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Applications of the Piqtr Model to Two-Dimensional Charged Particle Scattering

Zachary Temple

Published Jan 1, 2018• 0 citations
OpenAlex

Abstract

Heavy-ion charged particle collisions have important applications in fields such as astrophysics, biophysics, and plasma physics, and from a fundamental standpoint, provide valuable information about the few-body problem. Current theoretical atomic collision models work well for electron projectiles, but heavy-ion projectiles continue to present a challenge for even the most advanaced models. To help address these challenges, we have developed the Path Integral Quantum Trajectory (PIQTr) model for the calculation of time-dependent wave functions. The method is numerically exact and has been successfully applied to particles moving in one dimension. These results show a favorable scaling in computational requirements as the mass of the projectile increases, and in general, the model has been shown to work well for heavy ions. We have now extended our method to charged particles moving in two dimensions. Here we present time-dependent numerical results for various different charged particles and discuss applications to heavy-ion atomic collisions.

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Heavy-ion charged particle collisions have important applications in fields such as astrophysics, biophysics, and plasma physics, and from a fundamental standpoint, provide valuable information about the few-body problem. Current theoretical atomic collision models work well for electron projectiles, but heavy-ion projectiles continue to present a challenge for even the most advanaced models. To help address these challenges, we have developed the Path Integral Quantum Trajectory (PIQTr) model for the calculation of time-dependent wave functions. The method is numerically exact and has been successfully applied to particles moving in one dimension. These results show a favorable scaling in computational requirements as the mass of the projectile increases, and in general, the model has been shown to work well for heavy ions. We have now extended our method to charged particles moving in two dimensions. Here we present time-dependent numerical results for various different charged particles and discuss applications to heavy-ion atomic collisions.
The path integral technique is an alternative formulation of quantum mechanics that is based on a Lagrangian approach. In its exact form, it is completely equivalent to the Hamiltonian-based Schrödinger equation approach. We have used the path integral formalism to develop our Path Integral Quantum Trajectory (PIQTr) model for use in the study of charged particle dynamics. We will present results for several one-dimensional systems, and demonstrate the method's ability to analyze individual trajectories and their influence on the total probability amplitude. We will also show how the range of included trajectories can affect the time evolution of the wave function, resulting in interference fringes reminiscent of those observed with single slit Fresnel diffraction.
The path integral technique is an alternative formulation of quantum mechanics that is based on a Lagrangian approach. In its exact form, it is completely equivalent to the Hamiltonian-based Schrödinger equation approach. We have used the path integral formalism to develop our Path Integral Quantum Trajectory (PIQTr) model for use in the study of charged particle dynamics. One of the challenges of the PIQTr model is the need to integrate a highly oscillatory function. Here we explore the application of several integration techniques for use in the PIQTr model and benchmark their computational requirements.
The simulation of ion-atom collisions remains a formidable challenge due to the complex interplay between electronic and nuclear degrees of freedom. We present a hybrid quantum-classical computing framework for simulating time-dependent ion-atom collision dynamics, within which two variational quantum time evolution algorithms are implemented. To validate our framework, we simulate the charge-transfer dynamics and compute the corresponding cross sections for the proton-hydrogen collision system across an energy range 1--25 keV. Our results accurately reproduce the charge-transfer dynamics with high fidelity and exhibit very good agreement with available experimental and theoretical cross-section data across the entire energy range. These results highlight the accuracy and applicability of our hybrid quantum-classical framework for scattering cross-section calculations. Our work demonstrates an effective approach for mapping time-dependent many-body collision problems onto near-term quantum computing devices, and also provides promising directions for practical applications of universal quantum computing in the noisy intermediate-scale quantum era.
In this paper, we present a unitary transformation technique to solve the equation of quantum dynamics of a charged particle moving in a time-dependent electromagnetic field. The primary advantage of the technique lies in the connection of this technique with one of the essential features of the system which is the gauge. The gauge has been chosen in a way where the first three dimensional Hamiltonian transformed easily through this technique to a two dimensional coupled problem. This problem is central in several areas of physics, including solid-state and plasma physics, where the understanding of the behavior of charged particles in the presence of time-dependent electromagnetic fields is essential to explore the behavior of various phenomena. The complete solution of the Schrödinger equation is presented and the propagator is also determined using this technique. This technique is also used to shows how the problem can be reduced to a two-dimensional system of coupled harmonic oscillators, each subjected to a time-dependent linear potential. This reduction simplifies the analysis by transforming the problem into a form that can be mapped onto a well-established model, which has been previously studied using the path integral formalism.
Non-relativistic quantum scattering theory informs our broad understanding of nuclear collision processes. However, detailed theoretical insights into the dynamics of fusion, whereby the colliding nuclei form a compound nucleus, remain elusive with conventional approaches. Fundamentally, understanding the complex dissipative processes inherent to compound nucleus formation and when they occur during the collision remains challenging. This, for example, prevents a consistent description of heavy-ion fusion over a wide range of collision energies and hinders searches for new super-heavy elements. To understand these dynamics, we need time-dependent approaches to nuclear collisions. In this thesis, a non-relativistic, time-dependent wave packet theory for potential scattering of charged quantum particles (such as nuclei) has been developed. It formally extends the application of Tannor and Weeks's (1993) formulation of the scattering matrix theory to systems containing the long-range Coulomb potential, by utilising Dollard's (1964) Møller operators. Using the asymptotic localisation of wave packets, analytical theorems have been formulated that allow convenient numerical application of this method. Numerical tests illustrating the theory mark the first rigorous application of this method in nuclear physics. Further insights are gained through a novel analysis of the wave packet time-correlation function (which underlies the theory) to understand how different dynamical processes, such as the formation of long-lived, quasi-bound states, during scattering contribute to the scattering matrix. This provides guidance on using the time-correlation function to obtain fusion observables, along with the associated challenges. Furthermore, this analysis showed that time-correlation functions can be obtained from static calculations, opening numerous new possibilities for studying the dynamics of nuclear reactions, in general, using this approach. This work opens new avenues for studying the dynamics of compound nucleus formation and other dissipative processes in nuclear reactions.
Modeling many-body quantum systems with strong interactions is one of the core challenges of modern physics. A range of methods has been developed to approach this task, each with its own idiosyncrasies, approximations, and realm of applicability. However, there remain many problems that are intractable for existing methods. In particular, many approaches face a huge computational barrier when modeling large numbers of coupled electrons and ions at finite temperature. Here, we address this shortfall with a new approach to modeling many-body quantum systems. On the basis of the Bohmian trajectory formalism, our new method treats the full particle dynamics with a considerable increase in computational speed. As a result, we are able to perform large-scale simulations of coupled electron-ion systems without using the adiabatic Born-Oppenheimer approximation.