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free ebooks www.ebook777.comFrontiers in PhysicsDAVID PINES, EditorVolumes in Preparation:E. R. CaianielloCOMBINATORICS AND RENORMALIZA TlON IN QUANTUM FIELDTHEORYSTATISTICAL MECHANICSR. P. FeynmanA SET OF LECTURESPHOTON-HADRON INTERACTIONSG.E. Pake and T. L. EstleTHE PHYSICAL PRINCIPLES OF ELECTRON PARAMAGNETICRESONANCE. Second Edition, completely revised, reset, and enlargedR.P. FEYNMANCalifornia Institute of TechnologyNotes taken byR. Kikuchi and H. A. FeivesonHughes Aircraft CorporationEdited byJacob ShahamUniversity of Illinois, UrbanaTHE BENJAMIN/CUMMINGS PUBLISHING COMPANY, INC.ADVANCED BOOK PROGRAMRea ding , Massachu settsLondon · Amsterdam. Don Mills, Ontario · Sydney ·

free ebooks www.ebook777.comCONTENTSStatistical Mechanics : A Set of LecturesFirst printing, 1972Second printing, 1973Third printing, 1974Fourth printing, 1976Fifth printing, 1979Sixth printing, 1981Seventh printing, 1982International Standard Book NumbersClothboun d : 0-8053-2508-5Paperbound: 0-8053-2509-3Library of Congress Catalog Card Number: 72-1769Chapter1.1ChapterPrinted in the United States of wledgmentsxiiiIntroduction to Statistical MechanicsThe Partition FunctionDensity Matrices2.1 Introduction to Density Matrices2.2 Additional Properties of the Density Matrix .2.3 Density Matrix in Statistical Mechanics2.4 Density Matrix for a One-Dimensional Free Particle2.5 Linear Harmonic Oscillator2.6 Anharmonic Oscillator .2.7 Wigner's Function2.8 Symmetrized Density Matrix for N Particles2.9 Density Submatrix2.10 Perturbation Expansion of the Density Matrix2.11 Proof that F :S Fo (HHo o-Chapter33. 1972 by W. A. Benjamin, Inc.Philippines copyright 1972 by W. A. Benjamin, Inc.Published simultaneously in Canada.All rights reserved. No part of this publication may be reproduc d, stored i? a retrievalsystem, or transmitted, in any form or by any means, electromc, mechamcal, p oto copying, recording, or otherwise, without the prior written permission of the publisher,W. A. Benjamin, Inc. , Advanced Book Program, Reading, Massachusetts 01867,U.S.A.1Editor's Path IntegralsPath Integral Formation of the Density MatrixCalculation of Path IntegralsPath Integrals by Perturbation ExpansionVariational Principle for the Path IntegralAn Application of the Variation Theorem7278848688Classical System of N ParticlesIntroductionThe Second Virial CoefficientMayer Cluster ExpansionRadial Distribution Functionvii97100105111

viiiContentsfree ebooks www.ebook777.com4.5 Thermodynamic Functions .4.6 The Born-Green Equation for n24.7 One-Dimensional Gas4.8 One-Dimensional Gas with Potential of the Form e - 1xl4.9 Brief Discussion of 106.116.12Chapter78127130131136149Creation and Annihilation OperatorsA Simple Mathematical ProblemThe Linear Harmonic OscillatorAn Anharmonic Oscillator .Systems of Harmonic OscillatorsPhononsField QuantizationSystems of Indistinguishable Particles .The Hamiltonian and Other OperatorsGround State for a Fermion SystemHamiltonian for a Phonon-Electron SystemPhoton-Electron InteractionsFeynman Diagrams151154156157159162167176183185190192Spin Waves7.1 Spin-Spin Interactions7.2 The Pauli Spin Algebra .7.3 Spin Wave in a Lattice .7.4 Semiclassical Interpretation of Spin Wave7.5 Two Spin Waves .7.6 Two Spin Waves (Rigorous Treatment)7.7 Scattering of Two Spin Waves .7.8 Non-Orthogonality7.9 Operator Method7.10 Scattering of Spin Waves-Oscillator Theory5.1 Introducti.on .5.2 Order-Disorder in One-Dimension .5.3 Approximate Methods for Two Dimensions .5.4 The Onsager Problem5.5 Miscellaneous .111.211.311.411.511.611.711.811.911.10ixElectron Gas in a MetalIntroduction: The State Function rpSound Waves .Calculation of P(R)Correlation EnergyPlasma OscillationRandom Phase Approximation .Variational Approach. . . .Correlation Energy and Feynman Diagrams .H igher-Order Perturbation imental Results and Early TheorySetting Up the HamiltonianA Helpful Theorem .Ground State of a SuperconductorGround State of a Superconductor (continued)ExcitationsFinite Temperatures .'Real Test of Existence of Pair Stat s nd En rg; G PSuperconductor with Current . . . . .Current Versus FieldCurrent at a Finite TemperatureAnother Point of dityIntroduction: Nature of TransitionSuperfiuidity An Early ApproachIntuitive Derivation of Wave Functio s:Phonons and RotonsRotonsCritical Velocity .Irrotational Superfiuid FlowRotational of the SuperfiuidA Reasoning Leading to Vortex LinesThe A Transition in Liquid HeliumIndexPolaron Problem8.1 Introduction .8.2 Perturbation Treatment of the Polaron Problem.8.3 Formulation for the Variational Treatment8.4 The Variational Treatment .8.5 Effective MassContents221225231234241www.ebook777.comG r un d St te :312319321326330334335337339343351

