gain deeper insight into the algorithm or to simplify it in a controlled manner, as described in the last sections of this review. We initially discuss representative applications using benchmark cases that exemplify the essential features of perturbation theory while avoiding unnecessary technical complications. The methods presented involve direct approaches to solutions for quantum systems with small potential perturbations, with consideration of indirect and approximation techniques. Key cases examined include the particle in a potential well, the harmonic oscillator, and systems with polynomial or Dirac delta distribution perturbations, each chosen to highlight unique aspects of perturbative analysis. Starting with an overview of the unperturbed Schrödinger equation, we derive canonical solutions and then use perturbative techniques to approximate energy shifts and wavefunction corrections in both non-degenerate and degenerate settings. By combining theoretical discussion with case studies, this review underscores how perturbative methods are applied across a range of elementary quantum mechanical problems especially for applications in the qubit model, which, in many cases of interest, can be reduced to the explicit calculations shown here.
The authors, together with A. del Campo, developed an alternative method for determining the Feynman propagator [1] for a non-relativistic problem. One started with the time dependent Schroedinger equation for the problem. Carried out a Laplace transform with respect to time to get the equation for the energy dependent Green function and derived it explicitly. We then carried out the inverse Laplace transform in the energy to get Feynman propagator. In this paper we carry out the same programme for a relativistic problem associated with the one dimensional Dirac equation of a free particle and the Dirac oscillator proposed by Moshinsky and Szczepaniak [2] twenty years ago.