A Textbook of Quantum Mechanics PDF

A Textbook of Quantum Mechanics Second Edition About the Authors P M Mathews held the position of Professor, Senior Professor and Head of the Department of Theoretical Physics for nearly three decades, and is now retired. He has been Visiting Professor and Visiting Scientist several times at numerous distin- guished institutions (MIT, Cambridge, MA, USA; Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, USA; Royal Observatory of Belgium at Brussels, Paris Observatory, Paris, France). He is a fellow of the Indian National Science Acad- emy, of the Indian Academy of Sciences, and of the American Geophysical Union. His research interests are in Classical and Quantum Theory of Relativistic Fields, and Geoastronomy. He has published about 120 papers in international journals. Dr Mathews is a recipient of the Meghnad Saha Award for Theoretical Sciences. K Venkatesan completed his PhD at the Institute of Mathematical Sciences (Mat- science), Chennai under the guidance of the late Prof. Alladi Ramakrishnan. His PhD work involved pion-nucleon interactions and other processes related to strongly interacting particles and photons. Continuing at Matscience for a while, he contrib- uted articles to volumes of ‘Matscience Symposia in Theoretical Physics’ and wrote papers on problems related to Quantum Mechanics and Elementary Particle Physics. Dr Venkatesan has translated from German, along with Dr Achuthan, the famous article ‘Die Algemeinen Prinzipien der Wellenmechanik’ (General Principles of Wave Mechanics) by Wolfgang Pauli and Numerical Analysis for Engineers and Physicists by Rudolf Zurmuhl. His main areas of interest in Physics are Quantum Mechanics and Elementary Particle Physics. A Textbook of Quantum Mechanics Second Edition P M Mathews Retired Senior Professor and Head of the Department of Theoretical Physics University of Madras K Venkatesan Formerly Associated with Institute of Mathematical Sciences, Chennai Tata Mcgraw Hill Education Private Limited NEW DELHI McGraw-Hill Offices New Delhi New York St Louis San Francisco Auckland Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal Paris San Juan Santiago Singapore Sydney Tokyo Toronto Tata McGraw-Hill Published by the Tata McGraw Hill Education Private Limited, 7 West Patel Nagar, New Delhi 110 008. A Textbook of Quantum Mechanics Copyright © 2010, 1976 by Tata McGraw Hill Education Private Limited. No part of this pub- lication may be reproduced or distributed in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise or stored in a database or retrieval system without the prior written permission of the publishers. 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However, neither Tata McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither Tata McGraw- Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that Tata McGraw- Hill and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought. Typeset at ACEPRO India Private Limited, Chennai and Printed at Adarsh Printers, C-50-51, Mohan Park, Naveen Shahdara, Delhi-110032. Cover Printer: SDR Printers Contents Preface to the Second Edition xiii Preface to the First Edition xv Chapter 1 Towards Quantum Mechanics 1 A. CONCEPTS OF CLASSICAL MECHANICS 1 1.1 Mechanics of Material Systems 1 1.2 Electromagnetic Fields and Light 4 B. INADEQUACY OF CLASSICAL CONCEPTS 5 (i) Macroscopic Statistical Phenomena 5 1.3 Black Body Radiation; Planck’s Quantum Hypothesis 5 1.4 Specific Heats of Solids 8 (ii) Electromagnetic Radiation—Wave-Particle Duality 9 1.5 The Photoelectric Effect 9 1.6 The Compton Effect 10 (iii) Atomic Structure and Atomic Spectra 13 1.7 The Rutherford Atom Model 13 1.8 Bohr’s Postulates 13 1.9 Bohr’s Theory of the Hydrogen Spectrum 15 1.10 Bohr-Sommerfeld Quantum Rules; Degeneracy 16 1.11 Space Quantization 18 1.12 Limitations of the Old Quantum Theory 19 (iv) Matter Waves 20 1.13 De Broglie’s Hypothesis 20 1.14 The Motion of a Free Wave Packet; Classical Approximation and the Uncertainty Principle 23 1.15 Uncertainties Introduced in the Process of Measurement 26 1.16 Approximate Classical Motion in Slowly Varying Fields 27 1.17 Diffraction Phenomena: Interpretation of the Wave-Particle Dualism 28 1.18 Complementarity 30 1.19 The Formulation of Quantum Mechanics 31 1.20 Photons: The Quantization of Fields 33 vi Contents Chapter 2 The Schrödinger Equation and Stationary States 35 A. THE SCHRöDINgER EQUATION 35 2.1 A Free Particle in One Dimension 35 2.2 Generalization to Three Dimensions 37 2.3 The Operator Correspondence and the Schrödinger Equation for a Particle Subject to Forces 38 B. PHYSICAL INTERPRETATION AND CONDITIONS ON y 40 2.4 Normalization and Probability Interpretation 41 2.5 Non-normalizable Wave Functions and Box Normalization 42 2.6 Conservation of Probability 44 2.7 Expectation Values; Ehrenfest’s Theorem 46 2.8 Admissibility Conditions on the Wave Function 48 C. STATIONARY STATES AND ENERgY SPECTRA 51 2.9 Stationary States; The Time-Independent Schödinger Equation 51 2.10 A Particle in a Square Well Potential 53 2.11 Bound States in a Square Well: (E < 0) 54 2.12 The Square Well: Non-Localized States (E > 0) 59 2.13 Square Potential Barrier 61 2.14 One-Dimensional Delta-Function Well 65 2.15 Multiple Potential Wells: Splitting of Energy Levels; Energy Bands 67 Problems 74 Chapter 3 general Formalism of Wave Mechanics 75 3.1 The Schrödinger Equation and the Probability Interpretation for an N-Particle System 75 3.2 The Fundamental Postulates of Wave Mechanics 78 3.3 The Adjoint of an Operator and Self-Adjointness 85 3.4 The Eigenvalue Problem; Degeneracy 87 3.5 Eigenvalues and Eigenfunctions of Self-Adjoint Operators 88 3.6 The Dirac Delta Function 90 3.7 Observables: Completeness and Normalization of Eigenfunctions 91 3.8 Closure 93 3.9 Physical Interpretation of Eigenvalues, Eigenfunctions and Expansion Coefficients 94 Contents vii 3.10 Momentum Eigenfunctions; Wave Functions in Momentum Space 95 3.11 The Uncertainty Principle 99 3.12 States with Minimum Value for Uncertainty Product 101 3.13 Commuting Observables; Removal of Degeneracy 102 3.14 Evolution of System with Time; Constants of the Motion 104 3.15 Non-Interacting and Interacting Systems 106 3.16 Systems of Identical Particles 107 Problems 111 Chapter 4 Exactly Soluble Eigenvalue Problems 113 A. THE SIMPLE HARMONIC OSCILLATOR 114 4.1 The Schrödinger Equation and Energy Eigenvalues 114 4.2 The Energy Eigenfunctions 115 4.3 Properties of Stationary States 118 4.4 The Abstract Operator Method 120 4.5 Coherent States 125 B. ANgULAR MOMENTUM AND PARITY 127 4.6 The Angular Momentum Operators 127 4.7 The Eigenvalue Equation for L2; Separation of Variables 128 4.8 Admissibility Conditions on Solutions; Eigenvalues 129 4.9 The Eigenfunctions: Spherical Harmonics 131 4.10 Physical Interpretation 133 4.11 Parity 135 4.12 Angular Momentum in Stationary States of Systems with Spherical Symmetry 138 C. THREE-DIMENSIONAL SQUARE WELL POTENTIAL 141 4.13 Solutions in the Interior Region 142 4.14 Solution in the Exterior Region and Matching 143 D. THE HYDROgEN ATOM 146 4.15 Solution of the Radial Equation; Energy Levels 146 4.16 Stationary State Wave Functions 148 4.17 Discussion of Bound States 149 4.18 Solution in Terms of Confluent Hypergeometric Functions; Nonlocalized States 154 4.19 Solution in Parabolic Coordinates 156 viii Contents E. OTHER PROBLEMS IN THREE DIMENSIONS 160 4.20 The Anisotropic Oscillator 160 4.21 The Isotropic Oscillator 161 4.22 Normal Modes of Coupled Systems of Particles 162 4.23 A Charged Particle in a Uniform Magnetic Field 165 4.24 Integer Quantum Hall Effect 174 Problems 176 Chapter 5 Approximation Methods for Stationary States 178 A. PERTURBATION THEORY FOR DISCRETE LEVELS 178 5.1 Equations in Various Orders of Perturbation Theory 179 5.2 The Non-Degenerate Case 180 5.3 The Degenerate Case — Removal of Degeneracy 183 5.4 The Effect of an Electric Field on the Energy Levels of an Atom (Stark Effect) 186 5.5 Two-Electron Atoms 189 B. THE VARIATION METHOD 192 5.6 Upper Bound on Ground State Energy 192 5.7 Application to Excited States 193 5.8 Trial Function Linear in Variational Parameters 195 5.9 The Hydrogen Molecule 196 5.10 Exchange Interaction 200 C. THE WKB APPROxIMATION 200 5.11 The One-Dimensional Schrödinger Equation 201 5.12 The Bohr-Sommerfeld Quantum Condition 206 5.13 WKB Solution of the Radial Wave Equation 208 Problems 209 Chapter 6 Scattering Theory 211 A. THE SCATTERINg CROSS-SECTION: gENERAL CONSIDERATIONS 211 6.1 Kinematics of the Scattering Process: Differential and Total Cross-Sections 211 6.2 Wave Mechanical Picture of Scattering: The Scattering Amplitude 213 6.3 Green’s Functions; Formal Expression for Scattering Amplitude 214 B. THE BORN AND EIKONAL APPROxIMATIONS 217 6.4 The Born Approximation 217 6.5 Validity of the Born Approximation 219 6.6 The Born Series 221 6.7 The Eikonal Approximation 221 Contents ix C. PARTIAL WAVE ANALYSIS 223 6.8 Asymptotic Behaviour of Partial Waves: Phase Shifts 224 6.9 The Scattering Amplitude in Terms of Phase Shifts 226 6.10 The Differential and Total Cross-Sections; Optical Theorem 227 6.11 Phase Shifts: Relation to the Potential 228 6.12 Potentials of Finite Range 230 6.13 Low Energy Scattering 233 D. ExACTLY SOLUBLE PROBLEMS 237 6.14 Scattering by a Square Well Potential 237 6.15 Scattering by a Hard Sphere 238 6.16 Scattering by a Coulomb Potential 238 E. MUTUAL SCATTERINg OF TWO PARTICLES 241 6.17 Reduction of the Two-Body Problem: The Centre of Mass Frame 241 6.18 Transformation from Centre of Mass to Laboratory Frame of Reference 243 6.19 Collisions between Identical Particles 245 Problems 247 Chapter 7 Representations, Transformations and Symmetries 249 7.1 Quantum States; State Vectors and Wave Functions 249 7.2 The Hilbert Space of State Vectors; Dirac Notation 249 7.3 Dynamical Variables and Linear Operators 251 7.4 Representations 254 7.5 Continuous Basis—The Schrödinger Representation 257 7.6 Degeneracy; Labelling by Commuting Observables 259 7.7 Change of Basis; Unitary Transformations 260 7.8 Unitary Transformations Induced by Change of Coordinate System: Translations 262 7.9 