free ebooks www.ebook777.comEDITOR'S FOREWORDThe problem of communicating in a coherent fashion the recent developmentsin the most exciting and active fields of physics seems particularly pressing today.The enormous growth in the number of physicists has tended to make thefamiliar channels of communication considerably less effective. It has becomeincreasingly difficult for experts in a given field to keep up with the currentliterature ; the novice can only be confused. What is needed is both a consistentaccount of a field and the presentation of a definite "point of view" concerningit. Formal monographs cannot meet such a need in a rapidly developing field,and, perhaps more important, the review article seems to have fallen intodisfavor. Indeed, it would seem that the people most actively engaged in devel oping a given field are the people least likely to write at length about it.FRONTIERS IN PHYSICS has been conceived in an effort to improve thesituation in several ways. First, to take advantage of the fact that the leadingphysicists today frequently give a series of lectures, a graduate seminar, or agraduate course in their special fields of interest. Such lectures serve to sum marize the present status of a rapidly developing field and may well constitutethe only coherent account available at the time. Often, notes on lectures exist(prepared by the lecturer himself, by graduate students, or by postdoctoralfellows) and have been distributed in mimeographed form on a limited basis.One of the principal purposes of the FRONTIERS IN PHYSICS Series is to makesuch notes available to a wider audience of physicists.It should be emphasized that lecture notes are necessarily rough andinformal, both in style and content, and those in the series will prove no excep tion. This is as it should be. The point of the series is to offer new, rapid,more informal, and it is hoped, more effective ways for physicists to teachone another. The point is lost if only elegant notes qualify.A second way to improve communication in very active fields of physicsis by the publication of collections of reprints of recent articles. Such collectionsare themselves useful to people working in the field. The value of the reprintswould, however, seem much enhanced if the collection would be accompaniedby an introduction of moderate length , which would serve to tie the collectiontogether and, necessarily, constitute a brief survey of the present status of thexi

xiiEditor's Forewordfree ebooks www.ebook777.comfield. Again, it is appropriate that such an introduction be informal, in keepingwith the active character of the field.A third possibility for the series might be called an informal monograph,to connote the fact that it represents an intermediate step between lecturenotes and formal monographs. It would offer the author an opportunity topresent his views of a field that has developed to the point at which a sum mation might prove extraordinarily fruitful, but for which a formal monographmight not be feasible or desirable.Fourth, there are the contemporary classics-papers or lectures whichconstitute a particularly valuable approach to the teaching and learning ofphysics today. Here one thinks of fields that lie at the heart of much of present day research, but whose essentials are by now well understood, such as quantumelectrodynamics or magnetic resonance. In such fields some of the best pedago gical material is not readily available, either because it consists of papers longout of print or lectures that have never been published. . . -. - .------ACKNOWLEDGMENTSThis volume is based on a series of lectures sponsored by Hughes ResearchLaboratories in 1961. The notes for the majority of lectures were taken byR. Kikuchi and H. A. Feiveson.Others who took notes for one or more of the lectures were F. L. Vernon,Jr. , W. R. Graham, Jr. , R. W. Hellwarth, D. P. Devor, J. R. Christman,R. N. Byrne, and J. L. Emmett.The notes were edited by Dr. Jacob Shaham who also prepared the Index.The above words, written i n August, 1961, seem equally applicable today(which may tell us something about developments in communication in physicsduring the past decade). Richard Feynman contributed two lecture note vol umes ("Quantum Electrodynamics" and "The Theory of Fundamental Pro cesses") to the first group of books published in this series, and, with the publica tion of the present volume and the forthcoming publication of "Photon-HadronInteractions," it gives me special pleasure to welcome him back as a majorcontributor to FRONTIERS IN PHYSICS."Statistical Mechanics : A Set of Lectures" will be of interest to everyoneconcerned with teaching and learning statistical mechanics. In addition toproviding an elegant introduction to the basic concepts of statistical physics,the notes contain a description of some of the many original and profoundcontributions, (ranging from polaron theory to the theory of liquid helium)which Professor Feynman has made in this field.Urbana, JIIinoisJune, 1972DAVID

