UNIT-I 1.0 INTRODUCTION

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1 BLOCK-I Ths Block-I of Mechancs s devoted to the man topcs of Classcal Mechancs. Tll the tme of standardzaton of Basc Unts of measurement--m L T Kelvn Ampere and Mole -- the laws about the behavour of natural actvtes of bodes were vague and dspute orented (e.g. sun goes round the earth or the earth round the sun). Classcal Mechancs brought these actvtes under reproducble laws and expermental verfcaton. The laws ganed so frm a footng that smple on earth as well as complex n the unverse they remaned unchanged. In the core of these laws the justfcaton came through mathematcs whch helped abundantly. Some nferences can be quoted as: Entropy of the unverse always ncreases Unversal Gravtaton constant Gas constant Plank s Analogy: constant do not change Kepler s Ellptcal moton and Hamlton Prncple Hamlton s prncple functon: W(r t) = Ldt and Hamlton s characterstc functon: S(r) = p dr lead to the Plank s theory e. P k and E ω and hence p k k and E = ħw = hv. Check here k becomes k n Schrödnger s Equaton where nstead of fndng soluton to W solutons to are determned mantan valdty. Physcal concepts such as space tme force energy smultanety varatonal prncple etc. pervaded. It would not be wrong here to state that the classcal mechancs gave brth to rest of all felds of physcs UNIT-I. INTRODUCTION After understandng generalzed co-ordnates and Lagrangan notatons To study the D Alembert s Prncple velocty dependence of potental Raylegh dsspaton functon Two-body problem Kepler s problem scatterng n a central force feld all under Lagrangan dynamc laws.

2 . OBJECTIVIES: The objectve of ths unt s to look nto the generalzed moton n N-degrees of freedom and by the use of dfferental calculus & partal dervatves know the ntrcaces of actvtes that generate MOTION.. CLASSICAL MECHANICS It deals wth the representaton of the moton of pont-lke objects embedded n three-dmensonal Eucldean space. Broadly speakng Galleo s laws of fallng objects Kepler s laws of planetary moton etc. are termed as Knematcs and Newton laws of moton Gravtaton etc. are called dynamcs..3 LAGRANGIAN MECHANICS In general moton of an object s restrcted by varous factors. These are Called constrants and the moton becomes constraned moton. Smple example s the moton of gas molecules n a contaner are constraned by the walls of the contaner. Further the actvtes of such motons are descrbable mathematcally by some relatons whch are called constrant equatons. The general form of the unlateral constrants s represented as f r... r r... r t o (.) n n Where r and r are poston and velocty of the th partcle of the system ( =..n) at tme t. Smlar forms can also be wrtten for other type of moton n cyclc polar and other coordnates. Generally constrants are of Holonomc and Non-Holonomc:- The condtons of constrants can smply be represented as: - f r r... t) and for rgd body r r c.4 GENERALISED CO-ORDINATES j j n Holonomc class. ( Intal problem of holonomc constrants s resolved by the consderaton of generalzed coordnates. Let there be a system of N-partcles whch s free

3 from constrants. It would have 3N ndependent co-ordnates; called degree of freedom. Now f k ndependent constrants are present then n ndependent varables would be n= 3N-k. Thus the elmnaton of dependent co-ordnates from ndependent ones can easy be expressed as q q... q3n k t nstead of the old r r... r ).The process s called transformaton Eq. and expressed as ( N r r ( q q... q3nk t).. (.) r N r N ( 3 q q... q Nk t) Here t s to be noted that f the tme (t) dependence of the system s explct the Eqs. (.) defne the system as Rheonomc otherwse Scleronomc. These transformatons are Eqs. From the set of (rl) varables to the (ql) set or vce-versa from set (ql) to set (rl) are the man features of generalzed coordnate study. In short Eqs (.) of constrants can be nverted to obtan any ql as a functon of the (rl) varable and tme. On the bass of these velocty acceleraton momentum force potental energes etc. can be formulated for the system/objects. Let us take the example of a partcle revolvng around a fxed attractng centre on the perphery of a rollng dsk. The Cartesan coordnates (x y z) would need equvalent terms of sphercal polar coordnates.e. x r sn cos x rsn cos z r cos (3.) Then the rollng can be depcted as follows to under stand the angle θ and 3

4 z y γ α o ) x Fg:. Dsk rollng wth depcton of v n (x y z) coordnates (x y) are the coordnates of the centre of the dsk an angle of rotaton (about the axs of the dsk and (s the angle between the axs of the dsk and x-axs. Resolvng v one can get x vsn and vcos Combnng these condtons of moton two dfferental equatons of constrant are: and dx asn d dy acos d...(4.) These equatons cannot be ntegrated wthout nfact solvng the problem e. One cannot fnd the ntegratng factor f (x y θ ) that wll turn ether of the equaton nto perfect dfferental..5 D ALEMBERT S PRINCIPLE AND LAGRANGE S EQUATION One of the orgnal processes to solve the Newton s equaton of moton needed evaluaton of forces of constrants. Ths used to make the procedure lengthy and complcated. So a method was thought over to bypass these forces of constrants by ntroducng the concept of vrtual work D Alembert s prncple s 4

5 one that works on the condton for vanshng vrtual work done by constrant forces. Let us start wth a vrtual dsplacement of a system as a result of any arbtrary nfntesmal change of the co-ordnate r. It s consstent wth the force and constrants mposed on the system at a gven tme t for an nterval dt. If the system remans n equlbrum the total force on each of ts partcles vanshes.e. F =. The dot product F. δr whch s vrtual work should also be zero. The sum of these vanshng products can therefore be wrtten as F r...(5.) Further f appled force s denoted by F (a) and the force of constrant as f F F ( a) f...(6.) the equaton (5.) can be wrtten as ( a) r f r F...(7.) When a partcle s constraned to move on a surface (ref. fg..) f s perpendcular to the surface whle the vrtual dsplacement must be tangent to t and hence the vrtual work of appled forces vanshes. Therefore ( a) F r...(8.) Eq.(8.) represents the prncple of vrtual work. It can be noted that the coeffcents of r can no longer be set to zero.e. n general F (a). Snce δr are not completely ndependent but connected by the constrants. In order to equate the coeffcents to zero one has to transform the prncple of vrtual work nto a form nvolvng the vrtual dsplacements of q whch are ndependent. Eq. (8.) satsfes ths condton as t does not contan f but deals only wth statcs. For ths purpose D Alembert proposed a method. The equaton of moton 5

6 F P...(9.) (P s momentum of the partcle) Whch states that the partcles n the system wll reman n equlbrum under a force equal to the actual force plus a reversed effectve force ( P ) Further the Eq. (5.) s becomes ( F P ) r...(.) Makng the same justfcaton wth appled forces and forces of constrants one can express ( a) {( P ) r } f r F...(.) Snce for the systems under consderaton the vrtual work of the forces of constrants vansh the second term of Eq. (.) becomes zero- And thus ( F ( a) P ) r... (.) Ths s called D Alembert s prncple..6 VELOCITY DEPENDENT POTENTIALS AND RAYLEIGH S DISSIPATION FUNCTION Whle understandng generalzed co-ordnates the transformaton of rn to qn s enumerated n Eq. (.). The calculus of partal dfferentaton helps us to wrte v dr dt k r q k q dt k dr dt...(3.) Smlarly arbtrary vrtual dsplacement r can be wrtten n terms of actual vrtual dsplacement q wthout varaton of tme t as 6

7 7...(4.) j j q q r r It generates vrtual work F as follows j j j j j j q Q q q r F r F )...(5. Where Qj stands for j j q F r and termed as generalzed force. If we consder the momentum of th partcle as p p leads to force and also j j q v q r dt d The total Knetc energy T of the system can be represented wth Qj term as follows:...(6.) j j j j j q Q q T q T dt d Equaton (6.) also represents the D Alembert s Prncple n terms of generalzed fore Qj and partal dfferental of Knetc energy q j T. Now n a system of Cartesan co-ordnates the partal dervatve of T w r. t. qj vanshes. And f the constrants are holonomc any vrtual dsplacement qj need to hold for the separate coeffcents (Eq. 6.) to vansh. That s...(7.) j j j Q q T q T dt d

8 The equaton (7.) s another form of the Lagrange s equaton and s used to relate the scalar potental V wth force F as F V Through ths relaton the generalzed force Qj becomes Q j r F q r V q j...(8.) Whch s exactly the same for the partal dervatve of a functon V V( r r... rn t) w. r. t. q j. or one can wrte Q j...(9.) q Replacng Qj by ths n equaton (7.) one gets j d dt T q ( T V ) j q j...(.) A new functon named as Lagrangan L would replace (T-V) whch shall be L used as Lagrangan Equatons wherever needed. That s d dt L L q j q j...(.) Now we ntroduce another term generalzed potental U q j q j wth the help of (.) even f there s no potental functon V n usual sense: It can be represented as Eq. (7.)} Q j U q j d dt U q j...(.). e. L T U...(3.) Functon U has another name and s called velocty dependent potental. Recently Morgenstern and Szabo have used the name to ths functon as 8

9 Scherng potental for the specfc velocty-dependent potental that gves the Corols force n a rotatng coordnate system. It should be noted that f all the forces actng on the system under consderaton are dervable from a potental then Lagrange s equaton can always be wrtten n the form d dt L L q j q j Q j...(4.) Where L contans the potental of all the conservatve forces and Qj represents the force not arsng from a potental. Such a stuaton often occurs when there s a frctonal force arsng from any alled actvtes. In such cases the frctonal force becomes proportonal to the velocty of the partcle n such a way that ts x- component has the form F k fx x v x...(5.) Such type of frctonal forces can be derved n terms of a new functon denoted as J as J k x vx k y vy k zvz...(6.) Where the summaton over the partcle of the system spread over x y & z axs and correspondng planes ths J functon s named as Raylegh Dsspaton functon and symbolcally represented as F J v f...(7.) Thus the Lagrange Eq. takes one more form as d dt dl L q j q j J q j 9

10 .7 VARIATIONAL PRINCIPLE So for we used partal dfferental terms and used the sgn of summaton to determnng the overall actvty of a system. The vrtual acton played a key role n that. The nventon of ntegral and varatonal calculus helped n solvng the problems more precsely nstead of solvng the same problem by varous methods e.g. defnng a crcle etc. The ntegral Hamlton Prncple descrbes the moton of those mechancal systems for whch all forces other than the constrant forces are dervable from a generalzed scalar potental that may be a functon of coordnates veloctes and tme. In short here t s treated that all forces are generated from sngle functon. For monogenc systems Hamlton s Prncple can be stated as the moton of the system from tme t to tme t s such that the lne Integral I Ldt...(8.) Where L= (T-V) has a fxed value for the correct path of moton. I s referred as the acton or acton ntegral that s tme bound. Now f I s to be consdered n the relm of Varatonal Prncple we can wrte the Hamlton s Prncple as I L q q... q q q... q t dt...(9.) n Equaton (9.) s true only when system constrants are holonomc and t sets both a necessary and suffcent condton for Lagrange s equatons (.) We can derve Lagrange s Equatons from the above Hamlton s Prncple now as the calculus of varaton s known. Let f s a functon of many ndependent varables y and ther dervatves y. Here y and n y are consdered as the functons of parametrc varable x. Before proceedng further let us be acquanted wth a new functon J a lne ntegral Say we have a lnear functon f ( y y x) defned on the path y = y (x) between two values of x and x where y s a dervatve of y w. r. t. x (not t).e. dy y then J s gven by dx

11 ...(3.) x x dx x y y f J It has a fxed value relatve to paths dfferng nfntesmally from the correct functon y(x) From equaton (3.) we have varaton of J as )...(3. )... ( ) ( )... ( ) ( dx x x y x y x y x y f J Further consderng J as functon of a parameter whch denotes a possble set of curves y (x) we can wrte...(3.) ) ( ) ( ) ( ) ( ) ( ) ( x o x y x y x o x y x y Where ) ( ) ( o x y o x y.etc. are the solutons of extremum and are ndependent functons of x. The varaton of J s calculated by fndng.e. d J dx d y y f d y y f d J It gves...(33.) dx y y f x y f J Where the varaton dx y y Now snce y varables are ndependent the varaton y are also ndependent then the condton that J s zero (for extremum) requres that coeffcents of y should separately vansh.e.

12 f y d dx f 3... n y...(34.) Equaton (34.) s known as Euler s-lagrange s Dfferental Equatons. Ther solutons represent curves for whch the varaton of ntegral as gven n the equaton (33.) vanshes. When we consder the Hamlton s prncple for two ponts and on a surface the lne ntegral for varaton can be wrtten (see equatons and 33.) as I L q q t dt...(35.) Where x s transformed to t y to q and f y y x to L q q t.. By such a transformaton the Euler-Lagrange equaton (equaton 34.) becomes d dt L L q q... n...(36.) Whch are well known Lagrange s equaton derved from Hamlton s prncple for monogenc systems wth holonomc constrants.8 TWO-BODY CENTRAL FORCE PROBLEM Let us consder a mutual central force between two pont mosses m and m n terms of Lagrangan formulaton of a monogenc system comprsed by these mosses. The only force here would be due to nteracton potental U. We can show the two bodes wth co-ordnates as. R m r m

13 Fg coordnates of Two-Body Problem Where t s assumed that U s any functon of the vector between two partcles (r - r) or ther relatve veloctes r r or any hgher dervatve of these vectors Such a system would have sx degrees of freedom.e. sx ndependent generalzed coordnates. Out of these three would be of the radus vector to the centre of mass R and three of dfference vector r r r. The Lagrangan L can then be wrtten n terms of total Knetc energy T and potental U as Here L T R r Urr......(37.) T= (Sum of Knetc Energy + T of the moton of centre of mass) (The knetc energy of moton about the centre of mass) = m m R T If r and r are rad vectors of the two m m masses relatve to the centre of mass T m r m r Snce m m r r & r m m m m r m m T m m r...(38.) 3

14 Puttng these n equaton (37.) we get total Lagrangan as L m m m m r m m R r Ur...(39.) It can be seen n ths relaton that the three co-ordnates R are cyclc. The equatons of moton for r wll not have terms R or R. We can drop the frst term here and therefore the rest of the Lagrangan s exactly what would be expected f we had a fxed centre of force wth sngle partcle at a dstance r and havng a mass mm m m or m m...(4.) μ s called reduced mass Thus the central force moton of TWO BODIES about ther centre of mass can always be reduced to all equvalent one body problem..9 THE KEPLER S PROBLEM: INVERSE SQUARE LAW OF FORCE Kepler s problem s a central force problem wth the law of gravtatonal force gven by Newton. In short the Kepler s problem s the nverse of Newton s problem. We now restrct ourselves to conservatve central forces where the potental s V(r) a functon of r only so that the force s always along r. By the results of the precedng secton we need consder only the problem of a sngle partcle of mass m movng about a fxed center of force whch wll be taken as the orgn of the co-ordnate system. Snce potental energy nvolve only the radal dstance the problem has sphercal symmetry.e. any rotaton about any fxed axs can have no effect on the soluton. Hence an angle co-ordnates representng rotaton about a fxed axs must be cyclc. These symmetry propertes result n a consderable smplfcaton n the problem. Snce the system s sphercally symmetrc the total angular momentum vector 4

15 L r p The nverse square law s the most mportant of all the central force laws and t deserves detaled treatment. For ths case the force and potental can be wrtten as f k r V k. r...(4.) There are several ways to ntegrate the equaton for the orbt the smplest beng to substtute (4.) n the dfferental equaton for the orbt: mf d u u mk u d l u l...(4.) mk Changng the varable to y u the dfferental equaton becomes l d y y d whch has the mmedate soluton y Bcos where B and θ beng the two constants of ntegraton. In terms of the soluton s mk r l e cos ( )...(43.) It s nstructve to obtan the orbt equaton also from the formal soluton of u du u me ma u l l u...(44.) 5

16 Whle ths procedure s longer than the smple ntegraton of the dfferental equaton (4.) t has the advantage that the sgnfcant constant of ntegraton e s a automatcally evaluated n terms of the energy E and the angular momentum of the system. We can wrte ths equ n ths form du...(45.) me ma u l l where the ntegral s now taken as ndefnte. The quantty θ appearng n (45.) s a constant of ntegraton determned by the ntal condtons and wll not necessarly be the same as the ntal angle θo at tme t =. The ndefnte ntegral s of the standard form dx x x x arc cos q...(46.) Where q = β - 4αγ To apply ths to (45.) we must set q 4 me mk l l and the dscrmnant q s therefore q mk El l mk 6