Unitary Transformation Induced by Rotation of Coordinate System 265 7.10 The Algebra of Rotation Generators 266 7.11 Transformation of Dynamical Variables 268 7.12 Symmetries and Conservation Laws 269 7.13 Space Inversion 270 7.14 Time Reversal 274 Problems 276 Chapter 8 Angular Momentum 278 8.1 The Eigenvalue Spectrum 278 8.2 Matrix Representation of J in the | jm〉 Basis 281 x Contents 8.3 Spin Angular Momentum 283 8.4 Non-Relativistic Hamiltonian with Spin; Diamagnetism 288 8.5 Addition of Angular Momenta 291 8.6 Clebsch-Gordan Coefficients 294 8.7 Spin Wave Functions for a System of Two Spin – 12 Particles 300 8.8 Identical Particles with Spin 303 8.9 Addition of Spin and Orbital Angular Momenta 305 8.10 Spherical Tensors; Tensor Operators 307 8.11 The Wigner-Eckart Theorem 310 8.12 Projection Theorem for a First Rank Tensor 314 Problems 316 Chapter 9 Evolution with Time 318 A. ExACT FORMAL SOLUTIONS: PROPAgATORS 318 9.1 The Schrödinger Equation; General Solution 318 9.2 Propagators 319 9.3 Relation of Retarded Propagator to the Green’s Function of the Time-Independent Schrödinger Equation 321 9.4 Alteration of Hamiltonian: Transitions; Sudden Approximation 324 9.5 Path Integrals in Quantum Mechanics 326 9.6 Aharonov–Bohm Effect 331 B. PERTURBATION THEORY FOR TIME EVOLUTION PROBLEMS 335 9.7 Perturbative Solution for Transition Amplitude 335 9.8 Selection Rules 338 9.9 First Order Transitions: Constant Perturbation 339 9.10 Transitions in the Second Order: Constant Perturbation 342 9.11 Scattering of a Particle by a Potential 345 9.12 Inelastic Scattering: Exchange Effects 346 9.13 Double Scattering by Two Non-Overlapping Scatterers 349 9.14 Harmonic Perturbations 351 9.15 Interaction of an Atom with Electromagnetic Radiation 354 9.16 The Dipole Approximation: Selection Rules 356 9.17 The Einstein Coefficients: Spontaneous Emission 358 C. ALTERNATIVE PICTURES OF TIME EVOLUTION 359 9.18 The Schrödinger Picture: Transformation to Other Pictures 359 9.19 The Heisenberg Picture 361 9.20 Matrix Mechanics—The Simple Harmonic Oscillator 362 Contents xi 9.21 Electromagnetic Wave as Harmonic Oscillator; Quantization: Photons 364 9.22 Atom Interacting with Quantized Radiation: Spontaneous Emission 372 9.23 The Interaction Picture 376 9.24 The Scattering Operator 380 D. TIME EVOLUTION OF ENSEMBLES 381 9.25 The Density Matrix 381 9.26 Spin Density Matrix 383 9.27 The Quantum Liouville Equation 384 9.28 Magnetic Resonance 385 Problems 386 Chapter 10 Relativistic Wave Equations 388 10.1 Generalization of the Schrödinger Equation 388 A. THE KLEIN-gORDON EQUATION 389 10.2 Plane Wave Solutions; Charge and Current Densities 389 10.3 Interaction with Electromagnetic Fields; Hydrogen-Like Atom 390 10.4 Nonrelativistic Limit 392 B. THE DIRAC EQUATION 394 10.5 Dirac’s Relativistic Hamiltonian 394 10.6 Position Probability Density; Expectation Values 395 10.7 Dirac Matrices 396 10.8 Plane Wave Solutions of the Dirac Equation; Energy Spectrum 398 10.9 The Spin of the Dirac Particle 401 10.10 Significance of Negative Energy States; Dirac Particle in Electromagnetic Fields 404 10.11 Relativistic Electron in a Central Potential: Total Angular Momentum 406 10.12 Radial Wave Equations in Coulomb Potential 408 10.13 Series Solutions of the Radial Equations: Asymptotic Behaviour 410 10.14 Determination of the Energy Levels 412 10.15 Exact Radial Wave Functions; Comparison to Non-Relativistic Case 414 10.16 Electron in a Magnetic Field—Spin Magnetic Moment 417 10.17 The Spin Orbit Energy 418 Problems 421 Appendices A. Classical Mechanics 425 B. Relativistic Mechanics 429 xii Contents C. The Dirac Delta Function 435 D. Mathematical Appendix 437 E. Many-Electron Atoms 448 F. Internal Symmetry 453 G. Conversion between Gaussian and Rationalized MKSA Systems 455 Table of Physical Constants 458 Index 460 Preface to the Second Edition The primary motivation for the preparation of a new edition of the book was to extend its coverage to include a couple of quantum phenomena which are currently of considerable interest, but were not known at the time of publication of the first edi- tion: the Integer Quantum Hall Effect (IQHE) and the Aharonov–Bohm effect. The former effect is the appearance of quantum jumps in the Hall conductivity of a mate- rial under suitable conditions, as the strength of a uniform magnetic field in which it is placed crosses certain discrete values. In order to bring out the physics of this phenomenon, we have gone further, in this edition, with the solutions for the degen- erate eigenfunctions of a charged particle in a uniform magnetic field in Sec. 4.23, we have exhibited the quantization of the magnetic flux linked to each such eigen- function, established the linkage between the strength of the magnetic field and the number of degenerate states that can exist for each energy level (per unit area perpen- dicular to the magnetic field), and hence shown, in the new Sec. 4.24, how the IHQE arises. Similarly, the Path Integral approach to quantum mechanics, which provides an alternative description of the time evolution of a quantum system and illumines the process of passing to the classical limit, is presented now as the new Sec. 9.5 in Chapter 9. The next section, also new, applies the path integral approach to elucidate the Aharonov–Bohm effect (a phase shift of the wave function of a charged particle that is caused by a magnetic field in a region which the particle does not even enter). One other addition to the earlier edition is in Chapter 10, where the exact eigenfunc- tions of a Dirac electron in a Coloumb potential are derived. Passage to the non- relativistic limit reveals a singularity in one case that is not shared by the solutions obtained from the non-relativistic treatment in Chapter 4. Apart from these, a dozen or so new worked out problems, distributed among the various chapters, have been added to the body of worked examples already present in the first edition. The new examples include squeezed states of a harmonic oscillator, and rotation and vibration spectra of molecules, all in Chapter 4, bound states of and scattering cross-section for a particle in a delta function potential in 3 dimensions (Chapter 6), and helicity eigenstates (Chapter 10), among other things. Furthermore, a few passages have been rearranged and/or rewritten for greater clarity. The problems at the close of the chap- ters are left untouched, to continue as a challenge to the serious student! Our concentrated effort has been to make this classic text more up-to-date with a discussion of the latest developments in the subject, relevant as per latest curricula. The features of the book are given below: It covers important topics, namely, drawbacks of classical mechanics at the atomic level, Schrodinger equation, matter–wave dual nature, wave functions and wave mechanics, eigenvalue problems, scattering the- ory, Heisenbergs’ uncertainty principle, angular momentum theory, and relativistic wave equations. xiv Preface to the Second Edition New topics like Aharanov–Bohm effect, Quantum Hall effect, diamag- netism, path integrals, radial wave functions for a Dirac particle in the Coulomb potential are incorporated. Despite being a high-end subject area, the contents are lucidly explained throughout the book with relevant examples interspersed. Pedagogy includes more than 100 solved and unsolved problems with related figures and tables. The authors wish to express their indebtedness to Prof. K. Raghunathan, recently retired from the Department of Theoretical Physics, University of Madras, for his generous advice and participation in choosing and preparing the additional material for inclusion in the new edition, and to other members of the department who helped in various ways. We would like to thank the publishing team at Tata McGraw-Hill, especially Ms Vibha Mahajan, Ms Shalini Jha, Ms Smruti Snigdha, Ms Renu Upadhyay, Ms Dipika Dey and Ms Anjali Razdan. A note of acknowledgement is also due to P Mitra, Pro- fessor, Saha Institute of Nuclear Physics, Kolkata and Vivek Mittal, Senior Lecturer in Physics, Institute of Advanced Management and Research (IAMR), Ghaziabad, for their critical comments and suggestions on enhancing the presentation and organ- isation of many chapters at a finer level. We believe that enhancements made in this new edition will make it even more appealing than the earlier version to teachers and students of physics at the Master’s degree level (including those taking the 5-year integrated course), Honours students, and generally to those who desire to have a thorough grounding in quantum mechan- ics and its applications to various areas of the physical sciences. We look forward to receiving constructive criticism from students and teachers so that the present book can be further improved in future editions. P M Mathews K Venkatesan Publisher’s Note: Do you have a feature request? A suggestion? We are always open to new ideas and the best ideas come from you!. You may send your comments to tmh.sciencemaths [email protected] (kindly mention the title and author name in the subject line). Piracy-related issues may also be reported. Preface to the First Edition The aim of this book is to give a reasonably comprehensive introduction to the fundamental concepts, mathematical formalism and methodology of quantum mechanics, without assuming any previous acquaintance with the subject. Quantum mechanics provides a characterization of microscopic physical systems in terms of essentially mathematical objects (wave functions, or at a deeper level, vectors in a linear vector space), and a set of rules enabling the information contained in the mathematical representation to be translated into physical terms. It is possible (and currently not uncommon) to develop the subject starting with these ‘rules of the game’ as postulates. This approach is economical and adequate if the reader’s interest is limited to the use of quantum mechanics as a readymade and trustworthy tool for exploring the properties of specific systems. However, to one who is encountering the subject for the first time, the conceptual picture of physical