Morrison Modern Physics

for Scientists and Engineers
2. Auflage 2015
ISBN: 978-0-12-800828-7
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 448 Seiten

The second edition of Modern Physics for Scientists and Engineers is intended for a first course in modern physics. Beginning with a brief and focused account of the historical events leading to the formulation of modern quantum theory, later chapters delve into the underlying physics. Streamlined content, chapters on semiconductors, Dirac equation and quantum field theory, as well as a robust pedagogy and ancillary package, including an accompanying website with computer applets, assist students in learning the essential material. The applets provide a realistic description of the energy levels and wave functions of electrons in atoms and crystals. The Hartree-Fock and ABINIT applets are valuable tools for studying the properties of atoms and semiconductors. - Develops modern quantum mechanical ideas systematically and uses these ideas consistently throughout the book - Carefully considers fundamental subjects such as transition probabilities, crystal structure, reciprocal lattices, and Bloch theorem which are fundamental to any treatment of lasers and semiconductor devices - Clarifies each important concept through the use of a simple example and often an illustration - Features expanded exercises and problems at the end of each chapter - Offers multiple appendices to provide quick-reference for students

John Morrison received a BS degree in Physics from University of Santa Clara in California. During his undergraduate years, he majored in English, Philosophy, and Physics and served as the editor of the campus literary magazine, the Owl. Enrolling at Johns Hopkins University in Baltimore, Maryland, he received a PhD degree in theoretical Physics and moved on to postdoctoral research at Argonne National Laboratory where he was a member of the Heavy Atom Group. He then went to Sweden where he received a grant from the Swedish Research Council to build up a research group in theoretical atomic physics at Chalmers Technical University in Goteborg, Sweden. Working together with Ingvar Lindgren, he taught a graduate level-course in theoretical atomic physics for a number of years. Their teaching lead to the publication of the monograph, Atomic Many-Body Theory, which first appeared as Volume 13 of the Springer Series on Chemical Physics. The second edition of this book has become a Springer classic. Returning to the United States, John Morrison obtained a position in the Department of Physics and Astronomy at University of Louisville where he has taught courses in elementary physics, astronomy, modern physics, and quantum mechanics. In recent years, he has traveled extensively in Latin America and the Middle East maintaining contacts with scientists and mathematicians at the Hebrew University in Jerusalem and the Technion University in Haifa. During the Fall semester of 2009, he taught a course on computational physics at Birzeit University near Ramallah on the West Bank, and he has recruited Palestinian students for the graduate program in physics at University of Louisville. He speaks English, Swedish, and Spanish, and he is currently studying Arabic and Hebrew.

Autoren/Hrsg.

