Ideal Quantum Gases. from Statistical Physics using Mathematica James J. Kelly,

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1 Idel Quntum Gses from Sttisticl Physics using Mthemtic Jmes J. Kelly, 996- The indistinguishbility of identicl prticles hs profound effects t low tempertures nd/or high density where quntum mechnicl wve pckets overlp pprecibly. The occuption representtion is used to study the sttisticl mechnics nd thermodynmics of idel quntum gses stisfying Fermi- Dirc or Bose-Einstein sttistics. This notebook concentrtes on forml nd conceptul developments, liberlly quoting more technicl results obtined in the uxiliry notebooks occupy.nb, which presents numericl methods for relting the chemicl potentil to density, nd fermi.nb nd bose.nb, which study the mthemticl properties of specil functions defined for Ferm-Dirc nd Bose-Einstein systems. The uxiliry notebooks lso contin lrge number of exercises. Indistinguishbility In clssicl mechnics identicl prticles remin distinguishble becuse it is possible, t lest in principle, to lbel them ccording to their trjectories. Once the initil position nd momentum is determined for ech prticle with the infinite precision vilble to clssicl mechnics, the swrm of clssicl phse points moves long trjectories which lso cn in principle be determined with bsolute certinty. Hence, the identity of the prticle t ny clssicl phse point is connected by deterministic reltionships to ny initil condition nd need not ever be confused with the lbel ttched to nother trjectory. Therefore, in clssicl mechnics even identicl prticles re distinguishble, in principle, even though we must dmit tht it is virtully impossible in prctice to integrte the equtions of motion for mny-body system with sufficient ccurcy. Conversely, in quntum mechnics identicl prticles re bsolutely indistinguishble from one nother. Since the prticle lbels hve no dynmicl significnce, exchnging the coordintes or lbels of two identicl prticles cn chnge the wve function by no more thn n overll phse fctor of unit mgnitude. In this chpter we nlye the sometimes quite profound consequences of permuttion symmetry for systems of identicl prticles, but first we would like to provide qulittive explntion for the similrities nd differences between the clssicl nd quntum pictures. Perhps the most significnt difference between these pictures is found in the inherent fuiness of quntum trjectories. Contrry to clssicl ides, it is not possible even in principle to determine both the position nd momentum of prticle simultneously with rbitrry precision. The precisions with which conjugte vribles cn be determined simultneously re limited by the Heisenberg uncertinty reltion Dx D p x ÅÅÅÅ Ñ, whereby precise mesurements of one vrible cuse the uncertinty in its conjugte vrible to be quite lrge. Therefore, the set of clssicl phse points cnnot be determined with bsolute certinty; nor cn the evolution of the system be confined to shrp trjectories through phse spce. The trjectory for ech prticle is initilly brodened by the uncertinty product nd it becomes fuier nd more diffuse s it evolves, much s wve pcket would spred s it propgtes through dispersive medium. However, quntum wve pcket spreds nturlly even without dispersive medium.

2 IdelQuntumGses.nb Under some circumstnces these two pprently conflicting pictures cn in fct become quite similr. When the wve pckets re sufficiently compct nd the density of the system is sufficiently smll tht different wve pckets rrely overlp, we cn gin distinguish prticles by the trjectories followed by their wve pckets. Smll corrections my be necessry for quntum effects tht might occur when pir of wve pckets does overlp, but otherwise the clssicl description cn be quite ccurte under pproprite conditions. As we rgued when developing semiclssicl sttisticl mechnics, the pplicbility of the clssicl picture is governed by the therml wvelength l B = h ÅÅÅÅÅÅÅÅ p B = i k j Å p Ñ y mk B T { ê p B = H p mk B TL ê corresponding to the de Broglie wvelength for prticle with typicl therml momentum p B. From dimensionl rguments lone it is cler tht the width of the wve pcket is inversely proportionl to the momentum of the prticle nd is similr to the wvelength for the dominnt component of the wve pcket. The typicl momentum must be proportionl to the mss nd to the typicl velocity ner the pek of the Mxwell-Boltmnn distribution. Therefore, lrge mss or high temperture produce smll l B where the clssicl pproximtion becomes useful. Smll mss or low temperture result in lrge l B, which increses the overlp between wve pckets nd requires quntum description. Thus, the reltive importnce of fundmentlly quntum mechnicl behvior is guged by the quntum concentrtion n Q = ÅÅÅÅÅÅÅÅÅÅÅÅ gv l B which compres the volume occupied by wve pckets to the volume vilble to ech prticle. Here we use g to represent the multiplicity fctor for point in phse spce. This fctor typiclly tkes the vlues s + where s is the intrinsic ngulr momentum, but my be lrger if there re other internl degrees of freedom. The clssicl picture my be pproprite when n Q `, but s n Q pproches unity nonclssicl behvior is expected. It is importnt to recognie tht even when the clssicl description is useful, the quntum mechnicl indistinguishbility of identicl prticles continues to hve importnt consequences. One must still consider indistinguishbility when enumerting clssicl sttes in order to resolve the Gibbs prdox nd to obtin extensive thermodynmic potentils. Similrly, the grnulrity of phse spce is needed to estblish the scle for entropy. Permuttion symmetry An -body system is described by wve function of the form Y@q,, q D where q i denotes the full set of coordintes belonging to prticle i. Suppose tht the prticles re identicl nd hence indistinguishble from one nother. The Hmiltonin must then be invrint with respect to interchnge of ny pir of identicl prticles. Consider the ction of the prticle exchnge opertor X i, j defined by X i, j Y@ q i q j D = s Y@ q j q i D which exchnges the coordintes of prticles i nd j. A second ppliction of the exchnge opertor X i, j Y = s Y = Y must restore the wve function to its originl stte. The symmetry of the Hmiltonin under prticle exchnge requires X to be unitry nd hence s =. Therefore, the wve function must be either symmetric (s = ) or ntisymmetric (s =-) with respect to prticle exchnge.

