Introduction to Quantum Mechanics E-Book

A. C. Phillips

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: The Manchester Physics Series
• Sprache: Englisch

Seitenzahl: 346

Veröffentlichungsjahr: 2013

Beschreibung

Introduction to Quantum Mechanics is an introduction to the power and elegance of quantum mechanics. Assuming little in the way of prior knowledge, quantum concepts are carefully and precisely presented, and explored through numerous applications and problems. Some of the more challenging aspects that are essential for a modern appreciation of the subject have been included, but are introduced and developed in the simplest way possible. Undergraduates taking a first course on quantum mechanics will find this text an invaluable introduction to the field and help prepare them for more advanced courses. Introduction to Quantum Mechanics: * Starts from basics, reviewing relevant concepts of classical physics where needed. * Motivates by considering weird behaviour of quantum particles. * Presents mathematical arguments in their simplest form.

Leseprobe

Contents

Foreword

Editor’s preface to the Manchester Physics Series

Author’s preface

1 Planck’s constant in action

1.1 PHOTONS

1.2 DE BROGLIE WAVES

1.3 ATOMS

1.4 MEASUREMENT

2 The Schrödinger equation

2.1 WAVES

2.2 PARTICLE WAVE EQUATIONS

3 Position and momentum

3.1 PROBABILITY

3.2 POSITION PROBABILITIES

3.3 MOMENTUM PROBABILITIES

3.4 A PARTICLE IN A BOX I

3.5 EXPECTATION VALUES

3.6 QUANTUM STATES

4 Energy and time

4.1 THE HAMILTONIAN OPERATOR

4.2 NORMAL MODES OF A STRING

4.3 STATES OF CERTAIN ENERGY

4.4 A PARTICLE IN A BOX II

4.5 STATES OF UNCERTAIN ENERGY

4.6 TIME DEPENDENCE

5 Square wells and barriers

5.1 BOUND AND UNBOUND STATES

5.2 BARRIER PENETRATION

6 The harmonic oscillator

6.1 THE CLASSICAL OSCILLATOR

6.2 THE QUANTUM OSCILLATOR

6.3 QUANTUM STATES

6.4 DIATOMIC MOLECULES

6.5 THREE-DIMENSIONAL OSCILLATORS

6.6 THE OSCILLATOR EIGENVALUE PROBLEM

7 Observables and operators

7.1 ESSENTIAL PROPERTIES

7.2 POSITION AND MOMENTUM

7.3 COMPATIBLE OBSERVABLES

7.4 COMMUTATORS

7.5 CONSTANTS OF MOTION

8 Angular momentum

8.1 ANGULAR MOMENTUM BASICS

8.2 MAGNETIC MOMENTS

8.3 ORBITAL ANGULAR MOMENTUM

9 The hydrogen atom

9.1 CENTRAL POTENTIALS

9.2 QUANTUM MECHANICS OF THE HYDROGEN ATOM

9.3 SIZES AND SHAPES

9.4 RADIATIVE TRANSITIONS

9.5 THE REDUCED MASS EFFECT

9.6 RELATIVISTIC EFFECTS

9.7 THE COULOMB EIGENVALUE PROBLEM

10 Identical particles

10.1 EXCHANGE SYMMETRY

10.2 PHYSICAL CONSEQUENCES

10.3 EXCHANGE SYMMETRY WITH SPIN

10.4 BOSONS AND FERMIONS

11 Atoms

11.1 ATOMIC QUANTUM STATES

11.2 THE PERIODIC TABLE

11.3 WHAT IF?

Hints to selected problems

Further reading

Index

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A catalogue record for this book is available from the British Library

ISBN 13: 978-0-470-85323-8 (H/B) ISBN 13: 978-0-470-85324-5 (P/B)

To my sons:

JosephMichaelPatrickPeter

Foreword

Sadly, Tony Phillips, a good friend and colleague for more than thirty years, died on 27th November 2002. Over the years, we discussed most topics under the sun. The originality and clarity of his thoughts and the ethical basis of his judgements always made this a refreshing exercise. When discussing physics, quantum mechanics was a recurring theme which gained prominence after his decision to write this book. He completed the manuscript three months before his death and asked me to take care of the proofreading and the Index. A labour of love. I knew what Tony wanted—and what he did not want. Except for corrections, no changes have been made.

Tony was an outstanding teacher who could talk with students of all abilities. He had a deep knowledge of physics and was able to explain subtle ideas in a simple and delightful style. Who else would refer to the end-point of nuclear fusion in the sun as sunshine? Students appreciated him for these qualities, his straightforwardness and his genuine concern for them. This book is a fitting memorial to him.