objects (on which the mathematical base rests) would appear strange indeed, and we believe that an account of the experimental developments which compelled the adoption of such a picture, superseding the classical picture, is an essential part of an introduction to quantum mechanics. Accordingly we have devoted Chapter 1 to these developments and to a discussion of the compatibility of the quantum picture with our experience in the macroscopic domain. Chapter 2 is designed to exhibit the main characteris- tic features of quantum systems with the aid of simple examples and to show how these features arise from the conditions on the Schrödinger wave function. It also provides the motivating background to the basic postulates of quantum mechanics, which are formally stated in Chapter 3. In developing the general formalism based on these postulates in this Chapter, and in presenting exact solutions (Chapter 4) and approximate methods of solution of a variety of eigenvalue problems (Chapter 5) and scattering problems (Chapter 6), we have adhered to Schrödinger’s wave mechanical language for the most part. However, we have sought, from an early stage, to bring out the correspondence between quantum states and vectors in a Hilbert space, and to make it clear that the Schrödinger language is just one way of describing such vectors and operations on them. To reinforce this point we have presented solutions to certain problems (e.g. the harmonic oscillator) employing algebraic methods which do not use the Schrödinger representation at all. It is our hope that the student would find the idea of an abstract vector space underlying the Schrödinger description to be natural if not self-evident by the time the Dirac notation and representation theory are formally introduced in Chapter 7. Transformations generated by changes of coordinate frame, and the question of symmetrics under such transformations, are also dealt with in this chapter. These developments are directly utilized in Chapter 8 in the treatment of angular momen- tum theory: the representation of spin, the coupling of two angular momenta, tensor operators, etc. The time evolution of quantum systems is dealt with in Chapter 9. Parts A and B of this Chapter are elementary and may be read before Chapter 7. xvi Preface to the First Edition In the context of the Heisenberg picture of time evolution (which, along with other pictures, is discussed in Part C), we give an elementary introduction to the concept of the quantized electromagnetic field and use it to calculate the rates of emission and absorption of radiation by atoms. A brief account of the representation of states of ensembles by density matrices, and of their time evolution, makes up the last part of Chapter 9. The final chapter is devoted to relativistic quantum mechanics, with emphasis on the Dirac equation and the natural way in which spin and its manifesta- tions (magnetic moment, spin-orbit interaction) as well as the concept of the antipar- ticle emerge from it. We have included a large number of examples distributed through the text, espe- cially in the earlier chapters, to facilitate a quick grasp of the principal ideas and methods, and in some cases, to indicate their extensions or applications. Some sup- plementary material, as well as background material for ready reference, is given in the Appendices. In the preparation of the book for publication, we have received cheerful coop- eration and assistance from the members of the Department of Theoretical Physics, University of Madras, for which we are grateful. It is a special pleasure to thank Dr. M. Seetharaman who has read the manuscript critically and rendered valuable help in many other ways. The idea of writing a book of this kind grew out of courses given by the first author for over ten years at Madras University. The work on the book has been sup- ported by the University Grants Commission through a Fellowship awarded to the second author and financial assistance towards preparation of the manuscript. For this encouragement by the Commission, the authors are deeply grateful. P M Mathews K Venkatesan Towards QuanTum mechanics 1 A. CONCEPTS OF CLASSICAL MECHANICS The formulation of quantum mechanics in 1925 was the culmination of the search, that began around 1900, for a rational basis for the understanding of the submicro- scopic world of the atom and its constituents. At the end of the nineteenth century, physicists had every reason to regard the Newtonian laws governing the motion of material bodies, and Maxwell’s theory of electromagnetism, as fundamental laws of physics. There was little or no reason to suspect the existence of any limitation on the validity of these theories which constitute what we now call classical mechanics. However, the discovery of the phenomenon of radioactivity and of X-rays and the electron, in the 1890s, set in motion a series of experiments yielding results which could not be reconciled with classical mechanics. For the resolution of the appar- ent paradoxes posed by these observations and certain other experimental facts, it became necessary to introduce new ideas quite foreign to commonsense concepts regarding the nature of matter and radiation—concepts which were implicit in clas- sical mechanics and had an essential role in determining its consequences. It was this revolution in concepts, which led to the mathematical formulation of quantum mechanics, that had an immediate and spectacular success in the explanation of the experimental observations. Thus, to appreciate the part played by various discoveries in bringing about this revolution, it is necessary, from the very outset, to have a clear idea of the classical concepts. We, therefore, begin our studies with a brief discus- sion of these concepts. The rest of this chapter reviews the developments during the first quarter of this century which culminated in the establishment of the authority of quantum mechanics over the domain of microscopic phenomena. 1.1 MECHANICS OF MATErIAL SySTEMS There are two broad categories of entities which physicists have to deal with: mate- rial bodies, whose essential attribute is mass, and electromagnetic (and gravitational) fields which are fundamentally distinct from matter. 2 A Textbook of Quantum Mechanics The most basic concept in the mechanics of material bodies is that of the particle, a point object endowed with mass. This concept emerged from Newton’s observation that while the mass of a body has a central role in determining its motions, there is a wide variety of circumstances in which the size of the body is quite immaterial (e.g. in the motion of the planets around the sun). The idealization to point size was a convenient abstraction under such circumstances. Even when such an idealization of a body as a whole is obviously impossible, as when considering the internal motions of an extended object, one could imagine the object to be made up of myriads of minuscule parts, each little part being then visualized as an idealized particle. In this manner, the mechanics of any material system could be reduced to the mechanics of a system of particles. It was taken for granted that the motion of such particles, however large or small their intrinsic mass and size may be, can be pictured in just the same way as the motion of projectiles or other macroscopic objects of everyday experience. An essential part of this picture is the idea that any material object has a definite position at any instant of time. The particle idealization makes it possible to specify the position precisely, it being intuitively obvious that the position of a point object is perfectly well defined. The trajectory or path followed by the particle is then pictured by a sharply defined line, and the instantaneous position of the particle on. the trajectory, its velocity, and its acceleration are represented by vectors x(t), x(t).. and x (t) with definite numerical values for their components. The manner in which these vary with time is governed by Newton’s famous equation of motion. Newton’s equation for the ith member of a system of N particles, is given by.. mi x i = Fi (i = 1, 2, … N ) (1.1) The force Fi acting on the ith particle (of mass mi) is a function of the positions of all the particles (and possibly also of the velocities). The form of the function is determined by the nature of the interactions of the particles among themselves and with external agencies. Once these are specified, Eq. (1.1) which form a coupled system of second order differential equations can, in principle, be solved to obtain all the xi as functions of t. Since the general solution of a second order ordinary differ- ential equation contains two arbitrary constants, the general solution of the system of N such equations for vectors xi depends on 2N arbitrary constant vectors. These may be chosen so as to satisfy specified initial conditions. In particular, given the 2N vec-. tors {xi (t0)} and {xi (t0)}, i.e. the positions and velocities of all the particles at some instant t0, as initial conditions, the xi(t) are completely determined as functions of t for. all t. The velocities xi(t) are then obtained by differentiation of xi (t), thus determining completely the state of the system at an arbitrary time t. The meaning ascribed to the word state is a crucial aspect of the difference between classical and quantum mechanics. In the classical context, knowledge of the state of a system of particles means knowing the instantaneous values of all the dynami- cal variables (like position, momentum, angular momentum, energy, etc.). Since these are all functions of the position coordinates and velocities (or momenta) of the constituent particles, complete information about the state is implicit in the knowl- edge of just these quantities. One may, therefore, say that in classical mechanics, the Towards Quantum Mechanics 3 state of a system of particles at any time t is represented by the instantaneous values. {xi (t)} and {x i(t)} of the position coordinates and velocities of all the particles. The concluding statements of the last paragraph may now be