3 IdelQuntumGses.nb Experimentlly, the exchnge symmetry of mny-body systems is found to be intimtely relted to the intrinsic ngulr momentum (spin) of their constituents. Prticles with hlf-integrl spin live in ntisymmetric wve functions nd re clled fermions; prticles with integer spin live in wve functions symmetric under prticle exchnge nd re clled bosons. Although these reltionships cn be derived using reltivistic quntum field theory, for our purposes it is sufficient to regrd this distinction s n experimentl fct. We shll soon find tht the sttisticl properties of fermion nd boson systems re profoundly different t low tempertures. Fermions obey Fermi-Dirc (FD) sttistics, wheres boson obey Bose-Einstein (BE) sttistics. In the clssicl limit, both distributions reduce to the Mxwell-Boltmnn (MB) distribution. The indistinguishbility of identicl prticles ffects the number of distinct sttes very strongly. Consider system of two identicl nonintercting fermions. The wve function must be ntisymmetric wrt exchnge of the coordintes of the two prticles. Hence, the two-prticle wve function tkes the form Y q D = ÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!! H y n@q D y D -y D y DL where q nd q re prticle coordintes nd y n nd y m re single-prticle wve functions. otice tht the two-prticle wve function vnishes if one ttempts to plce both prticles in exctly the sme stte. Therefore, the Puli exclusion principle sttes tht it is not possible for more thn one fermion to occupy ny prticulr stte; occupncy by one fermion blocks occupncy by others. Consider system of identicl prticles for which single-prticle sttes re vilble. If, s in clssicl mechnics, the prticles re considered distinguishble, they my be lbeled s A nd B. On the other hnd, both prticles must hve the sme lbel, sy A, if they re considered indistinguishble. The tbles below enumerte ll distinct sttes for distinguishble clssicl prticles or for indistinguishble fermions nd bosons. Clssiclly, = 9 distinct microsttes re vilble, but only re vilble to fermions or 6 to bosons. Also notice tht the rtio between the probbility tht the two prticles occupy the sme stte to the probbility tht they occupy different sttes is.5 clssiclly,. for bosons, nd for fermions. Therefore, the requirements of permuttion symmetry severely limit the number of sttes, especilly for fermions. clssicl fermions bosons AB AA AB AA AB AA A B A A A A B A A B A A A A B A A B A A A A B A

