Semiclassical Statistical Mechanics

Transcription

1 Semclasscal Statstcal Mechancs from Statstcal Physcs usng Mathematca James J. Kelly, A classcal ensemble s represented by a dstrbuton of ponts n phase space. Two mportant theorems, equpartton and vral, are derved from ths concept. However, fnte classcal entropy requres dscretzaton of the phase space usng a volume obtaned by comparson wth quantum mechancal results for prototypcal systems. Applcatons are made to harmonc oscllators, deal gases, and gases of datomc molecules. Classcal phase space A classcal system wth f degrees of freedom s descrbed by generalzed coordnates and momenta whch satsfy the equatons of moton q = H ÅÅÅÅÅÅÅÅÅÅÅÅ p p =- H ÅÅÅÅÅÅÅÅÅÅÅÅ q where H = H@q, p,, q f, p f, td s the Hamltonan functon. Each pont n the 2 f dmensonal phase space represents a dfferent mcrostate of the system. It s useful to let x = 8q, p,, q f, p f < represent a pont n phase space such that x@td represents the moton of the system, ts trajectory, through phase space. Suppose that A = A@x, td s some property of the system that depends upon the phase-space varables and may also depend explctly upon tme as well. The total tme rate of change for ths property s then A ÅÅÅÅÅÅÅÅÅÅÅ t = A ÅÅÅÅÅÅÅÅÅ t f + = j ÅÅÅÅÅÅÅÅÅÅÅ A q k q + A ÅÅÅÅÅÅÅÅÅÅÅÅ p Thus, f we defne the Posson HD as HD = = j ÅÅÅÅÅÅÅÅ A ÅÅÅ k q we obtan the tme dependence A ÅÅÅÅÅÅÅÅÅÅÅ t = A ÅÅÅÅÅÅÅÅÅ t HD H ÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅÅÅ A p p p y { z = ÅÅÅÅÅÅÅÅ A t H y ÅÅÅÅÅÅÅÅÅÅÅÅ z q { f + = k j ÅÅÅÅÅÅÅÅÅÅÅ A q H ÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅÅÅ A p Therefore, f we assume that the Hamltonan does not depend explctly upon tme, H ÅÅÅÅÅÅÅÅÅÅÅÅ t = 0 ï ÅÅÅÅÅÅÅÅÅÅÅÅÅ H = 0 t p H y ÅÅÅÅÅÅÅÅÅÅÅÅ z q {

2 2 Semclasscal.nb we mmedately fnd that the energy E = H@ x@0dd s a constant of the moton for conservatve systems. States of specfed energy are confned to a 2 f - dmensonal hypersurface embedded n the 2 f dmensonal phase space. A classcal ensemble conssts of a set of ponts n phase space, wth each pont representng a system n a specfed f mcrostate. The number of ponts n a regon of phase space near x n a volume G = = q p s gven by r@x, td G, where the phase-space densty r@x, td s the classcal analog of the quantum mechancal densty operator. As each member of the ensemble moves through phase space along a trajectory specfed by Hamlton's equatons of moton, the phase space densty evolves n tme. Consder a smply-connected volume G bounded by a surface s. The number of systems wthn G s gven by N G = G Gr@x, td Usng the dvergence theorem, the rate at whch members of the ensemble leave volume G can be expressed n terms of the flux through the surface s as s ÿ j = G ÿ j s G where j =rv s the current based upon phase velocty v = 8q,, p f <. Thus, recognzng that members of the ensemble are nether created nor destroyed, the phase-space densty must satsfy a contnuty equaton of the form r ÅÅÅÅÅÅÅÅ + ÿj = 0 t The two contrbutons to the dvergence ÿhr v L = v ÿ r + r ÿv can be smplfed usng Hamlton's equatons of moton v f ÿ r = = ÿv f = = j ÅÅÅÅÅÅÅÅÅÅÅ r k q q ÅÅÅÅÅÅÅÅÅÅÅ t Therefore, Louvlle's theorem r ÅÅÅÅÅÅÅÅ t = r ÅÅÅÅÅÅÅÅ t + r ÅÅÅÅÅÅÅÅÅÅÅÅ p p y ÅÅÅÅÅÅÅÅÅÅÅÅ z HD t { f j q ÅÅÅÅÅÅÅÅÅÅÅ + p y 2 H ÅÅÅÅÅÅÅÅÅÅÅÅ z = j ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - 2 H y ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ z = 0 k q p { k q p p q { = + v ÿ r = ÅÅÅÅÅÅÅÅ r HD = 0 t states that the local densty n the phase-space neghborhood of a representatve pont s conserved. In flud mechancs, the total dervatve ÅÅÅÅÅÅ r = ÅÅÅÅÅÅ r + v ÿ r, known as a convectve dervatve, measures the tme dependence of the densty for a t t movng parcel of flud vewed from a pont that moves wth that parcel, rather than from a fxed locaton. Accordng to Louvlle's theorem, the phase-space densty for a system obeyng Hamlton's equatons of moton behaves lke an ncompressble flud. Therefore, as a classcal ensemble evolves ts locaton and shape n phase space may change but ts volume s conserved. The ensemble average of property A whch depends upon the phase-space varables s XA\ = Ÿ G r@x, td A@x, td ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Ÿ G r@x, td where the ntegral spans the entre phase space. Note that Louvlle's theorem ensures that the denomnator s constant. In order to obtan a statonary expectaton value we normally requre that nether A nor r depend explctly upon tme, but

3 Semclasscal.nb ths condton s not suffcent because as the phase pont moves along the trajectores specfed by ther equatons of moton the densty r@x, td evolves. Equlbrum requres equlbrum ï ÅÅÅÅÅÅÅÅ r = 0 HD = 0 t and s most smply acheved usng an ensemble r =r@hd based upon the Hamltonan. The canoncal ensemble canoncal ensemble : r Exp@-bHD clearly satsfes ths condton and, hence, s statonary. Alternatvely, the mcrocanoncal ensemble consstng of all states wthn the volume of phase space wthn a specfed range of energy mcrocanoncal ensemble : r = G - Q@dE -» E - H@xD»D s also statonary. Countng states n classcal phase space If classcal mcrostates were to correspond to mathematcal ponts n phase space, the total number of states compatble wth fnte ntervals of energy and volume would be nfnte for most systems. Thus, the calculaton of entropy, or other thermodynamc potentals, s problematcal n classcal statstcal mechancs. Sensble enumeraton of classcal mcrostates requres that a 2 f dmensonal phase space be dvded nto cells wth fnte volume d f where d has dmensons q p. Although we mght envson a classcal lmt dø0, such a lmtng procedure would yeld nfnte entropes. Fortunately, quantum mechancs provdes a natural dscretzaton of the phase space the uncertanty prncple lmts the precson wth whch conjugate coordnates and momenta can be specfed. Thus, we expect that d must be closely related to Planck's constant. Furthermore, f the parameter d s to reman small and fnte t must be a unversal constant so that entropes can be added meanngfully for systems of dfferent types. Hence, we are free to evaluate ths constant by comparson wth any convenent quantum system. The two examples below both suggest that d =h. Therefore, we normalze the dfferental phase-space volume element f q p G = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ h = usng a factor h - for each degree of freedom. Ths analyss s based upon an nverse correspondence prncple. In quantum mechancs one often use correspondence wth classcal physcs to determne the form of the Hamltonan, but here we use quantum mechancs to determne the normalzaton factor for the phase-space densty. Smlarly, we sometmes mx classcal descrptons for some aspects wth quantum mechancal descrptons of other degrees of freedom. For example, when we study the heat capacty for nonnteractng gases wth nternal rotatonal and vbratonal degrees of freedom we wll employ a classcal descrpton for the center of mass but quantum mechancal descrptons for the nternal exctatons of each molecule. These types of mxed models are descrbed as semclasscal. It s nterestng to observe that quantum statstcs s usually easer than classcal statstcs and that classcal statstcs usually does not make sense unless quantum mechancs s used to obtan fnte entropy.