re-expressed as follows: Given the state at any instant of time to, the state at any other time t is uniquely deter- mined by the equations of motion (1.1). It is the system of Eq. (1.1) taken together with the concept of the state as outlined above which constitutes the classical mechan- ics of particle systems. We have based our discussion of classical mechanics above on the Newtonian form of the equations of motion. As is well known, there are other equivalent formulations of classical mechanics, notably the Lagrangian and Hamiltonian formulations. We present a brief account of these in Appendix A. As we shall see later, the Hamiltonian formalism has a special role to play in the process of setting up the quantum mechani- cal equations of motion. But for the present we will content ourselves with the follow- ing general observations: In the Hamiltonian formalism, corresponding to each of the 3N independent coordinate variables of an N-particle system a canonically conjugate momentum variable is defined. The 3N coordinates and the 3N momenta conjugate to them are independent variables at any instant of time. The state of the system is completely specified by giving precise numerical values to these momenta (instead of velocities as in the Newtonian formulation) and coordinates. The Hamiltonian equa- tions of motion, which are a system of 6N first order differential equations for the coordinates and momenta, determine the manner in which the state changes with time. These 6N equations are of course equivalent to the 3N second order Eq. (1.1). It is worthwhile, at this point, to recall Einstein’s discovery that the Newtonian and equivalent forms of mechanics (with the conventional definitions for forces, momenta, energy, etc) hold good only if the speeds of all the particles are very small compared to c, the velocity of light in vacuo. Whenever velocities υ of the order of c are involved, the more general equations following from Einstein’s special theory of relativity (1904) have to be used. The mechanics of classical particles based on these generalized (relativistic) equations is called the relativistic classical mechanics. The Newtonian mechanics, being not valid when relativistic effects are present, is said to be non-relativistic. We will need to use the concepts and results of relativity theory at several points in our study. A summary of those aspects of the theory which are most relevant for our purposes is presented in Appendix B. While relativity theory made possible a consistent description of physical phe- nomena involving fast-moving objects or observers, it is quantum mechanics which provided the key to the understanding of the behaviour of very small objects like the atom and its constituents. The failure of classical mechanics when applied to the submicroscopic world of the atom arose not from any inadequacy of the form of the equations of motion as such but from investing the symbols occurring in them with the conventional meanings idealized and extrapolated from the mechanics of macroscopic objects. This will be particularly evident when we consider the Heisen- berg version of quantum mechanics. The essential step in the formulation of the new (quantum) mechanics was the abandonment of the underlying concepts of classical mechanics, like the mental picture of the path of a particle as a sharply defined line 4 A Textbook of Quantum Mechanics and the possibility of characterizing the instantaneous state by precise positions and velocities. The new concepts which replaced these are necessarily divorced from such intuitive pictures and therefore appear rather strange initially, but they have their own beauty (especially in the simplicity and elegance of the associated mathematical structures) which becomes evident with a little familiarity. 1.2 ELECTrOMAgNETIC FIELdS ANd LIgHT The discovery of the phenomena of interference and polarization of light early in the nineteenth century provided convincing evidence that light is a wave phenomenon. The nature of light waves was identified some decades later, following Maxwell’s formulation of the electromagnetic theory. It was then recognized that light consists of electromagnetic waves and presents just one manifestation of the general phenom- ena of electromagnetism, governed by the fundamental equations of Maxwell: 1 ∂E 4π  c ∂t = curl H − c j,   div E = 4πρ,  (1.2) 1 ∂H   = − curl E,  c ∂t  div H = 0  Here E ≡ E (x, t) and H ≡ H (x, t) stand for the electric and magnetic fields at x at time t, and ρ(x,t) and j (x,t) are the electric charge and current densities, respec- tively. At any time t, the vector fields e(x,t) and h(x,t) are implicitly assumed to have definite numerical values for their components at any point x. This seemingly self-evident supposition is the essential feature of classical electromagnetic theory. Since all properties like the energy density, momentum density, etc., of the electro- magnetic field are functions of the instantaneous values of e and h, specification of e and h for all x at t0 amounts to a complete description of the state of the classical electromagnetic field at that instant. Further, such a specification provides the initial conditions necessary for identifying a particular solution of the first order differen- tial Eqs (1.2). Therefore, if the state of the field at some instant t0 is given, the state at any other time can be determined uniquely (at least in principle), provided of course that the charge and current densities ρ and j are known as functions of x and t. It is the set of Eqs (1.2) together with the concept of e and h as ordinary vector fields amenable to simultaneous specification with arbitrary precision at any given time, that constitutes what is known as the classical electromagnetic theory. It is worthwhile at this point to notice the obvious but important fact that the classical pictures of material particles on the one hand, and of light (or more generally, electro- magnetic fields) on the other, are mutually exclusive. While the ideal particle is a point object, a field necessarily exists over a region of space. In particular, the purest form of light, which is monochromatic, is a simple harmonic wave with a definite wavelength, existing throughout space with a uniform energy density everywhere. It would be impossible within the framework of this classical picture to conceive of a particle Towards Quantum Mechanics 5 possessing wave-like properties or of light exhibiting particle-like behaviour. Yet, experiments in the early 1900s gave firm evidence for the existence of both these kinds of phenomena. Let us now go on to a discussion of these and other experimen- tal results which revealed the limitations on the validity of classical concepts. B. INAdEQUACy OF CLASSICAL CONCEPTS1 (i) Macroscopic statistical pheNoMeNa BLACk BOdy rAdIATION; PLANCk’S 1.3 QUANTUM HyPOTHESIS One of the very few things for which classical theory had been unable to offer an explanation till the end of the last century was the nature of the distribution of energy in the spectrum of radiation from a black body. By definition, a black body is one which absorbs all the radiation it receives. As is well known, the best practical real- ization is an isothermal cavity with a small aperture through which radiation from outside may be admitted. The cavity always contains radiation emitted by the walls, the spectrum of radiation being characterized by a function u(v) where u(v)dv is the energy (per unit volume of the cavity) contributed by radiation with frequen- cies between v and v + dv. It had been deduced from very general thermodynamical arguments that the form of the function u(v) depends only on the temperature t of the cavity. Efforts to deduce the actual functional form from classical theory led to the Rayleigh-Jeans formula, u(v) = const. v2. Except at low frequencies this law was in violent disagreement with experimental observations which showed u(v) falling off after reaching a maximum as v was increased. That was how matters stood until 1900, when Planck announced the discovery of a law which reproduced perfectly the experimental curve for u(v): 8πv 2. hv u( v ) = 3 hv/kT (1.3) c e −1 This formula contains, besides the Boltzmann constant2 k, a new fundamental con- stant h, with the value h = 6.626 × 10–27 erg-sec. (1.4) It is called planck’s constant. 1 For a fuller account, especially of experimental details, see for example, Max Born, atomic physics, 5th ed., Blackie and Sons, London, 1952; F. K., Richtmyer, E. H. Kennard and T. Lauritsen, introduction to Modern physics, McGraw-Hill New York, 1955. 2 A table of fundamental constants and other data of interest is provided at the end of the book. As each physical problem is considered, the student is urged to familiarize himself with the orders of magnitude of the numbers involved. 