4 4 IdelQuntumGses.nb -body wve functions Consider system of identicl prticles confined to volume V. Let q = 8q,, q < represent complete set of coordintes. The wve function for mny-body stte lbeled Y, where =8,, < denotes complete set of quntum numbers, will then hve the form Xq q» Y \ = q D = Let P denote prticulr permuttion opertor nd let s@pd = represent the signture for n even (+) or odd (-) permuttion. For exmple, P 8q, q, q < = 8q, q, q < would be n odd permuttion with s@p D =-, while P 8q, q, q < = 8q, q, q < would be n even permuttion with s@p D =+. Any permuttion of the coordintes or lbels chnges the wve function by t most phse fctor; it mtters not whether we choose to permute coordintes or lbels. Bosons re symmetric with respect to prticle exchnge, such tht bosons ï P i <D = Y@P 8q i <D = Y i <D = wheres fermions re ntisymmetric with respect to prticle chnge, such tht the phse of permuttion is even or odd ccording to the signture of the permuttion fermions ï P i <D = Y@P 8q i <D = Y i <D = H-L s@pd It will be useful to define bosons : d P = + fermions : d P = H-L s@pd It is useful to introduce complete orthonorml set of single-prticle wve functions such tht Xf» f b \ =d, b. Unsymmetried product wve functions for the -body systems cn then be defined by D D D = f i D i= Properly symmetried -body wve functions cn now be constructed by dding ll possible permuttions of the product wve functions with pproprite phses, such tht = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!!! d P = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!!! d P U P P Any F constructed from single-prticle eigenfunctions f i will now represent properly symmetried eigenfunction for system of nonintercting identicl prticles. Wve functions for intercting systems cn be constructed from superpositions of product wve functions, which serve s bsis vectors for the system. However, for bosons one must be creful to properly normlie the product wve functions.

5 IdelQuntumGses.nb 5 Occuption representtion We begin our investigtion of the thermodynmic consequences of permuttion symmetry by studying idel quntum fluids composed of identicl fermions or bosons under conditions where their mutul interctions my be neglected. Under these circumstnces the mny-prticle Hmiltonin my be expressed s summtion over independent identicl single-prticle contributions H = h i i= with eigenfunctions F n represented by the set of occuption numbers n =8n, n,, n < such tht H F n = E n F n E n = n = = = where is the number of prticles in the single-prticle orbitl f stisfying the eigenvlue problem h f = f The mny-body wve function F n = P H L P f i i= my then be constructed from properly symmetried product of n single-prticle wve functions where here the index i runs over ll occupied orbitls nd where the summtion over P includes ll permuttions of the prticle lbels with pproprite phses. Rther thn specifying the coordintes or quntum numbers for ech prticle in the system, the occuption representtion specifies how mny, but not which, prticles occupy ech single-prticle orbitl. This is the most nturl representtion of the sttes for system of identicl prticles nd voids the complictions of enforcing the permuttion symmetry nd evluting degenercy fctors. Although permuttion symmetry cn be enforced explicitly within the cnonicl ensemble or microcnonicl ensembles, the grnd cnonicl ensemble provides more efficient method. This method focuses upon the occupncy of prticulr orbitl s the relevnt subsystem insted of treting the prticles themselves s the subsystems. As subsystem the orbitl exchnges energy nd prticles with reservoir comprised of the prticles occupying ll other orbitls. Becuse the number of prticles occupying ech orbitl is indefinite, sttisticl quntity, this pproch nturlly involves the grnd cnonicl ensemble. The properties of the reservoir estblish temperture nd chemicl potentil which serve s Lgrnge multipliers tht enforce constrints upon the energy nd density of the entire system. The occupncy nd its dispersion cn be deduced esily. where The probbility P n for stte n =8n < in the occuption representtion cn be expressed s = Z - Exp@-b HE n -m n LD P n

6 6 IdelQuntumGses.nb n = E n = = = re the totl prticle number nd totl energy nd where Z@T, V, md = Exp@-bn H -mld n = n is the grnd prtition function. The temperture nd chemicl potentil re determined by the constrints upon energy nd density. The dependence upon volume is crried implicitly by the dependence of the single-prticle energy levels upon the sie of the system. The seprbility of the energy llows the prtition function for ech stte n to be expressed s product of prtition functions for ech orbitl. The restricted sum n indictes summtion over ll sets of occuption numbers 8n < with totl prticle number n. The grnd prtition function then sums over n s well, weighted by the fctor Exp@b m n D. The effect of this summtion over n is to remove the restriction upon sum over 8n <, such tht Z@T, V, md = 8n < Exp@-bn H -mld where ech sum over n is now independent, only restricted by the Puli exclusion principle for fermions. It is useful to define x ªbH -ml s the energy of orbitl reltive to the chemicl potentil in units of the therml energy k B T, such tht Z@T, V, md = 8n < Exp@-n x D The summtion nd product cn now be interchnged 8n < Exp@-n x D = Exp@-n x D n so tht Z = Z Z = Exp@-n x D n becomes product of independent single-orbitl grnd prtition functions. This result is quite nlogous to the fctorition of n -element cnonicl prtition function in terms of independent single-element fctors, now pplied to orbitls rther thn to prticles. ote tht there will normlly be n infinite number of Z fctors, but most will be unity. Similrly, the probbility P n lso fctors ccording to P n = P P = Z - Exp@-n x D where The men prticle number nd totl energy re now obtined s ensemble verges X\ = P n n = P n = n n n XE\ = n P n E n = n P n = n