free ebooks www.ebook777.comCHAPTER 1INTRODUCTION TO STATISTICAL MECHANICS1.1 THE PARTITION FUNCTIONThe key principle of statistical mechanics is as follows :If a system in equilibrium can be in one of N states, then the probability ofthe system having energy E" is ( l /Q) e E /kT, where-nk Boltzmann's constant, and T temperature. Q is called the partitionfunction.If we take I i) as a state with energy Ei and A as a quantum-mechanicaloperator for a physical observable, then the expected value of the observable is(A) ! L (iIA l i)e-EdkT Ii )QThis fundamental law is the summit of statistical mechanics, and the entiresubject is either the slide-down from this summit, as the principle is applied tovarious cases, or the climb-up to where the fundamental law is derived and theconcepts of thermal equilibrium and temperature T clarified. We will begin byembarking on the climb.If a system is very weakly coupled to a heat bath at a given "temperature,"if the coupling is indefinite or not known precisely, if the coupling has been onfor a long time, and if all the "fast" things have happened and all the "slow"things not, the system is said to be in thermal equilibrium.For instance, an enclosed gas placed in a heat bath will eventually erode itsenclosure; but this erosion is a comparatively slow process, and sometime beforethe enclosure is appreciably eroded, the gas will be in thermal equilibrium.Consider two different states of the system that have the same energy,Er E. The probabilities of the system being in states rand s are then equal.For if the system is in state r, any extremely small perturbation will cause thesystem to go into a different state of essentially the same energy, such as s. The1

2free ebooks www.ebook777.com1.1Introduction to statistical mechanicssH------Fig. 1.1PeEr)PeEr') ,same is true if the system is in state s. Since the system remains in contact withthe heat bath for a long time, one would expect states of equal energy to beequally likely. Also, states of different energies would be expected to havedifferent probabilities., Because two states of the same energy are equally probable, the probabilityof a state having energy. E is a function only of the energy; P P(E).Now consider a system, S, in equilibrium with a large heat bath, H (seeFig. 1 .1). Since experience shows that the behavior of a system in equilibrium isindependent of the nature of the heat bath, the bath may be assumed extremelylarge and its total energy E very great. Also, the possible energy levels of theheat bath may be assumed quasi-continuous.Let the energy levels of the heat bath be denoted by Hi' These levels aredistributed quasi-continuously. Let the energy levels of S be denoted by Ej Then Hi » Ej for all i, j. The bath plus the system can be thought of as a newsystem, T, which is also in thermal equilibrium.T has a definite energy, but as that energy is not fixed exactly (the bath is incontact with the outside world), we can assume that the energy may be anywherein the range Eo . If is sufficiently small, we can assume that the states ofthe heat bath are equally likely in the range Hi . Let I1(Hr) be the number ofstates per unit energy range in the heat bath H around energy H r.The probability, P eEr), that S is in a state with energy Er is proportional tothe number of ways S can have that energy. In other words, it is proportionalthe number of states of H that allow T to have energy into I1(Eo - Er)2 the range Eo . Then I1(Eo - Er)I1(Eo - Er,) e1nq(Eo-Er)-lnq(Eo-Er').e-P(Er-Er,)., PeEr') oc e-PEr'.Normalization requires that P (Ei) (l/Q) e-PE, where QErEnergy levels in a system S and a heat bath H.PeEr)PeEr')3Remember that Er «Eo. If it is true that (d/dE) In I1(E) peE) is almostconstant for E in the range under consideration, * then we can sayso------The partition function Li e-pE,.The fundamental law has just been shown to be quite plausible. But forthose who doubt the constancy of peE), let us consider some examples;First, assume that the heat bath consists of N independent harmonic oscil lators. The energy of the bath isNF L n/lwi'i 1where we assume that ni is very large, and we neglect the zero-point energy.2, we have theHow many states are there with energy less than F? If Nsituation. shown in Fig. 1.2.For N 2, the number of states i&proportional to the area of the triangle.Clearly the number of states with energy less than F for large F is proportionalto FN ; so the number of states per unit energy range is I1(F) oc (d/dF)FN oc FN-1 In I1(F)d In I1(F)dF constant (NN- 1F--1) In F,- 1N - sinceE r « E o.* The basic assumption made here is, that the system governing the probabilities hasa quasi-continuous spectrum in the region considered and for which there is no partic is only defined up to an additive constant e, we should haveSInce/(e1)/(e2) where/(e) is the probability. Definingwe obtainfee) gee - e2),g(e)g(e1 - e2)which is (uniquely) solved bythat is, e)/(e2 e) , g(O)g(el - e2 e),gee) g(O)e-PE(P constant),

