Foundations of Quantum Physics

by Burkhardt, Charles E.; Leventhal, Jacob J.

• ISBN13: 9780387776514
• ISBN10: 0387776516
• Format: Hardcover
• Copyright: 2008-10-01
• Publisher: Springer Verlag

Summary

Our book is meant to be a text for a first undergraduate course in quantum physics. Both of us have taught this course numerous times and have used several different texts, some of them excellent. Obviously, though, there are changes we would make and that is the reason we are writing this book.Two of the most widely used books for this course are the excellent texts by Griffiths (Prentice-Hall) and Gasiorowicz (Wiley), both of which we have used. Another, which neither of us has used but both of us are familiar with, is by Liboff (Addison-Wesley). While we find much to like about these books, there are matters of style, order of presentation and included subject matter that we obviously prefer. We do not wish our comments to be taken as criticism of other books but merely as a statement of our own preferences and the reasons that we believe our style and method can be helpful to students and instructors.We wish to avoid a modern trend in textbooks that is to condense and compress these texts into ever smaller and smaller size. We are not sure whether this trend is to lower the cost of the book or to make the book seem less formidable to the student, but we believe that a textbook should be more than a one or two semester acquaintance. This is especially true for a course such as quantum mechanics which is likely taken by physics majors who will use the subject for the rest of their careers. We therefore believe that a textbook can (and should) contain material that an instructor will choose to omit. In keeping with this theme, there are topics included that are not normally covered in introductory textbooks, not necessarily too advanced, but not usually covered. Perhaps in years to come this student, now working as a physicist, is interested in the subject that was skipped during the course. He/she knows where to find the material in a book that is quite familiar to him/her. Both of us have many such books in our personal libraries. It seems as if modern physics texts are not built to be long term reference books. This is just our observation and we would like our book to have 'œstaying power' and be long term companions.

