CLASSICAL AND QUANTUM MECHANICS WITH LIE ALGEBRAS

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How to see physics in its full picture? This book offers a new approach: start from math, in its simple and elegant tools: discrete math, geometry, and algebra, avoiding heavy analysis that might obscure the true picture. This will get you ready to master a few fundamental topics in physics: from Newtonian mechanics, through relativity, towards quantum mechanics. Thanks to simple math, both classical and modern physics follow and make a complete vivid picture of physics. This is an original and unified point of view to highlighting physics from a fresh pedagogical angle. Each chapter ends with a lot of relevant exercises. The exercises are an integral part of the chapter: they teach new material and are followed by complete solutions. This is a new pedagogical style: the reader takes an active part in discovering the new material, step by step, exercise by exercise. The book could be used as a textbook in undergraduate courses such as Introduction to Newtonian mechanics and special relativity, Introduction to Hamiltonian mechanics and stability, Introduction to quantum physics and chemistry, and Introduction to Lie algebras with applications in physics. Sample Chapter(s) 1 - Introduction to Newtonian Mechanics: Energy and Work Request Inspection Copy Contents: Introduction to Newtonian Physics: Introduction to Newtonian Mechanics: Energy-Work Angular Momentum and Its Conservation Stability in Geometrical Optics Towards Stability in Classical Mechanics: Poincare Stability in Classical Mechanics Cantor Set and Its Applications Is The Universe Infinite? Binary Trees and Chaos Theory The Binomial Formula and Quantum Statistical Mechanics: Newton's Binomial and Trinomial Formulas Applications in Quantum Statistical Mechanics Introduction to Relativity: Introduction to Special Relativity: Momentum-Energy Towards General Relativity: Spacetime and Its Coordinates Introduction to Quantum Physics and Chemistry: Introduction to Quantum Mechanics: Energy Levels and Spin Quantum Chemistry: Electronic Structure Introduction to Lie Algebras and Their Applications: Jordan Form and Algebras Design Your Lie Algebra Ideals and Isomorphism Theorems Exercises: Solvability and Nilpotency Nilpotency and Engel's Theorems Weight Space and Lie's Lemma and Theorem Cartan's Criterion for Solvability Killing Form and Simple Ideal Decomposition Hamiltonian Mechanics: Energy and Angular Momentum Lie Algebras in Quantum Mechanics and Special Relativity Appendix: Background in Calculus: Functions and Their Derivatives Polynomials and Partial Derivatives Matrices and Their Eigenvalues References Index Readership: Undergraduate and graduate students in Mathematics, Physics, and Chemistry.

Elementary Quantum Mechanics: With Problems and Solutions (1版)

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【簡介】 The impact of quantum mechanics on our day-to-day life is growing at an exponential rate, of which most people are unaware. For example, the toaster is a prime example of a quantum mechanical device. While quantum mechanics arose as a subject in physics, quantum mechanical thinking has had, and continues to have, a broad impact on mathematics. For example, operator theory as we know it today would not exist without von Neumann’s effort to put quantum mechanics on a solid mathematical foundation. This book is defined for undergraduate students of mathematics to obtain an introduction to quantum mechanical ideas, and the associated mathematics, without requiring an extensive background in physics.This book is focused around the following topics. 1) The Mathematical Structure of Quantum Mechanics. We develop the main mathematical structure of quantum theory in the setting of the accepted postulates of quantum mechanics. Dirac’s bra-ket notation for the required linear algebra aspects will be utilized, as well as the mathematical and physical implications arising from the mathematical aspects of self-adjoint linear operators, with emphasis on the finite dimensional setting. 2) Dynamics of a Quantum Particle. We introduce the Schrodinger equation and discuss its physical meaning and mathematical structure. The required partial differential equations theory is elementary, and will be developed from scratch. 3) Measurement, Time Evolution, Uncertainty, and the Harmonic Oscillator. This builds on the previously developed mathematics and culminates with a detailed discussion of the quantum mechanical workhorse example - the harmonic oscillator. 4) Quantum Mechanics of Angular Momentum. Most of the book is developed in the context of the (relatively) simple one dimensional quantum systems. This topic goes into three dimensions and illustrates how very interesting mathematics can yield novel physical explanations that are unobtainable with classical methods. 5) The Postulates of Quantum Mechanics, Measurement, Composite Systems, Tensor Products, and Entanglement. This is the area that most students have heard of - quantum entanglement, ＂action-at-a- distance＂ and Bell’s inequality. The mathematics will be at an elementary level (except possibly for tensor products, which are sadly neglected many linear algebra courses today, but which are literally exploding in a variety of current applications).

Quantum Measurement: Theory and Practice (1版)

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【簡介】 This book adopts a novel, physics-first approach to quantum measurement, using physical experiments as the basis to describe the underlying mathematical formalism. Topics covered include weak measurements, quantum measurement reversal, quantum trajectories and the stochastic path integral formalism. The theory of quantum measurement is also covered in detail, including discussion of how it can be tested and demonstrated in a laboratory: how to build quantum-limited amplifiers, fundamental noise limits imposed on measurement by quantum mechanics, and the design of superconducting circuits. This text is an excellent introduction for students with a basic understanding of quantum mechanics wanting to learn more about measurement theory, and the inclusion of a wide selection of end-of-chapter exercises make this book ideal for emerging courses on the topic. Key chapters introducing the foundations of quantum computing and the history of measurement theory are equally accessible to a broader, less specialised audience.