4 4 Semclasscal.nb à Example: harmonc oscllator Consder a one-dmensonal harmonc oscllator wth Hamltonan H = p ÅÅÅÅÅÅÅÅ p2 2 m + ÅÅÅÅ 2 kq2. è!!!!!!!!!!!! 2 me è!!!!!!!!!!!!! 2 E ê k q States wth energy E are found on an ellpse wth axes è!!!!!!!!!!!! 2 E ê k n q and è!!!!!!!!!!! 2 me n p. The mcrocanoncal ensemble of systems wth energy E s then the set of ponts on the ellpse, whch s nfnte n number. To obtan fnte results, we dscretze the set and clam that the multplcty W@ED s proportonal to the crcumference of the ellpse. Smlarly, the number of states S@ED wth energes E E s proportonal to the area of the ellpse. Thus, we clam S = ÅÅÅÅÅÅÅÅ p 2 E $%%%%%%%%%% ÅÅÅÅÅÅÅÅÅÅÅ S 0 k è!!!!!!!!!!! 2 me = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 p E m $%%%%%%% ÅÅÅÅÅÅ S 0 k E = 2 p ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ ws 0 where S 0 s the area of an elementary cell n phase space and where we have used the harmonc oscllator frequency w= è!!!!!!!!!! k ê m. Classcal physcs does not specfy the cell sze because classcal physcs clams that t s possble, at least n prncple, to specfy coordnates and momenta wth arbtrary precson, t would appear that one should take the lmt S 0 Ø 0. Thus, n that lmt entropy becomes nfnte because there are an nfnte number of states compatble wth the defnton of the ensemble. However, quantzaton of energy requres there to be a mnmum volume of phase space surroundng each of the fnte number of states n a specfed energy range. The smplest method for obtanng ths cell sze s to examne the densty of states g = ÅÅÅÅÅÅÅÅ S E = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 p ws 0 and to use the quantzaton of energy g = ÅÅÅÅÅÅÅÅÅÅÅÅ Ñ w ï S 0 = 2 p Ñ = h to recognze that the cell sze s governed by Planck's constant.

5 Semclasscal.nb 5 à Example: partcle n a box Consder a cubcal box wth volume V = L contanng N nonnteractng partcles wth mass m. Sngle-partcle wave functons take the form y Hx, y, zl Sn@k x xd Sn@k y yd Sn@k z zd where the requrement that y vanshes upon the walls mposes the quantzaton condton k = n p ÅÅÅÅÅÅÅÅÅÅÅÅ œ 8x, y, z< L where the n must be ntegers. Thus, we fnd that the spacng between adjacent wave numbers s p ê L such that each quantum state occupes a cell n k -space wth volume Hp ê LL. Assumng that nonrelatvstc knematcs apples, the snglepartcle energes become k = p2 ÅÅÅÅÅÅÅÅÅÅÅÅ 2 m = HÑ kl2 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 m =n2 0 where 0 = p2 Ñ 2 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 ml = h 2 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ 2 8 ml 2 s the basc unt of energy and n 2 = n 2 x + n 2 2 y + n z represents a dmensonless exctaton varable. Note that n but need not be an nteger. The states avalable to a partcle n the box can be represented by ponts on a three-dmensonal lattce n the space 8k x, k y, k z <. All dstnct states are represented by ponts wth the octant wth all n 0; negatve ntegers merely change the sgn of the wave functon and are redundant. The total number of states n the sphercal shell wth radus between k and k + k s then the volume of one octant of a sphercal shell dvded by the cell volume, such that G = ÅÅÅÅÅ 4 p k 2 k ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = ÅÅÅÅÅÅÅÅÅÅÅÅÅ V 8 Hp ê LL 2 p 2 k2 k Although ths dervaton was made for a cube, t can be shown that for a suffcently large volume V the result s ndependent of the shape of the enclosure. Therefore, t s convenent to express G n the more general form G = r p ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ =r H2 p x ï D = H2 p ÑL - where x = 8r, p < are the coordnates of a sngle partcle and r s the classcal phase-space densty for a sngle partcle. Therefore, the volume d for an elementary cell n classcal phase space s dentfed as beng Planck's constant, h = 2 p Ñ. Ths result s obvously related to the Hesenberg uncertanty prncple, whch states that there s a fnte precson wth whch a par of conjugate varables can be known smultaneously.

6 6 Semclasscal.nb Equpartton theorem A powerful theorem of classcal statstcal mechancs concerns the equpartton of energy among the harmonc degrees of freedom. Suppose that the energy E = E@q,, q f, p,, p f D can be separated n the form E D + p f D where x s any coordnate or momentum whch does not appear n E. Further, suppose that = b x a. Normally a =2 for a harmonc coordnate or momentum. If a s even, the ntegraton extends over - < x <, but for odd or nonntegral a the range of x must be lmted to ensure a convergent ntegral. The mean value of n the canoncal dstrbuton s êê = Ÿ GExp@-b D Exp@-bE D ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = Ÿ xexp@-b D ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Ÿ GExp@-b D Exp@-bE D Ÿ xexp@-b D = - ÅÅÅÅÅÅÅÅ b LogB xexp@-b DF If we now let y a ªbx a ï x = b -êa y we fnd xexp@-bb x a D = b -êa y Exp@-b y a D Assumng that the lmts of ntegraton wth respect to y do not depend upon b, vz. the lmts of x are (0, ) or H-, L, we obtan LogB xexp@-b DF =-a - Log@bD + LogB y Exp@-b y a DF ï Thus, the mean value of reduces to k B T ê a. Therefore, the equpartton theorem states: ÅÅÅÅÅÅÅÅ b LogB xexp@-b DF =- k B T ÅÅÅÅÅÅÅÅÅÅÅÅÅ a The mean energy êê contrbuted by each separable coordnate or momentum x whose contrbuton to the Hamltonan s proportonal to x a s êê = kb T ê a. For example, consder an deal gas consstng of nonrelatvstc partcles wth no nternal degrees of freedom. The total energy s then a sum of N quadratc contrbutons to the knetc energy arsng from the three ndependent momentum components for each partcle. Hence, the equpartton theorem predcts U = ÅÅÅÅ 2 Nk B T for a nonrelatvstc deal gas. The equpartton theorem also apples to nternal degrees of freedom. For example, consder a datomc molecule. Harmonc vbratons along the separaton between the two atoms contrbute two quadratc terms to the energy, one for the nternal potental and one for the nternal knetc energy. Hence, the equpartton theorem predcts an addtonal contrbuton of 2 HN k B T ê 2L to the nternal energy of the gas. Furthermore, datomc molecules possess two degrees of freedom correspondng to rotatons about axes perpendcular to the symmetry axs. Note that rotatons about a symmetry axs are forbdden by quantum mechancs because the physcal state s unchanged. Accordng to the equpartton theorem, these

