Quantum Physics: Wave-Particle Duality and Energy Quantization, Esquemas de Química Inorgânica

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MODERN

PHYSICS

Third edition

K e n n e t h S. K r a n e

D E P A R T M E N T O F P H Y S I C S

O R E G O N S T A T E U N I V E R S I T Y

JOHN WILEY & SONS, INC

PREFACE

This textbook is meant to serve a first course in modern physics, including relativity, quantum mechanics, and their applications. Such a course often follows the standard introductory course in calculus-based classical physics. The course addresses two different audiences: (1) Physics majors, who will later take a more rigorous course in quantum mechanics, find an introductory modern course helpful in providing background for the rigors of their imminent coursework in classical mechanics, thermodynamics, and electromagnetism. (2) Nonmajors, who may take no additional physics class, find an increasing need for concepts from modern physics in their disciplines—a classical introductory course is not sufficient background for chemists, computer scientists, nuclear and electrical engineers, or molecular biologists. Necessary prerequisites for undertaking the text include any standard calculus- based course covering mechanics, electromagnetism, thermal physics, and optics. Calculus is used extensively, but no previous knowledge of differential equations, complex variables, or partial derivatives is assumed (although some familiarity with these topics would be helpful). Chapters 1–8 constitute the core of the text. They cover special relativity and quantum theory through atomic structure. At that point the reader may continue with Chapters 9–11 (molecules, quantum statistics, and solids) or branch to Chapters 12–14 (nuclei and particles). The final chapter covers cosmology and can be considered the capstone of modern physics as it brings together topics from relativity (special and general) as well as from nearly all of the previous material covered in the text. The unifying theme of the text is the empirical basis of modern physics. Experimental tests of derived properties are discussed throughout. These include the latest tests of special and general relativity as well as studies of wave-particle duality for photons and material particles. Applications of basic phenomena are extensively presented, and data from the literature are used not only to illustrate those phenomena but to offer insight into how “real” physics is done. Students using the text have the opportunity to study how laboratory results and the analysis based on quantum theory go hand-in-hand to illuminate such diverse topics as Bose-Einstein condensation, heat capacities of solids, paramagnetism, the cosmic microwave background radiation, X-ray spectra, dilute mixtures of 3 He in 4 He, and molecular spectroscopy of the interstellar medium. This third edition offers many changes from the previous edition. Most of the chapters have undergone considerable or complete rewriting. New topics have been introduced and others have been rearranged. More experimental results are presented and recent discoveries are highlighted, such as the WMAP microwave background data and Bose-Einstein condensation. End-of-chapter problem sets now include problems organized according to chapter section, which offer the student an opportunity to gain familiarity with a particular topic, as well as general problems, which often require the student to apply a broader array of concepts or techniques. The number of worked examples in the chapters and the number of end-of-chapter questions and problems have each increased by about 15% from the previous edition. The range of abilities required to solve the problems has been

vi Preface

broadened, so that this edition includes both more straightforward problems that build confidence as well as more difficult problems that will challenge students. Each chapter now includes a brief summary of the important points. Some of the end-of-chapter problems are available for assignment using the WebAssign program (www.webassign.net). A new development in physics teaching since the appearance of the 2 nd^ edition of this text has been the availability of a large and robust body of literature from physics education research (PER). My own teaching style has been profoundly influenced by PER findings, and in preparing this new edition I have tried to incorporate PER results wherever possible. One of the major themes that has emerged from PER in the past decade or two is that students can often learn successful algorithms for solving problems while lacking a fundamental understanding of the underlying concepts. Many approaches to addressing this problem are based on pre-class conceptual exercises and in-class individual or group activities that help students to reason through diverse problems that can’t be resolved by plugging numbers into an equation. It is absolutely essential to devote class time to these exercises and to follow through with exam questions that require similar analysis and articulation of the conceptual reasoning. More details regarding the application of PER to the teaching of modern physics, including references to articles from the PER literature, are included in the Instructor’s Manual for this text, which can be found at www.wiley.com/college/krane. The Instructor’s Manual also includes examples of conceptual questions for in-class discussion or exams that have been developed and class tested through the support of a Course, Curriculum and Laboratory Improvement grant from the National Science Foundation. Specific changes to the chapters include the following: Chapter 1 : The sections on Units and Dimensions and on Significant Figures have been removed. In their place, a more detailed review of applications of classical energy and momentum conservation is offered. The need for special relativity is briefly established with a discussion of the failures of the classical concepts of space and time, and the need for quantum theory is previewed in the failure of Maxwell-Boltzmann particle statistics to account for the heat capacities of diatomic gases. Chapter 2 : Spacetime diagrams have been introduced to help illustrate relation- ships in the twin paradox. The application of the relativistic conservation laws to decay and collisions processes is now given a separate section to help students learn to apply those laws. The section on tests of special relativity has been updated to include recent results. Chapter 3 : The section on thermal radiation has been rewritten, and more detailed derivations of the Rayleigh-Jeans and Planck formulas are now given. Chapter 4 : New experimental results for particle diffraction and interference are discussed. The sections on the classical uncertainty relationships and on wave packet construction and motion have been rewritten. Chapter 5 : To help students understand the processes involved in applying boundary conditions to solutions of the Schr¨odinger equation, a new section on wave boundary conditions has been added. A new introductory section on particle confinement introduces energy quantization and helps to build the connection between the wave function and the uncertainty relationships. Time dependence of the wave function is introduced more explicitly at an

