1000 Solved Problems In Modern Physics - WordPress

1000 Solved Problems in Modern Physics

Ahmad A. Kamal1000 Solved Problemsin Modern Physics123

Dr. Ahmad A. Kamal425 Silversprings LaneMurphy, TX 75094, USAanwarakamal@yahoo.comISBN 978-3-642-04332-1e-ISBN 978-3-642-04333-8DOI 10.1007/978-3-642-04333-8Springer Heidelberg Dordrecht London New YorkLibrary of Congress Control Number: 2009943222c Springer-Verlag Berlin Heidelberg 2010 This 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.Cover design: eStudio Calamar SteinenPrinted on acid-free paperSpringer is part of Springer Science Business Media (www.springer.com)

Dedicated to my parents

PrefaceThis 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 solutions are neither pedantic nor terse. The approach is straight forward and step-bystep 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 revolutionary 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, statistical 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 deBroglievii

viiiPrefacewaves, 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 angular 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 applications 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. Problems 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 invariance 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, photoelectric 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, proprortional and G.M. counters, Accelerators which include Betatron, Cyclotron, SynchroCyclotron, 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-

Prefaceixmagnetic and weak, the conservation laws applicable to the three types of interactions, Gell-mann’s formula, the properties of quarks and classification into supermultiplets, the types of weak decays and Cabibbo’s theory, the neutrino oscillations,Electro–Weak interaction, the heavy bosons and the Standard model.AcknowledgementsIt 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, Maqsood 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 encouragement.Murphy, TexasFebruary 2010Ahmad A. Kamal

Contents1 Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.1 Basic Concepts and Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.1Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.2Fourier Series and Fourier Transforms . . . . . . . . . . . . . . . . .1.2.3Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . .1.2.4Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.5Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.6Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.7Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.8Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . .1.2.9Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.10Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.11Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.12Calculus of Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.13Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.14Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.1Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.2Fourier Series and Fourier Transforms . . . . . . . . . . . . . . . . .1.3.3Gamma and Beta Functions . . . . . . . . . . . . . . . . . . . . . . . . .1.3.4Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.5Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.6Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.7Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.8Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . .1.3.9Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.10Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.11Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.12Calculus of Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.13Statistical Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3.14Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7676872747785xi