7 IdelQuntumGses.nb 7 n = n P n is the men occupncy of orbitl. Recogniing tht ech single-orbitl probbility distribution is normlied, such tht D =, n the men occupncy for ech orbitl cn be evluted independently using n = n P n where the reltionship between the occupncies for vrious orbitls is governed by nd m nd where the totl density is governed by m. Thus, the thermodynmic properties of the system re determined by the men occupncies, which re obtined from the grnd prtition function ccording to n = kb T ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ln Z ÅÅÅÅÅÅÅ = - i m k j ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ G y = - ln Z m { x T,V where G =-k B T ln Z is the single-orbitl contribution to the grnd potentil G = G. The vrince in occupncy cn lso be obtined esily. Evluting the men squre prticle number in ech orbitl Xn \ = Z - Z ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ x = i k j ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ G y m { - k B T G ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ Å m = n - k B T G ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Å m we find tht the vrince becomes s n = XHn - n L \ = ln Z ÅÅÅÅ x = -k B T i k j ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ G y Å m { T,V = - n ÅÅÅÅÅÅÅÅÅÅÅÅÅ x Therefore, the occupncy vrince is relted to the derivtive with respect to chemicl potentil by s n = - n ÅÅÅÅÅÅÅÅÅÅÅÅÅ x Continuous pproximtion for uniform systems If the system is sufficiently lrge nd the spcing between single-prticle energy levels sufficiently smll, we cn replce discrete sums over sttes by integrls over energy weighted by the density of sttes. Thus, the totl number of prticles becomes = n ö ã D@D êê n@, b, md nd determines the dependence of the chemicl potentil on the temperture nd density of the system. If the prticle density is specified, the chemicl potentil cn be determined by numericl solution of this eqution. However, we will find tht for the Bose-Einstein distribution cre must be tken with the continuous pproximtion whenever one or more

8 8 IdelQuntumGses.nb sttes develops mcroscopic occupncy. Similrly, the eqution of stte is determined by the grnd potentil, which in the continuous pproximtion tkes the form G = -k B T ln Z ö -k B T D@D ln Z@, b, md where Z@, b, md = H -bh-ml L is the single-orbitl prtition function. Therefore, the grnd potentil becomes G = k B T D@D Log@ -bh-ml D with upper or lower signs for fermions or bosons, respectively. The density of sttes for gs confined to simple box of volume V is relted to the phse-spce density by D@D = D@k D k = gv k ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H pl where g is the intrinsic degenercy of stte in momentum spce. For exmple, if the only degree of freedom tht is needed to describe the internl stte of prticle is its spin s, then g = s + represents the totl number spin sttes which re degenerte in the bsence of n externl mgnetic field. If the system is sphericlly symmetric, we cn use D@k D k ö gv k k ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p Assuming tht nonreltivistic kinemtics re pplicble, the reltionship between energy nd momentum representtions of the density of sttes is given by = HÑ kl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ m ï ÅÅÅÅÅÅÅÅ k = ÅÅÅÅÅ k ï k k = ÅÅÅÅÅ J ÅÅÅÅÅÅÅÅÅÅÅÅ m ê ê Ñ such tht D@D = ÅÅÅÅÅÅÅÅ gv ÅÅÅÅÅ 4 p J m ê ÅÅÅÅÅÅÅÅÅÅÅÅ ê Ñ Systems with lower dimensionlity, or with reltivistic kinemtics, or with hrmonic oscilltor rther thn squre-well confinement potentils, re considered in the exercises. Sttistics of occuption numbers The occupncy of single-prticle sttes in mny-body system is sttisticl quntity subject to thermodynmic fluctutions. Since the number of prticles occupying ny stte is indeterminte, ech orbitl cn be visulied s subsystem in equilibrium with reservoir of energy nd prticles. Therefore, the sttistics of occupncy re susceptible to nlysis using the grnd cnonicl ensemble. The probbility distribution describing the occupncy of orbitl with single-prticle energy in n idel quntum fluid is

