Quantum Mechanics

A Simplified Approach

Shabnam Siddiqui

• Publisher: CRC Press
• Year: 2018
• Print ISBN: 9781032652443
• eBook ISBN: 9781351840033
• Topic: Technology & Engineering
• Subtopic: Quantum Theory
• Index: Technology & Engineering

Quantum mechanics is one of the most challenging subjects to learn. It is challenging because quantum phenomenon is counterintuitive, and the mathematics used to explain such a phenomenon is very abstract, and difficult to grasp. This textbook is an attempt to overcome these challenges. Every chapter presents quantum ideas step- by- step in a structured way with a comparison between quantum and classical concepts. It provides a clear distinction between classical and quantum logic. Conceptual questions are provided after every important section so that the reader can test their understanding at every step. Such an approach aids in preventing misconceptions. Problem solving is not restricted to solving differential equations and integration. But it requires to systematically and creatively analyze a problem, to apply the new and powerful concepts for finding a solution and to understand the physical meaning of the solution. The tutorials on special topics are an effort to teach problem solving by actively engaging the reader in a thinking process, to apply the concepts and to understand the physical meaning of the solution. The simulations are provided for some of the topics. The simulations aid in the visualization of the quantum phenomenon, and for meaningful understanding of the mathematics. This approach may lead to development of "quantum mechanical intuition "as well as learning mathematical techniques for problem solving. Most importantly, the book is not flooded with numerous topics that makes the reader confused and distracted, rather the most important topics are discussed at a deeper level. The understanding of quantum mechanics is incomplete without understanding the early ideas and experiments that lead to the development of the quantum theory. Thus, the first two chapters of the book are dedicated to such topics.

The key features of this book are:

• A simplified, structured, and step-by-step introduction to quantum mechanics. The simplification is attained through use of two-level system, step- by- step discussion of important topics in a simplified language at a deeper level, analogies, and visualization using illustrations and simulations
• A systematic arrangement of topics, and numerous worked- out examples. The presentation of the structure in the mathematical formalism of quantum mechanics provides clarity in understanding complicated and abstract mathematics. It also helps to understand the distinction between the quantum mechanical and classical approaches
• Conceptual questions at the end of every important section. The conceptual questions can be used in a classroom as a point of discussion between an instructor and students
• Tutorials on special topics. Simulations on special topics aid in the visualization of the physical phenomenon, and demonstration of the application of mathematics
• An in-depth discussion of the wave-particle duality, measurement problem, and their philosophical implications in Chapter 2 provides an understanding of the broader meaning of quantum mechanics

1

An Introductory History of Quantum Mechanics-I

1.1Classical View of an Electromagnetic Wave

The origin of quantum mechanics is linked to the discovery of then new laws of radiation by Max Planck in 1900. The explanation of the empirical facts of radiation by Planck using an extraordinarily powerful yet simple model lead to the discovery of the “quantum” nature of radiation. According to his model, radiation is emitted or absorbed in “bits” or packets of discrete energy. His formula for the intensity of radiation modeled the experimental results very well. His radiation model turned out to be a much closer simulation of the experimental results. However, this finding that radiation is emitted or absorbed in discrete “quanta” rather continuous energy flow was profoundly novel idea unexpected to classical physicists at that time. This new idea disturbed the foundations of the classical theory of radiation. At first, even Planck was skeptical about his own idea and spent time and effort in attempting to reconcile his model with the classical theory. However, he failed in his attempt, since there was no realistic means for him to reconcile quantum theory with the classical theory. Therefore, he published his novel theory in 1900. It took another five years for Planck’s new theory to be applied to the solution of a concrete physical problem. It was Albert Einstein in 1905 who extended Planck’s idea to an explanation of the specific heats of solids and of gases at low temperature and to the photoelectric effect. In both cases, Einstein was able to demonstrate that by applying Planck’s model of the quantization of energy, these phenomena can be readily explained in detail. However, this also marked the beginning of the understanding of light as a phenomenon with a discrete nature, which was unknown to classical physics at that time. Einstein was able to explain the photoelectric effect using Planck’s idea of quantization of energy and showed that light can be discrete as well. According to Maxwell’s theory, light is an electromagnetic wave, and thus is continuous. Well-known phenomenon such as diffraction and interference could only be explained by considering light as a continuous system such as an electromagnetic wave. This new idea about the discrete nature of light (consisting of particles) turned out to be contradictory to the Maxwell’s theory of light. But could light have a dual nature? Can it be both discrete and continuous? How can something be discrete and continuous at the same time? Similar paradoxes also arose in atomic physics. In 1913, Niels Bohr applied these new ideas to atomic physics by explaining the spectrum of the hydrogen atom. However, it turned out that there was a discrepancy between the calculated orbital frequency of the electrons and the frequency of the emitted radiation. Further, in 1924, Compton discovered that the frequency of X-rays changes as they collide with electrons. This could only be explained by considering X-rays as discrete particles colliding with electrons. All these experimental observations and the failure of classical physics to explain these new findings related to the atomic phenomenon paved the way for the development of a mathematical theory of quantum mechanics. The...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Author
• Fundamental Constants and Important Equations
• Chapter 1: An Introductory History of Quantum Mechanics-I
• Chapter 2: An Introductory History of Quantum Mechanics-II
• Chapter 3: Formalism
• Chapter 4: Applications of the Formalism-I
• Chapter 5: Applications of the Formalism-II
• Chapter 6: Perturbation Theory
• Chapter 7: Time-Dependent Perturbation Theory
• Chapter 8: Quantum Computer
• References
• Index