17 Wth these substtutons Eq. (45.) becomes arc cos l u mk El mk Fnally by solvng for u the equaton of the orbt s found to be r r mk cos( ) El l mk...(47.) whch agrees wth (43.) except that here e s evaluated n terms of E and e. The constant of ntegraton θ can now be dentfed from Eq. (47.) as one of the turnng angles of the orbt. It wll be noted that only three of the four constants of ntegraton appear n the orbt equaton and ths s always a characterstc property of the orbt. In effect the fourth constant locates the ntal poston of the partcle on the orbt. If we are nterested solely n the orbt equaton ths nformaton s clearly rrelevant and hence does not appear n the answer. Of course the mssng constant has to be suppled f t s desred to complete the soluton by fndng r and as functons of tme. Thus f one chooses to ntegrate the conservaton theorem for angular momentum mr d ldt by means of (47.) one must specfy n addton the ntal angle θo. Now the general equaton of a conc wth one focus at the orgn s C r ecos...(48.) where e s the eccentrcty of the conc secton. By comparson wth Eq. (47.) t follows that the orbt s always a conc secton wth the eccentrcty 7

18 e El mk...(49.) The nature of the orbt depends on the magntude of e accordng to the followng scheme: e e e e E : hyperbola E : parabola E : ellpse mk E l : crcle. Ths classfcaton agrees wth the qualtatve dscusson of the orbts based on the energy dagram of the equvalent one-dmensonal potental V. The condton for crcular moton appears here n a somewhat dfferent form but t can easly be derved as a consequence of the prevous condtons for crcularty. For a crcular orbt T and V are constant n tme and from the vral theorem E T V V V V Hence E k r...(5.) But from equaton: f l ( r ) 3 mr the statement of equlbrum between the central force and the effectve force we can wrte or k r l 3 mr 8

19 l r mk...(5.) Wth ths formula for the orbtal radus Eq. (63.) becomes mk E l s the above condton for crcular moton. In the case of ellptc orbts t can be shown the major axs depends solely on the energy a theorem of consderable mportance n the Bohr Theory of the atom. The sem-major axs s one half the sums of the two apsdal dstances r and r. By defnton the radal velocty s zero at these ponts and the conservaton of energy mples that the apsdal dstances are therefore the roots of the equaton l E mr k r or r k E l r me...(5.) Now the coeffcent of the lnear term n a quadratc equaton s the negatve of the sum of the roots. Hence the sem-major axs s gven by r r a k E...(53.) Note that n the crcular lmt Eq. (53.) agrees wth Eq (5.). In terms of the sem-major axs the eccentrcty of the ellpse can be wrtten e l mka...(54.) 9

20 Further from Eq. (54.) we have the expresson l mk a( e )...(55.) n terms fo whch the ellptcal orbt equaton (43.) can be wrtten a( e ) r e cos( )...(56.) From ths Eq. t follows that the two apsdal dstances (whch occur when θ - θ s and respectvely) are equal to [a(-e)] and [a(+e)] as s to be expected from the propertes of an ellpse.. SCATTERING IN A CENTRAL FORCE FIELD In ts one-body formulaton the scatterng problem s concerned wth the scatterng of partcles by a center of force. We consder a unform beam of partcles-whether electrons or - partcles or planets s rrelevant-all of the same mass and energy ncdent upon a center of force. It wll be assumed that the force falls off to zero for very large dstances. The ncdent beam s characterzed by specfyng ts ntensty I (also called flux densty) whch gves the number of partcles crossng unt area normal to the beam n unt tme. As a partcle approaches the center of force t wll be ether attracted or repelled and ts orbt wll devate from the ncdent straght lne trajectory. After passng the center of forces the force actng on the partcle wll eventually dmnsh so that the orbt once agan approaches a straght lne. In general the fnal drecton of moton s not the same as the ncdent drecton and the partcle s sad to be scattered. The cross secton for scatterng n a gven drecton σ (Ω) s defned by

21 numberof ( ) d partcle scattered nt osold angled ncdent nt ensty per unt tme...(57.) where dω s an element of sold angle n the drecton Ω. Often σ (Ω) s also desgnated as the dfferental scatterng cross secton. Wth central forces there must be complete symmetry around the axs of the ncdent beam; hence the element of sold angle can be wrtten d sn d...(58.) drectons known as the scatterng angle (Fg 3 where repulsve scatterng s llustrated). It wll be noted that the name cross secton s deserved n that σ (Ω) has the dmensons of an area. For any gven partcle the constants of the orbt and hence the amount of scatterng are determned by ts energy and angular momentum. It s convenent to express the angular momentum n terms of the energy and a quantty known as the mpact parameter s defned as the perpendcular dstance between the center of force and the ncdent velocty. If v s the ncdent speed of the partcle then l mvs s me...(59.) Fgure 3 scatterng of an ncdent beam of partcles by a center of force

22 Once E and s are fxed the angle For the moment t wll be assumed that dfferent values of s cannot lead to the same scatterng angle. Therefore the number of partcles scattered nto a sold angle σ (Ω) lyng between l to the number of the ncdent partcles wth mpact parameter lyng between the correspondng s and s +ds l s ds I sn d...(6.) Absolute sgns are ntroduced n ths Eq. because numbers of partcles must of ary n opposte drectons. If s s consdered as a functon of the energy and the correspondng scatterng angle s s E...(6.) s sn ds d...(6.) A formal expresso obtaned from the orbt equaton. Agan for smplcty we wll consder the case of purely repulsve scatterng. As the orbt must be symmetrc about the drecton of the perapss the scatterng angle s gven by...(63.) where ψ s the angle between the drecton of the ncomng asymptote and the perapss drecton. In turn ψ can be obtaned from Eq.

23 r dr r me mv r l l by settng r r...(64.) Fgure 4. Relaton of orbt parameters and scatterng angle n an example of repulsve scatterng. When θ = (the ncomng drecton) whence θ=-ψ when r = rm the dstance of closest approach. A trval rearrangement then leads to r m r dr me mv l l r...(65.) Expressng l n terms of the mpact parameter s (Eq.59.) the resultant expresson r m r r s dr V ( r) s E...(66.) or 3

24 u m s du V ( u) s u E...(67.) The above two equatons are rarely of use except for drect numercal computaton of the scatterng angel. However when an analytc expresson s by nspecton. An hstorcally mportant llustraton of such a procedure s the repulsve scatterng of charged partcles by a coulomb feld. The scatterng force feld s that produced by fxed charge (-Ze) actng on the ncdent partcles havng a charge (-Z e); so that the force can be wrtten as f ZZ' e r that s a repulsve nverse square law. The earler results can be taken over here wth no more change than wrtng the force constant as k ZZ' e...(68.) The energy E s greater than zero and the orbt s a hyperbola wth the eccentrcty gven by* El m( ZZ' e ) Es ZZ' e...(69.) where use has been made of Eq. (59.). If θ n Eq. ecos ' chosen to be perapss corresponds to θ= and the orbt equaton becomes r mk l s mzz' e r l cos...(7.) 4

25 * Note:- To avod confuson wth the electron charge e the eccentrcty wll temporarly be denoted by Є The drecton of the ncomng asymptote ψ s then determned by the condton r cos or by Eq. (63.) Hence usng cot sn Es cot ZZ' e The desred functonal relatonshp between the mpact parameter and the scatterng angle s therefore s ZZ' e E cot...(7.) so that on carryng through the manpulaton requred by Eq. (6.) we fnd that ZZ' e ( ) 4 E csc 4...(7.) Equaton (7.) gves the famous Rutherford scatterng cross secton orgnally derved by Rutherford for the scatterng of partcles by atomc nucle. Quantum mechancs n the non-relatvstc lmt yelds a cross secton dentcal wth ths classcal result. In atomc physcs the concept of a total scatterng cross secton σt s defned as 5

26 T d ( )sn d 4 where the varous terms have ther usual meanngs. SUMMARY: - In ths chapter we study sofar. In physcs classcal mechancs s one of the two major sub-felds of mechancs whch s concerned wth the set of physcal laws descrbng the moton of bodes under the acton of a system of forces. Later more abstract and general methods were developed leadng to reformulatons of classcal known as Lagrangan mechancs and Hamltonan mechancs. Lagrangan mechancs apples to systems whether or not they conserve energy or momentum and t provdes condtons under whch energy and momentum are conserved.in Lagrangan mechancs the trajectory of a system of partcles s derved by solvng the Lagrange equatons n one of two forms ether the Lagrange equatons of the frst knd whch treat constrants explctly as extra equatons often usng Lagrange multplers or the Lagrange equatons of the second knd whch ncorporate the constrants drectly by judcous choce of generalzed coordnates.. CHECK YOUR KNOWLEDGE Questons. What s D Alembert s prncple of vrtual work done by constrant forces?. Deduce the Lagranan equaton for a system of mutual central force between two pont masses. 3. Descrbe the Keplares nverse Square Law of force wth the help of Lagrangan Key - Mechancs.. Equaton and related explanaton. Two body central force problem and equaton. 39 & 4 3. Kepler s Problem and dervaton of equaton REFERENCE. Classcal Mechancs; - H. Goldsten Addson-Wesley Pub. Co. Inc. Massachusetts 95 (Norsa Pub. House Inda). Classcal Mechancs; - B. D. Gupta & S. Prakash KedarNath Ram Nath Meerut Classcal Mechancs; - Vo. I of Course of Theoretcal Physcs; - L.D. Landau and E.M. Lfshtz Pergamon Press Oxford 96. 6

27 UNIT-II. GENERAL CONCEPT In unt I we learnt the Lagrangan dynamcs and related equaton of moton of systems. For a set of n generalzed coordnates there exsts another equvalent representaton called Hamlton s dynamcs. The former were equatons of moton of second order total dfferental but the later (Hamltonan) are the frst order total dfferental equatons However a logcal connecton between the two shall always reman wth a brdge of Legendre s transformatons. Before we proceed further let us acquant our self wth (a) Hamlton s Functon (b) Hamlton s equaton of moton and (c). The propertes of Hamltonan and ts equatons of moton. HAMILTON S FUNCTION In Lagrangan of a system we represented L as Lq. q q... q t... n n Here q and t are passve varables and q as actve ones where =...n. So the generalzed momenta becomes and the dual functon of L as p L q...(.) where H H H p q L q q t...(.) q... q p... p t Hq p...(3.) n n t. HAMILTON S EQUATION OF MOTION From Eq. (3.) we get q H... n p...(4.) and the passve varable gves (see. here) 7

28 L H t t and L q H q... (5.) From Euler s-lagrange s equaton of moton we get L q d dt L q p Therefore (5.) we can wrte p H t...(6.) In ths way eq. (4.) and (6.) for q together are called Hamlton s equatons of moton..3 PROPERTIES OF THESE EQUATIONS a. The H-functon does not explctly depend on tme b. Snce we can wrte from eq. (3.) dh d H q..... qn p... pn t dt dt H t The Hamltonan s a constant of moton Agan from eq. (.) H p q L E s correct for conservatve systems where we wrte H=T+V=E but n general HH(t) and L L(t) and hence an exstence of constant of moton ntroduces a new ntegral called Jacob Integral and s defned as H a cons tan t J c. As mentoned earler Hamlton s equatons of moton (number) are frst order total dfferental equatons but wth no. of equatons as n unlke the n second order total dfferental equatons of Euler-Langrange s equatons of moton 8

29 d. These equatons are symmetrc n q and p except for a change of sgn for second set..4 HAMILTON EQUATION FROM VARIATION PRINCIPLE Hamlton s prncple was dscussed s Unt I and we had expressed the lne ntegral I by equaton by (9.) as follows I t Ldt t...(7.) whch refers to path s n confguraton space. It needs modfcaton. In place of only space t should be evaluated over the trajectory of system ponts n phase space confguraton. It means that both the q and p should be treated as ndependent coordnates of phase space. So eq. (7.) becomes I t t p q Hq p t dt...(8.) Ths helps u to determne Hamlton s canoncal equatons of moton. The varaton problem n a space of n-dmensons can be represented as t I f for whch the nd Euler-Lagrange equatons s t q q p p t dt...(9.) d f f d f f and dt q j q j dt p j p j Where j..n....(.) and ntegral f contans q j only through p term s Hamltonan H. Ths would q lead eq. (.) to p j H q j...(.) and 9

30 q j H p j...(.) Equatons (.) and (.) are exactly Hamlton s equatons of moton derved through varatonal prncple. These are sometmes not acceptable by physcst but logcally the Legendre transformaton procedure shows that the Lagrangan and Hamltonan formulatons and therefore ther respectve varatonal prncple have the same physcal content. Further one pont s elaborated here specfcally. In Euler-Langrange equatons the varatons of ndependent varables vansh at END POINTS. In phase space that would requre that q= and p= at end ponts whereas Hamlton s prncple requres only q= under the same crcumstances. The problem s solved by ntroducng a new functon f here whch s ndeed a functon of q j and wthout actual explct appearance of p j and smlarly p j wthout q j (equatons.). Only by broadenng the feld of ndependent varables from n to n quanttes t enabled to obtan equatons of moton as desred by Hamlton..5 EQUATIONS Of CANONICAL TRANSFORMATION A stuaton may occur n whch the Hamltonan s a constant of moton and all coordnates q are cyclc. Under these states the conjugate moments p are all constants.e. p and snce H cannot become explct functon of tme or cyclc coordnates t can be wrtten as H H..... n whch helps s wrtng Hamlton s equaton for q as q H w...(3.) where w are functons of only and constant n tme. These equatons have solutons as 3

31 q w t....(4.) where s constants of ntegratons. Further when moton of a partcle s n a plane and to be dscussed one can use generalzed coordnates ether of Cartesan type.e. q=x q=y or the plane polarzed type.e. q=r q=θ. Both choces are equally vald. The transformatons consdered here nvolve gong from one set of coordnate s q to a new set Q by transformaton equatons of the form Q Q qt...(5.) Such transformatons are known as pont transformatons. However n Hamltonan formulaton the momenta are also ndependent varables lke generalzed coordnates. Therefore the transformatons of ndependent q p to a new set of Q P should exst (nvertble) n the followng forms of equatons and Q Q P P q p t q p t...(6.) Thus the new coordnates are defnable not only n old coordnates but also n old momenta. Eq. (5.) may be sad to defne a transformaton of confguraton space and eq. (6.) a confguraton of phase-space. To make Q and P as canoncal coordnates a new functon K (Q P t) s thought over n such a way that the equatons of moton n Hamltonan form become. K Q P and P K Q...(7.) Now snce Q and P become canoncal coordnates they must satsfy a modfed Hamlton s prncple put n the form 3

32 t t PQ KQ P t dt...(8.) At the same tme the old canoncal coordnates satsfy the smlar prncple t t p q Hq p t dt...(9.) But t should not be confused that the ntegrands n both these expressons are equal snce the modfed form of Hamlton s prncple has zero varaton at end ponts the ntegrants can be connected by the followng expresson through whch both statements reman consstent q H PQ K...(.) p df dt here s a constant ndependent of the canoncal coordnates and tme and F s any functon of phase space. Wthout gong n complcated stuaton we understand that f transforms wll be called extended canoncal transforms and f = eq. (.) holds and smply called canoncal transforms. As a specal case f the equatons of Q P (eq. 6.) do not contan the tme explctly durng transformaton then they are called restrcted canoncal transformatons. To show how the generatng functon specfes the equatons of transformatons let F be gven by old and new generalzed coordnates then (=) eq.. becomes F F q Q...(.) t p q H df p Q K dt F P Q K t F F q Q q Q...(.) 3

33 Snce old and new coordnates q Q are separately ndependent eq. (.) shall hold only f the coeffcents of q and Q each vansh. It means F p q F F p and K H Q t...(3.) The equaton (.) gves us an dea of evolvng a generatng functon F from F. On smlar lnes nter-transformaton of all old varables q p t canoncally to the new varables Q P t wth ndvdual relatons mantanng cyclc nature and the degree of freedom we can nvolve: next and F q F F q P t Q P...(4.) F q p F p Q...(5.) 3 t p Q p F4 t p P...(6.) we reach to smlar type of relatons for K s terms of Hamltonan as K F F F3 F4 H K H ; K H and K H t t t t...(7.) Fnally we reach to the concluson that a sutable generatng functon does not have to conform to one of the FOUR general types for all degrees of freedom of the system. It s thus possble and for some canoncal transformatons necessary to use a generatng functon that s mxture of the four types. As an example say we want a canoncal transformaton wth two degrees of freedom to be defned by a generatng functon of the form F q p P Q t F F ' then we can wrte q p P Q t Q P q...(8.) ' p and the equatons of transformaton would be obtaned from the relatons F' p Q q q F' P P F' P F' Q...(9.) 33

