This chapter should be treated as an appendix. It is included for completeness and covers certain fundamentals of quantum mechanics that will aid the understanding of topics discussed in this textbook. The reader is urged to refer to many of the good introductory textbooks on the subject to learn or revisit the fundamental concepts [1, 2, 3, 4]. First, we briefly discuss three key ingredients of quantum mechanics, i.e., the blackbody radiation spectrum, the concept of the photon and de Broglie wavelength, and the related wave-particle duality concept. Next, we briefly restate the four major postulates of quantum mechanics used in the so-called orthodox or Copenhagen interpretation of quantum mechanics that is in vogue. We continue on to describe a generalized form of the Heisenberg Uncertainty Principle which is illustrated with a few familiar bound state problems (particle-in-a-box and one-dimensional harmonic oscillator) and a few spin related problems.
This chapter should be treated as an appendix. It is included for completeness and covers certain fundamentals of quantum mechanics that will aid the understanding of topics discussed in this textbook. The reader is urged to refer to many of the good introductory textbooks on the subject to learn or revisit the fundamental concepts [1, 2, 3, 4]. First, we briefly discuss three key ingredients of quantum mechanics, i.e., the blackbody radiation spectrum, the concept of the photon and de Broglie wavelength, and the related wave-particle duality concept. Next, we briefly restate the four major postulates of quantum mechanics used in the so-called orthodox or Copenhagen interpretation of quantum mechanics that is in vogue. We continue on to describe a generalized form of the Heisenberg Uncertainty Principle which is illustrated with a few familiar bound state problems (particle-in-a-box and one-dimensional harmonic oscillator) and a few spin related problems.
Assuming a background in basic classical physics, multivariable calculus, and differential equations, A Concise Introduction to Quantum Mechanics provides a self-contained presentation of the mathematics and physics of quantum mechanics. The relevant aspects of classical mechanics and electrodynamics are reviewed, and the basic concepts of wave-particle duality are developed as a logical outgrowth of experiments involving blackbody radiation, the photoelectric effect, and electron diffraction. The Copenhagen interpretation of the wave function and its relation to the particle probability density is presented in conjunction with Fourier analysis and its generalization to function spaces. These concepts are combined to analyze the system consisting of a particle confined to a box, developing the probabilistic interpretation of observations and their associated expectation values. The Schrödinger equation is then derived by using these results and demanding both Galilean invariance of the probability density and Newtonian energy-momentum relations. The general properties of the Schrödinger equation and its solutions are analyzed, and the theory of observables is developed along with the associated Heisenberg uncertainty principle. Basic applications of wave mechanics are made to free wave packet spreading, barrier penetration, the simple harmonic oscillator, the Hydrogen atom, and an electric charge in a uniform magnetic field. In addition, Dirac notation, elements of Hilbert space theory, operator techniques, and matrix algebra are presented and used to analyze coherent states, the linear potential, two state oscillations, and electron diffraction. Applications are made to photon and electron spin and the addition of angular momentum, and direct product multiparticle states are used to formulate both the Pauli exclusion principle and quantum decoherence. The book concludes with an introduction to the rotation group and the general properties of angular momentum.
The mathematical formalism of quantum theory in terms of vectors and operators in infinite-dimensional complex vector spaces is very abstract. The definitions of many mathematical quantities used do not seem to have an intuitive meaning, which makes it difficult to appreciate the mathematical formalism and understand quantum mechanics. This book provides intuition and motivation to the mathematics of quantum theory, introducing the mathematics in its simplest and familiar form, for instance, with three-dimensional vectors and operators, which can be readily understood. Feeling confident about and comfortable with the mathematics used helps readers appreciate and understand the concepts and formalism of quantum mechanics. This book is divided into four parts. Part I is a brief review of the general properties of classical and quantum systems. A general discussion of probability theory is also included which aims to help in understanding the probability theories relevant to quantum mechanics. Part II is a detailed study of the mathematics for quantum mechanics. Part III presents quantum mechanics in a series of postulates. Six groups of postulates are presented to describe orthodox quantum systems. Each statement of a postulate is supplemented with a detailed discussion. To make them easier to understand, the postulates for discrete observables are presented before those for continuous observables. Part IV presents several illustrative applications, which include harmonic and isotropic oscillators, charged particle in external magnetic fields and the Aharonov–Bohm effect. For easy reference, definitions, theorems, examples, comments, properties and results are labelled with section numbers. Various symbols and notations are adopted to distinguish different quantities explicitly and to avoid misrepresentation. Self-contained both mathematically and physically, the book is accessible to a wide readership, including astrophysicists, mathematicians and philosophers of science who are interested in the foundations of quantum mechanics.
This textbook provides ample opportunities for practice and real experimental demonstrations. Conceptual understanding and mastering key techniques are enhanced by rigorous derivations, numerous worked examples, more than 300 exercises, about 150 problems and 16 computer codes. The preface summarizes all of the key concepts and formulas, along with a detailed schedule for teaching. The first three chapters introduce the quantum idea, wave-particle duality, operators and measurement. The Noether theorem is invoked to introduce the Schrödinger equation, followed by applications to infinite and finite quantum wells, quantum tunneling, harmonic oscillators, Heisenberg equation of motion, uncertainty principle, blackbody radiation and photoelectric effect. Chapters 4 and 5 are on angular momentum, the hydrogen atom and time-independent approximate methods. Chapters 6 and 7 are on spin and time-dependent perturbation theory. Chapters 8, 9 and 10 are on molecular orbitals, energy bands, quantum transport, scanning tunneling microscopy, lattice vibrations, Berry phase and quantum computing. The book is intended for a one-semester or one-year course and is also appropriate for researchers in related fields.
This chapter starts with the historical development of quantum mechanics and explanations of key concepts behind fundamental discoveries such as Planck's hypothesis, the photoelectric effect, Bohr's atomic model, Compton scattering, the de Broglie hypothesis, the Pauli exclusion principle, the Schrödinger equation, Heisenberg quantum mechanics and the uncertainty principle, the Born interpretation of wave function, the Davisson and Germer experiment, and Dirac relativistic quantum mechanics, which made important contributions to the foundation of modern quantum mechanics. Concepts like the particle properties of radiation, the wave properties of matter, quantized energy and momentum, and the idea that one can no longer know exactly where a single particle like an electron is at any instant of time are brought out clearly through this chapter. A list of important quantum mechanical parameters along with solved examples and unsolved exercises on the key concepts are included towards the end.
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