Table of Contents

Preface	p. vii
Introduction	p. 1
Early Experiments	p. 1
The Photoelectric Effect	p. 1
The Franck-Hertz Experiment	p. 3
Atomic Spectroscopy	p. 5
Electron Diffraction Experiments	p. 7
The Compton Effect	p. 8
Early Theory	p. 10
The Bohr Atom and the Correspondence Principle	p. 10
The de Broglie Wavelength	p. 18
The Uncertainty Principle	p. 19
The Compton Wavelength Revisited	p. 21
The Classical Radius of the Electron	p. 23
Units	p. 24
Retrospective	p. 25
References	p. 25
Problems	p. 25
Elementary Wave Mechanics	p. 27
What is Doing the Waving?	p. 27
A Gedanken Experiment-Electron Diffraction Revisited	p. 27
The Wave Function	p. 28
Finding the Wave Function-the Schrodinger Equationo	p. 29
The Equation of Continuity	p. 32
Separation of the Schrodinger Equation-Eigenfunctions	p. 33
The General Solution to the Schrodinger Equation	p. 35
Stationary States and Bound States	p. 38
Characteristics of the Eigenfunctions [psi subscript n] (x)	p. 38
Retrospective	p. 43
Problems	p. 44
Quantum Mechanics in One Dimension-Bound States I	p. 47
Simple Solutions of the Schrodinger Equation	p. 47
The Infinite Square Well-the "Particle-in-a-Box"	p. 47
The Harmonic Oscillator	p. 56
Penetration of the Classically Forbidden Region	p. 69
The Infinite Square Well with a Rectangular Barrier Inside	p. 73
Retrospective	p. 77
References	p. 77
Problems	p. 78
Time-Dependent States in One Dimension	p. 83
The Ehrenfest Equations	p. 83
The Free Particle	p. 85
Quantum Representation of Particles-Wave Packets	p. 86
Momentum Representation of the Operator x	p. 90
The Dirac [delta]-function	p. 91
Parseval's Theorem	p. 93
The Harmonic Oscillator Revisited-Momentum Eigenfunctions	p. 94
Motion of a Wave Packet	p. 96
Case I. The Free Packet/Particle	p. 98
Case II. The Packet/Particle Subjected to a Constant Field	p. 101
Case III. The Packet/Particle Subjected to a Harmonic Oscillator Potential	p. 104
Retrospective	p. 108
Problems	p. 109
Stationary States in One Dimension II	p. 113
The Potential Barrier	p. 113
The Potential Step	p. 121
The Finite Square Well-Bound States	p. 123
The Morse Potential	p. 130
The Linear Potential	p. 139
The WKB Approximation	p. 145
The Nature of the Approximation	p. 145
The Connection Formulas for Bound States	p. 148
A Bound State Example-the Linear Potential	p. 155
Tunneling	p. 158
Comparison with a Rectangular Barrier	p. 162
A Tunneling Example-Predissociation	p. 163
References	p. 165
Problems	p. 165
The Mechanics of Quantum Mechanics	p. 169
Abstract Vector Spaces	p. 169
Matrix Representation of a Vector	p. 171
Dirac Notation for a Vector	p. 172
Operators in Quantum Mechanics	p. 173
The Eigenvalue Equation	p. 179
Properties of Hermitian Operators and the Eigenvalue Equation	p. 180
Properties of Commutators	p. 186
The Postulates of Quantum Mechanics	p. 189
Listing of the Postulates	p. 189
Discussion of the Postulates	p. 190
Further Consequences of the Postulates	p. 198
Relation Between the State Vector and the Wave Function	p. 200
The Heisenberg Picture	p. 202
Spreading of Wave Packets	p. 207
Spreading in the Heisenberg Picture	p. 207
Spreading in the Schrodinger Picture	p. 211
Retrospective	p. 216
References	p. 217
Problems	p. 217
Harmonic Oscillator Solution Using Operator Methods	p. 219
The Algebraic Method	p. 219
The Schrodinger Picture	p. 219
Matrix Elements	p. 224
The Heisenberg Picture	p. 227
Coherent States of the Harmonic Oscillator	p. 229
Retrospective	p. 236
Reference	p. 236
Problems	p. 237
Quantum Mechanics in Three Dimensions-Angular Momentum	p. 239
Commutation Relations	p. 240
Angular Momentum Ladder Operators	p. 241
Definitions and Commutation Relations	p. 241
Angular Momentum Eigenvalues	p. 242
Vector Operators	p. 247
Orbital Angular Momentum Eigenfunctions-Spherical Harmonics	p. 249
The Addition Theorem for Spherical Harmonics	p. 257
Parity	p. 259
The Rigid Rotor	p. 260
Another Form of Angular Momentum-Spin	p. 262
Matrix Representation of the Spin Operators and Eigenkets	p. 266
The Stern-Gerlach Experiment	p. 270
Addition of Angular Momenta	p. 273
Examples of Angular Momentum Coupling	p. 277
Spin and Identical Particles	p. 285
The Vector Model of Angular Momentum	p. 292
Retrospective	p. 294
References	p. 294
Problems	p. 294
Central Potentials	p. 297
Separation of the Schrodinger Equation	p. 298
The Effective Potential	p. 300
Degeneracy	p. 302
Behavior of the Wave Function for Small and Large Values of r	p. 304
The Free Particle in Three Dimensions	p. 305
The Infinite Spherical Square Well	p. 308
The Finite Spherical Square Well	p. 309
The Isotropic Harmonic Oscillator	p. 316
Cartesian Coordinates	p. 317
Spherical Coordinates	p. 319
The Morse Potential in Three Dimensions	p. 339
Retrospective	p. 343
References	p. 344
Problems	p. 344
The Hydrogen Atom	p. 347
The Radial Equation-Energy Eigenvalues	p. 347
Degeneracy of the Energy Eigenvalues	p. 352
The Radial Equation-Energy Eigenfunctions	p. 354
The Complete Energy Eigenfunctions	p. 361
Retrospective	p. 362
References	p. 362
Problems	p. 362
Angular Momentum-Encore	p. 365
The Classical Kepler Problem	p. 365
The Quantum Mechanical Kepler Problem	p. 367
The Action of A[subscript +]	p. 371
Retrospective	p. 372
References	p. 372
Problems	p. 372
Time-Independent Approximation Methods	p. 375
Perturbation Theory	p. 375
Nondegenerate Perturbation Theory	p. 375
Degenerate Perturbation Theory	p. 382
The Variational Method	p. 390
Problems	p. 393
Applications of Time-Independent Approximation Methods	p. 397
Hydrogen Atoms	p. 397
Breaking the Degeneracy-Fine Structure	p. 397
Spin-Orbit Coupling and the Shell Model of the Nucleus	p. 409
Helium Atoms	p. 411
The Ground State	p. 411
Excited States	p. 417
Multielectron Atoms	p. 422
Retrospective	p. 427
References	p. 428
Problems	p. 428
Atoms in External Fields	p. 431
Hydrogen Atoms in External Fields	p. 431
Electric Fields-the Stark Effect	p. 431
Magnetic Fields-The Zeeman Effect	p. 436
Multielectron Atoms in External Magnetic Fields	p. 442
Retrospective	p. 446
References	p. 446
Problems	p. 446
Time-Dependent Perturbations	p. 449
Time Dependence of the State Vector	p. 449
Two-State Systems	p. 452
Harmonic Perturbation-Rotating Wave Approximation	p. 452
Constant Perturbation Turned On at t = 0	p. 455
Time-Dependent Perturbation Theory	p. 457
Two-state Systems Using Perturbation Theory	p. 459
Harmonic Perturbation	p. 459
Constant Perturbation Turned On at t = 0	p. 462
Extension to Multistate Systems	p. 464
Harmonic Perturbation	p. 464
Constant Perturbation Turned On at t = 0	p. 465
Transitions to a Continuum of States-The Golden Rule	p. 465
Interactions of Atoms with Radiation	p. 468
The Nature of Electromagnetic Transitions	p. 469
The Transition Rate	p. 470
The Einstein Coefficients-Spontaneous Emission	p. 473
Selection Rules	p. 476
Transition Rates and Lifetimes	p. 480
References	p. 483
Problems	p. 483
Answers to Problems	p. 485
Chapter 1	p. 485
Chapter 2	p. 485
Chapter 3	p. 487
Chapter 4	p. 489
Chapter 5	p. 489
Chapter 6	p. 490
Chapter 7	p. 491
Chapter 8	p. 491
Chapter 9	p. 492
Chapter 10	p. 492
Chapter 11	p. 493
Chapter 12	p. 493
Chapter 13	p. 494
Chapter 14	p. 495
Chapter 15	p. 496
Useful Constants	p. 497
Energy Units	p. 499
Useful Formulas	p. 501
Greek Alphabet	p. 503
Acronyms	p. 505
[Gamma]-Functions	p. 507
Integral [Gamma]-Functions	p. 507
Half-Integral [Gamma]-Functions	p. 507
Useful Integrals	p. 509
Useful Series	p. 511
Taylor Series	p. 511
Binomial Expansion	p. 511
Gauss' Trick	p. 512
Fourier Integrals	p. 515
Commutator Identities	p. 519
General Identities	p. 519
Quantum Mechanical Identities	p. 519
Miscellaneous Operator Relations	p. 521
Baker-Campbell-Hausdorff (BCH) Formula	p. 521
Translation Operator	p. 522
Index	p. 525
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