Morrison, John

Inhaltsverzeichnis

1;Front Cover ;1
2;Modern Physics: for Scientists and Engineers;4
3;Copyright;5
4;Dedication;6
5;Contents;8
6;Preface;12
6.1;This New Edition;12
6.2;New Features;12
6.3;The Nature of the Book;13
7;Acknowledgments;16
8;Introduction;18
8.1;I.1. The Concepts of Particles and Waves;18
8.1.1;I.1.1. The Variables of a Moving Particle;18
8.1.2;I.1.2. Elementary Properties of Waves;20
8.1.3;I.1.3. Interference and Diffraction Phenomena;27
8.2;I.2. An Overview of Quantum Physics;30
8.3;Basic Equations;35
8.4;Summary;36
8.5;Suggestions for Further Reading;37
8.6;Questions;37
8.7;Problems;37
9;Chapter 1: The Wave-Particle Duality;40
9.1;1.1. The Particle Model of Light;40
9.1.1;1.1.1. The Photoelectric Effect;40
9.1.2;1.1.2. The Absorption and Emission of Light by Atoms;43
9.1.3;1.1.3. The Compton Effect;49
9.2;1.2. The Wave Model of Radiation and Matter;51
9.2.1;1.2.1. X-ray Scattering;51
9.2.2;1.2.2. Electron Waves;52
9.3;Suggestions for Further Reading;54
9.4;Basic Equations;54
9.4.1;Photoelectric Effect;54
9.4.2;Emission and Absorption of Radiation by Atoms;55
9.4.3;Wave Properties of Radiation and Matter;55
9.5;Summary;55
9.6;Questions;55
9.7;Problems;56
10;Chapter 2: The Schrödinger Wave Equation;58
10.1;2.1. The Wave Equation;58
10.2;2.2. Probabilities and Average Values;62
10.3;2.3. The Finite Potential Well;65
10.4;2.4. The Simple Harmonic Oscillator;69
10.4.1;2.4.1. The Schrödinger Equation for the Oscillator;71
10.5;2.5. Time Evolution of the Wave Function;72
10.6;Suggestion for Further Reading;75
10.7;Basic Equations;75
10.7.1;The Wave Equation;75
10.7.2;Solutions of Schrödinger Time-Independent Equation;75
10.7.3;Time Evolution of Wave Function;76
10.8;Summary;76
10.9;Questions;76
10.10;Problems;77
11;Chapter 3: Operators and Waves;80
11.1;3.1. Observables, Operators, and Eigenvalues;81
11.2;3.2. A Closer Look at the Finite Well;84
11.3;3.3. Electron Scattering;88
11.3.1;3.3.1. Scattering from a Potential Step;88
11.3.2;3.3.2. Barrier Penetration and Tunneling;92
11.4;3.4. The Heisenberg Uncertainty Principle;95
11.4.1;3.4.1. Wave Packets and the Uncertainty Principle;96
11.4.2;3.4.2. Average Value of the Momentum and the Energy;99
11.5;Suggestion for Further Reading;100
11.6;Basic Equations;100
11.6.1;Observables, Operators, and Eigenvalues;100
11.6.2;Electron Scattering;100
11.6.3;The Heisenberg Uncertainty Principle;101
11.7;Summary;101
11.8;Questions;102
11.9;Problems;102
12;Chapter 4: The Hydrogen Atom;104
12.1;4.1. The Gross Structure of Hydrogen;104
12.1.1;4.1.1. The Schrödinger Equation in Three Dimensions;104
12.1.2;4.1.2. The Energy Levels of Hydrogen;106
12.1.3;4.1.3. The Wave Functions of Hydrogen;107
12.1.4;4.1.4. Probabilities and Average Values in Three Dimensions;110
12.1.5;4.1.5. The Intrinsic Spin of the Electron;112
12.2;4.2. Radiative Transitions;113
12.2.1;4.2.1. The Einstein A and B Coefficients;113
12.2.2;4.2.2. Transition Probabilities;114
12.2.3;4.2.3. Selection Rules;118
12.3;4.3. The Fine Structure of Hydrogen;119
12.3.1;4.3.1. The Magnetic Moment of the Electron;119
12.3.2;4.3.2. The Stern-Gerlach Experiment;122
12.3.3;4.3.3. The Spin of the Electron;123
12.3.4;4.3.4. The Addition of Angular Momentum;124
12.3.5;4.3.5. * The Fine Structure;125
12.3.6;4.3.6. * The Zeeman Effect;127
12.4;Suggestion For Further Reading;129
12.5;Basic Equations;129
12.5.1;Wave Function for Hydrogen;129
12.5.2;Probabilities and Average Values;129
12.5.3;Transition Probabilities;129
12.5.4;Selection Rules;129