6 A Textbook of Quantum Mechanics The essential new ingredient in the derivation of the law was the following ad hoc hypothesis: The emission and absorption of radiation by matter takes place, not as a continu- ous process, but in indivisible discrete units or quanta of energy. The magnitude ε of the quantum is determined solely by the frequency of the radiation concerned, and is given by ε = hv (1.5) where h is Planck’s constant. This hypothesis was a revolutionary break from classical radiation theory based on Maxwell’s Eqs (1.2). According to the classical theory, oscillating charges are responsible for the emission (or absorption) of electromagnetic radiation with fre- quency equal to that of the charge oscillations. Emission or absorption takes place continuously at a rate determined by the parameters of the oscillating system. The success of Planck’s hypothesis was the first indication that one might have to look beyond classical theories for the understanding of at least some areas of physics. Figure1.1 shows the various regions of the electromagnetic spectrum viewed from both the wave and quantum points of view. Thermal Optical & γ-rays Nuclear γ-rays of Radio Waves Micro Waves Radiation Ultra Violet γ-rays Cosmic Origin Wavelength (cm) 106 103 1 10– 3 10– 6 10– 9 10– 12 Waves Frequency (Hz) 105 108 1011 1014 1017 1020 Photons Ergs 10–21 10–18 10–15 10– 12 10– 9 10– 6 (Energy) eV 10–9 10–6 10–3 1 10 3 106 Fig. 1.1 Characterization of electromagnetic waves and quanta in various regions of the Spectrum. Let us now examine briefly how Planck’s law (1.3) follows from his quantum hypothesis. We make use of the fact that the electromagnetic waves, which constitute the radiation in the cavity, can be analyzed into a superposition of normal modes characteristic of the cavity. In each normal mode, the fields vary with time in simple harmonic fashion, in unison throughout the cavity. Thus each normal mode is equiva- lent to a simple harmonic oscillator, and the radiation field forms an assembly or ensemble of such oscillators. The absorption (or emission) of radiation by the walls of the cavity is equivalent to a transfer of energy to (or from) the walls by (or to) the oscillators. As a result of such energy exchanges, which are continually taking place, the ensemble of radiation oscillators comes into thermal equilibrium at the tempera- ture t of the walls of the cavity. Under these conditions, different oscillators having a given frequency v have different energies at any given time, but their average energy Towards Quantum Mechanics 7 E (v) has a definite value determined by the temperature t. The energy of radiation in the frequency range v to v + dv is then simply the number of normal mode oscil- lators n(v)dv having frequencies within this range, multiplied by the average energy E (v) per oscillator. Thus if V is the volume of the cavity, V.u(v)dv = n(v) dv. E (v) (1.6) The counting of the oscillators is simply a geometrical problem, and it is easily shown (see end of Sec. 2.5) that 8πv 2V n( v ) = (1.7) c3 The determination of E (v) is done by applying the standard results of statistical mechanics to the ensemble of oscillators. Statistical mechanics tells us that an indi- vidual member of an ensemble in thermal equilibrium at temperature t has energy e with probability e – E/kT PE = (1.8a) ∑ e – E/kT E so that its average energy is E = ∑ EPE (l.8b) E The summations are to be taken over all values which the energy e (of any member of the ensemble) may take. The spectrum of permissible values of e is thus of crucial importance in determining E. It is here that Planck’s hypothesis comes into play. It implies that normal mode oscillators of the radiation field with the frequency v can have only the energy values e(v) = nhv, n = 0, 1, 2,... (1.9) This is because the field oscillator gets its energy by emission from the cavity walls (and loses energy through absorption by the walls) only in packets or quanta of mag- nitude hv. On substituting the values (1.9) for e(v) in Eqs (1.8), we immediately get3 ∞  ∑ nhv.e−nhv/kT  n=0  E (v) = ∞  e−nhv/kT  (1.10) ∑ n=0   hv  = hv/kT e −1  3 The denominator is a geometric series whose sum is D = (1– e–βhv)–1, where β = (1/kt ) The numerator is observed to be nothing but − ∂D/∂β = h ve−βhv (1 − e−βhv )−2. 8 A Textbook of Quantum Mechanics for the mean energy of a field oscillator. When the expressions (1.10) for E (v) and (1.7) for n(v) are employed in Eq. (1.6) we obtain the Planck distribution law (1.3). According to the classical theory, e(v) could have any value from 0 to ∞, and the same thing would effectively happen if the quantum hv in the above treatment had a vanishingly small magnitude. Therefore, passage to the limit h → 0 in Eq. (1.10) should lead to the value kt predicted by the equipartition theorem of classical statistical mechanics, and indeed it does. With this value [ E (v) = kt] substituted in Eq. (1.6), one gets the Rayleigh-Jeans law for u(v) which we have already seen to be incorrect. It appears, therefore, that the very small but nonzero value of the constant h is a measure of the failure of classical mechanics. This surmise is indeed confirmed by the mathematical formu1ation of quantum mechanics. Example 1.1 If quantum effects are to be manifested through a departure of E (v) from its classical value kt, the frequency should be high enough, so that (hv/kt) becomes compara- ble to unity. For room temperatures (t ≈ 300° K), (hv/kt) ≈ 1/6 for v = 1012 Hz. It is only when ‘oscillators’ of at least this frequency are involved, that quantum statistical effects become noticeable at room temperature. n 1.4 SPECIFIC HEATS OF SOLIdS That the success of Planck’s hypothesis was no mere accident became evident when precisely the same kind of ideas provided the solution for another puzzling prob- lem of classical physics. It is well known that atoms in solids execute oscillations about their mean positions due to thermal agitation. Each atom may be thought of as a three-dimensional harmonic oscillator and its mean thermal energy should be three times that of a simple (one-dimensional) harmonic oscillator. As we saw in the last paragraph, classical theory predicts the latter to be kt. Therefore, the thermal energy of a solid should be 3kt per atom, or 3rt = 3Nkt per gram-atom (containing N atoms where N is the Avogadro number, 6·022 × 1023). The atomic heat (i.e., the rate of increase of thermal energy with temperature, per gram-atom) then becomes 3r, a universal constant. Many solids do conform to this expecta- tion (at least approximately) at ordinary temperatures, as observed by Dulong and Petit empirically. But when the temperature is lowered sufficiently, the specific heat decreases instead of remaining constant, and indeed goes down to zero as t approaches 0 K. Einstein4 observed that this behaviour can be simply explained if it is postulated that the energy of oscillation of any atom in a solid can take only a discrete set of values—just like the energy of Planck’s field oscillators. More pre- cisely, it was proposed that the energy associated with each component (in the x, y, z directions) of the oscillation of an atom be constrained to take only one of the values nhv (n = 0, 1, 2,...), where v is now the frequency of oscillation of the atom. The mean energy per atom then has exactly the form (1.10), except for an extra 4 A. Einstein, ann. d. physik, 22, 180, 1907. Towards Quantum Mechanics 9 factor 3 coming from the three directions of motion. If it is assumed that all atoms have the same frequency of oscillation, one immediately obtains the atomic heat as   hv 2   3Nhv  d e hv /kT C=   = 3R .    (1.11)  hv/kT− 1  dT e  (e hv/kT −1)2  kT   It is evident that this formula has the desired property of a gradual decrease in c as the temperature is lowered. It was found in fact that the behaviour of specific heats of solids is rather well accounted for by this formula, with a suitable choice of v in each case. Einstein’s derivation is by no means the last word on the theory of specific heats of solids. But it suffices for the purpose of displaying one of the early manifestations of the inadequacy of classical concepts, namely the need for the supposition that harmonic oscillators—whether radiation oscillators as in Planck’s theory, or material oscillators as in Einstein’s theory—can take only discrete energy values. (ii) electroMagNetic raDiatioN—WaVe-particle Duality 1.5 THE PHOTOELECTrIC EFFECT We have already mentioned the success of Planck’s hypothesis, which makes it appear that in the process of emission or absorption, light behaves as if it were a particle-like bundle of energy. However, the evidence here for the quantum nature of energy exchange between radiation and matter is indirect. It was Einstein5 who drew attention to the fact that the phenomenon of photoelectric emission could be understood in terms of Planck’s hypothesis and provides a direct verification of the hypothesis. This phenomenon is the emission of electrons by many metals (especially the alkali metals) when irradiated by light. Applying Planck’s hypothesis, Einstein proposed that absorption of light of frequency v by electrons in the metal takes place as discrete quanta of energy hv. If hv exceeds the amount of energy W needed for an electron to escape from the surface of the metal, the electrons absorbing such quanta may escape with energies up to a maximum value emax = hv – W (1.l2) Available experimental data were in conformity with this equation, which was accurately confirmed by later experiments. On the basis of the classical theory of absorption of light, it would be practically impossible to understand the existence of a maximum electron energy related to the frequency v of the incident radiation, as well as the absence of any photoelectric emission (i.e. the inability of electrons to acquire the amount of energy required for escape) when v is below a definite value v0 ≡ W / h. In the classical theory there is no reason to expect any sensitive frequency dependence for the amount of energy which an electron in the metal could ultimately accumulate by gradual absorption from the incident light wave; nor can 5 A. Einstein, ann. d. physik, 17, 132, 1905. 