7 Semclasscal.nb 7 rotatons contrbute k B T for each molecule. Therefore, we predct that the specfc heat for a gas of datomc molecules should be datomc ï C V = ÅÅÅÅÅ 7 2 R based upon equpartton among translatonal, rotatonal, and vbratonal degrees of freedom. Although equpartton of the energy among ndependent degrees of freedom s an mportant property of classcal statstcal mechancs, uncrtcal applcaton of the theorem can lead to absurd results. Accordng to the theorem, the contrbuton of each ndependent degree of freedom to the nternal energy appears to be ndependent of the ampltude of the moton assocated wth that coordnate. Consder a gas of datomc molecules for whch the nternal sprng constant s enormous. Unless the knetc energes are large enough for collsons to excte ths stff sprng, the molecules mght as well be rgd. However, the equpartton theorem predcts that the contrbuton to the nternal energy of the gas s the same for small T as for hgh T whether or not the sprng can be excted wth apprecable ampltude. Ths contrbuton remans even as we make the sprng nfntely stff. Ths predcton of classcal statstcal mechancs leads to a serous paradox. Any classcal system can be subdvded nto arbtrarly small components nteractng wth each other. The bndng of adjacent parts can be descrbed by potental and knetc contrbutons to the total energy. Each of these degrees of freedom contrbutes equally to the specfc heat even f the bndng forces are so strong that these parts are rgdly bound. Therefore, classcal statstcal mechancs predcts the absurd result that all specfc heats are nfnte! The resoluton of ths paradox agan les n quantum mechancs. Because mcroscopc motons are quantzed, these motons can be excted n dscrete quanttes only, not wth ampltudes related to the exctaton energy. Unless suffcent energy s avalable, some degrees of freedom wll be dormant and wll not contrbute to the specfc heat. Only when k B T s much larger than the spacng between energy levels s classcal equpartton among avalable degrees of freedom acheved. Nevertheless, the theorem does provde lmtng values that are useful when the number of actve degrees of freedom can be enumerated. à Example: harmonc oscllators Consder a collecton of N ndependent classcal harmonc oscllators wth the same frequency w governed by the Hamltonan N H = = N H = j p 2 ÅÅÅÅÅÅÅÅÅÅÅÅ k 2 m + ÅÅÅÅÅ 2 m w2 2 q y z { = The separablty of the energy permts factorzaton of the canoncal partton functon, Z = Z N, where Z = ÅÅÅÅÅÅÅÅ q ÅÅÅÅÅÅÅÅÅÅÅÅÅ p Ä Exp h ÇÅ -b j ÅÅÅÅÅÅÅÅÅÅÅÅ p2 k 2 m + ÅÅÅÅÅ 2 m w2 q 2y É z { ÖÑ = k B T ÅÅÅÅÅÅÅÅÅÅÅÅÅ Ñ w Thus, the prncpal thermodynamc functons become Ä k B T É F = -Nk B T Log ÅÅÅÅÅÅÅÅÅÅÅÅÅ ÇÅ Ñ w ÖÑ U = - ÅÅÅÅÅÅÅÅ ln ÅÅÅÅÅÅ Z Å b = Nk B T ï C V = Nk B S = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ U - F = Nk Ä k B T É y U B j + Log ÅÅÅÅÅÅÅÅÅÅÅÅÅ z = Nk B J + LogB ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ T k ÇÅ Ñ w ÖÑ { N Ñ w FN m=-t k j ÅÅÅÅÅÅÅÅÅÅÅÅ S Ä y k B T É z =-k B T Log ÅÅÅÅÅÅÅÅÅÅÅÅÅ N { U ÇÅ Ñ w ÖÑ

8 8 Semclasscal.nb The pressure vanshes because n ths smple model there are no nteractons between oscllators and no dependence of the oscllator frequency upon the volume or densty of the system. The nternal energy s consstent wth the classcal equpartton theorem whch assgns ÅÅÅÅ 2 k B T to each harmonc coordnate or momentum that appears n the Hamltonan. In the hgh-temperature lmt the entropy per oscllator s proportonal to average number of quanta of exctaton, U ê N Ñ w, whch accordng to the equpartton theorem s equal to k B T ê Ñ w. Smlarly, at hgh temperature the chemcal potental s large and negatve. However, the low temperature lmt remans problematcal because the entropy appears to become negatve for k B T ê Ñ w -. Therefore, the classcal enumeraton of mcrostates fals as the average number of quanta per degree of freedom falls below unty. Clearly, we must employ quantum mechancs n the low temperature lmt where the thermal energy k B T becomes comparable to the spacng between energy levels. The lmtng value C V = Nk B apples to hgh temperatures but the heat capacty s suppressed at low temperature where nsuffcent energy s avalable to excte the modes counted by the equpartton theorem. Vral theorem We now consder a generalzaton of the equpartton theorem for classcal systems. The proof s usually presented usng the mcrocanoncal ensemble but s much easer usng the canoncal ensemble. Hence, we choose the latter confdent that the results are ndependent of ensemble n the thermodynamc lmt. The vral theorem s a generalzaton of the famlar work-energy theorem n Newtonan mechancs that apples n Hamltonan mechancs to generalzed coordnates and forces. To motvate ths generalzaton, consder the tme dependence of r ÿ p for a collecton of N nonrelatvstc classcal partcles wth coordnates r and momenta p, such that N ÅÅÅÅÅÅÅ t r ÿ p N = jr ÿ ÅÅÅÅÅÅÅÅÅÅÅÅÅ p + p = = k t ÿ r N y ÅÅÅÅÅÅÅÅÅÅÅ z = I r ÿ F + 2 K M t { = where F s the force actng on partcle and K = ÅÅÅÅÅÅÅÅ p2 r ÿ p must be constant n equlbrum, we fnd [ ÅÅÅÅÅÅÅ N t r ÿ p _ = ÅÅÅÅÅÅÅÅ = N t [ = 2 m = ÅÅÅÅ N r ÿ p _ = 0 ï V = [ = 2 p ÿ r ÅÅÅÅÅÅÅ s ts knetc energy. Snce the ensemble average of t r ÿ F _ = -2 K where K = X K \ s the average knetc energy and V s known as the Clausus vral. Alternatvely, for any partcle we can evaluate the long-term tme average of r ÿ F usng êêêêêêê r ÿ F T = ÅÅÅÅÅÅÅ T tr HtL ÿ F HtL = ÅÅÅÅÅÅ o T Hr ÿ p L T 0 - ÅÅÅÅÅÅÅ T T 2 tmv 0 where T s the observaton perod. Makng the perfectly reasonable assumpton that the coordnates and momenta reman fnte, the ntegrated term must vansh n the lmt T Ø, such that êêêêêêê r ÿ F êêêêêêê êêêêêêêêê êêê = -2 K ï [ r ÿ F _ = -2 K êêê Therefore, we obtan the same vral theorem V =-2 K from ether temporal or ensemble averagng.