12.5.5;The Fine Structure of Hydrogen;130
12.6;Summary;130
12.7;Questions;131
12.8;Problems;131
13;Chapter 5: Many-Electron Atoms;134
13.1;5.1. The Independent-Particle Model;134
13.1.1;5.1.1. Antisymmetric Wave Functions and the Pauli Exclusion Principle;135
13.1.2;5.1.2. The Central-Field Approximation;136
13.2;5.2. Shell Structure and the Periodic Table;137
13.3;5.3. The LS Term Energies;139
13.4;5.4. Configurations of Two Electrons;139
13.4.1;5.4.1. Configurations of Equivalent Electrons;140
13.4.2;5.4.2. Configurations of Two Nonequivalent Electrons;142
13.5;5.5. The Hartree-Fock Method;143
13.5.1;5.5.1. The Hartree-Fock Applet;144
13.5.2;5.5.2. The Size of Atoms and the Strength of Their Interactions;147
13.6;Suggestion for Further Reading;152
13.7;Basic Equations;152
13.7.1;Definition of Atomic Units;152
13.7.2;Atomic Unit of Distance;152
13.7.3;Atomic Unit of Energy;152
13.8;Summary ;152
13.9;Questions;153
13.10;Problems;153
14;Chapter 6: The Emergence of Masers and Lasers;156
14.1;6.1. Radiative Transitions;156
14.2;6.2. Laser Amplification;157
14.3;6.3. Laser Cooling;162
14.4;6.4. * Magneto-Optical Traps;162
14.5;Suggestions for Further Reading;165
14.6;Basic Equations;166
14.6.1;Hamiltonian of Outer Electron in the Magnetic Field of Nucleus;166
14.6.2;Total Angular Momentum of Electron and Nucleus;166
14.6.3;The z-Component of Magnetic Moment of Outer Electron;166
14.6.4;The Energy of the Outer Electron Due to the Magnetic Field B;166
14.7;Summary;166
14.8;Questions;166
14.9;Problems;167
15;Chapter 7: Statistical Physics;168
15.1;7.1. The Nature of Statistical Laws;168
15.2;7.2. An Ideal Gas;171
15.3;7.3. Applications of Maxwell-Boltzmann Statistics;173
15.3.1;7.3.1. Maxwell Distribution of the Speeds of Gas Particles;173
15.3.2;7.3.2. Black-Body Radiation;179
15.4;7.4. Entropy and the Laws of Thermodynamics;184
15.4.1;7.4.1. The Four Laws of Thermodynamics;186
15.5;7.5. A Perfect Quantum Gas;188
15.6;7.6. Bose-Einstein Condensation;192
15.7;7.7. Free-Electron Theory of Metals;194
15.8;Suggestions for Further Reading;199
15.9;Basic Equations;200
15.9.1;Maxwell-Boltzmann Statistics;200
15.9.2;Applications of Maxwell-Boltzmann Statistics;200
15.9.3;Entropy and the Laws of Thermodynamics;200
15.9.4;Quantum Statistics;201
15.9.5;Free-Electron Theory of Metals;201
15.10;Summary;201
15.11;Questions;202
15.12;Problems;203
16;Chapter 8: Electronic Structure of Solids;206
16.1;8.1. Introduction;206
16.2;8.2. The Bravais Lattice;207
16.3;8.3. Additional Crystal Structures;211
16.3.1;8.3.1. The Diamond Structure;211
16.3.2;8.3.2. The hcp Structure;211
16.3.3;8.3.3. The Sodium Chloride Structure;212
16.4;8.4. The Reciprocal Lattice;213
16.5;8.5. Lattice Planes;216
16.6;8.6. Bloch's Theorem;220
16.7;8.7. Diffraction of Electrons by an Ideal Crystal;224
16.8;8.8. The Bandgap;226
16.9;8.9. Classification of Solids;228
16.9.1;8.9.1. The Band Picture;228
16.9.1.1;Insulators;228
16.9.1.2;Semiconductors;228
16.9.1.3;Metals;229
16.9.1.4;Graphene;229
16.9.1.5;Carbon Nanotubes;231
16.9.2;8.9.2. The Bond Picture;231
16.9.2.1;Covalent Bonding;232
16.9.2.2;Ionic Bonding;233
16.9.2.3;Molecular Crystals;234
16.9.2.4;Hydrogen-Bonded Crystals;234
16.9.2.5;Metals;234
16.10;Suggestions for Further Reading;235
16.11;Basic Equations;235
16.11.1;Bravais Lattice;235
16.11.2;Reciprocal Lattice;236
16.11.3;Bloch’s Theorem;236
16.11.4;Scattering of Electrons by a Crystal;236
16.12;Summary;236
16.13;Questions;237
16.14;Problems;237