10 A Textbook of Quantum Mechanics one see why there should be a definite upper limit on the energy so absorbed. Other properties of the photoelectric emission are also equally difficult to understand on the classical picture, but are almost self-evident when this phenomenon is viewed as the instantaneous absorption of light quanta by the electrons with which they collide. For example, photoelectric emission starts instantly when light falls on the emitter, however weak the light intensity may be. (Classically, the electron would need some time to absorb enough energy to escape.) Under irradiation with mono- chromatic light, the rate of emission of electrons is directly proportional to the light intensity—which is exactly what would be expected on the quantum picture since the number of quanta (and hence, of the collisions with electrons) is evidently pro- portional to the intensity. We conclude, therefore, without further discussion that in the photoelectric effect, light behaves as a collection of corpuscles and not as a wave. At the same time, we know only too well that the phenomena of diffraction, etc, require light to be waves. How are we to escape the paradox created by the existence of two quite irreconcilable manifes- tations for one and the same physical entity? One possibility is to suppose that light propagates in the form of waves and therefore undergoes diffraction, etc. but assumes corpuscular character (in some unexplained manner) at the instant of absorption (or emission) by material objects. However, even this supposition, far-fetched as it is, was made untenable by the discovery of the compton effect6 in the scattering of X-rays. Example 1.2 It is an experimental fact that if at all there is any delay between the commencement of irradiation and the emission of photoelectrons, it is less than 10–9 sec. An electron requires, let us say, 5 × 10–12 ergs (about 3 eV) to escape from the irradiated metal. If this much energy is to be absorbed classically (in a continuous fashion), the rate of absorption must be at least 5 × 10–3 ergs/sec. If the light energy is continuously distributed over the wave front, the electron can only absorb the light incident within a small area near it, say 10–15 cm2 (i.e. of the order of the square of the interatomic distance). Therefore, the intensity of illumination required would be at least (5 × 10–3/10–15) = 5 × 1012 ergs/sec/cm2, that is half a million watts/cm2! Clearly, explanation of the photo-effect in classical terms is not feasible. n 1.6 THE COMPTON EFFECT That X-rays are electromagnetic waves (differing from light only in the considerably higher values of frequency) had become clear fairly soon after their discovery. Their wave nature was amply confirmed by the Laue photographs (1913) showing the dif- fraction of X-rays by crystals. Yet, barely ten years later, Compton had to invoke the extreme quantum picture to explain the fact that when monochromatic X-rays are scattered, part of the radiation scattered in any given direction has a definite wavelength higher than that of the primary beam. Assuming that X-rays of wave- length λ consist of a stream of corpuscles or quanta of energy E = hv = hc / λ , 6 A. H. Compton, phys. rev., 21, 483, 1923; 22, 409, 1923. Towards Quantum Mechanics 11 Compton theorized that when one of these quanta hits any free or loosely bound electron in the scatterer, the electron (being quite a light particle) would recoil. Its kinetic energy has to come from the energy of the incident quantum, and the latter would be left with an energy e9 < e after the collision (in which it gets scattered). The frequency v⬘ = E⬘/ h of the X-rays so scattered would therefore be less than ν and the corresponding wavelength λ9 > λ. On this picture, quantitative calculation of ∆λ = λ⬘ − λ can be made from considerations of energy and momentum con- servation in the collision. Since ‘the energy and momentum transported by electro- magnetic radiation are known to be related by a factor c, the momenta of the X-ray quantum before and after scattering are given by E hv h E⬘ hv⬘ h p= = = , and p⬘ = = = (1.13) c c λ c c λ⬘ The electron recoiling from the collision may have a velocity comparable to c, and therefore the relation between its energy W and its momentum p has to be taken as the relativistic one (see Appendix B): W = ( m2 c 4 + c 2 P 2 )1/2 (1.14) where m is the rest mass of the electron. The energy of the electron before colli- sion may be taken to be the rest energy W0 = mc 2 since the initial kinetic energy is relatively very small. The configuration of the scattering event is shown in Fig. 1.2 where, following current practice, the quantum of radiation is depicted by a wavy line and the electron by a straight line. The equations of conservation of the E′ p ′, p, E θ ϕ P, W Fig. 1.2 Compton scatttering. 12 A Textbook of Quantum Mechanics momentum components perpendicular and parallel to the direction of the incident quantum are p⬘ sin θ = P sin ϕ (1.15a) p − p⬘ cos θ = P cos ϕ. (1.15b) The energy conservation equation is E + W0 = E⬘ + W , or in view of Eqs (1.13) and (l.l4) cp + mc 2 = cp⬘ + ( m2 c 4 + c 2 P 2 )1/2 (1.15c) These equations are to be solved for p9 which is related to λ9. We can elimi- nate ϕ first by squaring Eqs (l.15a) and (1.15b) and adding. Then we get P 2 = p2 + p2 ⬘ − 2 pp⬘ cos θ. On substituting this expression for p2 in Eq. (1.l5c) and eliminating the square root, we obtain [c( p − p⬘) + mc 2 ]2 = m2 c 4 + c 2 ( p2 + p2⬘ − 2 pp⬘ cos θ) This simplifies to 2mc 2 ⋅ c( p − p⬘) − 2c 2 pp⬘ = − 2c 2 pp⬘ cos θ , whence 1 1 mc  −  = 1 − cos θ  p⬘ p  The use of Eqs. 1.13 enables us to write this finally in terms of λ, λ9 as h ∆λ ≡ λ⬘ − λ = (1 − cos θ) (1.16) mc This result was verified by Compton and his coworkers immediately after its derivation. The wavelength shift (h/mc) when the scattering is at 90° is called the compton wavelength associated with the electron: λ C = (h/mc) = 2.426 × 10−10 cm (1.17) This constant, which is characteristic of the X-ray scattering with wavelength shift (Compton effect) could not be reproduced by applying classical electromag- netic theory to the scattering process. The success of the theory presented above thus implies unequivocally that in the Compton effect, electromagnetic radiation mani- fests itself as a stream of corpuscles or quanta during propagation. The occurrence of corpuscular behaviour in emission or absorption processes (as in photoelectric effect) has been already referred to. It seems therefore that electromagnetic quanta are not merely some shadowy concept but have real physical existence. Such quanta are now known under the name of photons. The ability of radiation to manifest itself either as waves or as photons is referred to as the wave-particle dualism. Side by side with the developments leading to the recognition of the dual character of electromagnetic radiation, other equally startling discoveries were being made regarding the fundamental nature of material systems. We now turn our attention to these developments. Towards Quantum Mechanics 13 (iii) atoMic structure aND atoMic spectra 1.7 THE rUTHErFOrd ATOM MOdEL The first outlines of the structure of atoms, of which all matter is constituted, began to be discernible soon after Thomson’s discovery of the electron (1897). It became known then that an atom consists of a number of negatively charged electrons plus a positive residue which carries almost the entire mass of the atom. But it was only in 1911, with Rutherford’s7 analysis of the data on scattering of alpha particles by thin foils, that the picture of the atom became clearly defined. Rutherford came to the conclusion that the observed high proportion of alpha particles suffering large- angle scattering required that the heavy positive part of the atom be concentrated in a nucleus, whose size is extremely small compared to the dimension of the atom itself. This immediately suggested a structure for the atom resembling that of the solar system, with the electrons revolving in orbits around the nucleus (like planets around the sun). The Coulomb (electrostatic) attraction between each electron and the oppositely charged nucleus provides the force which holds the atom together. It was realized immediately that this picture of the atom encounters serious difficul- ties of principle in the context of classical theory. In fact, such a structure should not be stable at all, for the orbital motion of the electrons (which are charged particles) should cause them to emit radiation continuously. The consequent loss of energy should make the paths go spiralling inwards until the electrons ‘collapsed’ into the nucleus. During this process the frequency of the emitted radiation, which coincides with that of orbital motion, should be continually increasing. Obviously none of these things happens. The collapse of the kind envisaged does not take place; in fact, atoms have tremendous stability. Nor does the light actually emitted by atoms have the con- tinuum character demanded by the above picture. As is well known the most impor- tant feature of atomic spectra is the presence of very sharp, discrete, spectral lines which are characteristic of the emitting atom. In brief, acceptance of Rutherford’s nuclear model of the atom meant also recognition of a complete breakdown of the classical mechanism of radiation in the case of the atom. 1.8 BOHr’S POSTULATES This situation was tackled by Niels Bohr8 who adopted the Rutherford model of the atom, overcoming its unacceptable consequences by postulating that the classical theory of radiation does not apply to the atom. He enunciated the following further postulates concerning the dynamical behaviour of atoms: (i) The system of electrons and nucleus which constitute the atom cannot exist in any arbitrary state of motion allowed by the classical mechanics. The system 7 E: Rutherford, phil. Mag., 21, 669, 1911. 8 N. Bohr, phil. Mag., 26, 1, 1913. 