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arXiv:1712.00046 · Physical Review C

Scattering using real-time path integrals

Wayne Nicholas Polyzou, Ekaterina S. Nathanson

Published Jun 4, 2020• 2 citations

Abstract

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.

DOI: 10.1103/physrevc.101.064001

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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.

We propose a new path-integration calculation method to treat the time evolution of a wavefunction within the framework of the real-space finite-difference formalism, and also develop an effective scheme to compute the scattering wavefunction for an incident electron with arbitrary energy, in which an impulse wavefunction is adopted as an initial state of the time evolution. In this method, once the time evolution of the initial impulse wavefunction is calculated, all of the solutions in the scattering problem can be derived by means of Fourier analysis of the time-evolved wavefunction, which leads to a reduction of the calculation time. In order to test the applicability of our newly developed simulation procedures, we implemented simulations for the one-dimensional scattering problem. Each simulation showed the usefulness of the present scheme by yielding the steady scattering states in agreement with exact ones.

This chapter shows how scattering problems are formulated in the framework of path integrals. In quantum mechanics, the state of an isolated system evolves under the action of a unitary operator, as a consequence of the conservation of probabilities and, thus, of the norm of vectors in Hilbert space. Quantum evolution (that is, in real time) is introduced, after which a path integral representation of the scattering matrix is constructed. From this S matrix, the standard perturbative expansion in powers of the potential is recovered. Even the evolution of a free quantum particle is slightly non-trivial; in general, one observes a spreading of wave packets. Scattering is then characterized by the asymptotic deviations at infinite time from this free evolution and this leads to the definition of a scattering or S-matrix. An S-matrix is defined in the example of bosons and fermions. Various other semi-classical approximation schemes are then discussed.

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We introduce a new approach for calculating quantum time-correlation functions and time-dependent expectation values in many-body thermal systems; both electronically adiabatic and non-adiabatic cases can be treated. Our approach uses a path integral simulation to sample an initial thermal density matrix; subsequent evolution of this density matrix is equivalent to solution of the time-dependent Schrödinger equation, which we perform using a linear expansion of Gaussian wavepacket basis functions which evolve according to simple classical-like trajectories. Overall, this methodology represents a formally exact approach for calculating time-dependent quantum properties; by introducing approximations into both the imaginary-time and real-time propagations, this approach can be adapted for complex many-particle systems interacting through arbitrary potentials. We demonstrate this method for the spin Boson model, where we find good agreement with numerically exact calculations. We also discuss future directions of improvement for our approach with a view to improving accuracy and efficiency.