17;Chapter 9: Charge Carriers in Semiconductors;242
17.1;9.1. Density of Charge Carriers in Semiconductors;242
17.2;9.2. Doped Crystals;245
17.3;9.3. A Few Simple Devices;246
17.3.1;9.3.1. The p-n Junction;247
17.3.2;9.3.2. Bipolar Transistors;249
17.3.3;9.3.3. Junction Field-Effect Transistors;250
17.3.4;9.3.4. MOSFETs;251
17.4;Suggestions for Further Reading;251
17.5;Summary;252
17.6;Questions;252
18;Chapter 10: Semiconductor Lasers;254
18.1;10.1. Motion of Electrons in a Crystal;254
18.2;10.2. Band Structure of Semiconductors;256
18.2.1;10.2.1. Conduction Bands;256
18.2.2;10.2.2. Valence Bands;257
18.2.3;10.2.3. Optical Transitions;257
18.3;10.3. Heterostructures;259
18.3.1;10.3.1. Properties of Heterostructures;259
18.3.2;10.3.2. Experimental Methods;260
18.3.3;10.3.3. Theoretical Methods;262
18.3.4;10.3.4. Band Engineering;263
18.4;10.4. Quantum Wells;264
18.4.1;10.4.1. The Finite Well;265
18.4.2;10.4.2. Two-Dimensional Systems;265
18.4.3;10.4.3. *Quantum Wells in Heterostructures;266
18.5;10.5. Quantum Barriers;268
18.5.1;10.5.1. Scattering from a Potential Step;268
18.5.2;10.5.2. T-Matrices;270
18.5.3;10.5.3. Scattering from More Complex Barriers;271
18.6;10.6. Reflection and Transmission of Light;274
18.6.1;10.6.1. Reflection and Transmission by an Interface;275
18.6.2;10.6.2. The Fabry-Perot Laser;277
18.7;10.7. Phenomenological Description of Diode Lasers;278
18.7.1;10.7.1. The Rate Equation;279
18.7.2;10.7.2. Well Below Threshold;281
18.7.3;10.7.3. The Laser Threshold;281
18.7.4;10.7.4. Above Threshold;282
18.8;Suggestions for Further Reading;283
18.9;Basic Equations;283
18.9.1;Quantum Wells;283
18.9.2;Potential Barriers;284
18.9.3;Reflection and Transmission of Light by an Interface;284
18.9.4;Phenomenological Description of Diode Lasers;284
18.10;Summary;285
18.11;Questions;285
18.12;Problems;286
19;Chapter 11: Relativity I;288
19.1;11.1. Galilean Transformations;288
19.2;11.2. The Relative Nature of Simultaneity;291
19.3;11.3. Lorentz Transformation;293
19.3.1;11.3.1. The Transformation Equations;293
19.3.2;11.3.2. Lorentz Contraction;296
19.3.3;11.3.3. Time Dilation;297
19.3.4;11.3.4. The Invariant Space-Time Interval;300
19.3.5;11.3.5. Addition of Velocities;301
19.3.6;11.3.6. The Doppler Effect;302
19.4;11.4. Space-Time Diagrams;304
19.4.1;11.4.1. Particle Motion;305
19.4.2;11.4.2. Lorentz Transformations;308
19.4.3;11.4.3. The Light Cone;309
19.5;11.5. Four-Vectors;310
19.6;Suggestions For Further Reading;314
19.7;Basic Equations;315
19.7.1;Galilean Transformations;315
19.7.2;The Relativistic Transformations;315
19.7.3;Four-Vectors;316
19.8;Summary;316
19.9;Questions;316
19.10;Problems;317
20;Chapter 12: Relativity II;320
20.1;12.1. Momentum and Energy;320
20.2;12.2. Conservation of Energy and Momentum;323
20.3;12.3. * The Dirac Theory of the Electron;327
20.3.1;12.3.1. Review of the Schrödinger Theory;327
20.3.2;12.3.2. The Klein-Gordon Equation;329
20.3.3;12.3.3. The Dirac Equation;329
20.3.4;12.3.4. Plane Wave Solutions of the Dirac Equation;332
20.4;12.4. * Field Quantization;335
20.5;Suggestions For Further Reading;337
20.6;Basic Equations;337
20.6.1;Definitions;337
20.6.2;The Dirac Theory of the Electron;338
20.7;Summary;339
20.8;Questions;339
20.9;Problems;340
21;Chapter 13: Particle Physics;342
21.1;13.1. Leptons and Quarks;342
21.2;13.2. Conservation Laws;349
21.2.1;13.2.1. Energy, Momentum, and Charge;349
21.2.2;13.2.2. Lepton Number;350
21.2.3;13.2.3. Baryon Number;351
21.2.4;13.2.4. Strangeness;353
21.2.5;13.2.5. Charm, Beauty, and Truth;355