9 IdelQuntumGses.nb 9 P@n D = -n x Z = Z n -n x where m is the chemicl potentil estblished by the reservoir nd where it is very convenient to define reduced energy reltive to the chemicl potentil s x =bh -ml. The grnd prtition function Z sums over ll possible vlues of the occupncy n for orbitl. The men occupncy nd its vrince re given by n = P n = - ln Z x n s n = XHn - n L \ = - n ÅÅÅÅÅÅÅÅÅÅÅÅÅ x à Mxwell-Boltmnn distribution Although the semiclssicl Mxwell-Boltmnn (MB) distribution does not ctully pertin to ny rel quntum system, it is nevertheless instructive to compre the FD nd BE distributions to this clssicl model. Clssiclly, the degenercy for stte of identicl prticles described by occuption numbers n =8n < would be! ê n!, but we must divide by! to resolve the Gibbs prdox so tht g n = H n!l - becomes the proper sttisticl weight for the MB distribution. The grnd prtition function cn then be fctored, such tht Z = -n x n = n! = Exp@ -x D is recognied s n exponentil function with fncy rgument. Therefore, the grnd potentil, men occupncy, nd occupncy vrince become G = -k B T -x ï n = -x ï s n = n The single-orbitl occupncy probbility distribution cn now be expressed in form P@n D = ÅÅÅÅÅÅÅÅÅÅÅÅ n! -n x - -x = n n ÅÅÅÅÅÅÅÅÅÅÅÅÅ n! -n ê we recognie s the Poisson distribution, which is n pproximtion to the binomil distribution tht pplies to ensembles consisting of mny independent trils with low probbility for success in ech tril. Similrly, the MB distributions pplies when the totl number of prticles is lrge but the probbility tht ny prticulr single-prticle orbitl is occupied, nd hence its men occupncy, is smll. Thus, the rms fluctution in orbitl occupncy s n ÅÅÅÅÅÅÅÅÅÅÅÅ n = J ê ÅÅÅÅÅÅÅÅ n is chrcteristic of systems with sttisticl independence. à Fermi-Dirc distribution The exclusion principle restricts the occupncy n for fermions to either or. Thus, the grnd prtition function nd grnd potentil for the Fermi-Dirc (FD) distribution reduce to simply Z = + -x ï G =-k B T Log@ + -x D where x =bh -ml. Therefore, the men occupncy becomes

10 IdelQuntumGses.nb n ln Z = - = H x + L - x for the FD distribution. Using x = - n Å ï Z = - n n the probbility distribution P@n D for occuption numbers cn be expressed in the form - P@n = D = Z = - n P@n = D = Z - -x = n Therefore, we cn interpret n for the FD distribution s the probbility tht orbitl is occupied nd - n s the probbility tht it is empty. Similrly, the occupncy vrince is s n =- n x ÅÅÅÅÅÅÅÅ ÅÅÅÅÅ = ÅÅÅÅÅÅ = n x H x + L H - nl so tht the reltive rms fluctution in occupncy is s n ÅÅÅÅÅÅÅÅÅÅÅÅ n = J - n ê Å n Hence, the occupncy fluctutions re suppressed reltive to norml (uncorrelted) distribution by fctor of H - n L ê due to the influence of quntum correltions rising from permuttion symmetry. à Bose-Einstein distribution For bosons the occupncy of ech orbitl is unrestricted so tht the sum over n extends from ero to infinity. The grnd prtition function Z = -n x = H - -x L - n = is thus n infinite geometric series. The grnd potentil nd men occupncy re then G = k B T Log@ - -x D ï n = H x - L - The requirements tht the orbitl occupncies be positive definite nd tht Z converges necessitte m where is the lowest single-prticle energy. Since by convention =, we require - m for the Bose-Einstein (BE) distribution, wheres m is unrestricted for the FD distribution. Using x = + n Å ï Z = + n n we find tht the occupncy probbility distribution P@n D = -n x n n - n n + = ÅÅÅÅÅÅÅÅÅÅÅÅÅ Z H + = n n L n + J ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ + n is geometric distribution with constnt rtio