34 wth K F' H t...(3.) Transformaton termed canoncal can further be understood by some other examples solved n standard looks on classcal mechancs..6 THE POISSION BRACKET A relaton between a par of dynamcal varables of any holonomc system whch reman INVARIANT under any canoncal transformaton s called Posson Bracket (further t would be referred by PB only). PB can be used to construct new ntegrals of moton from the known ones. To understand PB say two dynamcal varables are lke u (p q t) and v (p q t). Then they are defned as n p q u v q p u p v q u v...(3.) where the suffx (p q) refers to the set (p q t) of ndependent varables wth n- number of degrees of freedom (DOF) w.r.t. whch the P B s evaluated. It may be noted that some authors defne P B as a negatve of the defnton (3.) wth each term havng the opposte sgn.7 SOME USEFUL INDENTITIES Throughout ths chapter u (p q t) v (p q t) and w (p q t) are assumed to be any three dynamcal varables pertanng to a holonomc system wth the number ODF=n whose generalzed coordnates and momenta are denoted by the set (q p). The followng denttes can be easly proved. () The PB of any two dynamcal varables s antcommutatve. 34

35 u v v u...(3.) As a corollary we have u u u u () If c s a constant that s not a functon of (p q t) them cu v u cv cu v...(33.) () The PBs also satsfy the dstrbutve property u v w u w v w and u vw u vw vu w...(34.) (v) The partal dervatve of any PB relaton can be shown to satsfy t u u t t u v u u...(35.) (v) A famous dentfy called Jacob s dentty s gven by u v w v w u wu v...(36.) (v) Let w w wn be a set of a dynamcal quanttes (all functons of p q t) and let F(w wn) be a dfferentable functon of w w wn. Then F F u Fw w... w u w u w... u w...(37.) F n n w w wn.8 ELEMENTARY PBs The PBs constructed out of the canoncal coordnates and momenta themselves are called elementary PBs. It s trval to show that q q p p and q p q q...(38.) j j j j j We also have 35

36 36 )...(39. q u p u and p u q u These equatons mply for Cartesan coordnates )...(4. u p u and u r u r p Thus by replacng u (q p t) n Eq. (39.) by the Hamltonan functon H (q p t) one obtans Hamlton s equatons of moton n terms of the PBs )...(4. p H p and q H q For a sngle partcle and n terms of Cartesan coordnates Eqs (4.) Take the followng form )...(4. p H p and r H r Now snce p q are explct functons of tme t s possble to nvert these relatons namely p = p (t) and q = q (t) to get t as a functon f p and q. Thus for t as a dynamcal varable expressed as t (p q) we can wrte...(43.) t t t p p t t q q t q H p t p H q t H t.9 POISSON S THEOREM-I The total tme rate of evoluton of any dynamcal varable u (p q t) s gven by )...(44. uh t u t u Proof: Startng wth the left-hand sde H u t u q H p u p H q u t u p p u q q u t u t u Thus f u s a constant of moton so that du/dt = then by Posson s theorem

37 37 H u t u Furthermore f u does not contan tme explctly that s u/t = then [u H] = s the requred condton for u to be a constant of moton Now t s easy to check for any gven Hamltonan of the form H=H (q p t) that one has...(45.) t H t H Apart from Hamlton s equatons of moton as obtaned from Eq. (4.). JACOBI-POISSION THEOREM-II If u and v are any two constants of moton of any gven holonomc dynamcal system ther PB (u v) s also a constant of moton. Ths s called Jacob-Posson s theorem of Posson s second theorem on the PB relatons. Proof: Consder )...(46. H v u v u v u t v u t Usng Eqs (35.) and (36.) we get t v u v t u H v t v u v H u v t u v u H u H v t u v t u v u t

38 Because both du/dt and dv/dt vansh by the requrement that u and v are both constants of moton Ths theorem has profound sgnfcance for determnng new constants of moton. To start wth f we have got any two ndependent constants of moton then a thrd one can be constructed from the PB of these two whch may result n ether a new (that s ndependent) constant of moton or trvally ether of the frst two. If the former s true we may make another par of new PBs and f we are lucky we can n ths way generate all the hdden constants of the moton. It should be remembered that a dynamcal system havng n degrees of freedom can have at the most n- ndependent constants of moton whch are functons of p and q s only and one constant of moton that must nvolve tme explctly. Examples Consder an sotropc -D harmonc oscllator that has the Hamltonan of the form H x y p p p p kx y...(47.) z y m Beng two dmensonal t has only three ndependent ntegrals of moton. One s of course the energy ntegral E snce H does not contan tme explctly. Agan snce the force feld here s a central one the angular momentum perpendcular to the x-y plane L xp y yp z s also conserved. To construct a thrd one note that H can be decomposed nto two parts each correspondng to a -D harmonc oscllator and conservng ts own energy component so that the dfference of energes of these two components s also a constant of moton. Therefore m z y p p kx y B ( say)...(48.) x y s a constant of moton 38

39 Now let us construct a new ntegral of moton say C from L and B usng the Posson Theorem II gven by C m L B xp yp B p p mk xy...(49.) y z z y C s obvously ndependent of L and B but not smultaneously of L B and H as H E B C 4 L...(5.) where k one can also check that m C L 4B and C B 4kL/ m We get back B and L respectvely and hence no further ndependent constants of moton can be generated n ths process. In order to fnd the nature of C one has to wrte down the complete soluton for x and y gven by t t y bsn t t...(5.) x asn where s the constant phase dfference between the two. The four constants of moton here are a b and t. One should remember that only three of these can have no explct tme dependence n ther expressons. Now from Eqs (49.) and (5.) we get C abm cos. The constants of moton of a -D sotropc harmonc oscllator are however best studed by the Runge-Lenz tensor of second rank defned by A j m p p j k x x j...(5.) 39

40 Snce we know that dp/dt = -kx are the equatons of moton of ths system they would mmedately satsfy daj/dt= whch means that Aj s are all constants of moton. For a -D sotropc harmonc oscllator as expected A A (=A) and A are the only three ndependent constants of moton. Therefore The trace of the A matrx= A+A = H = constant and L The determnant of the A matrx = AA A const. 4 Are just two other dependent constants of moton.. EULER S EQUATIONS OF MOTION FOR A RIGID BODY It s certan that after Newton s concepts about motons of a rgd body a detaled work on rgd body dynamcs and contnuum mechancs through the applcaton of calculus s undertaken by Leonhard Euler. Before we proceed to derve the Euler s equaton let us get acquanted wth the followng The angular momentum L of a partcle about a pont O n a rgd body s gven by L r p (where r & p have usual meanng) and torque or moment of force N by Therefore N r F r R N r d dt (mv) usng vector dentty we can wrte d dt d dt r mv vmv r (mv) N d dt ( r mv) dl dt...(53.) 4

41 Now consder ether an nertal frame whose orgn s the same pont or a system of space axes wth orgn at the centre of mass taken as and wrte dl dt s N...(54.) The subscrpt s s used because the tme dervatve s w. r. t. the axes that do not share the rotaton of the rgd body. However f we recall that n case of a and b behavng subscrpt for space and body axes as tme dervatves the standard notaton d dt a d dt b w L the eq. (54.) can be expressed as d dt s d dt b w L...(56.) (we consder b subscrpt for the body). Snce our am to fnd equaton of moton for the body only the subscrpt b n (56.) can be dropped. Hence eq. (56.) becomes dl w L N dt And for the th component t can be wrtten as...(57.) dl dt gk w j L k N...(58.) [Note: The change n any matrx n the tme dt s a change from the unt matrx and therefore corresponds to the matrx Є of the nfntesmal rotaton. Further by usng the ant symmetrc property of Є the permutaton symbol Єjk can be used where very necessary] 4

42 If now the body axes are taken as the prncpal axs relatve to the reference pont here o the angular momentum components are L= I W then the equatons (58.) can be wrtten as I dw dt jk w j w k I k N...(59.) Snce the prncpal moments of nerta are of course tme dependent. The eq. (59.) can be wrtten n the followng three forms IW WW I W W3W I W W W 3 3 3I I 3 N...( a) I 3 I N...( b) I I N...( c) 3...(6.) Equatons (6.) (a) (b) and (c) are termed as Euler s equatons of moton of a rgd body wth one pont fxed (here O) The Euler s equatons can also be represented n modfed forms as and dl dt dp F dt fxed fxed dl dt rot dp dt rot w L w P...(6.) where Г and F are torque and force (external) w L and P have usual meanngs and all these are expressed n terms of unt vectors frame wth dl dt rot 3 dl ˆ dt and dp dt rot 3 ˆ ˆj and kˆ of the rotatng body dp ˆ dt...(6.) It should be noted here that translatonal and rotatonal motons are dealt wth ndependently. Also f w s not parallel to L or P then n 4

43 general F dl dt and F dp dt. Further even f Г and F are zero ther tme depravtes need not vansh.e. L and P can change wth tme even n absence of Г and F smply one looks them from a rotatng body frame.. TORSION FREE SYMMETRIC RIGID BODY-APPLICATION OF EULER S EQUATION: We consder a freely rotatng rgd body that s one n the absence of any net external torque. Snce Г = Eq. (6.) becomes dl L dt...(63.) Takng a scalar product wth L one gets or or that s dl L. L dt. L dl L. dt d dt ( L) L const...(64.) Thus for Г= even though dl/dt = L ω (unless ω s parallel to L) t s seen that L or the magntude of L remans constant. Ths s not surprsng because the L vector s actually fxed n space (as Г=) but wth respect to the body frame only the drecton of L appears to change wthout of course brngng n any change n ts magntude. 43

44 Agan f we take a scalar product wth ω on both the sdes of Eq. (63.) we obtan or or dl dt L. dl. dt dt dt that s T = const. (65.) Therefore for freely rotatng rgd bodes ω and dl/dt are perpendcular to each other and the rotatonal knetc energy T s a constant of moton. Thus there are two constants of moton L and T for rgd bodes rotatng n any manner about a fxed pont. Let as assume that rgd body s rotatng freely that s wthout any external torque actng on t about a fxed pont say ts centre of mass. Hence the knetc energy of ts rotaton T s constant of moton. From the equaton T I j. Where T s constant and the summaton over j are mpled we can wrte I j T j T j...(66.) Ths equaton look lke the equaton of the ellpsod of nerta provded we dentfy ω / T wth x n Eq. of ellpsod of nerta that s x T...(67.) Now snce T s constant the surface of ths nertal ellpsod plotted n terms of the poston vector x as defned above must represent a surface of constant T and by defnton of x the drecton of x s the same as that of the nstantaneous 44

45 angular velocty ω. The dstance from the centre of ths nertal ellpsod to any pont on ts surface s smply ω / T. Ths nertal ellpsod s depcted n fg-. 3 T N P L ω T O T = Const Inertal ellpsod T Fg. Ponsot s geometrcal constructon for the moton of nertal ellpsod n the ω space Let us elaborate on ths new model of the nertal ellpsod. It s an nertal ellpsod because ts equaton contans all the elements of the nertal tensor. Ths nertal tensor s about the same pont as the fxed pont n the body about whch the free rotaton has been assumed to take place. If we fully expand Eq. 6. we can see that the space n whch ths equaton stands for an ellpsod s the space of nstantaneous angular velocty. The hgher the amount of rotatonal knetc energy the bgger s the sze of ths nertal ellpsod. For a gven constant knetc energy of rotaton the longest axs of the ellpsod (that s the hghest value of the nstantaneous angular velocty) should correspond to the drecton n the actual body of the smallest moment of nerta whch has to be one of the prncpal axes of the body about the pont under consderaton. Ths s sensble because f the body s rotatng about an axs correspondng to the lease moment of nerta t has to rotate very fast (whch means that t should pont along the longest radus vector of ths ellpsod) n order to mantan the same knetc energy of rotaton. So f we choose any arbtrary pont P on the ellpsod and say that ths pont represents the nstantaneous dynamcal state of the rotatng rgd body t would mmedately mean that the radus vector OP whch corresponds to unque drecton n the actual body (because the prncpal axes of MI of the body are n some defnte manner algned wth the geometrcal axes of ths ellpsod) s the drecton about whch the body s currently rotatng and the length of OP s smply ω / T. If the freely rotatng body changes the 45

46 drecton of ω wth tme as t usually should the pont P would be shftng on the surface of the ellpsod but t cannot go outsde ths surface because t mantans a constant rotatonal knetc energy. Now from the standard expresson for T T t I L j j A geometrcal nterpretaton of ths result n terms of the above nertal ellpsod would be that the angular momentum vector L T T x T...(68.) where x s defned through Eq. (63.) by Eq. (64.) We see that L s drected perpendcular to the surface of constant knetc energy depcted n the space of angular velocty (note that x ω). Snce the surface of the above knd of nertal ellpsod represents a constant T surface the normal to ths s a case of free rotaton; L s fxed n space n drecton and magntude. But wth respect to the body frame the drecton (but not the magntude) changes as the body rotates. As the body rotates ts nertal ellpsod must also rotate wth t keepng however ts orgn fxed. Snce the drectons of the normal and the radus vector at any arbtrary pont P on the ellpsod are not parallel the drectons of L and (ω) are also not parallel. Ths s all expected. For a sphercal top the moment of nerta tensor reduces to a sngle scalar the nertal ellpsod becomes sphercal and hence L and ω must pont n the same drecton. Let us now draw a tangent plane to the surface of the nertal ellpsod at the pont P (see Fg ). The perpendcular dstance of ths plane from the orgn of the nertal ellpsod s gven by ON OP cos( L) T L L T L const. 46

47 If ON s fxed for all tme the tangent plane s also fxed n space for all tme that s t must serve as an mmovable or nvarable plane for the dynamcal study of free rotaton. It does not change wth tme although the nertal ellpsod would change ts orentaton wth tme. In other words the ellpsod of nerta of any rgd body under free rotaton always touches a fxed plane known as the nvarable plane the perpendcular to whch drawn from the orgn ponts always toward the fxed drecton of the angular momentum of the body. Ths was the geometrcal pcture of a freely rotatng rgd body suggested by Ponsot n 834. The fact that both L and the nvarable plane are fxed n space has the followng mplcaton. Snce the nvarable plane has to meet tangentally the nertal ellpsod at some or other pont and snce ths pont of contact determnes the drecton of the nstantaneous angular velocty of the body (that s the ω vector) and snce ω changes wth tme ths pont of contact on the surface of the nertal ellpsod ought to change wth tme. However we know that the nvarable plane has to reman fxed n space for all tme and therefore the pont of contact can change only f the nertal ellpsod tself changes ts orentaton wth tme (keepng ts orgn fxed). Further snce ω changes contnuously wth tme the nertal ellpsod has to roll over the nvarable plane wthout slppng. Note that every pont on the nertal ellpsod corresponds to a dfferent drecton of ω wth respect to the body frame. As the ellpsod of nerta rolls wthout slppng over the nvarable plane the pont of contact P between the ellpsod of nerta and the nvarable plane ndcatng the drecton of nstantaneous ω wth respect to the body as well as the fxed plane traces a curve on the ellpsod of nerta called a polhode. Smlarly a curve s also traced on the nvarable plane and s called a herpolhode. 47

48 .3 SUMMARY :-Hamltonan mechancs: Operates n a phase space. Hamltonan mechancs s a reformulaton of classcal mechancs 833 by Irsh from The Hamltonan method dffers from the Lagrangan method n that nstead of expressng second-order dfferental constrants on an n-dmensonal coordnate space (where n s the number of degrees of freedom of the system) t expresses frst-order constrants on a n-dmensonal phase space. Hamlton's equatons provde a new and equvalent way of lookng at Newtonan physcs. Generally these equatons do not provde a more convenent way of solvng a partcular problem n classcal mechancs. Rather they provde deeper nsghts nto both the general structure of classcal mechancs and ts connecton to quantum mechancs as understood through Hamltonan mechancs as well as ts connecton to other areas of scence..4 CHECK YOUR PROGRESS Questons. Deduce Hamlton s Equaton of Moton and gve propertes of ths equaton.. What s meant by Posson Brackets from canoncal coordnates and momentum of a system. 3. State and Explan Possons Theorems I & II Key. Equaton (6) of unt II and related matter. Text Headngs.3. to.5 3 Text Headngs.6 &.7..5 REFERENCE Classcal Mechancs:- N.C. Rana and P.S. Jog Tata-McGraw-Hll Pub. Co. Ltd 99. Lagrangan Dynamcs; D.W. Wells 3 Analytcal Dynamcs; E.T. Whttaker Note : Ref. 5 & 6 can be searched on nternet for publshng agences and year 48