21.3;13.3. Spatial Symmetries;356
21.3.1;13.3.1. Angular Momentum of Composite Systems;356
21.3.2;13.3.2. Parity;357
21.3.3;13.3.3. Charge Conjugation;359
21.4;13.4. Isospin and Color;361
21.4.1;13.4.1. Isospin;361
21.4.2;13.4.2. Color;367
21.5;13.5. Feynman Diagrams;369
21.5.1;13.5.1. Electromagnetic Interactions;370
21.5.2;13.5.2. Weak Interactions;371
21.5.3;13.5.3. Strong Interactions;373
21.6;13.6. * The Flavor and Color SU(3) Symmetries;374
21.6.1;13.6.1. The SU(3) Symmetry Group;375
21.6.2;13.6.2. The Representations of SU(3);377
21.7;13.7. * Gauge Invariance and the Electroweak Theory;382
21.8;13.8. Spontaneous Symmetry Breaking and the Discovery of the Higgs;384
21.9;Suggestion for Further Reading;387
21.10;Basic Equations;388
21.10.1;Leptons and Quarks;388
21.10.2;Definition of Hypercharge and Isospin;388
21.10.3;Isospin;388
21.10.4;Feynman Diagrams;388
21.10.5;SU(3) Symmetry;388
21.11;Summary;389
21.12;Questions;389
21.13;Problems;390
22;Chapter 14: Nuclear Physics;392
22.1;14.1. Properties of Nuclei;392
22.1.1;14.1.1. Nuclear Sizes;393
22.1.2;14.1.2. Binding Energies;396
22.1.3;14.1.3. The Semi-Empirical Mass Formula;398
22.2;14.2. Decay Processes;400
22.2.1;14.2.1. a-Decay;401
22.2.2;14.2.2. The ß-Stability Valley;402
22.2.3;14.2.3. .-Decay;404
22.2.4;14.2.4. Natural Radioactivity;406
22.3;14.3. The Nuclear Shell Model;407
22.3.1;14.3.1. Nuclear Potential Wells;407
22.3.2;14.3.2. Nucleon States;408
22.3.3;14.3.3. Magic Numbers;410
22.3.4;14.3.4. The Spin-Orbit Interaction;410
22.4;14.4. Excited States of Nuclei;411
22.5;Suggestions for Further Reading;415
22.6;Basic Equations;415
22.6.1;Binding Energy;415
22.6.2;The Semi-Empirical Formula;415
22.6.3;Magic Numbers;415
22.7;Summary;415
22.8;Questions;416
22.9;Problems;416
23;Appendix A: Constants and Conversion Factors;420
23.1;Constants;420
23.2;Particle Masses;420
23.3;Conversion Factors;421
24;Appendix B: Atomic Masses;422
25;Appendix C: Introduction to MATLAB;428
25.1;Creating a Vector;428
25.2;Plotting Functions;429
25.3;Using Arrays in MATLAB;429
25.4;Using Functions in MATLAB;430
26;Appendix D: Solution of the Oscillator Equation;432
27;Appendix E. The Average Value of the Momentum;436
28;Appendix F. The Hartree-Fock Applet;438
29;Appendix G. Integrals that Arise in Statistical Physics;440
29.1;Reference;442
29.2;Further Reading;442
30;Index;444
31;Appendix AA. The Gradient and Laplacian Operators;450
31.1;The Gradient Operator;450
31.2;The Divergence of a Vector;450
31.3;The Laplacian of a Function;451
31.4;The Angular Momentum Operators;451
32;Appendix BB. Solution of the Schrödinger Equation in Spherical Coordinates;454
32.1;Separation of the Schrödinger Equation;454
33;Appendix CC. More Accurate Solutions of the Eigenvalue Problem;460
33.1;A 5-Point Finite Difference Formula;460
34;Appendix DD. The Angular Momentum Operators;468
34.1;Generalization of the Quantum Rules;468
34.2;Commution Relations;468
34.3;Spectrum of Eigenvalues;471
35;Appendix EE. The Radial Equation for Hydrogen;474
36;Appendix FF: Transition Probabilities for z-Polarized Light;476
37;Appendix GG: Transitions with x- and y-Polarized Light;480
38;Appendix HH: Derivation of the Distribution Laws;482
38.1;Maxwell-Boltzmann Statistics;482
38.2;Bose-Einstein Statistics;483
38.3;Fermi-Dirac Statistics;484
39;Appendix II: Derivation of Bloch's Theorem;486
40;Appendix JJ: The Band Gap;488
41;Appendix KK: Vector Spaces and Matrices;492
42;Appendix LL: Algebraic Solution of the Oscillator;496