11 IdelQuntumGses.nb + D ÅÅÅÅÅÅ P@n D = n + n between successive terms. The occupncy vrince is s n =- n x ÅÅÅÅÅÅÅÅ ÅÅÅÅÅ = ÅÅÅÅÅÅ x H x - L = n H + nl so tht the rms fluctution s n ÅÅÅÅÅÅÅÅÅÅÅÅ n = J + n Å n ê is enhnced by fctor of H + n L ê compred with norml (clssicl) system. Thus, fluctutions in the FD distribution re suppressed nd in the BE distribution enhnced by the occupncy of n orbitl. These correltions rise from the permuttion symmetry of the mny-body wve function even without interctions. The BE occupncy distribution should be fmilir from our studies of the Debye model nd the Plnck distribution. For these models we found tht the men number of qunt in mode with energy ws n = H b - L -, which hs the sme form s the Bose-Einstein distribution with m =. Recogniing tht qunt of the electromgnetic field re photons, n elementry prticle with spin, we must tret photons with the sme energy nd polrition s indistinguishble bosons stisfying Bose-Einstein sttistics. However, becuse the number of photons is truly indeterminte, the chemicl potentil for photon gs vnishes. Although there is no elementry prticle tht corresponds to quntied lttice vibrtions of crystl, these qunt behve identicl spinless prticles with vnishing chemicl potentil. In fct, there is wide vriety of quntied excittions tht re governed by BE sttistics with m =. à Comprison between FD, BE, nd MB occupncies The men occupncy n b, md for n orbitl with energy in system with inverse temperture b nd chemicl potentil m cn be expressed in the generic form êê b, md = H b H-mL +gl - where FD ï g = + MB ï g = BE ï g = - where g =, governs the sttisticl properties of the distribution. Adopting the usul convention tht the singleprticle energy spectrum strts t =, such tht, the requirement tht n êê be positive definite requires - m for the BE distribution, but m my ssume either sign for the MB nd FD distributions. These occupncy functions re compred below.

12 IdelQuntumGses.nb 5 Men Occupncy for FD, BE, MB Distributions 4 MB êê n BE FD b H -ml These occupncy functions converge in the limit -m ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k B T p ï nêê -b b, md = where the men occupncy êê n ` is very smll. Ordinrily one considers the clssicl limit to be synonymous with high temperture, but this comprison demonstrtes tht more rigorous description of the clssicl limit is "smll single-orbitl occupncy", which is needed for the Gibbs resolution of the indistinguishbility problem to become ccurte. It is not sufficient to require the temperture to be lrge, which is needed to pproximte discrete spectrum of energy sttes by continuous phse spce, but one must lso require the density to smll enough for the chemicl potentil to be lrge nd negtive such tht clssicl limit : m ` - k B T As T increses, m must become incresingly negtive to fulfill this condition for ny fixed. To guge the ccurcy of the clssicl pproximtion, we relte the chemicl potentil for clssicl idel gs to its density by summing the men number of prticles in ech momentum stte using the MB distribution. ÅÅÅÅÅÅ V = Ä É p ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H p ÑL Exp ÇÅ -bi j ÅÅÅÅÅÅÅÅ p ÅÅÅÅ k m -my { ÖÑ = l B - bm where l B = i k j Å p Ñ y mk B T { Thus, we find tht the chemicl potentil is closely relted to the quntum concentrtion: m =k B T LogB ÅÅÅÅÅÅ V l B F = k B T Log@n Q D This result pplies to nonreltivistic gs in three sptil dimensions. Clerly, m ` ï n Q ` lrge negtive chemicl potentil requires smll quntum concentrtion to minimie the overlp between wve pckets ssocited with the chrcteristic therml energy. The reltionship between chemicl potentil nd density is more complicted for the FD nd BE distributions, but reduces to the MB result in the clssicl limit n Q `. ê