viii Preface

Paul Lee, California State University, Northridge Jeff Loats, Metropolitan State College of Denver Jay Newman, Union College Stephen Pate, New Mexico State University David Roundy, Oregon State University Rich Schelp, Erskine College Weidian Shen, Eastern Michigan University Hongtao Shi, Sonoma State University Janet Tate, Oregon State University Jeffrey L. Wragg, College of Charleston Weldon Wilson, University of Central Oklahoma

I am also grateful for the many anonymous comments from students who used the manuscript at the test sites. I am indebted to all those reviewers and users for their contributions to the project. Funding for the development and testing of the supplemental exercises in the Instructor’s Manual was provided through a grant from the National Science Foundation. I am pleased to acknowledge their support. Two graduate students at Oregon State University helped to test and implement the curricular reforms: K. C. Walsh and Pornrat Wattasinawich. I appreciate their assistance in this project. The staff at John Wiley & Sons have been especially helpful throughout the project. I am particularly grateful to: Executive Editor Stuart Johnson for his patience and support in bringing the new edition into reality; Assistant Production Editor Elaine Chew for handling a myriad of complicated composition and illustration details with efficiency and good humor; and Photo Editor Sheena Goldstein for helping me navigate the treacherous waters of new copyright and permission restrictions. In my research and other professional activities, I occasionally meet physicists who used earlier editions of this text when they were students. Some report that their first exposure to modern physics kindled the spark that led them to careers in physics. For many students, this course offers their first insights into what physicists really do and what is exciting, perplexing, and challenging about our profession. I hope students who use this new edition will continue to find those inspirations.

Corvallis, Oregon Kenneth S. Krane August 2011 [email protected]

CONTENTS

Chapter

CASSINI INTERPLANETARY TRAJECTORY

VENUS SWINGBY 26 APR 1998

EARTH SWINGBY 18 AUG 1999 JUPITER SWINGBY 30 DEC 2000

VENUS SWINGBY 24 JUN 1999

LAUNCH 15 OCT 1997

ORBIT OF EARTH

ORBIT OF VENUS

DEEP SPACE MANEUVER 3 DEC 1990

ORBIT OF JUPITER (^) ORBIT OF SATURN

SATURN ARRIVAL 1 JUL 2004

1

THE FAILURES OF CLASSICAL

PHYSICS

Classical physics, as postulated by Newton, has enabled us to send space probes on trajectories involving many complicated maneuvers, such as the Cassini mission to Saturn, which was launched in 1997 and gained speed for its trip to Saturn by performing four ‘‘gravity-assist’’ flybys of Venus (twice), Earth, and Jupiter. The spacecraft arrived at Saturn in 2004 and is expected to continue to send data through at least 2017. Planning and executing such interplanetary voyages are great triumphs for Newtonian physics, but when objects move at speeds close to the speed of light or when we examine matter on the atomic or subatomic scale, Newtonian mechanics is not adequate to explain our observations, as we discuss in this chapter.

1.1 |^ Review of Classical Physics 3

physics offers either inadequate or incorrect conclusions. These situations are not necessarily those that originally gave rise to the relativity and quantum theories, but they do help us understand why classical physics fails to give us a complete picture of nature.

1.1 REVIEW OF CLASSICAL PHYSICS

Although there are many areas in which modern physics differs radically from classical physics, we frequently find the need to refer to concepts of classical physics. Here is a brief review of some of the concepts of classical physics that we may need.