4free ebooks www.ebook777.comIntroduction to statistical mechanicsThe partition function5\\\\.\,.,.··. .·F , . . \- .\ .,,,,. . .constant.-- ---- -- nlFig. 1.2States with energy less than F for two independent harmonic oscillators.For N large,p 1energy per oscillator peEr)peEr') W'Fe-Er/We-Er'/Was-t -----N-t 2m2m * 2nfz(integer)LProblem: Why is W independent of N? P;N P;N P;N2m2nxnfz'Le -PA,L e-PAj -------,--,--Now place SA and SB in loose contact with each other and consider the combinedsystem S SA SB' with combined energy Tk Ai Bj TPT( Tk) Assuming periodic boundary conditions,Px2222ny, nz, nX2 . nZN]PA(A1.)00.*We then get the same value for p as by our previous method.As a second example, consider the case of a heat bath consisting of N particlesin a box.F (JF)3N. It follows that p is roughly constant, and equals 1/(tW), where W energy per particle.Consider two independent systems SA and SB with energy levels Ai and Bj The probability of system SA having energy Ai is(Eo - Er)N-11](Eo - Er)(Eo - Er,)N-11](Eo - Er,)P; P;2 P;2 . . 22nfz [-- nx2 ,2Lmcircle with radius JF(Lm/nh). For N particles, we must use a hypersphere in 3Ndimensions, and the number of states with energy less than F is proportional toand plug in N directly. We getP;, P;, P;, If there were just two 1]'S, we would calculate p with the help of Fig. 1 .3.The number of states with energy less than F is roughly equal to the area of the1](Eo - Er)1](Eo - Er,)(1 - ErjNW)N-1( 1 - Er,jNW)N-1Number of states with energy less than F is roughly equal to the area of thecircle shown.where L is the length of the sides of the box. Then1where W energy per oscillator.Alternatively, we can go back to the equation--Fig. 1.3The probability of systeme-PT(A, Bj): ---, A -,---'" .i e -PT , '" .j e-PTBje-PTA,e-PTBjA'" .j e-PTBj' .i e -PT , '"PT(A.1 BJ.) -ST being such that system SA has energy Ai is