49 UNIT-III 3. HAMILTON-JACOBI THEORY In unt-ii we have already read about the transformed Hamltonan term K and related t wth generated functon as follows:- F K H t...(.) Here K would be zero f F satsfes the equaton H F t q p t...(.) It s convenent to take F as a functon of old coordnate q new constant momenta P and tme t and desgnate a new generatng functon F (q P t). Snce P F the Eq. (.) becomes q F F q... q... t q q H n...(3.) Ths equaton (3.) s known as Hamlton-Jacob Equaton and t consttutes a partal dfferental equaton of (n+) varables. It s customary to denote the soluton of Eq. (3.) by S and call t Hamlton s prncpal functon. Suppose there exsts a soluton of Eq. (3.) as q. q......(4.) F S S... n n n t where the quanttes n+ are (n+) ndependent constants of ntegraton. Now f S s the soluton of Eq. (4.) the n(s+) where s any constant must also be a soluton Ths means 49

50 S S q. q......(5.)... n n t hch can correlate the new momenta P= However when the old momenta can be expressed as p S q t q they become n-transformaton equatons. It s one part of the soluton. The other part whch would provde the new constant w coordnates appear as Q S q t...(6.) where s are another constants whch can be obtaned from the ntal condtons. Ths equaton can be used to fnd q q t j j. On dfferentatng p as gven above and substtutng ths qj we can get a smple but useful expresson as p=p ( t). These p and qj consttute a complete soluton of Hamlton Equaton. In bref we have establshed equvalence between n Canoncal Equatons of moton to the frst order partal dfferental equaton called Hamlton-Jacob equaton. In terms of Lagrangan we understand Hamlton s prncple as follows:- S S S Snce q the basc equatons (3.) and p as t q t q S t q gve rse to S t p q H L...(7.) or S s another form of the soluton of (3.) Ldt cons tant 3. HARMONIC OSCILLATOR AS AN EXAMPLE OF THE HAMILTON- JOCOBI METHOD Hamlton-Jacob method can be used to solve the moton of mechancal system taken as an example of Harmonc-oscllaton. Snce the Hamltonan term represent the energy of the system. We represent 5

51 E H m p m w q...(8.) where the angular velocty ω s gven by S Usng Eq. (7.) and snce p we can wrte q S m w m q q k w k beng a force constant. m S t...(9.) snce explct dependence of S on t s seen n the last term of Eq. (9.) we can wrte S(q t)=w (q )-t whch gve Eq. (8.) and (9.) as W m m q...(.) The ntegraton constant thus (defnes the total energy of the system E) helps S us to elmnate W and ω usng Eq. (3.) and we get H and fnally H=. t Next ntegrate Eq. (.) to fnd the values of W and S. substtute the partal dervatve of s wth n Eq. (6.) to reach the result S m dq mw q t...(.) or t w arc sn q mw...(.) The above equaton (.) gves q and earler quoted momentum relaton sn w mw t...(3.) p m cos w( t )...(4.) 5

52 Equatons (3.) and (4.) help us to connect and to the orgnal q and p at t = as follows m p m w q...(5.) and tan w mw q p...(6.) Fnally the complete soluton of H n terms of S wth the help of canoncal transformaton generates canoncal momentum dentfed as total energy of the system. That s S cos w( t ) dt t cos w( t ) dt...(7.) 3. HAMILTION S CHARACTERISTIC FUNCTION In prevous secton we have seen that S could be separated nto two parts: one nvolvng q only and the other t only. Such reparaton s always possble whenever old H does not nvolve t explctly. Under such consderatons H-J equaton for S becomes S t H q S q...(8.) The frst term here nvolves t and second q wth S. So a soluton of S can be thought over whch would separate t from t. That s assume S q twq t...(9.) Then equaton (8.) would generate W H q q But we known that 5

53 p W q and Q W p W...(.) The functon W s known as Hamlton s characterstc functon. It generates all canoncal transformatons n whch all the new coordnates are cyclc. It makes the canoncal equaton for momenta P as cyclc too. Therefore P K Q P...(.) Because the new H depends only on momenta we can wrte K Q when and Q K when wth the ntermedate solutons Q W W t or Q for...(.) The only coordnate that s smply a constant of moton s Q whch s equal to the tme-pulse a constant. Wthout gong much n detals the Hamlton s functon and the characterstc generatng functon W are related as follows 3.3 SEPARATION OF VARIABLES IN H-J EQUATIONS The coordnate qj s sad to be separable n H-J equaton when q from all qj s s taken as a separate coordnate from rest other. Mathematcally t means S q. q ;... ; t S q ;... ; t S' q... q ;... ;...(4.)... n n n n n t whch helps us to wrte H-J equaton n () separately for S and () for S. Now the transton from Hamlton s prncpal functon S to the characterstc functon would be 53

54 S q t S twq...(5.) Snce t s assumed that H s not an explct functon of tme the H-J equaton wth ths trval soluton becomes W S H q q t...(6.) Ths equaton can only hold f the two terms of Eq. (6.) are constant wth equal and opposte value to make ther addton zero. It means S t W & H q q......(7.) It ndcates that plays an mportant role n separaton of varables. To emphasse ths further suppose the cyclc coordnal q and the conjugate momenta p are constant and say denoted by then the H-J equaton for W becomes H q W W... q ; ;... n q qn...(8.) we now try to assume the separated soluton W W q ) W ( q... q...(9.) ( n then Eq(8.) becomes p W q fnally we get W=W +vq. Further to show the separaton specfcally t s convenent to desgnate q as non-cyclc temporarly and for > the conjugate p as... ;. Ths results n 3 n W n W ( q ) W ( q ) q n...(3.) 54

55 55 Here W s the soluton of the reduced H-J equaton whch appears as...(3.)... ; n q W q H In terms of qj and pj we can now express a relaton between W Wj and W as...(3.) ' j j j q W q f q W q H Ths Eq. (3.) can be nverted to solve for f as...(33.) ; ' ; j j j q W q g q W q f whch completes the purpose of separaton of varables n the form...(34.) ' j j j j q W q g and q W q f Here f s not a functon of any of q s except qj and g s ndependent of qj 3.4 ACTION ANGLE VARIABLE FOR COMPLETELY SEPARABLE SYSTEMS: Acton-angle varable s a good technque used to know the perodctes of a system wthout completng solvng the H-J equatons. The perodc system should be conservatve and orthogonally decomposable and the Hamlton s characterstc functon s completely separable n all ts varables. The acton angle s defned as...(35.). dq p J where s not summed over and ntegral s performed over a complete varaton n q. We have learnt that tme ndependent H-J equaton s

56 S S H q... qn;... E q q n...(36.) Its complete ntegral would be S where one of the {} s energy E then n... n n q. q... S q ;... n J S q dq S q dq...(37.) It means... n J J n... These n relaton s can be nverted to get J J J n... n... so that the complete ntegral can also be wrtten as ~ S q... q J... n J n where {J}s are constants of moton may be called acton Varables. The n S S S Q and p...(38.) P J q wth the new Hamltonan P Q Hp q EJ... K. J n whch does not depend on {Q}. If Q suppose the solutons of ths be Q v t where v E J also we should keep n mnd that P K Q K E and Q say v P J a cons tan t 56

57 Here β s also a phase constant of ntegraton Also Q s are the coordnate s conjugates to momenta J s and J s are the same dmenson as that of angular momentum or acton or Planck s constant: ћ. These Q s are bascally denoted by w s and are called angle varables gven by S w vt j whch can produce the general relaton k k q q J. J ; w... w... n n It s to be remembered that f the relaton E E J... J ) s known the ( n frequences of oscllaton s for all the co-ordnates n a system s gven by E J w v Therefore...(39.) If total N degrees of freedom are consdered then N-no. of frequences satsfy n N lnear equatons wth ntegral coeffcents. Ths s sad to be the state of n-fold degeneracy. In the case of cyclc motons nstead of N-fold (N-n) fold contnuum exsts and hence (N-n)-degrees of freedom we get by an approprate change of varables. 3.5 KEPLER S PROBLEM IN ACTION-ANGLE VARIABLES In prevous secton we have understood the mportance of Acton angle varables. Let us apply t to the Kepler s problem 57

58 58 Planetary motons become completely degenerate n all the three coordnates r θ and. The same thng can be sad that all bounded orbts of any two body system movng under any Inverse Square Law of an attractve central force are closed. Further we must assert on the pont that f energy E< & the moton s perodc the Hamltonan can be wrtten as...(4.) ) ( sn r V r p r p p H r Usng S r S r S r as a soluton to eq. (4.) the H-J equaton reduces to ) ( : ) (...(4.) sn sn : ) ( r p r V E r dr ds b and p d ds a r r The acton varables J n r θ and are gven by )...(4. sn d d S J...(43.) p d d S J And Jr = ( s r)dr = μ{e-vr }- /rdr

59 Some Puttng K V ( r) r 3 K E ( J J J r )...(45.) Snce ths relaton s symmetrc n Jr Jθ & J all the three frequences are same where V r V V k K G( M S m) E (46.) Ms s mass of the sun and m of any planet gong round t. Ths explans Kepler s Problem whch explans that f the central force contans terms lke non-nverse law (aganst Newton s law) the orbt would no longer be closed but would postvely satsfy the planer moton. Thus the planetary motons are completely degenerate n all the three co-ordnates r θ and. un_es thr ee 3.6 SUMMARY:- we study that the Hamlton Jacob theory s a general theory rch n analytc Geometrcal deas that unfes three apparently dsparate topcs: system of frst order dfferental equaton and the calculus of varaton. Ths gves the elementary treatment of Ths aspect of Hamlton jacob theory especally n relaton to calculus of varaton. Hamlton-Jacob theory provdes a powerful method for extractng the equatons of moton out of some gven systems n classcal mechancs. On occason t allows some systems to be solved by the method of separaton of varables. The value of the Hamltonan s the total energy of the system beng descrbed. For a closed system t s the sum of the knetc and potental energy n the system. 3.7 CHECK YOUR PROGRESS. Descrbe that Hamlton Jacob theory of n number of dynamc varables wth the help of partal dfferental equatons.. Apply the Hamlton Jacob method to solve the equaton of moton of mechancal system of harmonc oscllator. 3. Dscuss Kepler s problem n terms of acton Angle Varable s. Key:- 59

60 . Explanaton n Headng 3.. Explanaton n Headng Explanaton n Headng REFERENCES Hamlton-Jacob Theory n the calculus of varaton; H. Rund Quarterly Journal of Mechancs and Appled Mathematcs Vol. 7 (search on famous lbrary net) A Treatse on the Analytcal Dynamcs of Partcles and Rgd Bodes E. T. Whttaker Dover Pub. Inc. Ny Classcal Dynamcs; A Modern perspectve E. C. G. Sudarshan and N. Mukunda John Wley and Sons Inc. Ny Introducton to classcal Mechancs; R.G. Takwale and S. Purank Tata-McGraw-Hll Pub. Co. Ltd

61 M.Sc. (PREVIOUS) PAPER-II Block II Bascs of Statstcal Mechancs & Quantum Statstcs Unt wrter Dr V. G. Machve Edtor Dr Purnma Khare 6

62 Bascs of Statstcal Mechancs & Quantum Statstcs Unt-IV: Bascs of Statstcs Unt-V: Canoncal concept Partton Functons and Thermodynamcally propertes Unt-VI: Bose-Ensten and Ferm-Drac Statstcs and Applcaton to Gas Molecules 6

63 CONTENTS UNIT-IV 4. INTRODUCTION 4. OBJECTIVES 4.3 PHASE SPACE ENSEMBLE ENSEMBLE AVERAGE LIOUVILLE THEOREM 4.7 CONSERV ATION IN EXTENSION OF PHASE EQUATION OF MOTION AND LIOUVILLE THEOREM EQUAL A PRIORI PROBABILITY 8 4. STATISTICAL EQUILIBRIUM 8 4. MAXWELL VELOCITY DISTRIBUTION 4. EQUIPARTITION OF ENERGY 4.3 SUMMARY CHECK YOUR PROGRESS REFERNCES 7 UNIT -V 5. INTRODUCTION 4. OBJECTIVES 4.3 GRAND CANONICAL ENSEMBLE IDEAL GAS IN GRAND CANONICAL ENSEMBLE COMPARISON OF VARIOUS ENSEMBLE QUANTUM DISTRIBUTIONS USING OTHER ENSEMBLE GRAND CANONICAL ENSEMBLE CANONICAL PARTITION FUNCTION MOLECULAR PARTITION FUNCTION 4 63

64 4. TRANSLATIONAL PARTITION FUNCTION 4 4. ROTATIONAL PARTITION FUNCTION VIBRATIONAL PARTITION FUNCTION THERMODYNAMIC FUNCTION SUMMARY CHECK YOUR PROGRESS REFERNCES 68 UNIT-VI 6. INTRODUCTION 6. OBJECTIVES 6.3 BOSE-EINSTEIN DISTRIBUTION BOSE-EINSTEIN CONDENSATION THERMODYNAIC PROPERTIES BOSE EINSTEIN GAS LIQUID HELIUM SUPERFLUID PHASES OF 3 He FERMI-DIRAC DISTRIBUTION FUNCTION DEGENERACY FREE ELECTRONS IN METALS MAGNETIC SUSCEPTIBILITY OF FREE ELECTRONS 9 6. SUMMARY 6.3 CHECH YOUR PROGRESS 6.4 REFERNCES 3 64

65 UNIT-IV BASICS OF STATISTICS 4. INTRODUCTION Statstcs s the study of the collecton organzaton analyss and nterpretaton of data. It deals wth all aspects of ths ncludng the plannng of data collecton n terms of the desgn of surveys and experment. Ths chapter s dvdng n two parts. Part-I: Let us thnk of a molecule A n a chaotc state s not at rest. We fnd t at a partcular place and tme.to deal ths stuaton we take the help of PROBABILITY whch successfully deals wth random events. If n total of N trals the molecule can be seen n n trals the probablty P(A) s gven n by P( A) lm whch can never be more than one. Next t s to be n N remembered that () f two events are ndependents P(a b) = p(a) p(b) and () f the two & events are mutually exclusve P ( ) = P () + P (). It explans the multplcaton and summaton of probabltes to get resultant probablty. Part-II: To arrve at part-i we should have understand the bascs of permutaton (p) and combnaton (c) whch are nterconnected by the expresson n c r n n! pr where n are the choces of an object to put t nto r groups r! r!( n r)! at a tme. To be more exhaustve the number of permutatons of n DISTINCT objects taken r at a tme s n p r n! n( n )( n )...( n r ) ( n r)! 65

66 Now f the ORDER of arrangng these n objects s not mportant we have combnaton rather than permutaton.e. regard to order n whch they are related than 4. OBJECTIVES n c r when r taken at a tme WITHOUT n c n r r! pr. The total number of combnatons X number of permutaton n each combnaton = Total number of permutaton. Knowng Part-I and Part-II we can easly understand: (a) Phase space: In the moton of a pont lke object n-degrees of freedom can be bracketed n q q..qn coordnates and p... p p3... pn momenta. It helps to know whole of the system s moton n terms n-q s and np s. These values postvely vary as a defnte functon of tme. So t s regarded that ncoordnates at a pont n a space of n-dmeson s called Phase Space. (b) Ensemble: Probablty speaks about many events. A Group of collecton of smlar ndependent un-nteractng thought over systems s called ensemble. Any regon can be thought-over to contan a swarm of phase ponts. At that pont f number of molecules exst s N each wth energy E and the regon of the pont be defned by volume V then the magnary entty s termed N V E 4.3 PHASE SPACE ensemble Frst consder a very smple case. A bead of mass m moves freely and arbtrarly on a strng stretched along the x axs. It has one degree of freedom. The poston of the bead at tme t s x (t) and ts velocty V x x (or momentum mx ) at that nstant. The state of the bead at any nstant can be represented by a pont P n a hypothetcal two-dmensonal space called the phase space whose coordnates are x and px. As the bead moves on the strng the value of x changes. Under acceleratng forces px also changes. As a result the pont P traces a trajectory n the phase space wth the passage of tme (Fg..). p x 66

67 Fg.. (a) A bead sldng on a rng. (b) Phase space and phase lne for the bead. A molecule of an deal gas can be represented as a structure less partcle. Such a molecule has three translatonal degrees of freedom. Its phase space has sx dmensons whose Cartesan coordnates are x x x3 p p p3. It s called the space where stands for molecule. The nstantaneous translatonal state of the molecule s gven by the representatve pont n ths hypothetcal space. For a system of N molecules (gas) the nstantaneous state (Fg..a) s represented by a set of N ponts n the -space one for each molecule (Fg..b). It s a symbolc pcture of the space because t s not possble to dsplay a sx-dmensonal space. The total number of translatonal degrees of freedom s 3 N = 3N Followng Ehrenfest we can construct a phase space for all the molecules whch has 6N dmensons. It s called the r space where r stands for gas. It s spanned by 3N coordnate axes and 3N momentum axes. The 6N coordnates (x x x3 xn x N x3n pp p3 pn pn p3n) represent the postons and momenta of al the molecules (state of the system) at 67