Leseproben

Introduction

Every physical system can be characterized by its size and the length of time it takes for processes occurring within it to evolve. This is as true of the distribution of electrons circulating about the nucleus of an atom as it is of a chain of mountains rising up over the ages.

Modern physics is a rich field including decisive experiments conducted in the early part of the twentieth century and more recent research that has given us a deeper understanding of fundamental processes in nature. In conjunction with our growing understanding of the physical world, a burgeoning technology has led to the development of lasers, solid-state devices, and many other innovations. This book provides an introduction to the fundamental ideas of modern physics and to the various fields of contemporary physics in which discoveries and innovation are going on continuously.

I.1 The Concepts of Particles and Waves

While some of the ideas currently used to describe microscopic systems differ considerably from the ideas of classical physics, other important ideas are classical in origin. We begin this chapter by discussing the important concepts of a particle and a wave which have the same meaning in classical and modern physics. A particle is an object with a definite mass concentrated at a single location in space, while a wave is a disturbance that propagates through space. The first section of this chapter, which discusses the elementary properties of particles and waves, provides a review of some of the fundamental ideas of classical physics. Other elements of classical physics will be reviewed later in the context for which they are important. The second section of this chapter describes some of the central ideas of modern quantum physics and also discusses the size and time scales of the physical systems considered in this book.

I.1.1 The Variables of a Moving Particle

The position and velocity vectors of a particle are illustrated in Fig. I.1. The position vector r extends from the origin to the particle, while the velocity vector v points in the direction of the particle’s motion. Other variables, which are appropriate for describing a moving particle, can be defined in terms of these elementary variables.

Figure I.1 The position r and the velocity v of a moving particle of mass m. The point O denotes the origin, and r0 denotes the distance between the line of motion and the origin.

The momentum p of the particle is equal to the product of the mass and velocity v of the particle

=mv.

We shall find that the momentum is useful for describing the motion of electrons in an extended system such as a crystal.

The motion of a particle moving about a center of force can be described using the angular momentum, which is defined to be the cross product of the position and momentum vectors

=r×p.

The cross product of two vectors is a vector having a magnitude equal to the product of the magnitudes of the two vectors times the sine of the angle between them. Denoting the angle between the momentum and position vectors by ? as in Fig. I.1, the magnitude of the angular momentum vector momentum can be written

l|=|r||p|sin?.