14 A Textbook of Quantum Mechanics can exist only in certain special states characterized by definite discrete values of the total energy. These are stationary states, in any of which the atom can remain indefinitely without radiating. (ii) Emission of electromagnetic radiation takes place when (and only when) the atom ‘jumps’ from one of the stationary states with energy ei to another with energy ef < ei. The frequency of the radiation emitted is given by Ei − E f v= (1.18) h The first postulate extends the idea of discreteness of energy values, which originated with Planck, to the individual atom. The energy values associated with stationary states are called the energy levels of the atom. The second ·postulate incorporates the idea that emission of radiation takes place in discrete quanta, and adopts the Einstein relation between the energy of a quantum and the frequency of the associated radiation (first employed in the explanation of the photoelectric effect). It is an immediate consequence of these postulates that atomic spectra should consist of discrete lines, as observed. Equation (1.18) states that the frequencies of spectral lines of any atom are differences between ‘spectral terms’ (stationary state energies, divided by h) which are character- istic of the atom. This general property of atomic spectra had been already observed empirically and was known as the Rydberg-Ritz combination principle (1905). Thus it seemed certain that Bohr’s ideas were essentially sound. In fact, the supposition that the atom can have only discrete energy levels was directly verified from experiments by Franck and Hertz9 on the scattering of monoenergetic electrons by atoms. They found that as long as the electron energy was below a certain minimum value, the scat- tering was purely elastic, indicating that the atom is incapable of accepting energies less· than this amount. This behaviour is exactly what is demanded by the Bohr picture: an atom in its lowest energy level e0 must get an amount of energy (e1-e0) in order to go to the next permissible level e1. If energy exceeding this minimum is supplied to the atom, it can still take up only the exact amount (e1-e0), or (e2-e0) etc. The results of the Franck-Hertz experiment were indeed in accordance with this expectation. When the energy of the incident electrons was increased sufficiently, inelastic scattering with the absorption of discrete amounts of energy was found to take place. While the postulates stated above provide a frame-work for the understanding of the stability of atoms and the general features of atomic spectra, they do not indicate how the stationary states are to be identified or how the energy levels ei are to be determined for a particular system. But by supplementing these postulates by a quantum condition, Bohr was able to calculate the energy levels of the hydrogen atom and thus determine its spectral frequencies. His results were in agreement with the empirically deduced Balmer formula, v  1 1  = R  2 − 2  , n, m = 1, 2, … ; m > n (1.19) c  n m  9 J. Franck and G. Hertz, Verhandl. deut. phys. ges., 16, 457, 512, 1914. Towards Quantum Mechanics 15 where r is the so-called Rydberg constant. This was a spectacular triumph for the Bohr theory and inspired much of the later work which provided the guidance towards a more fundamental theory. Let us therefore consider it briefly before discussing the implications and limitations of Bohr’s theory in general. 1.9 BOHr’S THEOry OF THE HydrOgEN SPECTrUM The hydrogen atom consists of a proton (which is its nucleus) and a single electron. According to the Rutherford model, the electron moves in an orbit around the nucleus; the latter, being relatively very heavy, remains practically at rest. Since the force of attraction between the two is electrostatic and therefore obeys the inverse square law (just like the gravitational force in the problem of planetary motion) the possible orbits are circular or elliptical. Suppose the electron is moving in a cir- cular orbit of radius a with speed u (which is constant). In this orbit, the electrostatic attractive force e2/a2 is balanced by the centrifugal force mu2/a, e being the charge of the electron. This fact gives us the relation e2 mυ 2 = (1.20) a which can be used to eliminate u from the expression for the energy e: 1 e2 1 e2 E= mυ 2 − =− (1.21) 2 a 2 a Thus the total energy is half the potential energy. Classically, the orbital radius a can take any positive value, and therefore e can be anything from − ∞ to 0. However, Bohr’s first postulate asserts that only a special set from among these orbits is avail- able to the electron. Bohr proposed that these special orbits which characterize the stationary states are those in which the angular momentum l of the electron about the centre of the orbit (i.e. the position of the nucleus) is an integral multiple of ≡ h/2π. The angular momentum in a circular orbit is of course just the product of the linear momentum mu and the orbital radius a. Thus Bohr’s quantum condition is given by mυ a = n (1.22) The integer n whose values identify the various stationary states is called a quantum number. The radii of the allowed (‘quantized’) orbits are now obtained by eliminating u between the Eqs (1.20) and (1.22). For the nth orbit one has n2 2 a= (1.23) me 2 Substitution of this expression for a into Eq. (1.21) gives us the quantized energy levels of the hydrogen atom as: me 4 En = − , n = 1, 2, … (1.24) 2 n2 2 16 A Textbook of Quantum Mechanics The radius a = (h2/me2) of the first Bohr Orbit (n = 1), which belongs to the ground state, is known as the Bohr radius.” The state of the lowest energy (the ground state) corresponds to n = 1. The energies of the other ‘excited’ states increase with n, tend- ing to 0 as n → ∞. Knowing the energy levels, we can immediately obtain the frequencies of the hydro- gen spectrum, using Eq. (1.18). If the atom jumps from an initial state with the quantum numbers n = ni to another with n = nf < ni we find by substitution of the corresponding energies from Eq. (1.24) into (1.18), that the radiation emitted has the frequency me 4  1 1  v=  2 − 2  3  (1.25) 4π  n f ni  As already noted, this result agrees with the Balmer formula (1.19) and provides a theoretical value R = ( me 4 /4π c 3 ) for the Rydberg constant.10 We will see later that exactly the same formula follows from the quantum mechanical theory also, though the meaning of the quantum number n there is quite different. 1.10 BOHr-SOMMErFELd QUANTUM rULES; dEgENErACy While this first-ever theoretical derivation of an atomic spectrum was indeed an exciting event, it was clear even then that the quantum condition (1.22) used in the derivation could not be applied, in that form, in more complicated cases. However, it was soon realized that Eq. (1.22) is a special case of the condition ∫ p dq = nh (1.26) which had been employed already by Planck in connection with his theory of black body radiation.11 Here q is some generalized coordinate and p, the corresponding canonically conjugate momentum. The condition is applicable in the case of periodic motion only, and the integral is to be taken over one period, treating p as a function of the position q of the particle on the actual trajectory. Eq. (1.22) corresponds to choosing q to be the angular position ϕ (varying from 0 to 2π for one period) and p as its conjugate, the angular momentum (which is independent of ϕ, being a constant of the motion). Bohr postulated that the quantum condition (1.26) is applicable to any system with one degree of freedom. This quantum rule was further generalized by Sommerfeld to multiply-periodic systems, with many degrees of freedom, i.e. 10 This value is strictly correct only for an infinitely heavy nucleus which remains perfectly static. To indicate this fact explicitly, the notation r∞ is often used for the constant. To take into account the finite- ness of the mass (mN) of the nucleus (and the consequent motion of the nucleus) we have to replace the electronic mass m in the expression for r by the reduced mass m mN/(m + mN). The Rydberg constant for a finite nucleus is thus rN = r∞(1 + m/mN)-1. 11 Planck assumed that the emission and absorption of radiation in quanta hv was done by hypothetical har- monic oscillators capable of having only discrete energy values, nhv (n = 0, 1, 2,... ). He showed that this quantum condition on the energy levels of the oscillator was equivalent to quantizing the ‘action integral’ as in Eq. (1.26). For this reason, the name ‘quantum of action’ has been applied to Planck’s constant. Note that the value n = 0 does not make any sense in Bohr’s quantum condition, and had to be dropped. Towards Quantum Mechanics 17 systems which can be described by pairs of coordinate and momentum variables (q1, p1), (q2, p2),…,(qN, pN), each of which is periodic, with possibly different periods for different pairs. The generalized Bohr-sommerfeld quantum rule12 is given by: ∫ pr dqr = nr h, r = 1, 2, … N (1.27) where the integration in the case of each pair of conjugate variables is to be taken over one period of that particular pair. The quantum numbers nr take integral values. Bohr’s general postulates together with the quantum rule (1.27) constitute what is now known as the old Quantum theory..13 Details of the applications of the theory are of no particular interest now, since the theory itself has been superseded by the new quantum theory or Quantum Mechanics. But the quantization of the elliptical orbits of the hydro- gen atom deserves mention because it gave the first example of two general properties which persist in the quantum mechanical theory. One is the property of degeneracy of energy levels of systems possessing symmetries. Sommerfeld observed that the quan- tized elliptical orbits in a given plane are identified by two quantum numbers n9 and k, characterizing the radial and angular parts of the motion in the orbit. He found however that the energies associated with such quantum states depend only on the sum n = n9 + k of these quantum numbers, and are given by the Bohr formula (1.24). Thus for a given value of the total or ‘principal’ quantum number n, there are n different states (elliptical orbits of various eccentricities, corresponding to14 k = 1, 2,... n) all of which have the same energy en. We say that the energy level en is n-fold degenerate (when the quantum orbits in one plane alone are considered). It is now recognized that the degeneracy in the case of the hydrogen atom is due to the special nature or ‘symmetry’ of the distance- dependence of the electrostatic potential, and does not occur for other potentials (unless they have other symmetries, of course). The second general property exemplified in the Sommerfeld treatment is the removal of degeneracy (i.e., departure from equality of the energy values of the previously degenerate quantum states) when the symmetry is ‘bro- ken’. In the case of the hydrogen atom there is indeed a slight departure from symmetry. It is caused by the fact that the variation of the speed of the electron as it moves along an elliptical orbit induces corresponding changes in its mass as given by the theory of rela- tivity. Sommerfeld showed that this mass variation gives rise to a slow precession of the orbit in its own plane,15 and that because of this, the energy associated with each orbit is 12 W. Wilson, phil. Mag., 29,795, 1915; A. Sommerfeld, ann. d. physik, 51, 1, 1916. 