6free ebooks www.ebook777.com1.1Introduction to statistical mechanicsSimilarly,The partition function7where(1.4)We see that if two systems are placed in loose contact with each other, i nequilibrium they have the same p. Temperature has a similar property, and i nfact, by the way temperature i s conventionally defined, p l/kT where k isBoltzmann's constant. *From the basic principle of statistical mechanics, onceis known, all thermodynamic properties can be found. We define F, the Helm holtz free energy, so thatnF -kT In QS -kT In(1.1)( e-EnlkT) ,Lentropy, is defined as -kn(1.3)usince S - - C2PIP2CI C2,Pwhich is, indeed, the form of nee) for microscopic bodies (see pp. 3, 5).Also, it can easily be seen (from 1.7 below) that1 " (en2Q2 t,;.em)2e-P En Em) cv ( )vauaTaverage energy 1kT2"En -T2- Pnn ( )aavo1aFaT-T aTTaF F - TaFoTaFaT: aFaT a(I T)T(1.5)(1.6)av, SQ '; e-EnlkT( )G )v l En e-EnlkTaE nF TS. P(1.7) ,(1.8)a2FO T2(1.9) I-L ilPopli)e-Ei/kT.Q iOur alternative definition is equivalent to the first one ifoHov0, il Ii)so that U is a decreasing function of p. Suppose we mix two systems having differentP's, PI P2' After mixing there will be a common p, according toUI(PI) U2(P2) UI(P) InPOPnee) Ae"pressurenTo get a clearer idea of the nature of pressure, consider a possible alternativedefinition. The Hamiltonian operator for the system is dependent on the volume.Set H Hamiltonian H(V). We can take as a pressure operator -oH/aV.which is the well-known (experimental) formula.Equation ( *) has the unique solutiondU dP :;-aF/aT, UC/P and we recognize in C the usual (total) heat capacity, so thatCIk aTU[b e-EnlkT { ; - Q}]L( )T -k aQ But* Furthermore, suppose our spectrum has a density of states nee) which is relativelyconstant when e - pe, that isn(pe) n(e)q(p)Then U (P) (see 1.7)- (: \P(1.2)Pn In Pm(1.2) it can be seen thatFrom Eq.U2(P),whence, since the U's decrease, PI P P2, and energy flows from the low-P body tothe high-P body. This also fits into our intuitive perception of temperature.ButoEiavvlimv'. i.(V') - Ei(V)V' - V oEi avv i' IH'li') - iIHli)limv'.V' - Vwhere It) is the eigenvector of H' corresponding to I i), theith eigenvector of H.H' H(V')

free ebooks www.ebook777.com1.2Introduction to statistical mechanics8H' H (H' - H); so we can apply first-order perturbation theory tosay that i'l H' Ii' ) il H Ii) il H' - H Ii) for H ' - H O. ThenoEiIi(iIH' - Hli)' oH '-- mIl ) .- (l loVoVy' . yV' - VOEioexa. - ( - -) e-Ei(a.) /kT,1 1-1Q iand we can writeoEioex (iloHoexIi). - -(V, S).oUoVThat this definition is equivalent to the other two may be verified without diffi culty from the equation U F TS.The equation S - (oF/oT)v holds only at equilibrium, where F is defined.Away from equilibrium, S always increases with time. To see this, note thattime-dependent perturbation theory givesdPmdt Ln(IVmn l2P n - IV mnI2Pm) ,where IVn ml2 is the probability per unit time of transition from state n to state m,and IVn ml2 IVmn l2. ThendSdt-ksinceL(LidPidPi In PidtdtdPii dtThendSdtmn) Ldt iPi dPdtF -kT In Q-kT Innt-kT In (L e-PEn;)"iL e-PEn n -kTL e-P En; [ L e-PEn,] -kT In'r.,"1,"2, .In"i IFi The free energy of the whole system is the sum of the free energies of its non interacting parts.1.2 LINEAR HARMONIC OSCILLATORSA third definition of pressure isP9But each term i n the sum is negative, for the sign of Pn - Pm is opposite to thesign of (In Pm - In Pn ) . So dS/dt O.Before we start to make calculations, note that if we have a system that is acombination of several independent subsystems, with Etotal E L E ;i nnLi energies of the subsystems,Our two definitions are equivalent.Because H may be a function of the shape of the system, as well as its volume,our definition of pressure might depend on how the volume is changed. Ingeneral, for any parameter ex, there is a force that can be computed by(Force)Linear harmonic oscillators-k dPiIn Pi i dtConsider a system of harmonic oscillators i n thermal equilibrium. The partitionfunction Q, free energy F, and average energy of the system of oscillators can befound as follows: The oscillators do not interact with each other, but only withthe heat bath. Since each oscillator is independent, one can find Fi of the ithoscillator and thenF(M oscillators).QiQi LnL1 Fii e-E /kT(LlO)from quantum mechanics (n L e-A;w;(n 1/2) /kT 0, 1,2, . . . )(Ll 1)n(Ll2)LO.M F i - kT In QiUi liw·--f kT In (1- e-Aw;/kT)(Ll3)average energy of a single oscillator in thermal equilibrium EniQi n L e-E /kTIiw.e' -- Aw;/;k/TkT- e wliw·'21 - 0 Fi0 (1/T) T (Ll4)liw·liwi-' ---C- -2e w;/kT- 1(1.15 )

