This article discusses a set of Maple programs developed to reinforce understanding of basic concepts presented in an introductory quantum mechanics course. It was designed so that the solutions of the problems were analytically tractable as well as numerically obtainable. We note that there exist various approaches to a simple quantum mechanical problem. For example, for the problem of bound-states with a one-dimensional potential, the transfer matrix method, the Laplace transform method, and the Feynman path integral method can equally be applied. We believe that this type of program is extremely useful for igniting the student’s interest and widening his or her viewpoint. Several examples, such as the solution of various one-dimensional potential problems and the solution of the one-dimensional Schroedinger wave equation, are discussed in order to indicate how constructive reinforcement is established.
This article discusses a set of Maple programs developed to reinforce understanding of basic concepts presented in an introductory quantum mechanics course. It was designed so that the solutions of the problems were analytically tractable as well as numerically obtainable. We note that there exist various approaches to a simple quantum mechanical problem. For example, for the problem of bound-states with a one-dimensional potential, the transfer matrix method, the Laplace transform method, and the Feynman path integral method can equally be applied. We believe that this type of program is extremely useful for igniting the student’s interest and widening his or her viewpoint. Several examples, such as the solution of various one-dimensional potential problems and the solution of the one-dimensional Schroedinger wave equation, are discussed in order to indicate how constructive reinforcement is established.
Abstract This book provides the reader with a contemporary and comprehensive introduction to Quantum Mechanics. It is suitable for beginners as well as for more advanced university students. Quantum mechanics is presented in a pedagogical fashion, with a clear logical organization. The various concepts and methods are introduced first in elementary terms, and later developed into more precise formulations. Systematic studies of approximation methods and the discussion of a wide class of physical applications follow. Part I of the book, together with the opening sections of Part II, provide adequate material for an introductory course of one semester at most universities. The rest of the book might be used in an advanced course on Quantum Mechanics. The basic material is fairly standard, even though some discussions such as those on general systems with time-dependent Hamiltonians, on metastable systems, as well as the discussions in some of the Complement sections, may not be found in other textbooks. The book also contains many original observations or new ways of illustrating even well-known subjects. In fact, the authors wish to convey in this book the sense of wonder in the logical simplicity and at the same time the beauty of subtle and far-reaching consequences of Quantum Mechanics, to young physics students in particular. Problem sets are provided at the end of each chapter, to be solved either analytically or by numerical methods. The solutions to both types of problems are given as separate pdf files or as Mathematica notebooks (there are 88 of them), all together on a CD accompanying the textbook. The presence of such a collection of numerical analyses enriches the main text and is one of the characteristic features of the book. With the many interesting systems discussed, the book will also be a useful reference for researchers and teachers. It provides the reader with a unique, enjoyable and rather complete textbook of Quantum Mechanics, destined to set a new standard for many years to come.
This paper presents a way to use a computer in the teaching of introductory, stationary state, Schrödinger quantum mechanics. The method is based upon an easily explained computer method; a 20-line program in BASIC is presented to implement the strategy. Not only is this numerical technique completely general for all one-dimensional and all spherically symmetric, three-dimensional problems, but also it allows the student to be introduced to topics of interest such as asymptotic solutions. Several examples of the use of the computer technique are presented.
Quantum Mechanics I: The Fundamentals provides a graduate-level account of the behavior of matter and energy at the molecular, atomic, nuclear, and sub-nuclear levels. It covers basic concepts, mathematical formalism, and applications to physically important systems. This fully updated new edition addresses many topics not typically found in books at this level, including: Bound state solutions of quantum pendulum Morse oscillator Solutions of classical counterpart of quantum mechanical systems A criterion for bound state Scattering from a locally periodic potential and reflection-less potential Modified Heisenberg relation Wave packet revival and its dynamics An asymptotic method for slowly varying potentials Klein paradox, Einstein-Podolsky-Rosen (EPR) paradox, and Bell’s theorem Delayed-choice experiments Fractional quantum mechanics Numerical methods for quantum systems A collection of problems at the end of each chapter develops students’ understanding of both basic concepts and the application of theory to various physically important systems. This book, along with the authors’ follow-up Quantum Mechanics II: Advanced Topics, provides students with a broad, up-to-date introduction to quantum mechanics. Print Versions of this book also include access to the ebook version.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend
Introduction to the Maths and Physics of Quantum Mechanics details the mathematics and physics that are needed to learn the principles of quantum mechanics. It provides an accessible treatment of how to use quantum mechanics and why it is so successful in explaining natural phenomena. This book clarifies various aspects of quantum physics such as ‘why quantum mechanics equations contain “I”, the imaginary number?’, ‘Is it possible to make a transition from classical mechanics to quantum physics without using postulates?’ and ‘What is the origin of the uncertainty principle?’. A significant proportion of discussion is dedicated to the issue of why the wave function must be complex to properly describe our “real” world. The book also addresses the different formulations of quantum mechanics. A relatively simple introductory treatment is given for the “standard” Heisenberg matrix formulation and Schrodinger wave-function formulation and Feynman path integrals and second quantization are then discussed. This book will appeal to first- and second-year university students in physics, mathematics, engineering and other sciences studying quantum mechanics who will find material and clarifications not easily found in other textbooks. It will also appeal to self-taught readers with a genuine interest in modern physics who are willing to examine the mathematics and physics in a simple but rigorous way. Key Features: Written in an engaging and approachable manner, with fully explained mathematics and physics concepts Suitable as a companion to all introductory quantum mechanics textbooks Accessible to a general audience
Incorporating relativity into quantum mechanics presents challenges, one of which is the inconsistent possibility of multiple particles within a single-particle theory. This chapter describes the traditional, semi-analytic bound state in a 1-D box problem that requires a trial-and-error search algorithm. It examines techniques for finding the bound states for an arbitrary potential as a numerical eigenvalue problem and employs similar techniques for the relativistic hydrogen states of the Klein-Gordon equation. The chapter also employs a Monte Carlo algorithm to simulate the spontaneous decay of atoms or nuclei. It presents the numerical solution of the time-independent Schrodinger equation and also examines partial wave expansions, which leads to methods of computing the spherical harmonics and the associated Legendre functions, and ultimately the 3-D hydrogenic wave functions. The chapter deals with the Feynman path integral formulation of quantum mechanics, and its solution via the Metropolis algorithm.