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INTRODUCTION TO QUANTUM MECHANICS - Fisica

INTRODUCTION to QUANTUM MechanicsThe ManchesterPhysicsSeriesGeneralEditorsD. J. SANDIFORD: F. MANDL: A. C. PHILLIPSD epartmentof Physics and Astronomy,University of ManchesterPropertiesof Matter:B. H. Flowers and E. MendozaStatistical Physics:F. MandlSecondEditionElectromagnetism:I. S. Grant and W. R. PhillipsSecondEditionStatistics:R. J. BarlowSolidStatePhysics:J. R. Hookand H. E. HallSecondEditionQuantum MECHANICS :F. MandlParticle Physics:B. R. Martin and G. ShawSecondEditionThe Physics of Stars:A. C. PhillipsSecondEditionComputing for Scientists:R. J. Barlowand A. R. BarnettNuclear Physics:J. S. LilleyIntroduction to QUANTUM MECHANICS :A. C. PhillipsINTRODUCTION TOQUANTUM MECHANICSA. C. PhillipsDepartment of Physics andAstronomyUniversityof ManchesterCopyright#2003 by John Wiley & Sons Ltd,The Atrium, Southern Gate, Chichester,West Sussex, PO19 8SQ, EnglandNational 01243 779777 International ( 44) 1243 779777e-mail (for orders and customer service enquiries): our Home Page on rights reserved.

. provides an understanding of the power and elegance of quantum mechanics that will make more advanced texts accessible. Chapter 1 provides a qualitative description of the remarkable properties of quantum particles, and these properties are used as the guidelines for a theory of quantum mechanics which is developed in Chapters 2, 3 and 4.

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Transcription of INTRODUCTION TO QUANTUM MECHANICS - Fisica

1 INTRODUCTION to QUANTUM MechanicsThe ManchesterPhysicsSeriesGeneralEditorsD. J. SANDIFORD: F. MANDL: A. C. PHILLIPSD epartmentof Physics and Astronomy,University of ManchesterPropertiesof Matter:B. H. Flowers and E. MendozaStatistical Physics:F. MandlSecondEditionElectromagnetism:I. S. Grant and W. R. PhillipsSecondEditionStatistics:R. J. BarlowSolidStatePhysics:J. R. Hookand H. E. HallSecondEditionQuantum MECHANICS :F. MandlParticle Physics:B. R. Martin and G. ShawSecondEditionThe Physics of Stars:A. C. PhillipsSecondEditionComputing for Scientists:R. J. Barlowand A. R. BarnettNuclear Physics:J. S. LilleyIntroduction to QUANTUM MECHANICS :A. C. PhillipsINTRODUCTION TOQUANTUM MECHANICSA. C. PhillipsDepartment of Physics andAstronomyUniversityof ManchesterCopyright#2003 by John Wiley & Sons Ltd,The Atrium, Southern Gate, Chichester,West Sussex, PO19 8SQ, EnglandNational 01243 779777 International ( 44) 1243 779777e-mail (for orders and customer service enquiries): our Home Page on rights reserved.

2 No part of this publication may be reproduced, storedin a retrieval system, or transmitted, in any form or by any means, electronic,mechanical, photocopying, recording, scanning or otherwise, except under the termsof the Copyright, Designs and Patents Act 1988 or under the terms of a licence issuedby the Copyright Licensing Agency, 90 Tottenham Court Road, London, UK W1P 9HE,without the permission in writing of the Wiley Editorial OfficesJohn Wiley & Sons, Inc., 605 Third Avenue,New York, NY 10158-0012, USAW iley-VCH Verlag GmbH, Pappelallee 3,D-69469 Weinheim, GermanyJohn Wiley & Sons (Australia) Ltd, 33 Park Road, Milton,Queensland 4064, AustraliaJohn Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01,Jin Xing Distripark, Singapore 0512 John Wiley & Sons (Canada) Ltd, 22 Worcester Road,Rexdale, Ontario M9W 1L1, CanadaLibrary of Congress Cataloging-in-Publication DataBritish Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryISBN 0-470-85323-9 (Hardback)0-470-85324-7 (Paperback)Typeset by Kolam Information Services Pvt.

