1000 Solved Problems in Modern Physicsdownload.e-bookshelf.de/download/0000/0142/65/L-G-0000014265...1000 Solved Problems in Modern Physics. Ahmad A. Kamal 1000 Solved Problems in

1000 Solved Problems in Modern Physics

Ahmad A. Kamal

1000 Solved Problemsin Modern Physics

123

Dr. Ahmad A. Kamal425 Silversprings LaneMurphy, TX 75094, [email protected]

ISBN 978-3-642-04332-1 e-ISBN 978-3-642-04333-8DOI 10.1007/978-3-642-04333-8Springer Heidelberg Dordrecht London New York

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c© Springer-Verlag Berlin Heidelberg 2010This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevant protectivelaws and regulations and therefore free for general use.

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Dedicated to my parents

Preface

This book is targeted mainly to the undergraduate students of USA, UK and otherEuropean countries, and the M.Sc of Asian countries, but will be found useful for thegraduate students, Graduate Record Examination (GRE), Teachers and Tutors. Thisis a by-product of lectures given at the Osmania University, University of Ottawaand University of Tebrez over several years, and is intended to assist the students intheir assignments and examinations. The book covers a wide spectrum of disciplinesin Modern Physics, and is mainly based on the actual examination papers of UK andthe Indian Universities. The selected problems display a large variety and conform tosyllabi which are currently being used in various countries. The book is divided intoten chapters. Each chapter begins with basic concepts containing a set of formulaeand explanatory notes for quick reference, followed by a number of problems andtheir detailed solutions.

The problems are judiciously selected and are arranged section-wise. The solu-tions are neither pedantic nor terse. The approach is straight forward and step-by-step solutions are elaborately provided. More importantly the relevant formulas usedfor solving the problems can be located in the beginning of each chapter. There areapproximately 150 line diagrams for illustration.

Basic quantum mechanics, elementary calculus, vector calculus and Algebra arethe pre-requisites. The areas of Nuclear and Particle physics are emphasized as rev-olutionary developments have taken place both on the experimental and theoreticalfronts in recent years. No book on problems can claim to exhaust the variety in thelimited space. An attempt is made to include the important types of problems at theundergraduate level.

Chapter 1 is devoted to the methods of Mathematical physics and covers suchtopics which are relevant to subsequent chapters. Detailed solutions are given toproblems under Vector Calculus, Fourier series and Fourier transforms, Gamma andBeta functions, Matrix Algebra, Taylor and Maclaurean series, Integration, Ordinarydifferential equations, Calculus of variation Laplace transforms, Special functionssuch as Hermite, Legendre, Bessel and Laguerre functions, complex variables, sta-tistical distributions such as Binomial, Poisson, Normal and interval distributionsand numerical integration.

Chapters 2 and 3 focus on quantum physics. Chapter 2 is basically concernedwith the old quantum theory. Problems are solved under the topics of deBroglie

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waves, Bohr’s theory of hydrogen atom and hydrogen-like atoms, positronium andmesic atoms, X-rays production and spectra, Moseley’s law and Duan–Hunt law,spectroscopy of atoms and molecules, which include various quantum numbers andselection rules, and optical Doppler effect.

Chapter 3 is concerned with the quantum mechanics of Schrodinger andHesenberg. Problems are solved on the topics of normalization and orthogonalityof wave functions, the separation of Schrodinger’s equation into radial and angu-lar parts, 1-D potential wells and barriers, 3-D potential wells, Simple harmonicoscillator, Hydrogen-atom, spatial and momentum distribution of electron, Angularmomentum, Clebsch–Gordon coefficients ladder operators, approximate methods,scattering theory-phase-shift analysis and Ramsuer effect, the Born approximation.

Chapter 4 deals with problems on Thermo–dynamic relations and their applica-tions such a specific heats of gases, Joule–Thompson effect, Clausius–Clapeyronequation and Vander waal’s equation, the statistical distributions of Boltzmannand Fermi distributions, the distribution of rotational and vibrational states of gasmolecules, the Black body radiation, the solar constant, the Planck’s law and Wein’slaw.

Chapter 5 is basically related to Solid State physics and material science. Prob-lems are covered under the headings, crystal structure, Lattice constant, Electricalproperties of crystals, Madelung constant, Fermi energy in metals, drift velocity, theHall effect, the Debye temperature, the intrinsic and extrinsic semiconductors, thejunction diode, the superconductor and the BCS theory, and the Josephson effect.

Chapter 6 deals with the special theory of Relativity. Problems are solved underLorentz transformations of length, time, velocity, momentum and energy, the invari-ance of four-momentum vector, transformation of angles and Doppler effect andthreshold of particle production.

Chapters 7 and 8 are concerned with problems in low energy Nuclear physics.Chapter 7 covers the interactions of charged particles with matter which includekinematics of collisions, Rutherford Scattering, Ionization, Range and Straggiling,Interactions of radiation with matter which include Compton scattering, photoelec-tric effect, pair production and nuclear resonance fluorescence, general radioactivitywhich includes problems on chain decays, age of earth, Carbon dating, alpha decay,Beta decay and gamma decay.