9 Semclasscal.nb 9 Ths theorem can be generalzed to Hamltonan mechancs by replacng the coordnate r wth the generalzed coordnate or momentum x and replacng the force by the generalzed force H ê x j. Thus, we seek to evaluate the ensemble average [x H ÅÅÅÅÅÅÅÅÅÅÅÅ x j _ = H Ÿ G -bh x ÅÅÅÅÅÅÅ x j ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅ Ÿ G -bh where x represents any of the generalzed coordnates or momenta that appear n the Hamltonan H. The numerator can be ntegrated by parts wth respect to x j, such that G -bh x H ÅÅÅÅÅÅÅÅÅÅÅÅ = - ÅÅÅÅÅ x j b G j jhx -bh L x k j - x j -bh x y ÅÅÅÅÅÅÅÅÅÅÅÅ z x j { where G j excludes x j and where the ntegrated porton (surface terms) must be evaluated for the extreme values of x j. We assume that the surface terms vansh for extreme values of x j. For example, f x j s a poston coordnate for a system confned to a fnte volume the extreme values are on the walls where the confnement potental s nfnte and the Boltzmann factor vanshes. Smlarly, f x j s a momentum the extreme values represent nfnte knetc energes for whch the surface terms also vansh. Thus, we are left wth G -bh x H ÅÅÅÅÅÅÅÅÅÅÅÅ x j = k B T G -bh x ÅÅÅÅÅÅÅÅÅÅÅÅ x j = d, j k B T G -bh Therefore, we obtan the vral theorem n the more general form [x H ÅÅÅÅÅÅÅÅÅÅÅÅ _ = d, j k B T x j The vral theorem ncludes the equpartton theorem as a specal case. Suppose that the Hamltonan can be separated n the form H = j + H j where j = b x a j and where H j s ndependent of x j. We then obtan x H j ÅÅÅÅÅÅÅÅÅÅÅÅ = a j ï êêê k B T j = ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅ x j a as before. More generally, usng [q H ÅÅÅÅÅÅÅÅÅÅÅÅ _ = -Xq p q \ = k B T [p H ÅÅÅÅÅÅÅÅÅÅÅÅ _ = Xp q p \ = k B T we fnd N [ p q _ = Nk B T = N [ - = q p _ = Nk B T for N partcles n three dmensons wthout havng to assume that the Hamltonan s separable. à Example: quadratc forms Suppose that the Hamltonan can be expressed n the form

10 0 Semclasscal.nb H = HA j p 2 + B j q 2 j L j by means of an approprate canoncal transformaton. Clearly, jp j ÅÅÅÅÅÅÅÅÅÅÅÅÅ H + q H y j ÅÅÅÅÅÅÅÅ ÅÅÅÅ z = 2 H k p j q j { j for ths homogeneous second-order functon. Thus, the vral theorem takes the form XH\ = ÅÅÅÅÅ 2 fk B T where f s the total number of harmonc contrbutons to the Hamltonan (coordnates + momenta). Note that ths result s ndependent of the coeffcents, A j or B j. More generally, consder a Hamltonan of the general quadratc form H = ÅÅÅÅÅ 2 x M, j x j, j where each x s a generalzed coordnate or momentum and M, j s an f -dmensonal square matrx wth postve egenvalues. Usng the vral theorem we agan fnd x H j ÅÅÅÅÅÅÅÅÅÅÅÅ = 2 H ï XH\ = ÅÅÅÅÅ x j 2 fk B T j where f s the number of actve degrees of freedom. Ths result can be appled to the rotatonal Hamltonan for a rgd rotor H rot = ÅÅÅÅÅ 2, j w I, j w j ï XH rot \ = ÅÅÅÅÅ 2 k B T where I, j s the rotatonal nerta tensor and w are the three components of angular velocty wth respect to the center of mass. However, one must always exercse care n applcaton of classcal equpartton or vral theorems. Consder a datomc molecule composed of two ons surrounded by an electron cloud. The moments of nerta about axes perpendcular to the separaton between the ons are equal, I = I 2 = I, but the moment of nerta about the symmetry axs I ` I s much smaller because essentally all of the mass s concentrated n the ons and the nuclear rad, r N, are smaller than nteratomc separatons, R, by a factor of order 0-5, such that I ÅÅÅÅÅÅ ~ m e R 2 + m N r N ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = m e ÅÅÅÅÅÅÅÅÅÅÅ I m N R 2 2 m N + J r N ÅÅÅÅÅÅÅ R N2 ~ Hence, the energy scale for classcal rotaton about the separaton axs s about a factor of 0 hgher than the energy for the orthogonal axes. Recognzng that hgh energy modes are "frozen out" unless the temperature s hgh enough to permt full partcpaton, we expect to obtan a rotatonal contrbuton to the heat capacty of R for modest temperatures and an ncrease to ÅÅÅÅ R for hgh temperatures. Furthermore, there s a more rgorous quantum mechancal argument that excludes 2 rotatons about the symmetry axs. Quantum mechancally, rotaton about a symmetry axs may change the phase of a wave functon but does not lead to a new state of the system the rotated state s ndstngushable from the orgnal. Thus, there s no dynamcal varable assocated wth rotatons about a symmetry axs and no assocated contrbuton to the nternal energy or heat capacty. Therefore, n the absence of an electronc exctaton whch breaks the rotatonal symmetry, there are only two actve rotatonal degrees of freedom for a datomc molecule.

11 Semclasscal.nb à Example: nonrelatvstc deal gas We can evaluate Clausus' vral explctly for an deal gas n whch the only forces whch act on the partcles are the confnement forces at the walls. Snce those forces act only at the walls and are responsble for the pressure p upon the gas, we can wrte [ r ÿ F wall _ = -p A ÿ r where A s an outward element of surface area and r s the coordnate vector for a partcle at the surface of the contaner. The surface ntegral can be evaluated usng Gauss' law A ÿ r = r ÿr = V so that [ r ÿ F 2 wall _ = - pv = -2 U ï pv = ÅÅÅÅÅ U = Nk B T for an deal nonrelatvstc gas for whch U = K. à Example: ultrarelatvstc deal gas Consder a dlute gas that s so hot that the ultrarelatvstc lmt, = p c, apples. Although the Hamltonan does not separate n a convenent manner for use of the equpartton theorem, we can apply the vral theorem to each component a of the momentum p for partcle N H = = N p c = c Hp 2,x + p 2,y + p,z L ê2 ï = H ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p,a = cp,a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p ï [p,a such that the summaton over components gves the mean sngle-partcle energy as X \ = [ a p,a H ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ _ = k B T p,a Therefore, the nternal energy of an ultrarelatvstc gas becomes U = Nk B T ï C V = Nk B whch s larger by a factor of two than the correspondng nonrelatvstc result. H ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p,a _ = k B T Maxwell velocty dstrbuton Consder a gas that s suffcently dlute that ntermolecular nteractons may be neglected. The sngle-partcle energy may then be expressed n the form