3 Ltd., Pondicherry, IndiaPrinted and bound in Great Britain by Antony Rowe Ltd, Chippenham, WiltshireThis book is printed on acid-free paper responsibly manufactured from sustainableforestry, in which at least two trees are planted for each one used for paper my sons:JosephMichaelPatrickPeterThis page intentionally left blank ContentsForewordxiEditor's preface to the Manchester Physics SeriesxiiiAuthor's prefacexv1 PLANCK'S CONSTANT IN De Broglie Measurement10 The uncertainty principle11 Measurement and wave particle duality13 Measurement and non-locality16Pr o b l e m s1172 THE SCHRO DINGER Waves21 Sinusoidal waves21 Linear superpositions of sinusoidal waves22 Dispersive and non-dispersive Particle Wave Equations26A wave equation for a free particle27 Wave equation for a particle in a potential energy field29Pr o b l e m s2313 POSITION AND Probability35 Discrete random variables35 Continuous random Position Probabilities38 Two-slit interference38 The Born interpretation of the wave Particlein a o b l e m Modesof a of Certain Particlein a BoxII66A one-dimensional box66A of o b l e m states88 General Penetration94 Stationary stateanalysis of

4 Reflectionandtransmission95 Tunnellingthroughwidebarriers97 Tunnellingelectrons99 Tunnellingprotons100Pr o b l e m state125viiiContentsExcited states126 IsE0really the lowest energy?127 Mathematical properties of the oscillator eigenfunctions128Pr o b l e m s61287 OBSERVABLES AND Essential Position and Momentum138 Eigenfunctions for position138 Eigenfunctions for momentum139 Delta function Compatible Commutators142A particle in one dimension143A particle in three Constants of Motion146Pr o b l e m s71488 ANGULAR Angular Momentum Magnetic Moments158 Classical magnets158 QUANTUM magnets159 Magnetic energies and the Stern Gerlach Orbital Angular Momentum163 Classical orbital angular momentum163 QUANTUM orbital angular momentum164 Angular shape of wave functions164 Spherical harmonics169 Linear superposition171Pr o b l e m s81749 THE HYDROGEN Central Potentials179 Classical MECHANICS of a particle in a central potential179 QUANTUM MECHANICS of a particle in a central QUANTUM MECHANICS of the Hydrogen Atom185 Energy levels and Sizes and Radiative The Reduced Mass Relativistic

5 The Coulomb Eigenvalue Problem202 ContentsixPr o b l e m and Fermions222Pr o b l e m fieldapproximation230 Correctionsto thecentral ?241Pr o b l e m s11246 Hintsto selectedproblems249 Furtherreading262 Index263 Physicalconstants andconversionfactorsInsideBack CoverxContentsForewordSadly, Tony Phillips, a good friend and colleague for more than thirty years,died on 27th November 2002. Over the years, we discussed most topics underthe sun. The originality and clarity of his thoughts and the ethical basis of hisjudgements always made this a refreshing exercise. When discussing physics, QUANTUM MECHANICS was a recurring theme which gained prominence after hisdecision to write this book. He completed the manuscript three months beforehis death and asked me to take care of the proofreading and the Index. Alabour of love. I knew what Tony wanted and what he did not want. Exceptfor corrections, no changes have been was an outstanding teacher who could talk with students of all had a deep knowledge of physics and was able to explain subtle ideas in asimple and delightful style.

6 Who else would refer to the end-point of nuclearfusion in the sun as sunshine? Students appreciated him for these qualities, hisstraightforwardness and his genuine concern for them. This book is a fittingmemorial to MandlDecember 2002 This page intentionally left blank Editors' preface to theManchester Physics SeriesThe Manchester Physics Series is a series of textbooks at first degree level. Itgrew out of our experience at the Department of Physics and Astronomy atManchester University, widely shared elsewhere, that many textbooks containmuch more material than can be accommodated in a typical undergraduatecourse; and that this material is only rarely so arranged as to allow thedefinition of a short self-contained course. In planning these books we havehad two objectives. One was to produce short books: so that lecturers shouldfind them attractive for undergraduate courses; so that students should not befrightened off by their encyclopaedic size or price.