Chapter 8 is devoted to the static properties of nuclei such as nuclear masses,nuclear spin and parity, magnetic moments and quadrupole moments, the Nuclearmodels, the Fermi gas model, the shell model, the liquid drop model and the opticalmodel, problems on fission and fusion and Nuclear Reactors.

Chapters 9 and 10 are concerned with high energy physics. Chapter 9 coversthe problems on natural units, production, interactions and decays of high energyunstable particles, various types of detectors such as ionization chambers, propror-tional and G.M. counters, Accelerators which include Betatron, Cyclotron, Synchro-Cyclotron, proton and electron Synchrotron, Linear accelerator and Colliders.

Chapter 10 deals with the static and dynamic properties of elementary particlesand resonances, their classification from the point of view of the Fermi–Dirac andBose–Einstein statistics as well as the three types of interactions, strong, Electro-

Preface ix

magnetic and weak, the conservation laws applicable to the three types of interac-tions, Gell-mann’s formula, the properties of quarks and classification into super-multiplets, the types of weak decays and Cabibbo’s theory, the neutrino oscillations,Electro–Weak interaction, the heavy bosons and the Standard model.

Acknowledgements

It is a pleasure to thank Javid for the bulk of typing and suggestions and Maryamfor proof reading. I am indebted to Muniba for the line drawings, to Suraiya, Maq-sood and Zehra for typing and editing. I am grateful to the Universities of UK andIndia for permitting me to use their question papers cited in the text to CERN photoservice for the cover page to McGraw-Hill and Co: for a couple of diagrams fromQuantum Mechanics, L.I. Schiff, 1955, to Cambridge University Press for usingsome valuable information from Introduction to High Energy Physics, D.H. Perkinsand to Ginn and Co: and Pearson and Co: for access to Differential and IntegralCalculus, William A. Granville, 1911. My thanks are due to Springer-Verlag, inparticular Claus Ascheron, Adelheid Duhm and Elke Sauer for constant encourage-ment.

Murphy, Texas Ahmad A. KamalFebruary 2010

Contents

1 Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Basic Concepts and Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2.1 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2.2 Fourier Series and Fourier Transforms . . . . . . . . . . . . . . . . . 221.2.3 Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . 231.2.4 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.2.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.2.6 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.2.7 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.2.8 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . 261.2.9 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.2.10 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.2.11 Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.2.12 Calculus of Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.2.13 Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.2.14 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.3.1 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.3.2 Fourier Series and Fourier Transforms . . . . . . . . . . . . . . . . . 391.3.3 Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . . 421.3.4 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.3.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.3.6 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491.3.7 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.3.8 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . 571.3.9 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671.3.10 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681.3.11 Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721.3.12 Calculus of Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741.3.13 Statistical Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771.3.14 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

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2 Quantum Mechanics – I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.1 Basic Concepts and Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.2.1 de Broglie Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922.2.2 Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922.2.3 X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.2.4 Spin and μ and Quantum Numbers – Stern–Gerlah’s

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.2.5 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.2.6 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992.2.7 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.2.8 Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.3.1 de Broglie Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.3.2 Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032.3.3 X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.3.4 Spin and μ and Quantum Numbers – Stern–Gerlah’s

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112.3.5 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152.3.6 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202.3.7 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232.3.8 Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

3 Quantum Mechanics – II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.1 Basic Concepts and Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.2.1 Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.2.2 Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383.2.3 Potential Wells and Barriers . . . . . . . . . . . . . . . . . . . . . . . . . 1403.2.4 Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 1463.2.5 Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.2.6 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.2.7 Approximate Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523.2.8 Scattering (Phase-Shift Analysis) . . . . . . . . . . . . . . . . . . . . . 1533.2.9 Scattering (Born Approximation) . . . . . . . . . . . . . . . . . . . . . 154

3.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563.3.1 Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563.3.2 Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1623.3.3 Potential Wells and Barriers . . . . . . . . . . . . . . . . . . . . . . . . . 1683.3.4 Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 1993.3.5 Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2093.3.6 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2153.3.7 Approximate Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2293.3.8 Scattering (Phase Shift Analysis) . . . . . . . . . . . . . . . . . . . . . 2333.3.9 Scattering (Born Approximation) . . . . . . . . . . . . . . . . . . . . . 240

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4 Thermodynamics and Statistical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 2474.1 Basic Concepts and Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2474.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

4.2.1 Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2514.2.2 Maxwell’s Thermodynamic Relations . . . . . . . . . . . . . . . . . 2534.2.3 Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2554.2.4 Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

4.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2584.3.1 Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2584.3.2 Maxwell’s Thermodynamic Relations . . . . . . . . . . . . . . . . . 2664.3.3 Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2794.3.4 Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

5 Solid State Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2915.1 Basic Concepts and Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2915.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

