<p>This book presents a novel standpoint concerning contemporary physics, namely, quantum mechanics with a view toward algebraic geometry. As is well-known, algebraic geometry is the study of geometric objects delineated by polynomials, and the polynomial representations are ubiquitous in physics. For this reason, quantum mechanics is also an object of algebraic geometry. An example is the eigenvalue problem. It is a set of polynomial equations and has traditionally been the question of linear algebra. However, the modern method of computational algebraic geometry accurately unravels the information encapsulated in the polynomials. This approach shall not remain as a plaything. It has betokened an innovative style of electronic structure computation. The objects of this new method include the simultaneous determination of the wave-functions and the movements of nuclei, or the prediction of the required structure that shall show the desired property. Accordingly, this book explains the basic ideas of computational algebraic geometry and related topics, such as Groebner bases, primary ideal decomposition, Dmodules, Galois, class field theory, etc. The intention of the author is, nevertheless, not to give an irksome list of abstract concepts. He hopes that the readers shall use algebraic geometry as the active tool of the computations. For this reason, this book abundantly presents the model computations, by which the readers shall learn how to apply algebraic geometry toward quantum mechanics. The readers shall also see the modern computer algebra could facilitate the study when you would like to apply abstract mathematical ideas to definite physical problems.</p>

Quantum Mechanics built on Algebraic Geometry: Emerging Physics through Symbolic Computation

Science & Math

Kikuchi, Akihito

Mathematics

Eliva Press

This book is intended as an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration.The first part explains how to use computer algebra for applications in solid-state simulation, based on the GAP computer algebra package. Computer algebra enables us to easily obtain various group theoretical properties, such as the representations, character tables, and subgroups. Furthermore it offers a new perspective on material design, which could be executed in a mathematically rigorous and systematic way.The second part then analyzes the relation between the structural symmetry and the electronic structure in C60 (as an example of a system without periodicity). The principal object of the study was to illustrate the hierarchical change in the quantum-physical properties of the molecule, which correlates to the reduction in the symmetry (as it descends down in the ladder of subgroups).The book also presents the computation of the vibrational modes of the C60 by means of the computer algebra. In order to serve the common interests of researchers, the details of the computations (the required initial data and the small programs developed for the purpose) are explained in as much detail as possible. 

Computer Algebra and Materials Physics: A Practical Guidebook to Group Theoretical Computations in Materials Science (Springer Series in Materials Science, 272)

Chemistry

Springer

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