7 To achieve this, we have beenvery selective in the choice of topics, with the emphasis on the basic physicstogether with some instructive, stimulating and useful applications. Our secondobjective was to produce books which allow courses of different lengths anddifficulty to be selected with emphasis on different applications. To achievesuch flexibility we have encouraged authors to use flow diagrams showing thelogical connections between different chapters and to put some topics in starredsections. These cover more advanced and alternative material which is notrequired for the understanding of latter parts of each these books were conceived as a series, each of them is self-contained and can be used independently of the others. Several of them aresuitable for wider use in other sciences. Each Author's Preface gives detailsabout the level, prerequisites, etc., of that Manchester Physics Series has been very successful with total sales ofmore than a quarter of a million copies.

8 We are extremely grateful to the manystudents and colleagues, at Manchester and elsewhere, for helpful criticismsand stimulating comments. Our particular thanks go to the authors for all thework they have done, for the many new ideas they have contributed, and fordiscussing patiently, and often accepting, the suggestions of the wewouldliketo thankourpublishers,JohnWiley & Sons,Ltd,fortheirenthusiasticandcontinue dcommitment to SandifordF. PhillipsFebruary1997xivEditors'prefaceto theManchesterPhysicsSeriesAuthor's prefaceThere are many good advanced books on QUANTUM MECHANICS but there is adistinct lack of books which attempt to give a serious INTRODUCTION at a levelsuitable for undergraduates who have a tentative understanding of mathemat-ics, probability and classical book introduces the most important aspects of QUANTUM MECHANICS inthe simplest way possible, but challenging aspects which are essential for ameaningful understanding have not been evaded.

9 It is an INTRODUCTION toquantum MECHANICS the fundamental postulates of QUANTUM MECHANICS by consideringthe weird behaviour of QUANTUM relevant concepts in classical physics before corresponding conceptsare developed in QUANTUM mathematical arguments in their simplest an understanding of the power and elegance of QUANTUM mechanicsthat will make more advanced texts 1 provides a qualitative description of the remarkable propertiesof QUANTUM particles, and these properties are used as the guidelines for atheory of QUANTUM MECHANICS which is developed in Chapters 2, 3 and into this theory is gained by considering square wells and barriers inChapter 5 and the harmonic oscillator in chapter 6. Many of the concepts usedin the first six chapters are clarified and developed in chapter 7. Angularmomentum in QUANTUM MECHANICS is introduced in chapter 8, but becauseangular momentum is a demanding topic, this chapter focusses on the ideasthat are needed for an understanding of the hydrogen atom in chapter 9,identical particles in chapter 10 and many-electron atoms in chapter 10 explains why identical particles are described by entangled quantumstates and how this entanglement for electrons leads to the Pauli 7 and 10 may be omitted without significant loss of deal with concepts which are not needed elsewhere in the would liketo expressmythanksto students and colleaguesat theUniver-sityof Guisehelpfullycalculatedtheenergylevelsi n providerepresentationsof earlyversion of thefirstsixchapters , Iwould liketo thankFranzMandl forreadinganearlyversionof C.

10 PhillipsAugust 2002xviAuthor'spreface1 Planck's constant in actionClassical physics is dominated by two fundamental concepts. The first is theconcept of a particle, a discrete entity with definite position and momentumwhich moves in accordance with Newton's laws of motion. The second is theconcept of an electromagnetic wave, an extended physical entity with a pres-ence at every point in space that is provided by electric and magnetic fieldswhich change in accordance with Maxwell's laws of electromagnetism. Theclassical world picture is neat and tidy: the laws of particle motion account forthe material world around us and the laws of electromagnetic fields accountfor the light waves which illuminate this classical picture began to crumble in 1900 when Max Planck published atheory of black-body radiation; a theory of thermal radiation in equilibriumwith a perfectly absorbing body. Planck provided an explanation of the ob-served properties of black-body radiation by assuming that atoms emit andabsorb discrete quanta of radiation with energyE hn, wherenis the frequencyof the radiation andhis a fundamental constant of nature with valueh 6:626 10 34J s:This constant is now called Planck's this chapter we shall see that Planck's constant has a strange role oflinking wave-like and particle-like properties.


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