Mechanics

A particle of mass m moving with velocity v has a kinetic energy defined by

K = 12 mv^2 (1.1)

and a linear momentum (^)  p defined by

p  = m  v (1.2)

In terms of the linear momentum, the kinetic energy can be written

K =

p^2 2 m

When one particle collides with another, we analyze the collision by applying two fundamental conservation laws:

I. Conservation of Energy. The total energy of an isolated system (on which no net external force acts) remains constant. In the case of a collision between particles, this means that the total energy of the particles before the collision is equal to the total energy of the particles after the collision. II. Conservation of Linear Momentum. The total linear momentum of an isolated system remains constant. For the collision, the total linear momentum of the particles before the collision is equal to the total linear momentum of the particles after the collision. Because linear momentum is a vector, application of this law usually gives us two equations, one for the x components and another for the y components.

These two conservation laws are of the most basic importance to understanding and analyzing a wide variety of problems in classical physics. Problems 1–4 and 11–14 at the end of this chapter review the use of these laws. The importance of these conservation laws is both so great and so fundamental that, even though in Chapter 2 we learn that the special theory of relativity modifies Eqs. 1.1, 1.2, and 1.3, the laws of conservation of energy and linear momentum remain valid.

4 Chapter 1 |^ The Failures of Classical Physics

Example 1.

A helium atom ( m = 6. 6465 × 10 −^27 kg) moving at a speed of v He = 1. 518 × 106 m/s collides with an atom of nitro- gen ( m = 2. 3253 × 10 −^26 kg) at rest. After the collision, the helium atom is found to be moving with a velocity of v ′ He = 1. 199 × 106 m/s at an angle of θHe = 78. 75 ◦^ rela- tive to the direction of the original motion of the helium atom. ( a ) Find the velocity (magnitude and direction) of the nitrogen atom after the collision. ( b ) Compare the kinetic energy before the collision with the total kinetic energy of the atoms after the collision.

Solution ( a ) The law of conservation of momentum for this colli- sion can be written in vector form as (^)  p initial = (^)  p final , which is equivalent to

p (^) x ,initial = px ,final and py ,initial = py ,final

The collision is shown in Figure 1.1. The initial values of the total momentum are, choosing the x axis to be the direction of the initial motion of the helium atom,

p (^) x ,initial = m He v He and py ,initial = 0

The final total momentum can be written

px ,final = m He v ′ He cos θHe + m N v ′ N cos θN py ,final = m He v ′ He sin θHe + m N v ′ N sin θN

The expression for py ,final is written in general form with a + sign even though we expect that θHe and θN are on opposite sides of the x axis. If the equation is written in this way, θN will come out to be negative. The law of

x

x

N ( a )

( b )

He

v He

v ′N

v ′He

θ He θ N

y

y

FIGURE 1.1 Example 1.1. ( a ) Before collision; ( b ) after collision.

conservation of momentum gives, for the x components, m He v He = m He v ′ He cos θHe + m N v ′ N cos θN , and for the y components, 0 = m He v ′ He sin θHe + m N v ′ N sin θN. Solving for the unknown terms, we find

v ′ N cos θN =

m He( v He − v ′ He cos θHe) m N = {( 6. 6465 × 10 −^27 kg)[1. 518 × 10 6 m/s −( 1. 199 × 106 m/s)(cos 78. 75 ◦)]} ×( 2. 3253 × 10 −^26 kg)−^1 = 3. 6704 × 105 m/s

v ′ N sin θN = −

m He v ′ He sin θHe m N = −( 6. 6465 × 10 −^27 kg)( 1. 199 × 10 6 m/s) ×(sin78.75◦)( 2. 3253 × 10 −^26 kg)−^1 = − 3. 3613 × 10 5 m/s We can now solve for v ′ N and θN :

v ′ N =

( v ′ N sin θN)^2 + ( v ′ N cos θN)^2

(− 3. 3613 × 10 5 m/s)^2 + ( 3. 6704 × 10 5 m/s)^2 = 4. 977 × 105 m/s

θN = tan−^1

v ′ N sin θN v ′ N cos θN

= tan−^1

− 3. 3613 × 105 m/s

3. 6704 × 105 m/s

( b ) The initial kinetic energy is K initial = 12 m He v^2 He = 12 ( 6. 6465 × 10 −^27 kg)(1.518 × 10 6 m/s)^2 = 7. 658 × 10 −^15 J and the total final kinetic energy is K final = 12 m He v ′He^2 + 12 m N v ′N^2 = 12 ( 6. 6465 × 10 −^27 kg)( 1. 199 × 10 6 m/s)^2

• 12 ( 2. 3253 × 10 −^26 kg)( 4. 977 × 10 5 m/s) 2 = 7. 658 × 10 −^15 J Note that the initial and final kinetic energies are equal. This is the characteristic of an elastic collision, in which no energy is lost to, for example, internal excitation of the particles.