Franz MandlDecember 2002

Editors’ preface to the Manchester Physics Series

The Manchester Physics Series is a series of textbooks at first degree level. It grew out of our experience at the Department of Physics and Astronomy at Manchester University, widely shared elsewhere, that many textbooks contain much more material than can be accommodated in a typical undergraduate course; and that this material is only rarely so arranged as to allow the definition of a short self-contained course. In planning these books we have had two objectives. One was to produce short books: so that lecturers should find them attractive for undergraduate courses; so that students should not be frightened off by their encyclopaedic size or price. To achieve this, we have been very selective in the choice of topics, with the emphasis on the basic physics together with some instructive, stimulating and useful applications. Our second objective was to produce books which allow courses of different lengths and difficulty to be selected with emphasis on different applications. To achieve such flexibility we have encouraged authors to use flow diagrams showing the logical connections between different chapters and to put some topics in starred sections. These cover more advanced and alternative material which is not required for the understanding of latter parts of each volume.

Although these books were conceived as a series, each of them is self-contained and can be used independently of the others. Several of them are suitable for wider use in other sciences. Each Author’s Preface gives details about the level, prerequisites, etc., of that volume.

The Manchester Physics Series has been very successful with total sales of more than a quarter of a million copies. We are extremely grateful to the many students and colleagues, at Manchester and elsewhere, for helpful criticisms and stimulating comments. Our particular thanks go to the authors for all the work they have done, for the many new ideas they have contributed, and for discussing patiently, and often accepting, the suggestions of the editors.

Finally we would like to thank our publishers, John Wiley & Sons, Ltd, for their enthusiastic and continued commitment to the Manchester Physics Series.

D. J. SandifordF. MandlA. C. PhillipsFebruary 1997

Author’s preface

There are many good advanced books on quantum mechanics but there is a distinct lack of books which attempt to give a serious introduction at a level suitable for undergraduates who have a tentative understanding of mathematics, probability and classical physics.

This book introduces the most important aspects of quantum mechanics in the simplest way possible, but challenging aspects which are essential for a meaningful understanding have not been evaded. It is an introduction to quantum mechanics which

motivates the fundamental postulates of quantum mechanics by considering the weird behaviour of quantum particles

reviews relevant concepts in classical physics before corresponding concepts are developed in quantum mechanics

presents mathematical arguments in their simplest form

provides an understandingof the power and elegance of quantum mechanics that will make more advanced texts accessible.

Chapter 1 provides a qualitative description of the remarkable properties of quantum particles, and these properties are used as the guidelines for a theory of quantum mechanics which is developed in Chapters 2, 3 and 4. Insight into this theory is gained by considering square wells and barriers in Chapter 5 and the harmonic oscillator in Chapter 6. Many of the concepts used in the first six chapters are clarified and developed in Chapter 7. Angular momentum in quantum mechanics is introduced in Chapter 8, but because angular momentum is a demanding topic, this chapter focusses on the ideas that are needed for an understanding of the hydrogen atom in Chapter 9, identical particles in Chapter 10 and many-electron atoms in Chapter 11. Chapter 10 explains why identical particles are described by entangled quantum states and how this entanglement for electrons leads to the Pauli exclusion principle.

Chapters 7 and 10 may be omitted without significant loss of continuity. They deal with concepts which are not needed elsewhere in the book.

I would like to express my thanks to students and colleagues at the University of Manchester. Daniel Guise helpfully calculated the energy levels in a screened Coulomb potential. Thomas York used his impressive computing skills to provide representations of the position probabilities for particles with different orbital angular momentum. Sean Freeman read an early version of the first six chapters and provided suggestions and encouragement. Finally, I would like to thank Franz Mandl for reading an early version of the book and for making forcefully intelligent suggestions for improvement.

A. C. PhillipsAugust 2002

1

Planck’s constant in action

Classical physics is dominated by two fundamental concepts. The first is the concept of a particle, a discrete entity with definite position and momentum which moves in accordance with Newton’s laws of motion. The second is the concept of an electromagnetic wave, an extended physical entity with a presence at every point in space that is provided by electric and magnetic fields which change in accordance with Maxwell’s laws of electromagnetism. The classical world picture is neat and tidy: the laws of particle motion account for the material world around us and the laws of electromagnetic fields account for the light waves which illuminate this world.

This constant is now called Planck’s constant.

In this chapter we shall see that Planck’s constant has a strange role of linking wave-like and particle-like properties. In so doing it reveals that physics cannot be based on two distinct, unrelated concepts, the concept of a particle and the concept of a wave. These classical concepts, it seems, are at best approximate descriptions of reality.

1.1 PHOTONS

Photons are particle-like quanta of electromagnetic radiation. They travel at the speed of light c with momentum p and energy given by

(1.1)

The evidence for the existence of photons emerged during the early years of the twentieth century. In 1923 the evidence became compelling when A. H. Compton showed that the wavelength of an X-ray increases when it is scattered by an atomic electron. This effect, which is now called the Compton effect, can be understood by assuming that the scattering process is a photon–electron collision in which energy and momentum are conserved. As illustrated in Fig. 1.1, the incident photon transfers momentum to a stationary electron so that the scattered photon has a lower momentum and hence a longer wavelength. In fact, when the photon is scattered through an angle by a stationary electron of mass , the increase in wavelength is given by