12 2 Semclasscal.nb = p2 ÅÅÅÅÅÅÅÅÅÅÅÅ 2 m + s + V@r D where s s the nternal exctaton for state s and V@r D s an external potental. Internal modes of exctaton may nclude rotatonal or vbratonal motons wthn a molecule or exctatons of electrons. The external potental ncludes the confnement potental, defned to vansh wthn the enclosure and to become nfnte at ts walls, and may nclude the gravtatonal potental or nteractons wth an appled electromagnetc feld. The probablty P p D r p for a molecule n nternal state s to be found wthn a small volume r at poston r and wth momentum n a volume p centered on p s then expressed n the canoncal dstrbuton as P p Ä D r p Exp ÇÅ - b É ÅÅÅÅÅÅÅÅÅÅÅÅÅ p2 2 m ÖÑ Exp@-b sd Exp@-bV@r DD r p It s mportant to recognze that snce the three contrbutons to the energy are separable, the probablty s factorzable, such that P p D r p = HP D plhp D rl P s where P Ä D p Exp ÇÅ - b É ÅÅÅÅÅÅÅÅÅÅÅÅÅ p2 2 m ÖÑ p governs the momentum dstrbuton, P D r Exp@-bV@r DD r governs the densty, and P s Exp@-b s D governs the spectrum of nternal molecular exctatons. Frst, consder a free monatomc gas for whch s = 0 and for whch V@r D vanshes wthn the enclosure but s essentally nfnte at the walls so that there s no possblty for escape. The densty s then ndependent of poston and P r = V - such that the poston probablty ntegrated over volume s normalzed to unty. The normalzaton of P p s determned by the ntegral Ä p Exp - ÇÅ - b É ÅÅÅÅÅÅÅÅÅÅÅÅÅ p2 = H2 p mk B TL ê2 = p B 2 m ÖÑ Therefore, the basc probablty dstrbuton for momentum s P Ä D p = Exp ÇÅ - b É ÅÅÅÅÅÅÅÅÅÅÅÅÅ p2 p ÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 m ÖÑ p B It s often useful to express ths dstrbuton n several equvalent forms. If we are nterested only n the magntude of the momentum, we can use sotropy to replace the cartesan form p by the sphercal form Ä p Ø 4 p p 2 p ï P p = Exp ÇÅ - b É ÅÅÅÅÅÅÅÅÅÅÅÅÅ p2 4 p p 2 p ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 m ÖÑ p B and thus to determne the probablty P@pD p that the magntude of the momentum vector s found n the nterval Hp, p + pl. Alternatvely, we defne v to be the mean number of molecules per unt volume whose veloctes are found n the nterval Hv, v + vl, such that

13 Semclasscal.nb v = np p where n = N ê V s the partcle densty and p = mv s the nonrelatvstc momentum. Therefore, the customary form for the Maxwell velocty dstrbuton becomes Ä É v = n Exp ÇÅ - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ mv2 4 p v 2 v ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 k B T ÖÑ v B where v B = H2 p k B T ê ml ê2 s a scale parameter for the thermal dstrbuton of veloctes. Fnally, f we defne dmensonless varables u = ÅÅÅÅÅÅÅ v r= ÅÅÅÅÅÅÅÅ p v B p B the varous dstrbuton functons may be expressed n the followng dmensonless forms. P@r D r = Exp@-p r 2 D r D u = n Exp@-p u 2 D u P@rD r = Exp@-p r 2 D 4 p r 2 r u = n Exp@-p u 2 D 4 p u 2 u P@r x D r x = Exp@-p r 2 x D r x x D u x = n Exp@-p u 2 x D u x The Maxwell velocty dstrbuton s sketched below. A few mportant characterstcs of ths dstrbuton are the most probable speed u è, the mean speed êê u, and the rms speed u rms = Xu 2 \ ê2. The most probable velocty s determned by è D ã 0 ï u è = p -ê2 The mean and rms speeds are determned by the moments êê u = n - uf@ud u = ÅÅÅÅÅ 2 0 p u 2 rms = n - u 2 u = ÅÅÅÅÅÅÅÅ 0 2 p such that u è = 0.564, êê u = 0.67, u rms = 0.69 Due to the asymmetrc shape of the speed dstrbuton and ts long tal extendng to relatvely large speeds, the mean speed s about % larger than the most probable speed and the rms speed s 22% larger than u è. Successve moments wth hgher powers of u progressvely ncrease due to ther ncreasng emphass of the tal of the dstrbuton. Fnally, the wdth of the dstrbuton can be gauged by ts rms devaton, s u = HXu 2 \ - Xu\ 2 L ê2 = 0.48 u è. Thus, the spread of speeds s clearly a large fracton of the most probable speed.

14 4 Semclasscal.nb n è Maxwell Velocty Dstrbuton n êê n rms vêv B Insertng the scale parameter v B = è!!!!!!!!!!!!!!!!!!!!!! 2 p k B T ê m, we fnd that the moments of the speed dstrbuton v è = è!!!!!!!!!!!!!!!!! 2 k B T ê m ï ÅÅÅÅÅ 2 mvè2 = k B T êê 8 v = $%%%%%%%%%%%%%%%%%%%%%% ÅÅÅÅÅ p k B T ê m ï ÅÅÅÅÅ 2 mvêê = ÅÅÅÅÅ 4 p k B T v rms = è!!!!!!!!!!!!!!!!! k B T ê m ï ÅÅÅÅÅ 2 mv rms 2 = ÅÅÅÅÅ 2 k B T are all of order è!!!!!!!!!!!!!!! k B T ê m and the correspondng momenta are of order è!!!!!!!!!!!!! mk B T. The average knetc energy Z ÅÅÅÅÅ 2 mv2^ = ÅÅÅÅÅ 2 k B T corresponds to ÅÅÅÅ 2 k B T for each of the three ndependent translatonal degrees of freedom. (Note that the average knetc energy s obtaned from v rms and s larger that the most probably knetc energy.) The nternal energy of the gas s then the number of molecules tmes the average knetc energy per molecule, so that U = ÅÅÅÅÅ 2 Nk B T ï C V = ÅÅÅÅÅ 2 Nk B ndependent of speces or pressure. Obvously the molar gas constant s gven by R = N A k B where Avogadro's number N A s the number of molecules per mole and k B s Boltzmann's constant. Usng k B T º ÅÅÅÅÅ ev at room temperature, we estmate that the average knetc energy for an deal gas near 00 K s 40 about 0.07 ev ndependent of speces. For N 2, wth a molecular weght mc 2 ~ 28 µ 0 MeV, the rms velocty s about v rms ~ 2 µ 0-6 c ~ 600 m ê s. It s no concdence that ths speed s about twce the velocty of sound n ar, snce molecular speed sets the scale of velocty for mechancal dsturbances.

15 Semclasscal.nb 5 Entropy of a classcal deal gas Neglectng nteractons among the consttuents of a dlute deal gas, the energy of any state separates nto N ndependent knetc contrbutons of the form N E = = = p 2 ÅÅÅÅÅÅÅÅÅÅÅÅ 2 m where N s the number of partcles. We also neglect possble nternal exctaton energes wthn the consttuents. Hence, the partton functon reduces to Z = Z N where the sngle-partcle partton functon Z = V Ä p j ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k - 2 p Ñ Exp ÇÅ - b É ÅÅÅÅÅÅÅÅÅÅÅÅÅ p2 y z = V 2 m ÖÑ { k j mk ê2 B T y - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ z = V l 2 p Ñ 2 B { s obtaned by summng over cells of volume H2 p ÑL n classcal phase space. Because the three cartesan momenta, 8p x, p y, p z < provde three ndependent contrbutons of the same form, Z also factors such that Z = Z x Z y Z z = HZ x L. It s useful to defne the Boltzmann wavelength l B = h ÅÅÅÅÅÅÅÅ p B p B = H2 p mk B TL ê2 as the de Brogle wavelength for a partcle that carres a typcal thermal momentum (Boltzmann momentum) p B, such that l B = j 2 p Ñ2 y ê2 h ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅ z = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k mk B T { H2 p mk B TL ê2 It s also useful to defne a correspondng de Brogle or thermal densty as n B =l - B. When the partcle densty becomes comparable to the de Brogle densty, the wave packets of neghborng partcles overlap sgnfcantly, on average, and quantum effects become mportant. Thus, t s useful to defne a quantum concentraton n Q = ÅÅÅÅÅÅ N V l B = ÅÅÅÅÅÅÅÅ n n B as a gude to the relatve mportance of quantum effects for specfed temperature and densty. Clearly, for equal temperature and partcle densty, quantum effects are more mportant for lghter partcles because l B m -ê2 ï n Q m -ê2. We can now evaluate the Helmholtz free energy as F =-Nk B T Log@V l - B D The mechancal equaton of state s determned by evaluatng the pressure, such that p =- k j ÅÅÅÅÅÅÅÅÅÅÅ F y z ï pv = Nk B T V { T as expected. Smlarly, the nternal energy and heat capacty