This expression for the angular momentum may be written more simply in terms of the distance between the line of motion of the particle and the origin, which is denoted by r0 in Fig. I.1. We have

l|=r0|p|.

The angular momentum is thus equal to the distance between the line of motion of the particle and the origin times the momentum of the particle. The direction of the angular momentum vector is generally taken to be normal to the plane of the particle’s motion. For a classical particle moving under the influence of a central force, the angular momentum is conserved. The angular momentum will be used in later chapters to describe the motion of electrons about the nucleus of an atom.

The kinetic energy of a particle with mass m and velocity v is defined by the equation

E=12mv2,

where v is the magnitude of the velocity or the speed of the particle. The concept of potential energy is useful for describing the motion of particles under the influence of conservative forces. In order to define the potential energy of a particle, we choose a point of reference denoted by R. The potential energy of a particle at a point P is defined as the negative of the work carried out on the particle by the force field as the particle moves from R to P. For a one-dimensional problem described by a variable x, the definition of the potential energy can be written

P=-?RPF(x)dx.

  (I.1)

As a first example of how the potential energy is defined we consider the harmonic oscillator illustrated in Fig. I.2(a). The harmonic oscillator consists of a body of mass m moving under the influence of a linear restoring force

=-kx,

  (I.2)

where x denotes the distance of the body from its equilibrium position. The constant k, which occurs in Eq. (I.2), is called the force constant. The restoring force is proportional to the displacement of the body and points in the direction opposite to the displacement. If the body is displaced to the right, for instance, the restoring force points to the left. It is natural to take the reference position R in the definition of the potential energy of the oscillator to be the equilibrium position for which x = 0. The definition of the potential energy (I.1) then becomes

(x)=-?0x(-kx')dx'=12kx2.

  (I.3)

Figure I.2 (a) A simple harmonic oscillator in which a mass m is displaced a distance x from its equilibrium position. The mass is attracted toward its equilibrium position by a linear restoring force with force constant k. (b) The potential energy function for a simple harmonic oscillator.

Here x' is used within the integration in place of x to distinguish the variable of integration from the limit of integration.

If one were to pull the mass shown in Fig. I.2(a) from its equilibrium position and release it, the mass would oscillate with a frequency independent of the initial displacement. The angular frequency of the oscillator is related to the force constant of the oscillator and the mass of the particle by the equation

=k/m.

or

=m?2.

Substituting this expression for k into Eq. (I.3), we obtain the following expression for the potential energy of the oscillator

(x)=12m?2x2.

  (I.4)

The oscillator potential is illustrated in Fig. I.2(b). The harmonic oscillator provides a useful model for a number of important problems in physics. It may be used, for instance, to describe the vibration of the atoms in a crystal about their equilibrium positions.

As a further example of potential energy, we consider the potential energy of a particle with electric charge q moving under the influence of a charge Q. According to Coulomb’s law, the electromagnetic force between the two charges is equal to

=14p?0Qqr2,

where r is the distance between the two charges and ?0 is the permittivity of free space. The reference point for the potential energy for this problem can be conveniently chosen to be at infinity where =8 and the force is equal to zero. Using Eq. (I.1), the potential energy of the particle with charge q at a distance r from the charge Q can be written

(r)=-Qq4p?0?8r1r'2dr'.

Evaluating the above integral, one finds that the potential energy of the particle is

(r)=Qq4p?01r.

An application of this last formula will arise when we consider the motion of electrons in an atom. For an electron with charge -e moving in the field of an atomic nucleus having Z protons and hence a nuclear charge of Ze, the formula for the potential energy becomes

(r)=-Ze24p?01r.

  (I.5)

The energy of a body is defined to be the sum of its kinetic and potential energies

=KE+V.

For an object moving under the influence of a conservative force, the energy is a constant of the motion.

I.1.2 Elementary Properties of Waves

We consider now some of the elementary properties of waves. Various kinds of waves arise in classical physics, and we shall encounter other examples of wave motion...