5.2.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2945.2.2 Crystal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2945.2.3 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2955.2.4 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2975.2.5 Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

5.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2995.3.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2995.3.2 Crystal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015.3.3 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3035.3.4 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3095.3.5 Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

6 Special Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3136.1 Basic Concepts and Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3136.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

6.2.1 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3196.2.2 Length, Time, Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3206.2.3 Mass, Momentum, Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 3236.2.4 Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3266.2.5 Transformation of Angles and Doppler Effect . . . . . . . . . . 3286.2.6 Threshold of Particle Production . . . . . . . . . . . . . . . . . . . . . 330

6.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3326.3.1 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3326.3.2 Length, Time, Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3386.3.3 Mass, Momentum, Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 3426.3.4 Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3516.3.5 Transformation of Angles and Doppler Effect . . . . . . . . . . 3556.3.6 Threshold of Particle Production . . . . . . . . . . . . . . . . . . . . . 365

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7 Nuclear Physics – I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3697.1 Basic Concepts and Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3697.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

7.2.1 Kinematics of Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3827.2.2 Rutherford Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3837.2.3 Ionization, Range and Straggling . . . . . . . . . . . . . . . . . . . . . 3857.2.4 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3877.2.5 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3887.2.6 Pair Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3907.2.7 Cerenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3907.2.8 Nuclear Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3907.2.9 Radioactivity (General) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3917.2.10 Alpha-Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3937.2.11 Beta-Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

7.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3947.3.1 Kinematics of Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3947.3.2 Rutherford Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3997.3.3 Ionization, Range and Straggling . . . . . . . . . . . . . . . . . . . . . 4047.3.4 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4077.3.5 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4117.3.6 Pair Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4147.3.7 Cerenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4157.3.8 Nuclear Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4167.3.9 Radioactivity (General) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4177.3.10 Alpha-Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4227.3.11 Beta-Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

8 Nuclear Physics – II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4278.1 Basic Concepts and Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4278.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

8.2.1 Atomic Masses and Radii . . . . . . . . . . . . . . . . . . . . . . . . . . . 4348.2.2 Electric Potential and Energy . . . . . . . . . . . . . . . . . . . . . . . . 4358.2.3 Nuclear Spin and Magnetic Moment . . . . . . . . . . . . . . . . . . 4358.2.4 Electric Quadrupole Moment . . . . . . . . . . . . . . . . . . . . . . . . 4358.2.5 Nuclear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4368.2.6 Fermi Gas Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4378.2.7 Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4378.2.8 Liquid Drop Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4388.2.9 Optical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4398.2.10 Nuclear Reactions (General) . . . . . . . . . . . . . . . . . . . . . . . . . 4408.2.11 Cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4428.2.12 Nuclear Reactions via Compound Nucleus . . . . . . . . . . . . . 4438.2.13 Direct Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4438.2.14 Fission and Nuclear Reactors . . . . . . . . . . . . . . . . . . . . . . . . 4448.2.15 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

Contents xv

8.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4478.3.1 Atomic Masses and Radii . . . . . . . . . . . . . . . . . . . . . . . . . . . 4478.3.2 Electric Potential and Energy . . . . . . . . . . . . . . . . . . . . . . . . 4498.3.3 Nuclear Spin and Magnetic Moment . . . . . . . . . . . . . . . . . . 4508.3.4 Electric Quadrupole Moment . . . . . . . . . . . . . . . . . . . . . . . . 4518.3.5 Nuclear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4548.3.6 Fermi Gas Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4568.3.7 Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4578.3.8 Liquid Drop Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4588.3.9 Optical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4608.3.10 Nuclear Reactions (General) . . . . . . . . . . . . . . . . . . . . . . . . . 4628.3.11 Cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4688.3.12 Nuclear Reactions via Compound Nucleus . . . . . . . . . . . . . 4698.3.13 Direct Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4708.3.14 Fission and Nuclear Reactors . . . . . . . . . . . . . . . . . . . . . . . . 4718.3.15 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

9 Particle Physics – I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4859.1 Basic Concepts and Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4859.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

9.2.1 System of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4889.2.2 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4899.2.3 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4899.2.4 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4919.2.5 Ionization Chamber, GM Counter and Proportional

Counters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4939.2.6 Scintillation Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4959.2.7 Cerenkov Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4969.2.8 Solid State Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4979.2.9 Emulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4979.2.10 Motion of Charged Particles in Magnetic Field . . . . . . . . . 4979.2.11 Betatron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4989.2.12 Cyclotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4999.2.13 Synchrotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5009.2.14 Linear Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5019.2.15 Colliders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

9.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5039.3.1 System of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5039.3.2 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5049.3.3 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5059.3.4 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5089.3.5 Ionization Chamber, GM Counter and Proportional

Counters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5129.3.6 Scintillation Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

xvi Contents

9.3.7 Cerenkov Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5189.3.8 Solid State Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5209.3.9 Emulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5209.3.10 Motion of Charged Particles in Magnetic Field . . . . . . . . . 5219.3.11 Betatron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5249.3.12 Cyclotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5249.3.13 Synchrotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5279.3.14 Linear Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5309.3.15 Colliders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