68 a gven tme. In the Γ space the nstantaneous state of the whole system (gas of N molecules) s gven by a sngle representatve pont (or phase pont). Fg..c. The notaton [x] [p] stands for the 3N coordnate axes and 3N momentum axes. Fg... (a) A gas contanng N molecules (system). (b) Space for the system. (c)γ space and the representatve pont for the entre: system. In general f f ndependent poston coordnates and f momentum coordnates are requred to fully specfy the state of a system then the system s sad to possess f degrees of freedom. Any set of f generalzed coordnates q q.qf (Cartesan polar or some other convenent set) can be used to unquely determne the confguraton of the system. The correspondng generalzed momenta are p p.pf. The Γ space s then a conceptual Eucldean space havng f rectangular axes [q] [P]. The mcroscopc state of the whole system s specfed by a representatve pont n ths space Wth the lapse of rme some or all of the f coordnates take on dfferent values (Fg..3) As a result the representatve pont traces a phase lne (or phase trajectory) n the accessble phase space (Fg..4) Each pont on the phase lne represents one such possble mcroscopc state. A pont n the phase space s accessble f t corresponds to the 68

69 physcal specfcaton of the system under observaton. For example the states of the crystallne form of sodum are naccessble at very hgh temperature. The system s lkely to pass through coordnates take on all possble all the accessble states. In ths sense the f Fg:.3 A few possble states of the system (gas contanng N molecules). For convenence only few molecules are shown. V and E stand for Volume and E nergy values. Fg.4 Phase space and a porton of the phase lne We can say that they are randomzed. The phase lne tends to fll the accessble phase space. The measurement of macroscopc varables (lke P V T etc.) nvolves takng tme averages over an approprate porton of the phase lne of the system. 69

70 4.4 ENSEMBLE Each phase pont on the phase lne of a sngle system develops out of the prevous pont n tme accordng to the laws of mechancs. Gbbs replaced ths tme dependent pcture by a statc pcture n whch the entre phase lne exsts at one tme (Fg..5 a). Then each phase pont represents a separate system wth the same macroscopc propertes (N V E) as the system of nterest but a dfferent mcroscopc state. In other words we 'magne' a large number M (M ) of systems smlar n structure to the system of nterest but sutably randomzed n the accessble unobservable mcroscopc states. Instead of takng the tme average we take an average over ths artfcally constructed group exstng smultaneously at one tme. Such a group of replcas or collecton of smlar nonnteractng ndependent magned systems s called an ensemble by Gbbs (Fg..5b). Fg..5 (a) The ensemble (a small porton) at one tme. Any regon lke A wll appear to contan a swarm of phase ponts. (b) Schematc 'lattce' representaton of an ensemble of M (M ) magnary systems at one tme each wth same N V E. tme average of some property of a system n equlbrum s same as the nstantaneous ensemble average. Ths s known as the ergodc hypothess All the members of an ensemble whch are dentcal n features lke N V E are referred to as elements. These elements though dentcal n structure (same macroscopc state) are randomzed n the sense that they dffer from one another 7

71 n the coordnates and momenta of the ndvdual molecules that s the elements dffer n ther unobservable mcroscopc states. The varous elements beng magnary do not nteract wth each other. Each element behaves ndependently n accordance wth the laws of mechancs (classcal or quantum). A clear dfference exsts between the actual system of nterest and an element of the ensemble. The system s the physcal object about whch we ntend to make predctons. The elements of the ensemble are mental copes of t to enable us to use the probablty theory. Thus an ensemble of systems conssts of randomzed 'mental' pctures of the system of nterest that exst smultaneously. It s to be vewed as an ntellectual exercse to mtate and represent at one tme the states of the actual system as developed n the course of tme. It s easer to compute the statstcal behavour of such a sutably chosen ensemble than to study the behavour of any partcular complex system. Results so obtaned enable us to predct the probable behavour of the system of nterest. An ensemble average s the average at a fxed tme over all the elements n an ensemble. It s dffcult to prove the exact equvalence of the ensemble average and the tme average over a sngle system. However one can hope that the former would closely approxmate the latter f the followng essental condtons are satsfed: 4.5 ENSEMBLE AVERAGE (A) Average Values Consder the smple case of a set of N ponts dstrbuted arbtrarly along a lne. If x() s the dstance of the th pont from the orgn then the average dstance x from the orgn s gven by x N N x( )....(.) If the lne s dvded nto cells and N s the number of ponts n the th cell 7

72 located at x() then we can wrte x N N x( )....(.) If the dstrbuton s known n the form of a contnuous functon N(x) x N xn ( x) dx N N( x) dx...(.) In general f R(x) s any arbtrary property of the ponts R R( x) N( x) dx N( x) dx...(.3) Generalzaton to hgher dmensons s straghtforward. (B) Densty of Dstrbuton n the Phase Space The use of ensembles n statstcal mechancs s guded by the followng factors:. The am s to know only the number of systems or elements that would be found n dfferent states that s n dfferent regons of the Γ space at any tme. All the elements beng smlar n structure we need not dstngush between them.. The number of elements n the ensemble s so large (M ) that there s a contnuous change n ther number n passng from one regon of the phase space to another. We can therefore descrbe the condton of an ensemble by a densty D wth whch the phase ponts are dstrbuted n the f space. It s called the dstrbuton functon (or densty of dstrbuton or probablty densty). In an ensemble of systems of f degrees of freedom D s a functon of f poston and momentum coordnates q q... qf pp.pf whch correspond to the f axes n the phase space. It can also depend on tme t explctly. The reason 7

73 s that although we are free to fx the dstrbuton at any gven tme t we have as yet no assurance for the dstrbuton to reman same. If t remans same the partcular dstrbuton would be one of equlbrum. We shall dscuss ths later on. Thus n general DD( q... q f p... p f t) D( q p t)....(.4) Consder a small regon A of the r space such that the poston coordnates le between q and q + dq..... qf and qf + dqf and the momenta le between p and p + dp..pf and p f + dpf (Fg.. 5a). The hyper volume of ths regon s D dq dq f dp dp f dqdp...(.5) By the defnton of densty the number of systems or elements dm lyng n the specfed nfntesmal regon stuated at the phase pont ql.pf at the nstant t s dm D( q p t) d...(.6) If M s the total number of elements n the phase space then at every nstant t M D d...(.7) where the ntegraton s over the whole phase space. As suggested by (.3) the ensemble average of a quantty R (q p) s defned by R R( q p) D( q p t) d D( q p t) d M R( q p) D( q p t) d...(.8) If a system s selected at random from the ensemble the probablty of selectng one whose phase pont les n the small regon at the pont q...pf s smply ρdγ where 73

74 D Dd D M d...(.9) In terms of p (q p f) called the normalzed densty of dstrbuton R R( q p) ( q p t) dq dp ( q p t) dqdp R d...(.) The ensemble average (.) gves the average value of the physcal quantty R for the actual system of nterest. The macroscopc average propertes (lke N V E) of a system n thermo 'dynamc equlbrum do not change wth tme. Therefore our ensemble representng t must be such that the ensemble averages are tme ndependent. Ths s a reasonable requrement. It follows that to construct a sutable ensemble we should study the behavour of ρ (or D) wth tme. 4.6 LIOUVILLE THEOREM In 838 Louvlle showed n another connecton how to use the classcal Hamlton equatons of moton q H p H p (... f ) q...(.) where H s the Hamltonan (total energy expressed as a functon of q's and p's) to obtan a statement about dd/dt. A knowledge of H at every pont n the phase space yelds the element of the trajectory passng through the equaton (.). The unqueness theorem for systems of ordnary nonlnear dfferental equatons* mples that through every phase pont n Γ space there passes one and only one trajectory unquely determned by (.). Consequently no two trajectores can ever cross n Γ space. Consder at any pont q.qf p pf stuated n the Γ space a small regon of hyper-volume dγ = dql... dqf dp.. dpf (Fg..5a). At any nstant the number 74

75 dm of phase ponts n ths regon s gven by (.6) DM D q p t dq... dp... dq...(.) f Ths number wll n general change wth tme due to the flow of phase ponts. The change wll occur when the number of phase ponts enterng the hypervolume through anyone face s dfferent from the number leavng the opposte face. Let us consder two faces normal to the q axs wth coordnates qand q+dq (Fg..6). The number of phase ponts enterng the frst face (q = constant) n tme dt s gven by D q dt dq dq dp dp f f where q s the component of velocty n the drecton of q axs of representatve ponts at q pf. The number of phase ponts leavng the opposte face (q +dq=constant) n tme dt s D q D dq q q q D dt p dq... dq dp... dp...(.3)( A). f f 75

76 76 after neglectng the hgher dfferentals. Subtractng the latter expresson from the former and agan neglectng the second order dfferental we get the net number of phase ponts enterng the hyper-volume dγ n tme dt as.. d dt q q D q q D A smlar expresson exsts for the P coordnate.. d dt p p D p p D The total rate of change wth tme n the number of phase ponts (dm) /t n regon dγ s obtaned by summng the net numbers of phase ponts enterng the hyper-volume through all the faces labelled by q.pf. )...(.3)( B d dt p p D q q D p p q q D dt t DM f From equatons of moton (.)...(.4) q p H p q H p p q q From (.-4) we get for the rate of change of densty D/t at the fxed phase pont (q p) under consderaton...(.5) f p q p p D q q D t D If we take nto consderaton the full dependence of the densty D (q p t) on the coordnates momenta and tme (.5) can be expressed as

77 dd dt q p D t q p f D dq q dt f D dp p dt...(.6) Here dd/dt s the total tme dervatve of D (q...pf t). It gves the rate of change of D n the neghbourhood of any selected movng phase pont (q's and p's changng) nstead of n the neghbourhood of a fxed pont n the Γ space. The relaton (.5) or (.6) s known as Louvlle theorem. Usng (.9) we can wrte (.6) as d dt...(.7) The smplcty of (.5) depends on the use of (.4) whch n turn depends on the choce of conjugate coordnates and momenta for constructng the Γ space. Use of vcoctes n place of momenta would have led to a more complex result. From (.6) we note that D can vary wth tme under two separate condtons. () There s an explct dependence on tme ( D/ t)q p The densty can vary wth tme at a gven pont n Γ space. () There s an mplct dependence as some or all of the coordnates of the system vary wth tme and the phase pont wanders n the r space. Ths mplct dependence of the densty n the vcnty of a selected movng phase pont s descrbed by the two terms under the summaton sgns n (.6). The tme rate of change of D due to changes n q alone s gven by D/ q multpled by the component of velocty n q drecton that s by ( D/ q) (dq / dt). If we consder changes n all q's f D dq q dt D t p t' where the suffxes suggest that the varaton s beng consdered wth respect to q. Smlarly f D p p dt D t q t 77

78 and (.6) can be wrtten as dd dt D t q p D t p t D t q t Ths clearly shows that the total rate of change of densty dρ/at n the vcnty of any selected phase pont of a system as t moves through the Γ space s zero. Followng Gbbs ths s known as the prncple of the conservaton of densty n phase. It mples that the densty of a group of phase ponts remans constant along ther trajectores n the Γ space that s t does not dsperse. The dstrbuton of representatve ponts moves n phase space lke an ncompressble flud 4.7 CONSERV ATION OF EXTENSION IN PHASE From equaton (.6) to obtan one more mportant prncple of statstcal mechancs. Consder a small regon of hypervolume ΔΓ n the phase space whch s small enough for the densty D (or ρ) to be taken as unform throughout ts extenson. From (. 6) M D d( M ) dd d( ) D dt dt dt d D ( ) dt...(.8) Here ΔΓ s the hypervolume of a closed regon bounded by a (f-) dmensonal hypersurface n a f dmensonal phase space (Fg..7). The phase ponts lyng on ths hypersurface form a movable boundary whch changes ts shape and moves about n the Γ space due to the 'flow' of the phase ponts. The phase ponts can nether enter nor leave through ths boundary because wth every pont on each phase lne a defnte phase velocty s assocated. 78

79 Fg..7 Moton of phase volume n Γ space The phase ponts on the boundary form a knd of contnuous skn whch permanently encloses the phase ponts of the hypervolume ΔΓ under consderaton. Also these ponts can nether be created nor destroyed because each pont represents a defnte element n the ensemble. Consequently the number ΔM of phase ponts enclosed n the regon ΔΓ must reman constant. d (ΔM)/dt = and (.8) becomes d( ). dt...(.9) Ths means that the volume ΔΓ or extenson-n-phase n Γ space bounded by a movng surface and contanng a defnte number of phase ponts does not change wth tme n spte of the dsplacements and dstortons. Every fnte arbtrary extenson-n-phase can be regarded as composed of nfntesmal parts and so ths result can be generalzed. Followng Gbbs the result expressed n (.9) s called the prncple of conservaton of extenson n-phase. 4.8 EQUATION OF MOTION AND LIOUVILLE THEOREM The Posson bracket {a. b} s defned by a db a db a b q dp p dq...(.) The canoncal equatons of moton (.) and total tme dervatve dd/dt can be 79

80 wrtten as H H q H q p H p...(.) p q dd dt D t D dt D dq q dt D q dh dp D p D dp p dt dh dq D t D H...(.) The last result s the equaton of moton n terms of Posson bracket. The Louvlle theorem (.6) dd/dt = can now be expressed as D t D H....(.3) The mplct dependence of D(q p t) on tme s thus gven by the Posson bracket. We shall deal only wth those ensembles for whch D does not depend on tme explctly dd/dt =. Therefore the statonary ensemble descrbed by D(qp) or ρ(qp) s the same for all tmes t ( q p) H....(.4) 4.9 EQUAL A PRIORI PROBABILITY In the ensemble have large number of elements (replcas of the orgnal system) dstrbuted n dfferent Possble.(accessble) mcroscopc states but characterzed by the same macroscopc varables N V E. Ths s all the knowledge we can clam about them. To ths extent we haw no reason to prefer one mcroscopc state over the other. A fundamental postulate of statstcal mechancs s that a macroscopc system ll equlbrum s equally lkely to be ll any of ts accessble mcroscopc states satsfyng the 8

81 macroscopc condtons of the system. It s called tle postulate of equal a pror probablty. We have no drect proof for t. It s reasonable and does not contradct any known laws of mechancs. It leads to results that agree wth the observatons. In classcal mechancs every pont n the phase space represents a possble mcroscopc state of the system. Therefore t s reasonable to say that the number of states n a gven regon of phase space s proportonal to the hypervolume dγ = dq... dqf dp...dp f of that regon. As classcally the possble states form a contnuum the constant of proportonalty cannot be fxed. (In the quantum theory we can thnk of the phase space as subdvded nto cells of volume h' each where h s Planck's constant. The constant of proportonalty nvolves h as a result of quantal pcture of dscrete states). 4. STATISTICAL EQUILIBRIUM A statstcal ensemble s defned by the dstrbuton functon ρ whch characterzes t. In general there are seven constant ndependent addtve ntegrals of moton n mechancs: the energy the three components of the momentum vector and the three components of the angular momentum vector. Usually energy s the only constant known. For a large system the total momentum and angular momentum have zero value or can be reduced to zero by a sutable choce of the coordnate system. We shall therefore consder only those ensembles that are functons of the energy and so useful n thermodynamcs. The energy E s a constant of the moton for a conservatve system. Let us take ρ as a functon of energy whch n turn can be expressed as a functon of q and p (E)...(.5) d d E dq de q ' d d E dp de p '...(.6) 8

82 The Louvlle theorem (.5) becomes t q p d de E q q E p p...(.7) By hypothess E = E(q p) and de/dt = so that de dt E E q p q p...(.8) From (.7 8) p t q p....(.9) n agreement wth the condton (.4) for the statonary ensemble. Obvously for such an ensemble ( E) H....(.3) Thus an ensemble characterzed by (.5) s n statstcal equlbrum. Such ensembles enable us to apply statstcal mechancs to thermodynamcs where we are nterested n a system for whch the total energy H(q p) = E s conserved q... p cons tan....(.3) E f t Locus of phase ponts correspondng to (.3) forms a (f-)-dmensonal hypersurface called an energy surface or erotc surface n the Γ space (Fg..8a). We can magne a famly of such energy surfaces constructed n the Γ space (Fg..8b). Each energy surface dvdes the phase space n two parts one of lower and the other to hgher energy. Clearly two surfaces of constant energy cannot ntersect. 8