16 6 Semclasscal.nb U =- k j ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ln Z y Å z =-N j ln Z y ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ z b { V k b { C V = k j ÅÅÅÅÅÅÅÅÅÅÅÅ U y z = ÅÅÅÅÅ T { V 2 Nk B V = ÅÅÅÅÅ 2 Nk B T are also n agreement wth results famlar from classcal thermodynamcs. (Note that here N s the number of partcles rather than the number of moles, so that the molar gas constant s smply R = N A k B where N A s Avogadro's number.) The entropy s obtaned by dfferentatng the free energy wth respect to temperature, whereby S =- k j ÅÅÅÅÅÅÅÅÅÅÅ F y z = Nk B JLog@V l B D + ÅÅÅÅÅ T { V 2 N HncorrectL Although ths theory has predcted both the mechancal equaton of state and the heat capacty correctly, the free energy and entropy suffer from the serous flaw that nether s extensve. Suppose that the volume s dvded nto two equal halves by an magnary wall. Each half contans half the partcles and contrbutes entropy V Ø ÅÅÅÅÅÅ V 2, N Ø ÅÅÅÅÅÅ N 2 ï S Ø ÅÅÅ S Å 2 - ÅÅÅÅÅ 2 Nk B ln 2 Therefore, ths calculaton suggests that the entropy of the gas s reduced by the amount N k B ln 2 upon ntroducton of an magnary wall. Ths absurd result s known as the Gbbs paradox. Examnaton of our expressons for entropy and free energy reveals that the Gbbs paradox could be avoded f the volume were to be replaced by V ê N under the logarthm. A smlar result can be obtaned by subtractng k B Log@N!D = k B HN Log@ND - NL, whereby S Ø S - k B Log@N!D ï S = Nk Ä É V B jlog ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k ÇÅ N l B ÖÑ + ÅÅÅÅÅ 5 y z 2 { s known as the Sackur-Tetrode formula after the people who frst used Planck's constant to dscretze the phase space. Ths revsed entropy s now extensve n the sense that S scales wth N and V accordng to V ØaV, N ØaN ï S ØaS The physcal orgn of the Gbbs paradox concerns the ndstngushablty of dentcal partcles. Suppose that a mcrostate s descrbed by the set of occupaton numbers x = 8n < where labels a cell n the phase space for a sngle partcle and n s the number of partcles whose momentum and poston are found wthn that cell. The total number of permutatons of N partcles among the states belongng to the confguraton 8n < suggests the multplcty functon g x = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ N! n! where the numerator s the total number of permutatons of the N partcles and the denomnator represents the ndstngushable permutatons among the partcles wthn each cell of phase space. If we assume that t s possble, at least n prncple, to dstngush among these varous states, then the confguraton x should be assgned multplcty g x. On the other hand, f we assume that t s mpossble, even n prncple, to dstngush between dentcal partcles, the confguraton x should be assgned multplcty. The Gbbs prescrpton conssts of reducng the number of dstnct states by the overall factor N!. However, the actual factor g x depends upon the occupaton numbers. Evdently, the Gbbs prescrpton s applcable when the densty s small and the temperature hgh enough for the average occupaton number Xn \ ` to be small, such that n ` ï ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ N! n! º N! Thus, the Gbbs prescrpton for a classcal partton functon for N dentcal nonnteractng partcles reads

17 Semclasscal.nb 7 n Q ` ï Z N = Z N ÅÅÅÅÅÅÅÅÅÅÅÅÅ N! Wth ths prescrpton we fnd S = Nk Ä É V B jlog ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k ÇÅ N l B ÖÑ + ÅÅÅÅÅ 5 y z = Nk B J ÅÅÅÅÅ 5 2 { 2 - ln n QN F =-Nk B T Ä É V jlog ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅ k ÇÅ N l B ÖÑ + y z =-Nk B T H - ln n Q L { provded that n Q `. Thus, the Gbbs prescrpton s vald when the quantum concentraton s small and fals when n Q approaches unty. When n Q ~, we must employ a correct quantum mechancal treatment for the wave functon of N dentcal partcles wth due regard to the proper partcle-exchange symmetry of that wave functon. Fnally, we note that the chemcal potental m = k j ÅÅÅÅÅÅÅÅÅÅÅÅ F Ä É y V z = -k B T Log ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ N { T,V ÇÅ N l B ÖÑ = k B T ln n Q s large and negatve for a classcal deal gas wth small quantum concentraton. Recall that fugacty z s defned by m = k B T Log@zD. Thus, the fugacty for a classcal deal gas m z = ExpB ÅÅÅÅÅÅÅÅÅÅÅÅÅ k B T F = n Q reduces to the quantum concentraton and s small and postve at hgh temperatures where the classcal approxmaton apples. More generally, usng the Gbbs prescrpton for the partton functon for N dentcal ndstngushable partcles, the free energy and chemcal potental take the forms such that Z N = Z N ÅÅÅÅÅÅÅÅÅÅÅÅÅ N! F = -Nk B T J - ï F = -Nk B T JLogB Z ÅÅÅÅÅÅÅ N F + N, m = -k B T LogB Z ÅÅÅÅÅÅÅ N F m ÅÅÅÅÅÅÅÅÅÅÅÅÅ k B T N If we assume that the translatonal degrees of freedom can be factored out, such that Z = V l - B z@td where z@td represents the partton functon for nternal degrees of freedom, then Ä É Z = V l - V z@td B z@td ï m = -k B T Log ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÇÅ N l B ÖÑ Thus, the fugacty takes the form z = N l B ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ V z@td For an deal gas one assumes that the densty s small compared wth z@tdêl B such that the chemcal potental s large and negatve and the fugacty s small. If the densty s large, the fugacty appears to be large and the chemcal potental to be postve, but cauton must be exercsed wth respect to both the classcal approxmaton of nonoverlappng wave packets and the assumpton of neglgble nteractons. When we treat translatonal degrees of freedom classcally and nternal degrees of freedom (z@td) quantum mechancally, the model can be descrbed as semclasscal. For example, for a gas of elementary partcles wth spn s, the nternal partton functon reduces to z Øg = 2 s +, where g s the degeneracy of a sngle-partcle momentum state, and s ndependent of temperature. The quantum concentraton s then generalzed to read