10 Particle Physics – II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53510.1 Basic Concepts and Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53510.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544

10.2.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54410.2.2 Strong Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54610.2.3 Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55010.2.4 Electromagnetic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 55110.2.5 Weak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55210.2.6 Electro-Weak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 55610.2.7 Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556

10.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55810.3.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55810.3.2 Strong Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56310.3.3 Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57210.3.4 Electromagnetic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 57710.3.5 Weak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57810.3.6 Electro-weak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 59010.3.7 Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592

Appendix: Problem Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

Chapter 1Mathematical Physics

1.1 Basic Concepts and Formulae

Vector calculus

Angle between two vectors, cos θ = A.B|A||B|

Condition for coplanarity of vectors, A.B × C = 0

Del

∇ = ∂

∂xi + ∂

∂yj + ∂

∂zk

Gradient

∇φ =(∂φ

∂xi + ∂φ

∂yj + ∂φ

∂zk

)

Divergence

If V (x, y, z) = V1 i + V2 j + V3k, be a differentiable vector field, then∇.V = ∂

∂x V1 + ∂∂y V2 + ∂

∂z V3

Laplacian

∇2 = ∂2

∂x2+ ∂2

∂y2+ ∂2

∂z2(Cartesian coordinates x, y, z)

∇2 = 1

r2

∂

∂r

(r2 ∂

∂r

)+ 1

r2 sin θ

∂

∂θ

(sin θ

∂

∂θ

)+ 1

r2 sin2 θ

∂2

∂Φ2

(Spherical coordinates r, θ,Φ)

∇2 = ∂2

∂r2+ 1

r

∂

∂r+ 1

r2

∂2

∂θ2+ ∂2

∂z2(Cylindrical coordinates r, θ, z)

Line integrals

(a)∫

C φ dr

1

2 1 Mathematical Physics

(b)∫

C A . dr(c)

∫C A × dr

where φ is a scalar, A is a vector and r = xi + y j + zk, is the positive vector.

Stoke’s theorem

∮C

A . d r =∫∫

S(∇ × A) . n ds =

∫∫S(∇ × A) . ds

The line integral of the tangential component of a vector A taken around a simpleclosed curve C is equal to the surface integral of the normal component of the curlof A taken over any surface S having C as its boundary.

Divergence theorem (Gauss theorem)

∫∫∫V

∇ . A dv =∫∫

SA.n ds

The volume integral is reduced to the surface integral.

Fourier series

Any single-valued periodic function whatever can be expressed as a summation ofsimple harmonic terms having frequencies which are multiples of that of the givenfunction. Let f (x) be defined in the interval (−π, π ) and assume that f (x) hasthe period 2π , i.e. f (x + 2π ) = f (x). The Fourier series or Fourier expansioncorresponding to f (x) is defined as

f (x) = 1

2a0 +

∑∞n=1

(a0 cos nx + bn sin nx) (1.1)

where the Fourier coefficient an and bn are

an = 1

π

∫ π

−πf (x) cos nx dx (1.2)

bn = 1

π

∫ π

−πf (x) sin nx dx (1.3)

where n = 0, 1, 2, . . .If f (x) is defined in the interval (−L , L), with the period 2L , the Fourier series

is defined as

f (x) = 1

2a0 +

∑∞n=1

(an cos(nπx/L) + bn sin(nπx/L)) (1.4)

where the Fourier coefficients an and bn are

1.1 Basic Concepts and Formulae 3

an = 1

L

∫ L

−Lf (x) cos(nπx/L) dx (1.5)

bn = 1

L

∫ L

−Lf (x) sin(nπx/L) dx (1.6)

Complex form of Fourier series

Assuming that the Series (1.1) converges at f (x),

f (x) =∑∞

n=−∞ Cneinπx/L (1.7)

with

Cn = 1

L

∫ C+2L

Cf (x)e−iπnx/L dx =

⎧⎪⎨⎪⎩

12 (an − ibn) n > 012 (a−n + ib−n) n < 0

12 ao n = 0

(1.8)

Fourier transforms

The Fourier transform of f (x) is defined as

�( f (x)) = F(α) =∫ ∞

−∞f (x)eiαx dx (1.9)

and the inverse Fourier transform of F(α) is

�−1( f (α)) = F(x) = 1

2π

∫ ∞

−∞F(α)ei∝x dα (1.10)

f (x) and F(α) are known as Fourier Transform pairs. Some selected pairs are givenin Table 1.1.