83 Fg..8 (a) Ergodc surface. (b) Ergodc surfaces n the f-dmensonal phase space. The representatve pont of a conservatve system remans always on the same ergodc surface. The ensemble for a conservatve system at one tme wll populate one such ergodc surface. Values of q p have not been specfed n (.9). Therefore (.9) wll hold good for any pont n the Γ space. 4. MAXWELL VELOCITY DISTRIBUTION The canoncal ensemble s applcable both to macroscopc and atomc subsystems. When appled to a sngle molecule of mass m n a volume V we can wrte as P( E) Vd p exp( F / kt) exp ( p / mkt)( V / h 3 ) dp dp x y dp...(.3) z It gves the probablty of fndng the molecule n the momentum range dpx dpy dpz at (px py pz). The probablty of fndng the molecule n the velocty range dcx dcy dcz at (cx cy cz) can be expressed as 3 3 m( cx cy cz )/ kt( Vm / h dcxdcydcz exp ( F / kt) exp ) Let us N Z now evaluate exp (F/kT /Z For N= the relaton Z gves Z=z=V/λ 3 ). N 83

84 Therefore the Maxwell dstrbuton of veloctes s gven by dn( c x c y c ) dc dc dc z x y z m kt 3/ exp m( c c c )/ kt dc dc dc...(.33) x y z x y z or the Maxwell speed dstrbuton by m dn( c) dc kt 3/ exp ( mc / kt) 4 c dc...(.34) Where c c x c y c. The quantty dn(c)dc s the probablty that a partcle has z ts speed n dc at c. / 3/ Usng mc c dc () d / m we get the probablty dstrbuton that a molecule has translatonal knetc energy between Є+d Є dn( ) exp( / kt)( ) / kt f ( ) d( ). kt kt / d( ) kt Fg.9 Maxwell energy dstrbuton The Boltzmann factor e -Є/kT and the densty of states (Є/kT) / are shown by dashed curves. 84

85 In Fg.9 we show the plot of ths energy dstrbuton. The Boltzmann Factor exp (-Є/kT) decreases and the term representng the densty of states (Є/kT) / ncreases wth energy so that the total dstrbuton has a maxmum. 4. EQUIPARTITION OF ENERGY For the case N= U E Z Z N 3 kt...(.35) Thus the average energy assocated wth each varable lke p(= or 3) whch contrbutes a quadratc term to the energy has the value /kt. Ths can be verfed by drect calculaton. Suppose the Hamltonan of a system of partcles s a quadratc functon of the q s and the p s H ( a p b q ). For the partcular term a a p the average energy s p exp( a p / kt) dp akt a a kt a kt a exp( a p exp( a p / kt) dp Inexp( a p / kt) dp / kt) dp exp( a p / kt) dp / a In( kt / a ) kt....(.36) The same result s obtaned for a term lke b q. Thus each term n the H whch depends quadratcally on a p or a q contrbutes a mean energy of kt/ (theorem of equpartton of energy). 85

86 4.3 SUMMARY. Phase space: In the moton of a pont lke object n-degrees of freedom can be bracketed n q q..qn coordnates and p... p p3... pn momenta know as phase space and denoted by product of coordnate and momentum hyper volume dγ = dq... dqf dp...dp f. -Each phase space has sx dmensons whose Cartesan coordnates are x x x3 p p p3. It s called the space. 3-These elements though dentcal n structure (same macroscopc state) are randomzed n the sense that they dffer from one another n the coordnates and momenta of the ndvdual molecules that s the elements dffer n ther unobservable mcroscopc states 4-Densty of a group of phase ponts remans constant along ther trajectores n the Γ space that s t does not dsperse. The dstrbuton of representatve ponts moves n phase space lke an ncompressble flud. 5-By equpartton of energy mean energy s kt/. 4.4 CHECK YOU RE PROGRACE. Explan the terms phase space and ensemble for a system comprsng of a bead of mass n movng n a strng.. State and explan Louvlle Theorem wth the help of equaton of Hamlton equaton. Hence wrte the equaton of moton n lght of above equaton. 3. Dscuss Maxwells Velocty dstrbuton of canoncal ensembles for macroscopc and atomc subsystems. 4. Explan the theorem of equpartton of Energy wth the help of Hamltonan equaton Key-..3 &.4..6 &

87 4.5 REFERNCES. Statstcal Mechancs & Dynamcs; -H. Eyrng E.M. Eyrng & D. Henderson John Wley NY Statstcal Mechancs;- R. P. Feynman Benjamn Readng and Massachusetts Statstcal Mechancs; K. Huang John Wley NY Equlbrum Statstcal Mechancs; F.C. Andrews John Wley NY Statstcal Mechancs; R. Kubo North-Holland Amsterdam Statstcal Physcs; F. Mandl John Wley London 97. Fundamentals to Understand: 7. The Prncples of Statstcal Mechancs; R.C. Tolman Oxford UP Promnent Features: Oxford Statstcal Mechancs; B.K. Agrawal and Melvng Esner Wley Eastern Ltd: New Age Internatonal Ltd. New Delh. 87

88 UNIT V GENERAL CANONICAL CONCEPT 5.: INTRODUCTION A statonary ensemble descrbed by (q p) when behaves unformly and wthout change for all tmes then t q p and hence {(E)H} s also zero. These facts amounts to a smple choce as = constant for E = E and zero otherwse. An ensemble havng these features of densty dstrbuton s termed as Mcrocannoncal Ensemble. When we represent E as E (q.pf) the locus of ths expresson forms (f-) dmensonal hyperspaces. These hyperspaces are called energy surfaces or ergodc surfaces. Mcrocannoncal ensemble s properly used for an deal system (N V E) because the energy of such a system s constant. 5.: OBJECTIVE Now pror to understand Grand Canoncal Ensemble let us change the noton of the condton of constancy of Energy E to Ec = E +Er and a natural change n N to Nc = N + Nr c whch would be a functon of Ec Nc Es Nr. In such a state of affars f the probablty n the ensemble of fndng the system (s) s assumed as Pn then Pn (EN N) becomes equal to a constant C multpled by an exponental term. Ths C (becomes equal r (Ec Nc). The constancy of C and B when becomes ndependent of and N the dstrbuton s called Grand Canoncal. In the full text of ths unt a table s gven whch speaks about the dfferences elaborately. 88

89 5.3 GRAND CANONICAL ENSEMBLE In gong from the mcrocanoncal ensemble to the canoncal ensemble we relaxed the condton of constant energy E. Ths smplfed the calculatons n thermodynamcs where the exchange of energy s a common phenomenon. In chemcal processes the number of partcles vares. In quantum processes also partcles are beng created and destroyed. Therefore t would be useful to relax the condton of constant total number of partcles N as well. In the canoncal ensemble the subsystem could exchange energy but not partcles wth the reservor. We now consder the grand canoncal ensemble n whch the subsystem (s) can exchange energy as well as partcles wth the reservor (r) Fg (.). The varable N s replaced by the varable the chemcal potental per partcle. The composte system (c) s agan represented by a mcrocanoncal ensemble Fg.b because the total energy Ec and the total number of partcles Nc are fxed E c E E r...(.) E c N N r...(.) c ( Ec Nc Es Nr ) s ( Es Ns ) r ( Er Nr )...(.3) The phase space now depends on the number of partcles N n the quantum state of the system because t affects the number of dmensons. A partcular quantum state of the system s denoted by N. We wsh to fnd PN the probablty n the ensemble of fndng the system (s) n a gven state when t contans N = N partcles. Note: - The fgures n ths unt start from. and onwards 89

90 Fg.. (a) Isolated composte system (c) consstng of the system (s) and the reservor (r) n thermal and materal contact..(b) Grand canoncal ensemble of replcas n thermal and materal contact (exchange of energy and partcles across the conducton and porous walls) formng a lattce. Energy E=EN. Followng the arguments smlar to those leadng to canoncal dstrbuton P N ( E N) ( E EN N N) N r c c...(.4) Snce (s) s very small compared to (r) EN«Ec and N«Nc. [(s) & (c) mentoned n Fg. ln r ( E c EN N c N) ln ( E N ln r ( E N r r c r c ln r ( E ) Er N r ) Nr Nc r N r ) Er Ec E N N...(.5) The dervatves are evaluated for Er=Ec and Nr=Nc and so are constants characterzng the reservor (r) and denote them as 9

91 ln Er r Er Ec ln N r r Nr Nc...(.5) Where the chemcal potental represents Gbbs free energy per partcle then (.5) reads ( E EN N N) ( E N )exp r c c r c c ( E N )...(.6) N Snce ΔΓ( EcNc) s just a constant ndependent of and N (.4) can be wrtten as P N ( EN N) C exp N ( E N )...(.7) C B r ( Ec Nc )...(.8) Where C and B are constants ndependent of and N.Ths s called the grand canoncal dstrbuton. The constant C n (.7) s determned by the normalzaton condton N P N ( EN N) Cexp ( EN N)...(.9) N Then P N ( E N exp ( E N) N N) N exp ( E N)...(.) N Where s called the grand partton functon. It s the sum of the canoncal partton functons Z (N) for ensembles wth dfferent N s wth weghtng factors exp (βn) n Z( N)exp( N) Z( N) exp( E N )...(.) Consder a grand canoncal ensemble of M (M ) elements (Fg. b). The state of each element s characterzed by the energy EN and the number N of partcle 9

92 n t. The statstcal weght ΩgM of the ensemble assocated wth a partcular macrostate {mn} s gm M! m N...(.) mn! N To fnd the most probable macrostate {mn} we maxmze ΩgM {mn} subject to the constrants whch are generalzatons of (.) N N N m m N N N m M E N N E N c c...(.3) Where Nc s the total number of partcles n the ensemble. The result s m N M P N N exp ( E N exp ( E N) N N ) '...(.4) As expected wth the dentfcatons β=/kt and α = -/kt for the Lagrange multplers. Here α (or) s determned by the last condton n (.5). The dentfcaton of β follows from the fact that we get back the canoncal dstrbuton f assume N to have a fxed value. We can agan defne entropy by S k N P N ln P N....(.5) Substtutng (.) n (.5) and notng that EN s a functon of V alone S ke k ln k N...(.6) 9

93 ds kde k / kt P N P N ( de N N P N dv ) ( E N V ) dv k dn...(.7) We now rewrte (.6) as g U TS N P( T V ) V ktln ( T V )......(.8) where E=U and Ωg s the grand canoncal potental whch determnes the entre thermodynamcs. In partcular F U TS G U TS PV N S ( g / T ) g V N Note that from (.) droppng the suffxes N ( g / ) V T...(.9) P( N) exp( g E N / kt)...(.) exp( g / kt) exp( N / kt) exp N E( N) / kt d( N)...(.) Remember s a canoncal partton functon and Ωg s statstcal weght. 5.4 IDEAL GAS IN GRAND CANONICAL ENSEMBLE We can wrte for an deal gas Z( N) N! h 3N N V mkt N! h exp E( N) 3N N z N!' kt d( N)...(. N e N kt Z( N) N ( e kt z) N! N exp( ze kt ) exp( z )...(.3) 93

94 where we have used the seres expanson e x x ( x n / n!) and ntroduced the convenent notaton for the absolute actvty (or fugacty) kt e...(.4) It follows that g ktln ktz kt e kt mkt h 3 V...(.5) When S g ( ) kt V V ( mkt) h 3 3 e kt 5 kt...(.6) N g V T V ( mkt) h 3 3 e kt g kt...(.7) The relaton (.6) s the Sackur-Tetrode equaton. From (.7) we get the chemcal potental per partcle for an deal gas ktln ( V N)( mkt) 3 h 3 ktln( z N) ktln( n n )...(.8) where n N V s the concentraton of partcles and n 3 s called quantum Q concentraton. Thus ncreases wth ncrease n n. We see from (.4) that for an deal gas s drectly proportonal to the concentraton. The pressure s gven by (.5) P ( V ) ' kt N V g T...(.9) whch s the perfect gas law 94

95 From (.9 ) for the deal gas P N ( EN N) 3 N! h N exp ( E N N) kt....(.3) Let us take the sum of PN over for a gven N (.). Ths gves the probablty PN that a volume V of the deal gas at equlbrum wll happen to have N molecules n t rrespectve of the energy of the subsystem P N N e N e. e kt kt. N kt N! h exp( E 3N Z( N) N! h N 3N kt) exp E( N) kt d( N)...(.3) or P N exp ( g exp( N ) N) kt Z( N) N z N N z N! exp( N )( N ) N! N...(.3) where n the end we have used (.7). Ths s the Posson dstrbuton whch exhbts a maxmum near N N. Thus the bar can be dropped from N and (.8) can be taken as a proper defnton of Ωg. 5.5 COMPARISON OF VARIOUS ENSEMBLES We have shown that all the three ensembles mcrocanoncal canoncal and grand canoncal are applcable n prncple to the determnaton of the thermodynamc propertes of a system. The three ensembles are compared n Table.. As far as thermodynamc calculatons are concerned t s smply a matter of convenence whch method s followed. All of them gve equvalent 95

96 results. Usually the most convenent from the pont of vew of factorzablty of the partton functon s the grand canoncal ensemble. It s possble to construct other ensembles as the need arses. An example s of T-P dstrbuton wth ndependent parameters (T P N) and volume varable. The relaton between the grand canoncal ensemble and the canoncal ensemble s n some sense smlar to the relaton between the canoncal and mcrocanoncal ensembles. The descrpton of a subsystem by means of the mcrocanoncal dstrbuton gnores fluctuatons n ts total energy whle the canoncal dstrbuton takes t nto account. However the latter gnores the fluctuatons n the number of partcles (that s t s mcrocanoncal wth respect to the number of partcles) whereas the grand canoncal dstrbuton takes ths nto account (that s t s canoncal both as regards energy and the number of partcles). If we neglect fluctuatons n N we have Ωg +N=F and the dstrbuton (.) reduces to P(E) = exp [(F-E)/RT] and C the normalzaton condton speakes exp (F/RT) = /Z = C 5.6 QUANTUM DISTRIBUTIONS USING OTHER ENSEMBLES Canoncal Ensemble The canoncal partton functons for a system are Z exp E( q p) kt d ( classcal )...(.34) Z exp( E kt) ( quantum )...(.35) For a system (N V T) the canoncal dstrbuton s exp( E ) P Z...(.36) 96

97 where β=/kt and P s the probablty for a gven system n the ensemble to be n the state We use (.36) to descrbe a system (N V T) consstng of non-nteractng bosons or fermons. The wave functons occupaton numbers and macroscopc constrants are [4 = wave functon BE = Bose-Ensten and FD = Ferm-Drac] ( s) ( A) ( n n ( n n... n...)... n...) ( BE)...(.37) ( FD) n 3... or ( all BE) ( FD)...(.37) n N n E E ( BE FD)...(.38) The canoncal dstrbuton (.36) s P( n n... n...) exp ( n exp( n n kt) kt)'...(.39) where the summaton n the denomnator s over the set of all n satsfyng (.37). The mean value of n s gven by the average over the ensemble n nj n P{ n j} kt ln Z...(.4) where Z exp ( n kt) n n exp( n kt)...(.4) the evaluaton of Z and therefore of n s dffcult because of the restrctons on n. We do ths here for the FD case. 97

98 The constrants n = or and n=n can be ncorporated n the defnton of Z by usng the Kronecker delta (not to be confused wth the Drac delta functon) ( n) for n... for n...(.4) whch can be wrtten as N n e n ( n) e d e d...(.43) where s some arbtrary real number. Then Z n N e N e f d n j e N n exp( n exp ( kt) ( N kt) n n ) d...(.44) N f ( ) e ( b e ) b exp kt...(.45) The absolute value of f () has a sharp peak at =. We can choose such that df/d= at =. Then the phase of f () wll not change rapdly at = and most of the contrbuton to Z wll occur around =. For convenence we work wth the slowly varyng functon n f () such that d d ln f ( )...(.46) Then N s determned by b N b '...(.47) 98

99 and f ( ) exp ln f ( ) exp ln f () d d ln f () exp( an ) f ( ) d a N d d d ln f ( ) ln f ( )......(.48) It follows that N e Z N e ( an) exp( an f () ) d......(.49) and from (.4) n kt ln Z ( kt) b b ktn ln a b b b b kt ln a...(.5) b b neglectng the small quantty kt(ln a). Thus n exp ( kt) ' ( FD)...(.5) The dervaton s very smple and drect f we use the grand canoncal ensemble due to greater factorzablty of the grand partton functon. 99

100 4.7 GRAND CANONICAL ENSEMBLE The grand partton functon (.48) s N N e N Z( N ) exp n n... n n... N exp N n ( n n ( ) n...) exp ( n n...)...(.5) In the grand canoncal ensemble the lmtaton to a specfc value of N s removed as reflected n the summaton N If we frst sum over all the n n.. for fxed N then sum over all values of N from to the result of ths double summaton s equvalent to summng over all values of n n.. ndependently of each other. Every term n the frst case of double sum appears once and once only n the second case of summng each n ndependently. Ths s easly checked. Therefore n n...exp { exp[ ( ) n ]} n ( n n...) exp ( n n...) n exp[ ( ) n...(.53) We see that the grand partton functon s easly factorzed. The reason behnd t s that the occupaton numbers n n..are not constraned to add up to a fxed number. As a result the statstcal dstrbuton for each sngle-partcle state s ndependent of the presence of other sngle-partcle states.