18 8 Semclasscal.nb n Q = N l B ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅ gv However, for a gas of atoms or molecules wth more complcated nternal structure, the nternal partton functon depends upon temperature. Thus, z@td s a measure of the number of actve nternal degrees of freedom. Ideal gas of datomc molecules Consder a gas of nonnteractng datomc molecules. These molecules consst of two massve ons surrounded by a cloud of much lghter electrons. Due to the dfference n mass between the ons and the electrons, the moton of the electrons s much faster than that of the ons. Hence, the electron cloud can adjust rapdly to any on moton. Therefore, we can approxmate nteractons wth the electronc cloud by an effectve potental that depends upon the separaton between the ons. To a frst approxmaton, ths potental s harmonc for small-ampltude vbratons about the equlbrum separaton between the ons. Alternatvely, the system can rotate as a rgd object wth moment of nerta I º mr 2 where m s the on mass and R ther separaton. The rotatonal energy { can be estmated by assumng that the rotatonal nerta s approxmately I º mr 2 ï { H{ + L Ñ2 { º ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 mr 2 where m s the atomc mass, R s the equlbrum separaton between ons, and { s the rotatonal angular momentum. Thus, for N 2 we obtan 2 H97 ev ÅL 2 º ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ º 0.7 µ 2 H28 µ 0 9 evl HÅL ev Ths energy s qute small, so we mght expect rotatonal degrees of freedom to reach the equpartton lmt even for temperatures as low as a few tens of kelvns. The characterstc energy of molecular vbratons s much larger, typcally n the neghborhood of 0.5 ev. Electronc exctatons are generally characterzed by energes of several ev. Thus, the characterstc temperature for exctaton of the electron cloud s much larger than the energy of onc vbratons, whch n turn s very much larger than the energy of rotatons. Therefore, we may assume that, to a good approxmaton, these modes are decoupled from one another, such that H = H p + H rot + H vb + H e ï Z = Z p Z rot Z vb Z e where H p, H rot, H vb, and H e represent ndependent translatonal, rotatonal, vbratonal, and electronc degrees of freedom. The partton functon, Z = Z N ê N! can then be factored. The partton functon for translatonal moton has already been developed for the deal gas. In the remander of ths secton we present smple approxmatons for the rotatonal and vbratonal factors, Z rot and Z vb.

19 Semclasscal.nb 9 à Rotatons We assume that rotatons of a datomc molecule can be adequately descrbed by the rgd rotor model wth constant rotatonal nerta and that electronc motons are suffcently fast that azmuthal symmetry s mantaned. The energy levels of a rgd rotor are then { H{ + L Ñ2 { º ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 I where I s the rotatonal nerta and { s a nonnegatve nteger descrbng the orbtal angular momentum. The projecton of angular momentum upon any axs s restrcted to nteger values between -{ and +{. Hence, the degeneracy of energy level { s g { = 2 { +. If the two ons are dfferent (heteronuclear molecule), then { can assume any nonnegatve ntegral value, but f the two ons are dentcal the values of { are restrcted to ether even or odd ntegers by the requrements of exchange symmetry. For smplcty we consder here the heteronuclear case and relegate the homonuclear cases to the exercses. The partton functon for rotatons of a heteronuclear molecule can now be wrtten n the form Z rot = {=0 H2 { + L ExpB-{ H{ + L T rot ÅÅÅÅÅÅÅÅÅÅÅÅ T F where k B T rot = Ñ 2 ê 2 I determnes the temperature scale relevant to rotatonal degrees of freedom. In the low temperature lmt the molecules are found near the ground state wth overwhelmng probablty because the Boltzmann factor s a rapdly decreasng functon of {. Thus, only a few contrbutons to the partton functon are relevant and we may approxmate the rotatonal partton functon by T ` T rot ï Z rot º + Exp@-2 T rot ê TD The rotatonal contrbutons to the nternal energy and heat capacty are then approxmately T ` T rot ï -2 T rotêt U rot =-N ln Z rot ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ º Nk B T 2 T rot ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ b T + -2 T rotêt T ` T r ï C rot = Nk B J 2 T 2 rot ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ T N -2 T rotêt ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H + -2 T rotêt L 2 Ths system shares the low-temperature y 2 -y behavor characterstc of any two-state system because at low temperature the spacng between quantum levels reduces the system to bnary form, where here y = 2 T rot ê T. In the hgh temperature lmt T p T rot ï Z rot º { { ÅÅÅÅÅÅÅÅ ÅÅÅÅ 0 { Exp@-b {D = ÅÅÅÅÅÅÅÅ T ÅÅÅÅ T rot the spacng between levels s much smaller than the energy of thermal fluctuatons such that the spectrum s effectvely contnuous and the sum may be replaced by an ntegral. Thus, we fnd the essentally classcal result T p T rot ï U rot = Nk B T ï C rot = Nk B Ths result could have been obtaned from the equpartton theorem provded that quantum symmetry prncples are respected n countng the degrees of freedom. Each rotatonal degree of freedom make a quadratc contrbuton to the energy and hence yelds a contrbuton of k B ê 2 to the heat capacty. Classcally, we mght have expected there to be three rotatonal degrees of freedom per molecule. However, the precedng result suggests that only two degrees of freedom are actve even n the hgh temperature lmt. The dfference s due to the fact that rotatons about a symmetry axs do not produce observable changes of the quantum state and hence do not correspond to true dynamcal degrees of freedom. Therefore, an axally symmetrc rotor possesses only two rotatonal degrees of freedom.

20 20 Semclasscal.nb The analyss of the ntermedate temperature range, T ~T rot, s more complcated because the summaton cannot be performed n closed form. The fgure below s taken from rotvb.nb where numercal summaton of the rotatonal partton functon s performed for a heteronuclear datomc molecule. Interestngly, we fnd that the heat capacty overshoots the equpartton value and approaches the lmt from above. Rotatonal Heat Capacty 0.8 C r o t êhnk BL TêT rot à Vbratons We have already analyzed the vbratonal partton functon several tmes. Expressng the vbratonal energy as Ñ w=k B T vb, we fnd Z vb = -ntvbêt = n=0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - -T vbêt U vb = Nk B T vb ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ T vbêt - C vb = Nk B J T 2 vb ÅÅÅÅÅÅÅÅÅÅÅÅ T N -T vbêt ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H - -T vbêt L 2 wth lmtng behavors T p T vb ï C vb º Nk B T ` T vb ï C vb º Nk B J T 2 vb ÅÅÅÅÅÅÅÅÅÅÅÅ T N -T vbêt à Heat capacty The fgure below, taken from rotvb.nb, shows the combned translatonal, rotatonal, and vbratonal heat capactes, C V = ÅÅÅÅ 2 Nk B + C rot + C vb, usng T rot = 5 and T vb = 40 kelvn for HCl. Note that a logarthmc temperature scale s need because the rotatonal and vbratonal temperatures dffer by more than two orders of magntude. Thus, for room temperature we expect the molar heat capacty to be about ÅÅÅÅ 5 R, representng equpartton among translatonal and rotatonal modes, but vbratonal exctatons are frozen out. Full partcpaton of vbratonal modes requres temperatures 2 above about 0 4 kelvn.