Table 1.1f (x) F(α) f (x) F(α)

1

x2 + a2

πe−aα

ae−ax a

α2 + a2

x

x2 + a2−π iα

ae−aα e−ax2 1

2

√π

ae−α2/4a

1

x

π

2xe−ax2

√π

4a3/2αe−α2/4a

Gamma and beta functions

The gamma function Γ(n) is defined by


Γ(n) =∫ ∞

0e−x xn−1 dx (Re n > 0) (1.11)

Γ(n + 1) = nΓ(n) (1.12)

If n is a positive integer

Γ(n + 1) = n! (1.13)

Γ

(1

2

)= √

π ; Γ

(3

2

)=

√π

2; Γ

(5

2

)= 3

4

√π (1.14)

Γ

(n + 1

2

)= 1.3.5 . . . (2n − 1)

√π

2n(n = 1, 2, 3, . . .) (1.15)

Γ

(−n + 1

2

)= (−1)n2n√π

1.3.5 . . . (2n − 1)(n = 1, 2, 3, . . .) (1.16)

Γ(n + 1) = n! ∼=√

2πn nne−n (Stirling’s formula) (1.17)

n → ∞

Beta function B(m, n) is defined as

B(m, n) = Γ(m)Γ(n)

Γ(m + n)(1.18)

B(m, n) = B(n,m) (1.19)

B(m, n) = 2∫ π/2

0sin2m−1 θ cos2n−1 θ dθ (1.20)

B(m, n) =∫ ∞

0

tm−1

(1 + t) m+ndt (1.21)

Special funtions, properties and differential equations

Hermite functions:

Differential equation:

y′′ − 2xy′ + 2ny = 0 (1.22)

when n = 0, 1, 2, . . . then we get Hermite’s polynomials Hn(x) of degree n, givenby

Hn(x) = (−1)nex2 dn

dxn

(e−x2

)(Rodrigue’s formula)


First few Hermite’s polynomials are:

Ho(x) = 1, H1(x) = 2x, H2(x) = 4x2 − 2

H3(x) = 8x3 − 12x, H4(x) = 16x4 − 48x2 + 12 (1.23)

Generating function:

e2t x−t2 =∑∞

n=0

Hn(x)tn

n!(1.24)

Recurrence formulas:

H ′n(x) = 2nHn−1(x)

Hn+1(x) = 2x Hn(x) − 2nHn−1(x) (1.25)

Orthonormal properties:

∫ ∞

−∞e−x2

Hm(x)Hn(x) dx = 0 m = n (1.26)

∫ ∞

−∞e−x2{Hn(x)}2 dx = 2n n!

√π (1.27)

Legendre functions:

Differential equation of order n:

(1 − x2)y′′ − 2xy′ + n(n + 1)y = 0 (1.28)

when n = 0, 1, 2, . . . we get Legendre polynomials Pn(x).

Pn(x) = 1

2nn!

dn

dxn(x2 − 1)n (1.29)

First few polynomials are:

Po(x) = 1, P1(x) = x, P2(x) = 1

2(3x2 − 1)

P3(x) = 1

2(5x3 − 3x), P4(x) = 1

8(35x4 − 30x2 + 3) (1.30)


1√1 − 2t x + t2

=∑∞

n=0Pn(x)tn (1.31)



x P ′n(x) − P ′

n−1(x) = n Pn(x)

P ′n+1(x) − P ′

n−1(x) = (2n + 1)Pn(x) (1.32)


∫ 1

−1Pm(x)Pn(x) dx = 0 m = n (1.33)

∫ 1

−1{Pn(x)}2 dx = 2

2n + 1(1.34)

Other properties:

Pn(1) = 1, Pn(−1) = (−1)n, Pn(−x) = (−1)n Pn(x) (1.35)

Associated Legendre functions:


(1 − x2)y′′ − 2xy′ +{

l(l + 1) − m2

1 − x2

}y = 0 (1.36)

Pml (x) = (1 − x2)m/2 dm

dxmPl (x) (1.37)

where Pl (x) are the Legendre polynomials stated previously, l being the positiveinteger.

Pol (x) = Pl (x) (1.38)

and Pml (x) = 0 if m > n (1.39)


∫ 1

−1Pm

n (x)Pml (x) dx = 0 n = l (1.40)

∫ 1

−1{Pm

l (x)}2 dx = 2

2l + 1

(l + m)!

(l − m)!(1.41)

Laguerre polynomials:


xy′′ + (1 − x)y′ + ny = 0 (1.42)

if n = 0, 1, 2, . . . we get Laguerre polynomials given by


Ln(x) = ex dn

dxn(xne−x ) (Rodrigue’s formula) (1.43)

The first few polynomials are:

Lo(x) = 1, L1(x) = −x + 1, L2(x) = x2 − 4x + 2

L3(x) = −x3 + 9x2 − 18x + 6, L4(x) = x4 − 16x3 + 72x2 − 96x + 24 (1.44)


e−xs/(1−s)

1 − s=∑∞

n=0

Ln(x)sn

n!(1.45)


Ln+1(x) − (2n + 1 − x)Ln(x) + n2Ln−1(x) = 0

x L ′n(x) = nLn(x) − n2 Ln−1(x) (1.46)

