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Modeling of physical systems (Quantum Mechanics) learning goal

Carl Wieman

Published Sep 17, 2009• 0 citations
OpenAlex

Abstract

Learning Goal: Be able to describe a physical situation that would correspond to simple potential energy curves. Given a simple physical system, be able to draw the relevant potential energy curve needed to model dynamical behaviour. Explain why models are useful in physics.

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Given the knowledge of its kinematical class, a quantum system is completely characterized by the specification of a ground state. This provides a quantum formulation more general than the potential-based approach, handling situations where the ground state is a smooth function whereas the potential would be singular. Many results may be obtained directly from the ground state without needing to reconstruct the associated potential. Representations of the quantum algebra, generated by the ground state, describe all quantum states of the system and this approach is also useful as a tool to model observed physical phenomena and to find out what interactions are at play. The basic formalism and a variety of examples are included to illustrate the power of the ground state approach as a modelling tool for quantum systems.
Quantum Mechanics can be an abstract and complex subject. Students often complain of confusion, struggle, and frustration as they try to master the topic. The goal of this book is to reduce the complexity and clarify the abstractions with concrete visual examples driven by simple python programs. It is assumed that the reader is concurrently taking a course in quantum mechanics, or self-studying quantum mechanics, but is looking for supplementary material to help with understanding and visualizing how quantum mechanics works. The focus of this book is writing python programs to visualize the underlying behavior of the mathematical theory. The background needed to understand quantum mechanics is differential equations, linear algebra and modern physics. We need a strong foundation in differential equations and linear algebra because the behavior of quantum systems is governed by equations that are written in terms of these concepts. Modern physics includes concepts such as special relativity and quantum phenomena like the photoelectric effect and energy quantization that the theory of quantum mechanics seeks to explain. This book is also not an introduction to the python programming language, or to numpy, or even to VPython. However its programming examples start simply and grow more complex as the chapters progress, so deep expertise in any of these is not a pre-requisite. Key features: · Provides an accessible and practical guide to the abstractions in quantum mechanics with concrete visual examples driven by simple python programs. · Contains few derivations, equations, and proofs. · For complete beginners of python programming, appendix B serves as a very brief introduction to the main concepts needed to understand the code in this book.
In this paper we investigate a new approach to reduced dimensionality descriptions of quantum mechanical systems resident in a bath. We study physical situations in which the coupling between the system and the bath is slowly varying. Our method involves an operator expansion of the Feynman propagator following the Zassenhaus theorem. From this general expansion we are able to derive an especially simple special case in which the coupling is a slowly varying function of the position operators of the system and the bath. From the approximate propagator after tracing over bath degrees, we are able to derive a short time propagator which yields both a form for efficient numerical calculation and an effective Schrödinger equation for the evolution of the system under the average influence of the bath. This theory is then applied to tunneling rearrangement in mixed crystals of benzoic acid. We find that independent of potential energy perturbations, dynamic system bath couplings increase the rate of tunneling. A central goal of this type of approach is to model the increasingly complex experimental data for large (often biological) systems.
A simplified scheme is used to discuss the idea of the mathematical formalism of physical theory. The physical theory is associated with a certain mathematical structure which makes it possible to define objects, i.e., the subject of investigation, and some of their characteristics. The task of the theory is to find the characteristics of an object from its input data that implicitly define it, on the basis of the laws from the nature of the basic structure. It stands to reason that the task includes the finding of the indicated laws dictated by the structure, the elucidation of the general properties of the objects, the establishment of the relationship between the characteristics, etc. The finiteness of the models used to illustrate the given scheme is brought about by the replacement of the space or the space-time continuum of "real" physics by a finite set.
The Physics of Quantum Mechanics aims to give students a good understanding of how quantum mechanics describes the material world. It shows that the theory follows naturally from the use of probability amplitudes to derive probabilities. It stresses that stationary states are unphysical mathematical abstractions that enable us to solve the theory's governing equation, the time-dependent Schroedinger equation. Every opportunity is taken to illustrate the emergence of the familiarclassical, dynamical world through the quantum interference of stationary states. The text stresses the continuity be
Quantum computing defines the use of quantum mechanics in the computation process. Given that Moore's law is about to reach its physical limits, it is natural to search for new technologies and computation models. The need for mechanisms and verification processes for infinite quantum computation will draw the attention of the research community. Potential applications of the proposed automata concern the verification of quantum systems and the design of appropriate quantum circuits. In the present work we define a quantum version of ω-automata, the periodic quantum automata that are able to efficiently recognize ω-languages of the form (amb)ω, where m is a finite number. We present our definitions through examples and figures. Even if our definition is simple, yet it has several interesting and useful properties. For example, they seem to be space efficient for periodic ω-languages of the form (amb)ω.