21 Semclasscal.nb 2 4 Heat Capacty CV êhnkbl T Problems ô Invarance of the phase-space volume Prove that q p s nvarant wth respect to canoncal transformatons of the generalzed coordnates and momenta. ô Semclasscal nternal entropy Consder a system of N dentcal ndstngushable nonnteractng partcles and assume that the sngle-partcle partton functon can be expressed n the form Z = V l B - z@td where l B = H2 p Ñ 2 ê mk B TL ê2 s the thermal wavelength and z@td s the partton functon for nternal exctatons. Evaluate the entropy and dscuss the nternal contrbuton. Express the low temperature behavor of the nternal entropy n terms of the energes and degeneraces of the lowest few levels.

22 22 Semclasscal.nb ô Langmur adsorpton theorem Consder a plane crystal surface whch contans N s stes at whch a molecule from the surroundng gas can be adsorbed. The crystal, whch has neglgble volume tself, s placed at the bottom of a contaner of volume V whch contans a gas at temperature T consstng of N g molecules of mass m. We seek to determne the equlbrum fracton, N b ê N s, of the avalable stes that hold an adsorbed molecule, where N b s the total number of molecules adsorbed onto the surface. We assume that only one molecule may occupy each ste and that the energy whch bnds a molecule to a ste s b, whch s ndependent of the concentraton of flled stes. a) Recall that the free energy for an deal gas s gven by F g = N g k B T Ä k jlog N g l É ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÇÅ V ÖÑ - y z wth l= h 2 y j ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ z { k 2 pmk B T { and determne the chemcal potental of the gas, m g, as a functon of ts pressure and temperature. b) Determne the canoncal partton functon, Z b, TD, that governs the adsorbed molecules, takng care to account to the number of ways that N b molecules may be dstrbuted among N s stes. c) Calculate the free energy, F@N b, TD, for the adsorbed molecules. You may assume that both N b and N s are very large, but do not assume that N b ` N s. Then compute the chemcal potental, m b, for the bound gas. d) State the equlbrum condton and from t derve the Langmur adsorpton theorem, and determne p N b ÅÅÅÅÅÅ Å N s = p ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p + p e) Alternatvely, obtan the grand partton functon, Z md, for the system of adsorpton stes and deduce the mean number of adsorbed molecules, N êêê b. Show that both canoncal and grand canoncal methods yeld the same result. ê2 ô Relatvstc deal gas The classcal Hamltonan for an deal relatvstc gas takes the form N H = mc 2 j $%%%%%%%%%%%%%%%% + J p%%%%%% y ÅÅÅÅÅÅÅÅÅÅÅ mc N2 - z k { = where m s the rest mass and where the rest energy has been subtracted so that the Hamltonan represents the total knetc energy of N partcles. a) Show that the sngle-partcle partton functon takes the form Z Exp@xD where x = mc 2 ê k B T s the rato between rest and thermal energes and where

23 Semclasscal.nb 2 = 0 pp 2 ExpA-x è!!!!!!!!!!!! + p 2 E Although Mathematca 4.0 cannot evaluate the defnte ntegral for drectly, the change of varables p Ø è!!!!!!!!!!!! y 2 - provdes mmedate gratfcaton. b) Derve the mechancal equaton of state. c) Derve symbolc expressons for the nternal energy and sochorc heat capacty. Use Mathematca to produce a log-lnear plot of C and nterpret the low and hgh temperature lmts. ô Morse potental The potental energy between the atoms of a hydrogen molecule can be approxmated by a Morse potental of the form V HrL = V 0 H Exp@-2 Hr - r 0 Lêa 0 D - 2 Exp@-Hr - r 0 Lêa 0 DL where r s the dstance between nucle, V 0 = 4.4 ev, r 0 = 0.8 Þ, and a 0 = 0.5 Þ. Sketch the potental and calculate the characterstc rotatonal and vbratonal temperatures. Also estmate the temperature for whch anharmonc effects become mportant. ô Gas wth quartc nteracton An deal nonrelatvstc gas conssts of N dentcal molecules n volume V. Each molecule has one actve nternal mode of exctaton descrbed by the sngle-partcle Hamltonan H = P 2 ÅÅÅÅÅÅÅÅÅÅÅ Å 2 M + p 2 ÅÅÅÅÅÅÅÅ 2 m + gx 4 where M s the mass, P s the center-of-mass momentum, p s the nternal momentum conjugate to the nternal dsplacement varable x, and where m and g are postve constants. Use semclasscal statstcs. a) Compute the Helmholtz free energy and evaluate the mechancal equaton of state. b) Evaluate the nternal energy and sochorc heat capacty. How does ths result compare wth a harmonc nternal potental? c) Calculate the entropy for ths system and provde a qualtatve explanaton of ts dependences upon the parameters of the sngle-partcle Hamltonan. Recall: G@zD = 0 t -t t z- ô Classcal rgd rotor The Hamltonan for a rgd rotor s

24 24 Semclasscal.nb 2 L H rot = ÅÅÅÅÅÅÅÅÅÅÅ 2 I where L = I w s the angular momentum about a prncpal axs wth moment of nerta I. The orentaton of the body relatve to a space-fxed frame s descrbed by three Euler angles Hf, q, yl, where 0 f 2 p s a rotaton about the z` axs, 0 q p s a rotaton about the ỳ axs, and 0 y 2 p s a rotaton about the z` axs. One can show that the angular veloctes take the form w =q Sn@yD -f Sn@qD Cos@yD w 2 =q Cos@yD +f Sn@qD Sn@yD w =y +f Cos@qD a) Deduce the canoncal momenta 8p q, p f, p y < and express H rot n canoncal form. b) Evaluate the rotatonal partton functon, entropy, and nternal energy for N ndependent rgd rotors assumng that all orentatons are dstngushable. c) Suppose that for an nteger n > rotaton through an angle 2 p ê n about some axs produces an ndstngushable state for a body wth n-fold symmetry about that axs. For example, the H 2 O molecule has a 2-fold symmetry. How are the entropy and heat capacty affected by dscrete symmetres? = ô Heat capacty for polyatomc gases Suppose that a gas conssts of N dentcal nonnteractng molecules whch each contan n atoms. a) Separate the n degrees of freedom per molecule nto translatonal, rotatonal, and vbratonal modes and determne ther contrbutons to the classcal heat capacty. You wll need to dstngush lnear molecules wth axal symmetry from other structures wthout axal symmetry. b) Wrte a general expresson for the semclasscal partton functon for methane, CH 4, n terms of the prncpal moments of nerta, I, and vbratonal frequences, w j. Be sure to consder the dscrete symmetres for ths tetrahedral structure. ô Ortho- and para-hydrogen A proper treatment of the rotatonal thermodynamcs of an deal gas of homonuclear molecules must carefully consder the exchange symmetry of the nuclear part of the wave functon. Suppose that the two dentcal nucle are fermons, as n the H 2 molecule. Snce the nuclear wave functon must then be antsymmetrc wth respect to exchange, antsymmetrc spn must be combned wth symmetrc spatal wave functons and symmetrc spn must be combned wth antsymmetrc spatal wave functons. The spatal wave functon s even or odd wth respect to exchange accordng to whether the orbtal angular momentum { s even or odd. a) Show that there are s A H2 s A + L antsymmetrc and Hs A + L H2 s A + L symmetrc spn wave functons for two dentcal nucle wth spn s A. Thus, demonstrate that the rotatonal partton functon for a homonuclear datomc molecule wth fermon nucle s