Orthonormal properties:∫ ∞

0e−x Lm(x)Ln(x) dx = 0 m = n (1.47)

∫ ∞

0e−x {Ln(x)}2 dx = (n!)2 (1.48)

Bessel functions: (Jn(x))

Differential equation of order n

x2 y′′ + xy′ + (x2 − n2)y = 0 n ≥ 0 (1.49)

Expansion formula:

Jn(x) =∑∞

k=0

(−1)k(x/2)2k−n

k!Γ(k + 1 − n)(1.50)

Properties:

J−n(x) = (−1)n Jn(x) n = 0, 1, 2, . . . (1.51)

J ′o(x) = −J1(x) (1.52)

Jn+1(x) = 2n

xJn(x) − Jn−1(x) (1.53)


ex(s−1/s)/2 =∑∞

n=−∞ Jn(x)tn (1.54)


Laplace transforms:

Definition:A Laplace transform of the function F(t) is

∫ ∞

0F(t)e−st dt = f (s) (1.55)

The function f (s) is the Laplace transform of F(t). Symbolically, L{F(t)} =f (s) and F(t) = L−1{ f (s)} is the inverse Laplace transform of f (s). L−1 is calledthe inverse Laplace operator.

Table of Laplace transforms:

F(t) f (s)

aF1(t) + bF2(t) a f1(s) + b f2(s)aF(at) f (s/a)eat F(t) f (s − a)F(t − a) t > a0 t < a

e−as f (s)

11

s

t1

s2

tn−1

(n − 1)!

1

snn = 1, 2, 3, . . .

eat 1

s − asin at

a

1

s2 + a2

cos ats

s2 + a2

sinh at

a

1

s2 − a2

cosh ats

s2 − a2

Calculus of variation

The calculus of variation is concerned with the problem of finding a function y(x)such that a definite integral, taken over a function shall be a maximum or minimum.Let it be desired to find that function y(x) which will cause the integral

I =∫ x2

x1

F(x, y, y′) dx (1.56)


to have a stationary value (maximum or minimum). The integrand is taken to bea function of the dependent variable y as well as the independent variable x andy′ = dy/dx . The limits x1 and x2 are fixed and at each of the limits y has definitevalue. The condition that I shall be stationary is given by Euler’s equation

∂F

∂y− d

dx

∂F

∂y′ = 0 (1.57)

When F does not depend explicitly on x , then a different form of the aboveequation is more useful

∂F

∂x− d

dx

(F − y′ ∂F

∂y′

)= 0 (1.58)

which gives the result

F − y′ ∂F

∂y′ = Constant (1.59)

Statistical distribution

Binomial distribution

The probability of obtaining x successes in N -independent trials of an event forwhich p is the probability of success and q the probability of failure in a single trialis given by the binomial distribution B(x).

B(x) = N !

x!(N − x)!px q N−x = C N

x px q N−x (1.60)

B(x) is normalized, i.e.

∑N

x=0B(x) = 1 (1.61)

It is a discrete distribution.The mean value,

〈x〉 = N p (1.62)

The S.D.,

σ =√

N pq (1.63)


Poisson distribution

The probability that x events occur in unit time when the mean rate of occurrenceis m, is given by the Poisson distribution P(x).

P(x) = e−mmx

x!(x = 0, 1, 2, . . .) (1.64)

The distribution P(x) is normalized, that is∑∞

x=0p(x) = 1 (1.65)

This is also a discrete distribution.When N P is held fixed, the binomial distribution tends to Poisson distribution

as N is increased to infinity.The expectation value, i.e.

〈x〉 = m (1.66)

The S.D.,

σ = √m (1.67)

Properties:

pm−1 = pm (1.68)

px−1 = x

mpm and px+1 = m

m + 1px (1.69)

Normal (Gaussian distribution)

When p is held fixed, the binomial distribution tends to a Normal distribution as Nis increased to infinity. It is a continuous distribution and has the form

f (x) dx = 1√2πσ

e−(x−m)2/2σ 2dx (1.70)

where m is the mean and σ is the S.D.The probability of the occurrence of a single random event in the interval m − σ

and m + σ is 0.6826 and that between m − 2σ and m + 2σ is 0.973.

Interval distribution

If the data contains N time intervals then the number of time intervals n between t1and t2 is

n = N (e−at1 − e−at2 ) (1.71)

where a is the average number of intervals per unit time. Short intervals are morefavored than long intervals.


Two limiting cases:

(a) t2 = ∞; N = Noe−λt (Law of radioactivity) (1.72)

This gives the number of surviving atoms at time t .

(b) t1 = 0; N = No(1 − e−λt ) (1.73)

For radioactive decays this gives the number of decays in time interval 0 and t .Above formulas are equally valid for length intervals such as interaction lengths.

Moment generating function (MGF)

MGF = Ee(x−μ)t

= E

[1 + (x − μ)t + (x − μ)2 t2

2!+ . . .