101 In (.53) the sum n extends over the values n= 3.. for the bosons (symmetrc wave functon) and the values n= for the fermons(antsymmetrc wave functon). Usng the n n expanson ( x ) x x we get for the bosons ( s) { exp[ ( ) n ]} n exp[ ( )] for all ( BE)...(.54) For the fermons the result s obtaned drectly ( A) nor exp[ ( ) n exp[ ( )] ( FD)...(.55) From (.8 ) and the defnton of grand potental E TS N. we have for bosons g ( s) g kt kt n ( s) n{ exp[ ( )]} s g...(.56) s g kt n{ exp[ ( )]}...(.57) and N ( g / ) V T ( g / ) n...(.58

102 n ( ( s) g / ) kt { n[ exp[ ( )]} exp[ ( )] kt exp[ ( ) ( BE)...(.59) exp[ ( )] Ths agrees wth most probable dstrbuton of BE provded g= (sngle-partcle state) and... kt (.6) Note that depends upon temperature For fermons the grand potental s ( A) g ktn ( A) kt n( exp[ ( )) ( A) g...(.6) and the dstrbuton s n ( A) g ( ) kt { n[ exp[ ( )]} exp[ ( )] kt exp[ ( ) ( FD)...(.6) exp[ ( )] n agreement wth FD provded g= (sngle-partcle state) and =-β=-/kt. We can magne the th sngle-partcle state to be our system and the remanng sngle-partcle states to be the heat and partcle reservor. Exchange of energy and partcles occurs on collson. The sum n (.35) s over all mcrostates or quantum states of the system. The energes of the varous sngle-partcle states E are not necessarly dfferent. Then the sum wll nvolve many repeated equal terms gven by the degeneracy.

103 To nclude the degeneracy term g the partton functon (.35) can be expressed n an equvalent way as a sum over the dfferent energy levels and the degeneraces of the levels Z gexp( E ) levels ( all E dffrent )...63 The form (.35) s used for smplcty but for actual fnal calculatons one must remember to nclude g f the sum s over the energy levels For an deal gas e / N exp( / kt) ( classcal lm t) ( V / N) 5.6 M 3 T / P where P s the pressure n atmospheres and M s the molecular weght n atomc mass unts. The rght sde of (.65) s about 6 for ar at NTP -4 for electrons at room temperature and for helum gas at K atm. Thus the classcal statstcs can be used for ar fals for electrons and nearly fals for helum gas under above condtons. When the classcal dstrbuton fals we are n a regon where the dstrbuton s degenerate. 5.8 CANONICAL PARTITION FUNCTION Even when quantum effects are neglgble (classcal lmt) the approprate language for the descrpton of a system s provded by the quantum theory. To characterze a system we should therefore specfy the energy egen values and the correspondng wave functons for the system as a hole. Consder a system A composed of two nonnteractng dstngushable (localzed) atoms a b. The one partcle wave functons and energy levels are u u... u...and Є Є... Є... respectvely. The egen value equatons are H au (a) = Є a u(a) and H bu (b) = Є b u(b). For the whole system 3

104 H H a H b...(.66) ( j) u ( a) u ( b)...(.67) j E( j) a bj...(.68) The double ndex (j) denotes a sngle state of the composte system. The one partcle canoncal partcle functon s Z exp( )... (.69) and the canoncal partton functon for the whole system s J exp( E A ) exp ( a Z )... (.7) A j bj We can wrte Z as Z exp( a ). j exp( bj ) z a. z b... (.7) Where the summatons extend over all quantum states of the ndvdual atoms a and b. Snce a and b are dentcal Єa= Єb and za= zb. Then Z z...(.7) and generalzng t to a system of N dentcal but dstngushable partcles Z z N ( dentcal dstngushable partcles )....(.73) If a and b are ndstngushable partcles the wave functon must be symmetrzed (or antsymmetrzed). Then the! Ways of obtanng E (j) n Є a+ Є bj 4

105 and Є aj+ Є b correspond to only one wave functon ψa=u(a) uj(b)+ u(b) uj(a) for the symmetrc case. Consequently n the sum (.7) for a gven E (j) A we have! Terms n the summaton whch dffer only n the partcle labels. For example the two terms exp [-β (Є a+ Є b)] and exp [-β (Є a+ Є b)] correspond to the same energy E () A. Because our sum should contan only one term for each dstnct ψa the summaton s too large by a factor of!. Therefore for ndstngushable partcles we should replace (.7) by Z z!...(.74) and (.) by Z z N! N ( ndstngushable partcles )...(.75) n agreement wth prevous results. Thus the Boltzmann countng appears as a natural consequence of the symmetry of wave functons n quantum theory. The use of (.75) descrbes a Boltzmann gas. The molecules are dentcal (ether bosons or fermons). When many more partcle states than partcles are avalable (.7) the dfference between bosons and fermons can be neglected and (.64) used along wth (.75). 5.9 MOLECULAR PARTITION FUNCTIONS For dstngushable partcles formng a system we have to use (.73). The dstngush ablty of dentcal partcles arses when the classcal lmt rav>λ holds and the partcles can be treated as localzed. Another example s that of partcles constraned to occupy fxed lattce stes as n a crystal. Asde from localzaton any other measurable property of the partcles such as ther nternal state may be used to dstngush the partcles or ther states. For example 5

106 6 consder a system of two datomc molecules. The system can be found n two dstnct states ) ( ) ( ) ( ) ( b u a u b u a u II I where u(a ) means the molecule occupyng the translatonal state a s n the vbratonal state. In ths case the occupaton of dstnct translatonal quantum states provdes as much ground for dfferentaton as was provded by the occupaton of dstnct spatal postons n a crystal. In molecular system varous nternal degrees of freedom are only weakly coupled to the external degrees of freedom and each other. Therefore the total energy E of the system can be expressed as the sum of translatonal (t) vbratonal (v) rotatonal (r) electronc (e) and nuclear (n) energes...(.76) ) ( ) ( ) ( ) ( ) ( n E e E r E v E t E E To ths approxmaton we can wrte the total partton functon ZT for the system as...(.77) ). ( ) ( ) ( ) ( ) ( )) ( exp( )) ( exp( )) ( exp( ) ( ) ( ) ( ) ( ) ( exp ) exp( n Z e Z r Z v Z t Z r E v E t E n E e E r E v E t E E Z k k j j T Usng (.73.75) approprately for a gass of N dentcal molecules )...(.78 ) ( ) ( ) ( ) ( )) (!)( (/ N N N N N T n z e z r z v z t z N Z where z represents the partton functon for a sngle molecule. For the translatonal moton we have used (.75) because each molecule s free to move about n the whole volume and so the states cannot be dstnctly labelled as n the case of solds. The factor (/N!) multplyng [z (t)] N s approprate only when there s no degeneracy (all molecules occupy dfferent quantum states). If the

107 occupaton number of a gven state ψ s much less than effects arsng from symmetry mposed lmtatons on n are unmportant and classcal statstcs apples. Ths requres a large number of avalable system s states for a gven number of partcles. Then the use of (.78) and MB statstcs wth correct Boltzmann countng for the translatonal states s a good approxmaton. Excepton wll arse for example for helum at about K whch remans a gas at ths low temperature and the number of accessble translatonal states becomes comparable to the number of atoms N. then we must use the BE statstcs. Under normal condtons Z exp( ) ( states)...(.79) where the sum s over all the allowed quantum states of the molecule or equvalently Z g exp( ) ( dfferent ) ( levels)...(.8) where g s the degeneracy of the level and the sum s over the energy levels Є of the molecule. To a good approxmaton E Total ( t) ( v)... j...(.8) z T z( t) z( v) z( r) z( e) z( n)....(.8) so that (.78) smply reads Z T z N T N!...(.83) 5. TRANSLATIONAL PARTITION FUNCTION The classcal value of z(t) s gven by an deal gas n canoncal Ensemble as 3 z( t) V / h (mkt )...(.84) 7

108 To derve ths result n quantum theory consder the translatonal energy levels of a molecule enclosed n a box n x ml x x n x (.85) where Lx s the sde length of the rectangular box along the x-axs. Smlar expressons exst for the moton parallel to y and z axes. The translatonal partton functon s where z( t) N x exp( n x x ) N y exp( n y y ) N z exp( n z z )...(.86 ) x ( L x kt) etc...(.87) For ordnary temperatures and large sze of the box the quantty x s much less then unty. Hence x n x changes slowly as we vary nx. For m= -3 g 3 and Lx=cm we get ( h ml )( ) erg. The correspondng x x characterstc temperature for the translatonal moton s t ( x k) 4 K At temperatures / Tthe energy levels are closely space and we can replace the summaton by ntegraton t N exp( n x x ) exp( n x ) dn / x x x x...(.88) Usng V=Lx Ly Lz and 3/ / V x Lx ); Z( t) where 3 x y z m / / N n agreement wth (.). We have Z( t) [ z( t) N! ] and F(t) s gven by F( t) NRT ln Z NRT N 8

109 Thus we get for the system from E Z Z ln Z relaton d ln Z( t) E( t) d V d ln ( z( t)) d N! N and the entropy S(t) as d ln d 3N 3 N d ln 3N...(.89) d as contrbuton of translaton part V 5 S( t) NK ln 3 N NK 5. ROTATIONAL PARTITION FUNCTION For a datomc molecule the rotatonal energy levels are gven by ( r) ( / I) J( J ) J...(.9) Where I s the moment of nerta of the molecule and J s the rotatonal quantum number. The degeneracy of each level of a two-dmensonal rotator s g(r) = J+ snce there are J+values of the z component of angular momentum. Therefore Z( r) for j (J ) exp[ ( / I) J( J )]...(.9) I mr ~ 3 ( 8 ) 39 g cm ( r)~ 5 erg r / k~k when r / T we can replace the sum by ntegraton Z ( r) exp[ ( / I) J( J )] (J ) dj IKT T r r IK...(.9) where we have neglected unty n comparson to J and used 9

110 xe dx (/ a) The rotatonal moton about the axs (thrd rotatonal degree of freedom) s not counted at ordnary T because t nvolves the exctaton to hgher states of electronc angular momentum. Such exctatons requre large amounts of energy as the moment of nerta about ths axs s very small. The total wave functon ψt of any molecule ncludng datomc molecules can be expressed as T nternal t e v r n t ( molecule ) The ψe specfes an electronc state. The product ψe ψv specfes a vbratonal level because such levels are assocated wth some partcular electronc state. Smlarly a rotatonal level s specfed only when the product ψe ψv ψr s gven. We call ψv and ψr by themselves the vbratonal and rotatonal egen-functons respectvely. The ψr depends only on the magntude of the nter-nuclear dstance and so s no changed by reflecton n the orgn. For symmetry consderatons t s enough to examne e r The ψe s descrbed wth respect to asset of coordnates rgdly attached to the nucle. It s unaltered f the nucle rotate. We shall gnore nuclear spn here. For a smple rotator (heteronuclear) the ψr are smply sphercal harmoncs YJM whose party s (-) J. Ths means under a reflecton n the orgn the ψr changes sgn only for odd J. If the ψe of the ground state s even and does not change n chemcal reactons and phase transformaton the ψ s symmetrc for J even and antsymmetrc for J odd. For homonuclear molecules the nterchange of nucle can change the sgn of. Then each rotatonal level carres two symmetry labels: plus or mnus wth respect to a reflecton n orgn of all the partcles symmetrc or antsymmetrc

111 wth respect to an nterchange of nucle. A proper quantum treatment of the ndstngushablty of lke partcles gves the rule: symmetrc lnear molecules can have ether even ( 4 ) values or odd ( 3 5..) values of J but not both Therefore when the atoms n a datomc molecule are a lke the allowed energy levels wll yeld a summaton term just half as bg as f the atoms were dfferent. In such a case we must dvde the sum over all states by the symmetry number n=. We therefore should wrte z( r) T ( ) ( lnear molecule ) n r...(.93) where n= for an asymmetrcal molecule and n= for a symmetrc lnear molecule. The symmetry number s just the number of ndstngushable orentatons of the molecule. Any molecule can only exst n /n of the rotatonal energy levels. Thus n= for HO 3 for NH3 and for methane. For an deal gas of datomc molecules we have for T > r z( r) F( r) U ( r) kt N z( r) T ( n r ) NkT ln T ( ) C ( r) d U ( r) / dt v ( ln Z / T ) NkT n d dt r N N kt ln Z( r) Nk...(.94) T Note that Z(r) does not exst for a monatomc gas. For datomc gases F=F(t)+F(r) U=U(t)+U(r)=5 NkT+NkT=5 Nkt and Cv=5 Nk where F(t) s gven by F Z NRT ln NRT. Smlarly for Entropy N S( r) k ln Z( r) U( r) / T lt k Nkln ln so that for one mole of gas S ( r) Rln( IT / ) (.95)...(.96)

112 5. VIBRATIONAL PARTITION FUNCTION A datomc molecule has one degree of freedom assocated wth the vbratonal moton of the nucle along the axs jonng them. The vbraton of the massve atomc nucle s drven by the force provded by the electronc molecular dstrbuton whch acts as a lnear sprng yeldng a harmonc moton. The Hamltonan for a lnear harmonc oscllator of frequency v s H ( x p x ) ( p x / m) k s x k s m( v) The egenvalues for ths system are x ( n ) hv n (.97) It s obvously unreasonable to ntegrate to fnd z(v). Therefore the vbraton partton functon for the molecule s Z( v) n exp[ ( n ) hv ] exp( hv / )[( exp( hv) exp( hv)... ] exp( hv / )[( exp( hv)] [snh( hv / )].(.98) If we count our energes above the zero-pont vbratonal energy hv = constant as t represents vbratonal moton of the molecule n ts ground stete we get Z( v) exp( hv).(.99)

113 At room temperature (3k) a typcal value s ~ 3 v 6 s hv / k ~ 3K and so z v ~ For the whole system consstng of N (dstngushable) ndependent oscllators Z( v) [ Z( v)] N F( v) KT ln[ Z( v) N NkT ln[( exp( hv)] d ln z( v) U( v) kt N dt Nhv exp( hv) and; C v N( hv) exp( hv) ( v). kt exp( hv) )'...(.) At low temperatures (kt hv) these quanttes tend exponentally to zero F( v) NKT exp( hv) C v ( v) Nk( hv / kt) exp( hv) At hgh temperatures (kt hv) F( v) N( hv) NkTln( kt) NkT ln( hv) C v ( v) Nk. The contrbuton to entropy s S ( v) k ln Z( v) U( v) / T 3

114 Nk ln exp( hv) N T Nk hv kt exp( hv).(.) In c.g.s. unts h 8 Ik 4.3 I 4 r v hv k v.8 where r s the characterstc temperature of vbraton. For oxygen molecule r =.7 k v =hv/k=3k so that at 73K for one g-mole 3 z( t) : z( r) : z( v) 3.4 : 65.3:.6 S ( t) : S( r) : S( v) 35.9:8.5:.4(Cal/g-mole K) The values of parameters for a few datomc molecules are gven n Table. 4

115 Table. The values of parameters for some datomc molecules Molecule r (K) I(g.cm ) v (K) v( rad / s) H X X 4 HCL 5..65X X 4 N.9.4X X 4 O..94X X 4 CL.3.6X- 8.6X 4 Table. Comparson of the varous ensembles Ensemble (macroscopc) Mcrocanonc al (VNE) Canoncal (VNT) Dstrbuton Partton functon Thermodynamc functon ( Contact wth surroundn g P E) ( E E ) or n ) -TS=-kT ln none P l Ce ( l E l kt Z e d E / F=E-TS=-kT ln Z thermal Grand canoncal (VμT) T-P dstrbuton (TPN) P N Ce ( B E ) N P( dv ) C e[ E ( v) PV ] dv e N E / kt Y e e N / kt d( N). Z( T V N) dv g E TS N Thermal+ materal ktln PV / RT G = kt ln Y Thermal+pr essure (movable wall) 5