13 IdelQuntumGses.nb Degenerte Fermi gs According to the FD distribution, the men occupncy for single-prticle stte with energy is êê b, md = H b H-mL + L - At T = the rgument of the exponentil is - for <m or is + for >m. Hence, the occupncy is unity for ll sttes with below m nd is ero for sttes with >m. Therefore, the completely degenerte Fermi gs t T = is described by froen distribution in which ll orbitls below the chemicl potentil re occupied nd ll orbitls bove m re vcnt. For fixed density, the chemicl potentil is function of temperture lone. By convention, we define the Fermi energy, F, to be the chemicl potentil the system hs t T = for given density. Similrly, the Fermi temperture, T F, is defined by T = ï m = F = k B T F It is then useful to express nd m is units of F nd to define the reduced temperture s t =T ê T F. Also notice tht =m@td ï êê n =.5 such tht the chemicl potentil for finite temperture is equl to the energy for which the occupncy is one hlf. The reltionship between the chemicl potentil nd temperture is determined by fixing the verge density of the system. At T =, ll sttes below F hve unit occupncy so tht the totl number of prticles in completely degenerte nonreltivistic Fermi gs is given by = D@D êê n@d = ÅÅÅÅÅÅÅÅÅÅÅÅÅ gv 4 p J ÅÅÅÅÅÅÅÅÅÅÅÅ m Solving for F in terms of the prticle density r = ê V, we find F = ÅÅÅÅÅÅÅÅÅÅÅÅ Ñ i m k j ÅÅÅÅÅÅÅÅÅÅÅÅÅ 6 p g k F = i k j ÅÅÅÅÅÅÅÅÅÅÅÅÅ 6 p r y ê g { r y ê { = HÑ k FL m Ñ ê F ê = ÅÅÅÅÅÅÅÅÅÅÅÅÅ gv 6 p J ÅÅÅÅÅÅÅÅ m F ÅÅÅÅÅÅÅÅÅÅÅÅ Ñ The fct tht the Fermi momentum k F scles with r ê is consequence of the fct tht the wve numbers for singleprticle sttes scle with V -ê. For lter purposes it will be useful to observe tht the density of sttes cn lso be expressed in the forms D@D = ÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ê ÅÅÅÅÅ = J ÅÅÅÅÅÅÅÅ ê ê F F D@kD k = k k ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ = i k j ÅÅÅÅÅÅÅ k y k F { k F by scling ccording to either F or k F, s pproprite. The verge single-prticle energy is ê

14 4 IdelQuntumGses.nb êê Ÿ = D@D êê n@, m, TD ÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Ÿ D@D êê n@, m, TD which for completely degenerte Fermi gs reduces to T = ï êê = Ÿ F ê ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ F = ÅÅÅÅÅ Ÿ ê 5 F Hence, the totl internl energy for degenerte Fermi gs with prticles is simply T = ï U = ÅÅÅÅÅ 5 F Similrly, the pressure for ny nonreltivistic idel gs, clssicl or quntum, is simply two-thirds of the energy density. Thus, we find tht the pressure T = ï p = ÅÅÅÅÅ 5 r F = ÅÅÅÅÅÅÅÅÅÅÅÅ Ñ i 5 m k j ÅÅÅÅÅÅÅÅÅÅÅÅÅ 6 p y ê r 5ê g { is finite, scling with r 5ê, even t T =. Contrry to the behvior of clssicl idel gs, for which the pressure vnishes t T =, Fermi gs retins substntil pressure t bsolute ero. This dditionl pressure, which my be clled degenercy pressure, is due to the exclusion principle which, by preventing more thn g prticles from occupying the sme energy level, compels fermions to occupy distribution of energy levels bove the ground stte. Hence, the energy density is proportionl to r 5ê nd the pressure seeking to reduce tht energy density cn be quite lrge. Even without explicit interctions between prticles, correltions due to ntisymmetrition of the wve function produce n effective repulsion between fermions. This is purely quntum effect with no clssicl counterprt. à Exmple: conduction electrons Conduction electrons in metl move within n verge potentil tht confines them to the volume of the mteril, but mutul interctions cn often be neglected. The dispersion reltion for electrons moving within uniform men field cn then be expressed in the form = HÑ kl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ m + U@k D where U@k D is momentum-dependent potentil describing the interction with the lttice. It is convenient to express U s function of k rther thn k becuse it must be n even function to stisfy rottionl nd reflection symmetries. Recogniing tht moment ner the Fermi surfce ply dominnt role in the thermodynmics of nerly degenerte Fermi gs, it is useful to expnd the potentil bout the Fermi momentum k F, such tht U@k D º U + U Ñ Hk - k F L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + m where U = U@k F D U = m ÅÅÅÅÅÅÅÅÅÅÅÅ Ñ i j U@k D y k k { k =k F Combining the kinetic nd potentil terms, the dispersion reltion now tkes the form