]

= 1 + 0 + μ2t2

2!+ μ3

t3

3!+ . . . (1.74)

so that μn , the nth moment about the mean is the coefficient of tn/n!.

Propagation of errors

If the error on the measurement of f (x, y, . . .) is σ f and that on x and y, σx and σy ,respectively, and σx and σy are uncorrelated then

σ 2f =

(∂ f

∂x

)2

σ 2x +

(∂ f

∂y

)2

σ 2y + · · · (1.75)

Thus, if f = x ± y, then σ f = (σ 2

x + σ 2y

)1/2

And if f = xy then σ f

f =(σ 2

x

x2 + σ 2y

y2

)1/2

Least square fit

(a) Straight line: y = mx + cIt is desired to fit pairs of points (x1, y1), (x2, y2), . . . , (xn, yn) by a straight line

Residue: S = ∑ni=1(yi − mxi − C)2

Minimize the residue: ∂s∂m = 0; ∂s

∂c = 0

The normal equations are:

m∑n

i=1x2

i + C∑n

i=1xi −

∑n

i=1xi yi = 0

m∑n

i=1xi + nC −

∑n

i=1yi = 0


which are to be solved as ordinary algebraic equations to determine the bestvalues of m and C .

(b) Parabola: y = a + bx + cx2

Residue: S = ∑ni=1(yi − a − bxi − cx2

i )2

Minimize the residue: ∂s∂a = 0; ∂s

∂b = 0; ∂s∂c = 0

The normal equations are:

∑yi − na − b

∑xi − c

∑x2

i = 0∑xi yi − a

∑xi − b

∑x2

i − c∑

x3i = 0∑

x2i yi − a

∑x2

i − b∑

x3i − c

∑x4

i = 0

which are to be solved as ordinary algebraic equations to determine the bestvalue of a, b and c.

Numerical integration

Since the value of a definite integral is a measure of the area under a curve, it followsthat the accurate measurement of such an area will give the exact value of a definiteintegral; I = ∫ x2

x1y(x)dx . The greater the number of intervals (i.e. the smaller Δx is),

the closer will be the sum of the areas under consideration.

Trapezoidal rule

area =(

1

2y0 + y1 + y2 + · · · yn−1 + 1

2yn

)Δx (1.76)

Simpson’s rule

area = Δx

3(y0 + 4y1 + 2y2 + 4y3 + 2y4 + · · · yn), n being even. (1.77)

Fig. 1.1 Integration bySimpson’s rule andTrapezoidal rule


Matrices

Types of matrices and definitions

Identity matrix:

I2 =(

1 00 1

); I3 =

⎛⎝ 1 0 0

0 1 00 0 1

⎞⎠ (1.78)

Scalar matrix:

(a11 00 a22

);

⎛⎝a11 0 0

0 a22 00 0 a33

⎞⎠ (1.79)

Symmetric matrix:

(a ji = ai j

);

⎛⎝a11 a12 a13

a12 a22 a23

a13 a23 a33

⎞⎠ (1.80)

Skew symmetric:

(a ji = −ai j

);

⎛⎝a11 a12 a13

−a12 a22 a23

−a13 −a23 a33

⎞⎠ (1.81)

The Inverse of a matrix B = A−1 (B equals A inverse):if AB = B A = I and further, (AB)−1 = B−1 A−1

A commutes with B if AB = B AA anti-commutes with B if AB = −B A

The Transpose (A′) of a matrix A means interchanging rows and columns.

Further, (A + B)′ = A′ + B ′

(A′)′ = A, (k A)′ = k A′ (1.82)

The Conjugate of a matrix. If a matrix has complex numbers as elements, and ifeach number is replaced by its conjugate, then the new matrix is called the conjugateand denoted by A∗ or A (A conjugate)

The Trace (Tr) or Spur of a matix is the num of the diagonal elements.

T r =∑

aii (1.83)


Hermetian matrix

If A′ = A, so that ai j = a ji for all values of i and j . Diagonal elements of anHermitian matrix are real numbers.

Orthogonal matrix

A square matrix is said to be orthogonal if AA′ = A′ A = I , i.e. A′ = A−1

The column vector (row vectors) of an orthogonal matrix A are mutually orthog-onal unit vectors.

The inverse and the transpose of an orthogonal matrix are orthogonal.The product of any two or more orthogonal matrices is orthogonal.The determinant of an Orthogonal matrix is ±1.

Unitary matrix

A square matrix is called a unitary matrix if (A)′ A = A(A)′ = I , i.e. if (A)′ = A−1.The column vectors (row vectors) of an n-square unitary matrix are an orthonor-

mal set.

The inverse and the transpose of a unitary matrix are unitary.The product of two or more unitary matrices is unitary.The determinant of a unitary matrix has absolute value 1.

Unitary transformations

The linear transformation Y = AX (where A is unitary and X is a vector), is calleda unitary transformation.

If the matrix is unitary, the linear transformation preserves length.