13 Discussion of the old quantum theory and many of its applications may be found in L. Pauling and E. B. Wilson, introduction to Quantum Mechanics, McGraw-Hill, New York, 1935; A. Sommerfeld, atomic structure and spectral lines, 3rd ed., Methuen and Co., London, 1934. 14 In the Bohr-Sommerfeld theory the angular momentum in a stationary state was identified as kh but quantum mechanics shows that it is given by lh, where l takes the same values as (k-1) for given n, namely l = 0, 1,... , n–1. In the discussion of space quantization below, we use this quantum number l in preference to k. 15 More precisely, the orbit no longer closes on itself, but remains very nearly elliptical for each revolution, with the direction of the major axis changing slightly (in the plane of the ellipse) from one revolution to the next. This change of orientation of the ellipse at a steady rate is called precession. 18 A Textbook of Quantum Mechanics changed by a very small amount depending on k. Consequently, the nth Bohr level gets split into n closely-spaced levels—there is no more degeneracy. The spectral lines also then get split, developing what is called a fine structure. Sommerfeld’s calculation gave complete agreement with the observed fine structure. 1.11 SPACE QUANTIzATION In giving the degree of degeneracy of the nth Bohr level (i.e. the number of quan- tum states belonging to this level) as n, and in asserting that the relativistic mass variation removes this degeneracy, we have taken account of orbits in any one plane only. To put this in another way, only orbits with a specific direction for the angular momentum vector (which is normal to the plane of the orbit) were considered. Actu- ally there is a further degeneracy associated with the possibility of various orienta- tions for the angular momentum vector with respect to any fixed axis. Such an axis may be defined, for instance, by the direction of some externally applied field. From the consideration of the quantum condition (1.27) in axially symmetric situations it was inferred that the direction of angular momentum should be quantized. To be more specific, the component of angular momentum parallel to the axis has to be mh with the quantum number m taking integer values only. This is called quantization of direction or space quantization. When the quantum number characterizing the magnitude of the angular momentum has a value l, the values which m can take are limited to m = l, l - 1, … , - l + 1, - l. The existence of space quantization was experimentally demonstrated in a very direct and beautiful fashion by Stern and Gerlach.l6 They exploited the fact that an atom with nonzero angular momentum has a magnetic moment µ in the same direc- tion as the angular momentum vector L; for, the orbital motion of the electron (with which the angular momentum is associated in the Rutherford-Bohr picture) also pro- duces a magnetic moment since the electron is a charged particle. If such an atom is placed in a magnetic field h, the field exerts a torque µ × h tending to turn the direction of µ and hence that of L too into alignment with the field h. Now, it is well known that any torque acting on an angular momentum vector has a gyroscopic effect. Therefore L precesses around the direction of h, keeping a constant angle θ to h all the time. This is all that happens if h is a uniform field. If it is not uniform, there is also a net force ( µ.∇) h on the atom. Thus, if a coordinate system is chosen with z-axis in the direction (Fig. 1.3) of h and if the field strength h increases in the z-direction, we have a situation where µz and lz are constants for the atom, and there is also a force µ z (∂H/∂ z ) on the atom, acting in the z-direction. If a beam of atoms is shot through the field in a direction perpendicular to the field, say along the x-direction, the above force causes the individual atoms to be deflected up or down (i.e., in the positive or negative z-directions) by amounts proportional to their respective values of µz. Therefore, if the values of µz form a continuous range (in accordance with classical concepts) the beam would widen out into an expanding strip in the x-z plane. On the other hand, if there is space quantization, so that lz (and hence µz ) can take only a discrete set of values, the beam of atoms would split into a number 16 O. Stern and W. Gerlach, Z. physik, 8, 110, 1922. Towards Quantum Mechanics 19 EM Stream of Atoms z EM x S 0 Fig. 1.3 Schematic diagram of the Stern-Gerlach experiment showing splitting of the beam of atoms between the pole-pieces (EM) of electromagnet. of distinct diverging beams, each of which is characterized by a specific value of µz. Impinging on a plane perpendicular to the x-axis, these beams would leave spots spaced out along the z-direction (instead of a continuous line which would appear if there were no space quantization). It is the appearance of such distinct spots in the Stern-Gerlach experiment which gave direct confirmation of the idea of space quantization. In this experiment a fine, well-collimated beam of silver atoms was passed through an inho- mogeneous field created by an electromagnet with specially shaped pole pieces. One of these had a ridge along the middle, and facing this was a hollowed-out channel in the other pole piece. The net result was a concentration of field lines (high intensity) near the former and dispersal (low intensity) near the latter. It was found that when the atomic beam was made to pass between the pole pieces, traversing their whole length, the beam split into two—one part deflected upwards, and the other downwards. While this confirmation of space quantization was a success for the old Quantum theory, it must be mentioned that the appearance of just two values for µz was cor- rectly explained only after the discovery of the spin of the electron a couple of years later. Unlike l, which can have only integral values, the quantum number s charac- terizing the angular momentum associated with the spinning motion has the value 1 , and the component of spin in any direction can have just the two values + 1 2 2 or − 12. Deferring further discussion of this subject, we return now to our main theme: the progression of concepts from the classical to the quantum mechanical. 1.12 LIMITATIONS OF THE OLd QUANTUM THEOry We have seen above how the Old Quantum Theory has been able to provide the expla- nation of the spectrum of the hydrogen atom, including its fine structure. Recognition of the quantum character of the magnitude and direction of angular momentum remains as one of its finest achievements. However, despite these and other very considerable successes of the Old Quantum Theory, it is quite obvious that it is not really a funda- mental theory and is, in any case, only of limited applicability. The scope of the Bohr- Sommerfeld quantum rules is restricted to periodic or multiply-periodic motions; they have nothing to say about situations where other kinds of motion are involved. Even in the Franck-Hertz experiment which gave direct support to Bohr’s concept, the behav- iour of the electrons scattered by the atoms is outside the purview of the Old Quantum Theory. The limitations of the theory were greatly mitigated by skilful exploitation of 20 A Textbook of Quantum Mechanics the idea that the results of quantum theory should tend to those of the classical theory under circumstances where the quantum discontinuities are negligibly small. This idea, which places powerful constraints on the quantum theory, played a considerable role in the developments of the decade preceding the birth of quantum mechanics. A for- mal enunciation of the idea, under the name correspondence principle, was given by Bohr.17 In considering the quantum mechanics in later chapters we will have occasion to discuss its correspondence with classical mechanics in certain aspects of their math- ematical structures as well as in the sense of a passage to the limit h → 0. Looking back on the essentials of the Old Quantum Theory, we see that its fundamen- tal shortcoming is that it is a peculiar hybrid of quantum concepts grafted on to classical mechanics. The existence of discrete stationary states is experimentally well substanti- ated. But as long as the classical picture of well-defined particle orbits is retained, it remains incomprehensible why certain orbits should be completely stable and others not allowed to exist at all. This perplexing question was responsible in part for the ultimate realization that particle states at the microscopic level are not describable in terms of well-defined orbits, but must be pictured in terms of some kind of waves. Example 1.3 Show that the frequency (en+1 - en)/h of the line emitted by the quantum transition from level (n + 1) to n in the Bohr model of the hydrogen atom is, in the limit of large n, the same as if the radiation was being emitted classically by an electron moving in the Bohr orbit associated with either of these levels. The velocity u of motion of the electron in a Bohr orbit may be obtained by eliminating a between Eqs. (1.20) and (1.22). One finds that u = e2/(nh̄). The orbital frequency of the elec- tron is evidently vcl = u/(2πa). Since a = n2h̄2/(me2) from Eq. (1.23), we have υ me 4 vcl = =. (E1.1) 2πa 2πn3 3 If this electron were emitting radiation classically, the radiation would have the same fre- quency ν. According to quantum concepts, radiation is associated with transitions between quantum states. For a transition between adjacent Bohr orbits, the frequency of the radiation emitted is given by hv = (en+1 - en). Thus 1 me 4  1 1  me 4 v=  −   ≈ (E1.2) h 2 2  n2 ( n + 1)2  2π 3n3 for n >> 1. The equality of this expression to the frequency νcl for classical emission provides an illustration of Bohr’s correspondence principle. n (iv) Matter WaVes 1.13 dE BrOgLIE’S HyPOTHESIS The suggestion that matter may have wave-like properties was first put forward in 1924–25 by Louis de Broglie.18 He argued that if light (which consists of waves according to the classical picture) can sometimes behave like particles, then it should 17 N. Bohr, Nature, 12

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