10free ebooks www.ebook777.com1.3Introduction to statistical mechanicsF i Fi Li [IiW2 i kT - e-firodkT)]'U i Ui i [IiW.2 ' efiro,)kTIiW.]- n;,Ui t)IiWi', efirodkT U)Ui :::::; k TU.Iiw;/2 In ( 1 '-' '-'-(1. 1 6)Because of the periodic boundary condition,aAx nx- 1(n,n· 11and similarly.Returning to the example on p. 3 we see that for very high temperaturewe indeed have(andW:::::;11. an, according toIt is customary to define an averageThusBlackbody radiation'?!s.ckz2n - ny, 2n nz a dkxb dkyc dkzdnx,dny,2n2n2n dnzd3k (abc) dkx dky dkzd3n dnx dny dnz abc (2n)3(2n)3·k,k dkkd3k2 d3n 2-.(2n)3(abc)W kc c[-IiW(k)]) 2 d3kkbT(fffkbT (2n)3 hw/2f[-IiW(k)]) 4nk2 dk .!. 2 kb T (kbT (2n?U 2 f hW(k) [ -liw(k)/kbTJ 4nk2 dk[-liw(k)/kbTJ (2n)3 . Iiw/kbT Iikc/kbT.- 1/13, which is independent of»Note that the contribution to F of the ith oscillator is negligible ifterm). A t low temperatures the high-frequency modes are(except for the"frozen out" and do not contribute to the specific heat.IiWi kT For each there are two possible polarizations. Thus, the number of modesisper unit volume with wave number between and -1.3BLACKBODY RADIATIONI n dealing with blackbody radiation, our point of view will be as follows: In acavity (blackbody), there are a great number of modes of oscillation. Thenumber of modes per unit volume per frequency bandwidth is given by classicalconsiderations. Each mode, however, behaves as an independent quantumterm is neglected. That is, En harmonic oscillator, except that thebecause it leads to infinite energy when thereWe want to get rid of theisare an infinite number of modes. A Hamiltonian that eliminates thehw/2 Iiw/2i 1(p2i Wi2qi2) - IiW2'iH "2hw/2nliw.Nowwhere is the velocity of light. Also, the number of modes is solarge that the sum over the modes can be replaced by an integral.VInLet xIn V1 - exp.( 1 . 1 7)term has been omitted.)(Remember that theFrom symmetry,VWith the above assumptions, we can obtain an expression for the energyper unit volume per unit frequency.First we will find the number of modes per unit volume per frequency (orwave number).Assume a gigantic box of dimensions a, b, and c. The demand is made thatthe waves be periodic at the walls of the box. 1 - expexp 1 - exp ( 1 . 1 8)Then! number of waves/cm in x-directionAxAx number of waves in box (x-direction).( 1 . 1 9)