Rank of a matrix

If |A| = 0, it is called non-singular; if |A| = 0, it is called singular.A non-singular matrix is said to have rank r if at least one of its r -square minors

is non-zero while if every (r + 1) minor, if it exists, is zero.

Elementary transformations

(i) The interchange of the i th rows and j th rows or i th column or j th column.(ii) The multiplication of every element of the i th row or i th column by a non-zero

scalar.(iii) The addition to the elements of the i th row (column) by k (a scalar) times the

corresponding elements of the j th row (column). These elementary transfor-mations known as row elementary or row transformations do not change theorder of the matrix.


Equivalence

A and B are said to be equivalent (A ∼ B) if one can be obtained from the other bya sequence of elementary transformations.

The adjoint of a square matrix

If A = [ai j ] is a square matrix and αi j the cofactor of ai j then

adj A =

⎡⎢⎢⎣α11 α21 · · · αn1

α12 α22 · · · · · ·· · · · · · · · · · · ·· · · · · · · · · αnn

⎤⎥⎥⎦

The cofactor αi j = (−1)i+ j Mi j

where Mi j is the minor obtained by striking off the i th row and j th column andcomputing the determinant from the remaining elements.

Inverse from the adjoint

A−1 = ad j A

|A|

Inverse for orthogonal matrices

A−1 = A′

Inverse of unitary matrices

A−1 = (A)′

Characteristic equation

Let AX = λX (1.84)

be the transformation of the vector X into λX , where λ is a number, then λ is calledthe eigen or characteristic value.From (1.84):

(A − λI )X =

⎡⎢⎢⎢⎣

a11 − λ a12 · · · a1n

a21 a22 − λ · · · a2n... · · · · · · · · ·

an1 · · · · · · ann − λ

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

x1

x2...xn

⎤⎥⎥⎥⎦ = 0 (1.85)


The system of homogenous equations has non-trivial solutions if

|A − λI | =

⎡⎢⎢⎢⎣

a11 − λ a12 · · · a1n

a21 a22 − λ · · · a2n... · · · · · · · · ·

an1 · · · · · · ann − λ

⎤⎥⎥⎥⎦ = 0 (1.86)

The expansion of this determinant yields a polynomial φ(λ) = 0 is called thecharacteristic equation of A and its roots λ1, λ1, . . . , λn are known as the charac-teristic roots of A. The vectors associated with the characteristic roots are calledinvariant or characteristic vectors.

Diagonalization of a square matrix

If a matrix C is found such that the matrix A is diagonalized to S by thetransformation

S = C−1 AC (1.87)

then S will have the characteristic roots as the diagonal elements.

Ordinary differential equations

The methods of solving typical ordinary differential equations are from the book“Differential and Integral Calculus” by William A. Granville published by Ginn &Co., 1911.

An ordinary differential equation involves only one independent variable, whilea partial differential equation involves more than one independent variable.

The order of a differential equation is that of the highest derivative in it.The degree of a differential equation which is algebraic in the derivatives is the

power of the highest derivative in it when the equation is free from radicals andfractions.

Differential equations of the first order and of the first degree

Such an equation must be brought into the form Mdx+Ndy = 0, in which M and Nare functions of x and y.

Type I variables separable

When the terms of a differential equation can be so arranged that it takes on theform

(A) f (x) dx + F(y) dy = 0

where f (x) is a function of x alone and F(y) is a function of y alone, the process iscalled separation of variables and the solution is obtained by direct integration.

(B)∫

f (x) dx +∫

F(y) dy = C

where C is an arbitrary constant.


Equations which are not in the simple form (A) can be brought into that form bythe following rule for separating the variables.

First step: Clear off fractions, and if the equation involves derivatives, multiplythrough by the differential of the independent variable.

Second step: Collect all the terms containing the same differential into a singleterm. If then the equation takes on the form

XY dx + X ′Y ′ dy = 0

where X, X ′ are functions of x alone, and Y,Y ′ are functions of y alone, it may bebrought to the form (A) by dividing through by X ′Y ′.

Third step: Integrate each part separately as in (B).

Type II homogeneous equations

The differential equation

Mdx + Ndy = 0

is said to be homogeneous when M and N are homogeneous function of x and yof the same degree. In effect a function of x and y is said to be homogenous in thevariable if the result of replacing x and y by λx and λy (λ being arbitrary) reducesto the original function multiplied by some power of λ. This power of λ is calledthe degree of the original function. Such differential equations may be solved bymaking the substitution

y = vx

This will give a differential equation in v and x in which the variables are sepa-rable, and hence we may follow the rule (A) of type I.

Type III linear equations

A differential equation is said to be linear if the equation is of the first degree in thedependent variables (usually y) and its derivatives. The linear differential equationof the first order is of the form

dy

dx+ Py = Q

where P , Q are functions of x alone, or constants, the solution is given by

ye∫

Pdx =∫

Qe∫

Pdx dx + C

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