A Mathematical Journey To Quantum Mechanics
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A Mathematical Journey to Quantum Mechanics
Author: Salvatore Capozziello
language: en
Publisher: Springer Nature
Release Date: 2021-09-27
This book provides an itinerary to quantum mechanics taking into account the basic mathematics to formulate it. Specifically, it features the main experiments and postulates of quantum mechanics pointing out their mathematical prominent aspects showing how physical concepts and mathematical tools are deeply intertwined. The material covers topics such as analytic mechanics in Newtonian, Lagrangian, and Hamiltonian formulations, theory of light as formulated in special relativity, and then why quantum mechanics is necessary to explain experiments like the double-split, atomic spectra, and photoelectric effect. The Schrödinger equation and its solutions are developed in detail. It is pointed out that, starting from the concept of the harmonic oscillator, it is possible to develop advanced quantum mechanics. Furthermore, the mathematics behind the Heisenberg uncertainty principle is constructed towards advanced quantum mechanical principles. Relativistic quantum mechanics is finally considered.The book is devoted to undergraduate students from University courses of Physics, Mathematics, Chemistry, and Engineering. It consists of 50 self-contained lectures, and any statement and theorem are demonstrated in detail. It is the companion book of "A Mathematical Journey to Relativity", by the same Authors, published by Springer in 2020.
A Mathematical Journey Through Differential Equations Of Physics
Mathematics is the language of physics, and over time physicists have developed their own dialect. The main purpose of this book is to bridge this language barrier, and introduce the readers to the beauty of mathematical physics. It shows how to combine the strengths of both approaches: physicists often arrive at interesting conjectures based on good intuition, which can serve as the starting point of interesting mathematics. Conversely, mathematicians can more easily see commonalities between very different fields (such as quantum mechanics and electromagnetism), and employ more advanced tools.Rather than focusing on a particular topic, the book showcases conceptual and mathematical commonalities across different physical theories. It translates physical problems to concrete mathematical questions, shows how to answer them and explains how to interpret the answers physically. For example, if two Hamiltonians are close, why are their dynamics similar?The book alternates between mathematics- and physics-centric chapters, and includes plenty of concrete examples from physics as well as 76 exercises with solutions. It exploits that readers from either end are familiar with some of the material already. The mathematics-centric chapters provide the necessary background to make physical concepts mathematically precise and establish basic facts. And each physics-centric chapter introduces physical theories in a way that is more friendly to mathematicians.As the book progresses, advanced material is sprinkled in to showcase how mathematics and physics augment one another. Some of these examples are based on recent publications and include material which has not been covered in other textbooks. This is to keep it interesting for the readers.
A Mathematical Journey
Author: Pasquale De Marco
language: en
Publisher: Pasquale De Marco
Release Date: 2025-03-17
Embark on an intellectual journey into the realm of logic, where reasoning and argumentation unveil the secrets of truth and validity. Discover the foundations of logic, unraveling the intricacies of propositions, truth values, and logical connectives. Explore the power of deductive and inductive reasoning, crafting sound arguments and evaluating their validity. Witness the practical applications of logic in various fields, from philosophy and mathematics to law and everyday life. Delve into the fascinating world of propositional logic and predicate logic, uncovering the mechanisms that govern the validity of logical statements. Explore the intricacies of truth tables, logical equivalence, and the laws of logic, appreciating the elegance and rigor of logical reasoning. Investigate the realm of set theory and functions, encountering fundamental concepts like sets, elements, and set operations. Discover the properties of functions and relations, witnessing their patterns and underlying structures. Witness the practical applications of these concepts in diverse disciplines, from computer science and mathematics to economics and social sciences. Venture into the frontiers of mathematical logic, where Gödel's incompleteness theorems challenge our understanding of truth, provability, and the foundations of mathematics itself. Explore the intricate world of recursive functions and the halting problem, contemplating the limits of computability and the boundaries of human knowledge. Encounter the vibrant world of non-classical logics, where fuzzy logic, modal logic, and intuitionistic logic offer alternative perspectives on truth, possibility, and necessity. Expand your understanding of the diverse ways in which we can reason and draw conclusions. Throughout this captivating exploration, you will uncover the beauty, power, and far-reaching applications of logic. With each chapter, you will gain a profound understanding of logical reasoning, its role in shaping our world, and its ability to expand the boundaries of human knowledge. This book is an indispensable resource for students, researchers, and anyone seeking to master the art of logical reasoning and unlock the secrets of truth and validity. If you like this book, write a review!