12free ebooks www.ebook777.com1 .4Introduction to statistical mechanicsThe preceding results can be summarized and put in a more familiar form byreplacing nw with hv.The number of modes/unit volume between k and k dk is2.4nk2 dk(2n)3k2 dkn28nv2 dvc3since k 2nv/c. The average energy of an oscillator offrequency v is hv/ ehY/kTThus, the energy per unit volume between v and v dv isdUU Y dv V 8nv2 hv dv.c3 ehY/kT 1l.(1.20)Uy dv uT4is the Stefan Boltzmann Law.For T very large (kT » hv),U dv Y8nv2hv dvc3 1 (hv/kT)1 8nv2kT dv.c3(1.21)This is the Rayleigh-Jeans Law.Let F* F/ V and U*U/ V. cv au* 4uT3aTis the heat capacity of a gas of photons in equilibrium with the container.If an oscillator (or mode) is excited to the Nth level, that is, if E n W iN,one says that there are N photons with energy nwi . Since photons are defined asthe degree of excitation of a mode, one cannot consider a permutation of photonsas a new state. That is, photons are indistinguishable. This point is the basis ofquantum statistics and will be dealt with in a later section.1.4 VIBRATIONS INA132. Find the normal modes of the system. There are as many normal modes asthere are degrees of freedom; namely, 3(AN), where (AN) is the number ofatoms in the crystal. The modes behave as independent quantum oscillators.3. Given the modes, calculate F, the free energy.4. From F, calculate Cv and any other thermodynamic quantities of interest.Once F is found, we can quickly calculate U, the total energy of the system.In the following derivation, we will calculate Cv directly from U withoutalways writing down F first.Method of Labeling-This is the Planck Radiation Law, and is the same as Eq. ( 1 .18).fVibrations in a solidConsider a crystal with A atoms per unit cell. For convenience, assume that theunit cell is a rectangular solid of dimensions a, b, c, along three mutually per pendicular axes, x, y, z. Let the origin be at the "center" of a cell. This cell canbe denoted by the triplet (0,0,0). The cell to its right along the x-axis is denotedby (1, 0,0),and so on. Thus, any cell can be denoted by a vector N nxa nyb nzc where a ai,b bj,c ck. If there are A atoms/cell, 3A additionalcoordinates must be given to locate each atom. Let IX denote one of these 3Acoordinates.Call the displacement from equilibrium of the coordinate in the Nth cellZrz,N' (We either consider m mass equal to 1 or absorb it in Z)Zrz,N M is the displacement of an atom in a cell close to N; if, for example,there are two atoms Al' A per cell, the displacement of A 1 in the x direction is2denoted by Zl,N, that of A (in same cell) by Z4,N' and that of Al in an adjacent2cell by Zl,N 1' 1 (1,0,0) or (0,1,0), or (0,0,1).Normal ModesT (1/2)V YeO) L",NLparticlesand directions( )aZ",N z oZ;,N (1/2) L Z;,N",NZ",N(1 .23)SOLIDWe want to find the specific heat of a solid. However, en route to the specificheats, we will derive important results that will prove useful in many other cases;chief among these results is the calculation of the normal modes of a crystal.The program will be as follows:1. Consider the solid to be a crystal lattice of atoms, each atom behaving as anharmonic oscillator. These oscillators are, of course, coupled.(1.22)Assume that the electrons in the crystal always have time to adjust them selves to the configuration with lowest energy, even when the crystal is vibrating.In this configuration, there is no net force on the nuclei when the Z" N arezero, so( )OZ",Nz o 0 .

141.4free ebooks www.ebook777.comIntroduction to statistical mechanicsThe additive constant, YeO), will not affect our answers, so we might as welldrop it. Leta number that depends on the relative positions of the cells of the two atoms, andp .not on their absolute positions. Note thatNeglecting higher orders, take(1.24)For low temperatures it is not too unreasonable to neglect higher orders,because the separation between atoms is of the order of 1 A and at roomtemperature the vibrations have amplitude of the order of 0. 1 A. But we shouldnot be too surprised if experiment shows our idealization to be false. In matterssuch as the one under consideration, the general approach is to make idealiza tions and then try to find corrections to our assumptions that will give betterresults.In order to motivate the procedure that we will use for finding the vibrationsof a solid, let us consider the classical problem of vibrations of coupled oscil lators. Let the Hamiltonian bep,2 - i Hi 2MiLet the ath mode have frequencyWa15so that afor the motion of the ath mode, with a ) independent of time. ThenC ,(OZa'NO; 'N M)Z OCa C M Vibrations in a solid 2IC'. . q.q., , ij'J ' l'2wa,a) j aJThe classical problem of vibrations of coupled oscillators has just been re duced to the problem of finding the eigenvalues and eigenvectors of the real'we must solve the equationsymmetric matrix, IICijll. In order to get theThen thethata a) O.( 1 . 26)(the eigenvectors) can be found. It is possible to choose the a a) so'L a a)a P) ibap.The general solution for qi isqi L Caq a ,aQa Cae-iw t,, j a Qa a ba).Qa QJ"H La Ha,QjHa tP; tw;Q;.Qa Cae-iw t.where the Ca are arbitrary constants. If we takeFrom this it follows thatLa a a)Qa.I.P;waw2bijl-det IICij a(j)q. ,where the q; are the coordinates of the amount of displacement from equilibrium, MJl; is the momentum, and C;j Cji are constants. To eliminate theconstants Mi, let C.a ( )I.J l ) ·a( a ( )a ! )'-! I. l «,IMaking the change of variables,we get qiLi aF)qi' w

This volume is based on a series of lectures sponsored by Hughes Research Laboratories in 1961. The notes for the majority of lectures were taken by R. Kikuchi and H. A. Feiveson. Others who took notes fo r one or more of the lectures were F. L. Vernon, Jr., W. R. Gr