116 5.3 Thermodynamc Functons: Entropy: The connecton between statstcal mechancs and thermodynamcs s provded by entropy. Let Ωr(E) denote the volume n space enclosed by the ergodc surface of energy E ( E ) r E( q p) E dq... dq f dp... dp f '...(.) The volume of the shell (E) between the ergodc surfaces E and E+ whch s occuped by the mcrocanoncal shell s gven by r ( E) ( E) r ( E A) r ( E) g( E) E EH ( q p) E f f dq... d q d p ( q p)...(.3) where (q p) = f E < H(q p) < E+ and otherwse. The quantty (E)= Ωr(E)/dE s called the densty of states of the system. The entropy of a system n statstcal equlbrum s defned by ( V E) ln ( E)...(.4) Ths defnton s the same n quantum statstcal mechancs except that (E) s then calculated n quantum mechancs. The dmenson of Ωr(E) or of s (length momentum) = (acton). If s magned to be dvded nto elementary cells of volume h where the Planck constant h has the dmenson of acton then /h f s dmensonless. Defnng ln ( / h f ) ln f ln h...(.5) 6

117 we fnd that the change n entropy = ln s ndependent of the unt as t should be. If (.5) s to be acceptable t should exhbt the well-known propertes of the thermodynamc entropy S. To show ths we consder the equlbrum between two systems (N V E ) and (N V E ) n thermal contact. Composte System Consder two systems (N V E ) and (N V E ) that are solated from each other Fg..(a). We can construct the mcrocanoncal ensemble for each taken alone Fg.. (b) N V E (a) N V E [P ] E + E [P ] E + E (b) Fg. (a) Two solated systems. (b) Mcrocanoncal ensembles for the two systems. The coordnates and momenta of the molecules n the two systems are denoted by (q p ) and (q p ) respectvely. P or = constant for energy between E and E + = otherwse P or = constant for energy between E and E + = otherwse 7

118 Let (E ) and (E ) be the volumes occuped by the two ensembles n ther respectve spaces. Now magne the two systems to be brought n thermal contact and thereby form a composte system n equlbrum Fg. (b). The mcro Conductng Wall N V N V E = E + E Fg (.3) Composte system. The total energy s E = E + E. Canoncal ensemble for the composte system s defned by P or constant for energy between otherwse (E E'') and E' E'' Δ...(.6) Ths ensemble wll contan all mental copes of the composte system for whch: a. N partcles wth coordnates and momenta (q p ) are contaned n the volume V. b. N partcles wth coordnates and momenta (q p ) are contaned n the volume V c. E=E +E the total energy les between E and E+. d. n n are the numbers of molecules of the two knds that occupy the varous cells j nto whch the μ spaces for dfferent knds of molecules are dvded. The space of the composte system s the product space of the phase spaces of the ndvdual parts ( E' E'') ( E') ( E'')...(.7) 8

119 Extensve Property of Entropy From (.44) and (.4) we at once get the addtve property of entropy ln ln ln...(.8) In (.7) (E E ) s the accessble regon of space for the composte system when the system has an energy E. State of Equlbrum s the State of Maxmum Entropy From the basc postulates of statstcal mechancs the equlbrum state s the one for whch s maxmum. Therefore the entropy of a system n equlbrum should really be defned by ( V E) ln ( ) MP'...(.9) where the suffx denotes assocated wth the most probable dstrbuton. However the dstncton between the orgnal defnton (.4) and (.9) s unmportant for large systems. Ths s so because depends on ln whch s a very slowly varyng functon of. For example for a system of N ~ partcles ( )MP N! MP N ln N-N N ln N ln ~ 46. If n measurng we make an error of as much as the error n entropy s only = ln 3 MP. For the composte system at equlbrum we can wrte (.7) as ( E' E') ( E') ( E'') ( E') ( E E')...(.) The r and so s a rapdly ncreasng functon of energy E for a large system. The physcal reason s that the energy E as t ncreases can be dstrbuted n many more ways over the mcroscopc degrees of freedom of the system. Ths rapdly ncreases the number of accessble mcrostates. 9

120 For an estmate let us consder the case of an deal gas f = 3N. Then (.) can be expressed as ( E ) V r E N dp... dp 3N '...(.) where V N dq dq3n dxdxdx3dxdxdx3... dx NdxNdx3N because the energy of an deal gas does not depend on the postons of molecules. Snce E 3N p m ths ntegral s just the volume contaned n the 3N-dmensonal hyper-sphere of radus (me) /. The volume of an n-dmensonal hyper-sphere of radus R s V ( R) R n n n n n / / n / ( n ) /( n)!....(.) Therefore Ωr(E) can be wrtten as ( E) const. E r 3N /...(.3) whch for N~ certanly vares very ra;dly wth E. In fact t does not matter whether we wrte r~e / or r~e. Snce =g(e) for a gven we have g(e)~ E - ~ E. ) as f f r ( E ) r ( E ) const. E ( E ) For ~ /E and ~ const. E. f const. E E const. E f f ( e )....(.4) ~ f / E

121 Thus s practcally the volume of the whole hyper-sphere of radus (me) / ~r(e). It follows that n (.) the term (E ) ncreases rapdly wth E and the term (E-E ) decreases rapdly wth E (Fg.4 a). As a result or the combned densty of states has a sharp maxmum at say E ' E' ' E E' and (.) becomes ( E E') ( E') ( E'') maxmum...(.5) Ths mples Fg..4 (b) ( E') ( E'') maxmum...(.6) The value by the condton E ' whch corresponds to the maxmum of ln ( E ') s determned E (E-E ) (E ) ln. ln ln ln Ē E E Ē E (a) (b) Fg.4 (a) The combned densty of states wth a sharp maxmum at E ' (sold curve). (b) The dependence of ( ') and ln ( E ') on E (dotted curves) E E showng a unque maxmum (sold curve) at some value Ē for ( E '). E ln ( E E') ' E ( E E') ' ( E E') E...(.7) or

122 ln ( E') ln ( E'') ( ) ( E E' E'' E' E''). ' E E'' If we defne a quantty by then (.8) can be wrtten ( V E) E ' ' '' '' ( E ) ( E ) E E E '...(.8)...(.9)...(.) Ths s k the fundamental condton whch determnes the partcular value E = Ē (and so also the value E =E - Ē = Ē ) for whch the composte system occupes the maxmum volume n space that s the system s n equlbrum (or most probable state). Ths s also the state of maxmum entropy. Ths dscusson gves meanng to the quantty of an solated system. Thus of an solated system s the parameter that determnes the equlbrum between one part of the system and another. Prncple of Entropy Increase Let us dscuss the problem of the approach to thermal equlbrum. We have just seen that (E E ) has a sharp maxmum (fg..4 a) at E = Ē. It means the system l almost always has energy close to Ē and the system has an energy close to Ē = E - Ē when n contact. Suppose and are ntally separate. Ther mean energes at equlbrum are Ē and Ē respectvely. If they are brought n thermal contact an exchange of energy takes place untl the equlbrum of the composte system s reached. For ths the fnal energes Ēf and Ēf of the system must be such that E ' f E ' and E '' f E E ' E "...(.)

123 So that becomes maxmum or (.6) s satsfed. Thus the exchange of energy goes on untl the total entropy becomes maxmum. It mples that the fnal probablty can never be less that the ntal probablty ( E ) ( E ) ( E ) ( E ' '' ' '' f f )...(.) The world entropy s derved from the Greek word whch roughly means evoluton (Gr. en n +trope turnng). It ndcates the turn or drecton taken by the process. We have shown that the defnton (.) of entropy n statstcal mechancs based on mcrocanoncal ensemble possesses all the basc propertes of the thermodynamc entropy S. The mcrocanoncal ensemble for the composte system s defned n the shall over a range of values (E E ). However at equlbrum the most probable partton of energy (Ē Ē ) prevals for a macroscopc system. It mples that almost all the elements of the ensemble have the values (Ē Ē ). The followng defntons of are equvalent up to addtve constants: ln ( E) ln ( n ) ln g( E) lnw ln ( E)...(.) N! The last relaton s called Boltzmann s defnton of entropy and ( n ) n! n we get ln N! N ln N ln n! n ln n 3

124 At equlbrum (n) has ts maxmum value and n s gven by the Maxwell - Boltzmann dstrbuton. Therefore apart from a constant N ln N N ln N N E) Entropy The entropy s defned by S k[ N _ ln ( N _/ N) N _ n ( ) ln ( N )] N where N- N+ depend on E through n. In the approxmaton...(.3)...(.4 ) n W N ln n N S ~ k ln W k N ln n N ( E / ) N k ln N...(.5) A plot of S/Nk aganst E/ЄN s shown n Fg.5 S / Nk T= + ln T= - S E > S E < T= + T= E / ЄN Fg.5 Negatve temperature n a two-level system. The slope S/E gves the sgn of T. 4

125 The case of zero moment n = gves S = kn ln n the Strlng approxmaton (Nlarge). Ths s just the entropy of all possble arrangements ( = N ) because each partcle can have two possbltes parallel or ant-parallel moment regardless of how the other partcles are. It shows that n calculatng entropy no sgnfcant error s made f we assume that the entre accessble phase space has the propertes of the most probable condton of the system wth large N. In other words for large N the entropy s nsenstve to the precse specfcaton of the condton of the system n the neghbourhood of the most probable condton. Negatve Temperature The temperature s defned by the slope of the S versus E curve of fg (.5) T S E V ' N S n k N n ln N n k N ln N...(.6 ) Where for N+ N- and S. At the conventonal absolute zero all the partcles are n the ground state (Є-) n = N+ - N- = N < gvng T = + and S = (complete order) at E = -N Є. As the temperature s rased or energy s gven to the system the populaton N+ n the upper level (Є+) grows T > tll we have n = gvng S = Nk ln (maxmum dsorder) and T = +. If more energy s gven to the system the upper level becomes more populated than the lower level N+ N- n > and we get a decrease n S (more order) and T <. Thus the system s no longer normal n ts behavour. The negatve temperature T- corresponds to hgher energes than postve temperature T+. If T- and T+ systems are brought nto thermal contact energy wll flow from T- to T+ that s T- s hotter than T+. A maser or a laser s a devse based on such a T- system. As more and more energy s gven to the system we get n = N+ - N- = N = N > that s all the partcles are n the upper states S = (complete order) and 5

126 T = - (Fg.4). The dfference between the two zeros (+ -) s that we approach the frst of them from the sde of postve and the second from the sde of negatve temperatures. The possble temperature ranges are from + through + - to - + and - concdng wth each other. For a system to have negatve temperature t must () have a fnte upper lmt to the energy spectrum () be n nternal thermal equlbrum and () have negatve temperature states solated from those states that are at postve temperature. Specfc Heat From (.6) we get wth E = Є N n N N n N exp kt exp E kt N N exp exp kt kt exp kt exp E kt' N N exp exp kt kt exp kt exp E kt...(.7) These relatons for N-/N and N+/N gve the probabltes of fndng any one partcle n the states - Є and Є respectvely. The energy E and the specfc heat C are gven by E n ( N N ) N tanh ( / kt)...(.8 ) C de kt N k dt cos h kt 6

127 E N k kt exp exp E E kt kt...(.9) These quanttes are plotted n Fg..6 (a). The specfc heat of the peaked form Fg..6 (b) s called the Schottky anomaly. It s observed when a body has a gap E n ts energy states..6 N + / N E ЄN - ½ C / Nk kt Є 4 6 kt Є Fg..6 (a) Energy and (b) specfc heat and the fractonal populaton N+/N n the upper state as functons of kt/є for a two-level system. 5.4 Summary -Mcrocannoncal ensemble s properly used for an deal system (N V E) because the energy of such a system s constant. -Canoncal Ensemble n whch the number of partcles N system volume V and temperature T are constrants. 3-In the canoncal ensemble the subsystem could exchange energy but not partcles wth the reservor 4-The canoncal partcle functon s 7

128 Z exp( ) 5-Grand canoncal ensemble n whch the subsystem (s) can exchange energy as well as partcles wth the reservor (r) 6 In Grand canoncal partton functon. It s the sum of the canoncal partton functons Z (N) for ensembles wth dfferent N s wth weghtng factors exp (βn) 7-Total partton fucton of molecules are product of translatonal vbratonal rotatonal z( t) z( v) z( r) z T 5.5 CHECK YOR PROGRACE. Descrbe Grand Canoncal Ensembles of on deal gas and obtan deal gas Laws from t.. Gve the comparson of mcro canoncal canoncal and grand canoncal ensembles of thermodynamc propertes of a system. 3. Dscuss Extensve Property of Entropy n a thermodynamc system of ensembles Key REFERNCES -Statstcal Mechancs & Dynamcs; -H. Eyrng E.M. Eyrng & D. Henderson John Wley NY Statstcal Mechancs;- R. P. Feynman Benjamn Readng and Massachusetts Statstcal Mechancs; K. Huang John Wley NY Equlbrum Statstcal Mechancs; F.C. Andrews John Wley NY Statstcal Mechancs; R. Kubo North-Holland Amsterdam Statstcal Physcs; F. Mandl John Wley London 97. Fundamentals to Understand: 8

129 7- The Prncples of Statstcal Mechancs; R.C. Tolman Oxford UP Promnent Features: Oxford Statstcal Mechancs; B.K. Agrawal and Melvng Esner Wley Eastern Ltd: New Age Internatonal Ltd. New Delh. 9

130 UNIT-VI BOSE-EINSTEIN AND FERMI-DIRAC DISTTIBUTION 6. INTRODUCTION: These dstrbutons fall under the realm of quantum statstcs as aganst the classcal Maxwell s dstrbuton whch lays more stress on the veloctes of partcles /members: If we assure only two members the three statstcs can be pctursed as follows. Probablty of Dstrbuton Classcal Mech. a b a b b a b a Here members a & b are dstngushable n ther cells and can have four possble occupancy Bose Ensten Statstcs ٠٠ ٠ ٠ ٠٠ Symmetrc State Here a and b are not dstnct and thus take the shape of dots beng nondstngushable. Hence the mportance of cells arses. Photons mesons etc are examples whch have zero or ntegral spns. Paul Excluson Prncple does not apply. Such members are called Bosons. Ferm-Drac statstcs ٠ ٠ Antsymmetrc state Identcal ndstngushable partcles whch obey Paul Excluson Prncple. Thus the mportance of cell arses. The member partcles can be electrons. Protons neutrons etc. and possess half-ntegral spns. Such members are called Fermons. 6. OBJECTIVE The Bose-Ensten and Ferm-Drac statstcs are the two basc theores whch are wdely applcable n studyng the materal at atomc and subatomc scales. In solvng the Schrödnger equatons n quantum mechancs these statstcs are 3

131 requred. Therefore n the present text we would descrbe these statstcs derves these functons and study ther applcatons n smple problems. 6.3 BOSE-EINSTEIN DISTRIBUTION For an deal BE gas of N molecules n a volume V the most probable number of partcles wth energy Є s g g n ( )....(3.) exp( ) exp ( ) / kt ' para - -βµ and g = degeneracy of the th level. The N g n exp ( ) exp ( ) ( levels) g.(3.) where the sum s over the energy levels as we have ncluded g n eq (3.). Equvalently we can replace g by n (3.) and then sum over the quantum states n (3.). We must have n because the number of partcles n a level cannot be negatve. Therefore for a boson gas at all temperatures (Є - µ) must be greater than zero for all Є that s exp ( / kt) or....(3.3) We can replace the sum by an ntegral n (3.) by usng n place of g the densty of states g( ) d V (m) h d 3 / / 3...(3.4) Then (3.) becomes 3

132 N V (m) 3 h g( ) d e e 3/ / d e Note:- Take n as (a) (b) etc agan. n ~ all through. The fgures n ths unt III are numbered as V F ( 3 3 / 3 / ) Vn F ( )...(3.5) Where η = the absolute actvty (or fugacty for a gas) nω = l/λ3 = quantum concentraton (concentraton assocated wth one atom n a cube of sde equal to λ) e h / (mkt) '...(3.6) and wth x / kt F 3/ ( ) / x / x e dx / x x x dx x e ( e e /...) 3... n 3 / 3 / 3 3 / n n...(3.7) The F3/(η ) s a specal case of the general class of functons F ( ) s n n s n...(3.8) 3

133 llustrated n Fg. (3.). At the lmtng value η = (or µ = ) the dervatve of F3/(η ) dverges but the seres (3.7) converges Fg. 3. (a) The functons F3/ 5/ - curve C. (b) The functon F3/(η ). Remember that η = e- F 3/ ( ) n n 3/ / ( ) 3 3 /.6...(3.9) where ζ s the Remann.zeta functon. Ths s the maxmum possble value of F3/(η ) due to (3.3) and (3.6). We can defne a mnmum temperature T called the crtcal temperature at whch η has the maxmum value then (3.5) becomes V mkt N.6 V 3 h 3 /.6...(3.) If we have one mole of gas so that N s Avogadro number 33