15 IdelQuntumGses.nb 5 º + HÑ kl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ m eff where = U - U k F ÅÅÅÅÅÅÅÅ is n energy shift nd m m eff ÅÅÅÅÅÅÅÅÅÅÅ Å m = H + U L - is the effective mss for electrons ner the Fermi surfce. Therefore, the primry effect of the men field is to shift the energy scle nd lter the effective mss. In metls one usully finds tht U > ï m eff > m such tht the effective mss for conduction electron with momentum ner k F ppers to be incresed by its ttrction to the lttice. Thus, one cn tret conduction electrons s n idel Fermi gs of prticles with effective mss m eff. Using r = 8.5 µ cm - nd m eff =.9 m e, we obtin Fermi energy F = 5.7 ev for copper. The corresponding Fermi temperture, T F º 6 µ 4 kelvin, is so lrge tht the electron gs cn be treted s lmost completely degenerte t room temperture. The corresponding pressure within this gs is then p º.7 µ 5 tmospheres! This lrge outwrd pressure must be blnced by the ttrctive electrosttic forces binding the electrons to the lttice. The confinement of the electron gs is ccomplished by nerly uniform potentil rther thn by wlls t the boundries of its volume. evertheless, since the interctions with the lttice cn be represented s smooth men field in which ll of the electrons move more or less independently of ech other, the nonintercting Fermi gs model is still pproprite. à Exmple: nucler mtter The tomic nucleus consists of protons nd neutrons, which re both spin ÅÅÅÅ prticles of equl mss. It is useful to tret protons nd neutrons s sttes of the sme prticle, the nucleon, differing only in n internl quntum number clled isospin. ucler mtter is theoreticl system consisting of equl numbers of protons nd neutrons with the Coulomb interction turned off. Thus, the intrinsic degenercy fctor for momentum sttes in nucler mtter is g = 4. The density of nucler mtter is bsed upon the centrl density of lrge nuclei, which is pproximtely constnt t.6 fm -, where femtometer (fm) is -5 m. Using these vlues, one obtins F = 7 MeV, which corresponds to n enormous pressure of 4 µ 7 tmospheres tht must be blnced by the men potentil generted by the mutul interctions mong nucleons. Obviously the nme strong force for the nucleon-nucleon interction is well deserved! It ws mjor ccomplishment of nucler physics in the 95s to demonstrte tht the Fermi gs model provides resonble strting point for studying nucler structure despite the strength of this interction. Thermodynmics of nerly degenerte Fermi gses To develop the thermodynmics of nerly degenerte Fermi gses, we must determine the reltionship between chemicl potentil nd density for rbitrry temperture. The totl number of prticles is obtined by integrtion of the men occupncy over the density of sttes, such tht = D@D êê n@, m, TD ï ê F = ÅÅÅÅÅ ê Å ÅÅÅÅÅÅÅÅ b H-mL + provides n eqution tht cn be used to determine m given the temperture nd F, which depends only n density nd fundmentl constnts. It is useful to express energy nd temperture in dimensionless form using t =T ê T F nd to introduce the fugcity = bm, such tht

16 6 IdelQuntumGses.nb x ê x ÅÅÅÅÅÅ - x + ã ÅÅÅÅÅ t-ê provides n eqution tht determines the temperture dependence of the fugcity. In chemistry fugcity is often clled bsolute ctivity. The clssicl limit defined by lrge negtive m corresponds to smll, while lrge corresponds to the low-temperture limit of nerly degenerte Fermi gs. Recogniing tht other thermodynmic functions depend upon integrls of similr form, it is customry to define fmily of Fermi functions using f = ÅÅÅÅÅÅÅÅÅÅÅÅÅ G@nD x n- x ÅÅÅÅÅÅ - x + where the normlition fctor is chosen to ensure tht f Ø when Ø. Thus, the chemicl potentil is obtined from the solution of the eqution GB ÅÅÅÅÅ 5 F tê f ã or f ã ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 4 è!!! p t-ê. It is importnt to recognie tht T F H ê VL ê is function only of density, such tht ÅÅÅÅÅÅ V Tê f represents the temperture dependence of density t constnt or the dependence of chemicl potentil on temperture nd density. umericl methods for solution of the density eqution re developed in fermi.nb. Similrly, the internl energy cn be expressed in terms of Fermi functions using U = êê = Ÿ D@D êê n@, m, TD ÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Ÿ D@D êê = k B T Ÿ x x ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ê ÅÅÅÅ - x + G@5 ê D ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ = k n@, m, TD Ÿ x x ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ê B T ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ G@ ê D such tht the therml eqution of stte becomes U ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅ Å k B T = ÅÅÅÅÅ f f - x + f f ext, the mechnicl eqution of stte for nonreltivistic idel gs is simply pv = ÅÅÅÅ U, whereby pv ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅ Å k B T = f 5ê@D f Thus, the pressure nd energy density scle ccording to p ÅÅÅÅÅÅ U V T5ê f Finlly, the entropy S = ÅÅÅÅÅÅÅÅÅÅÅÅ U + pv - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ m T ï ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ S = ÅÅÅÅÅ 5 k B f f - Log@D cn be expressed in terms of nd ; hence, isentropic processes t constnt lso require constnt. Therefore, using constnt, ï VT ê = constnt, pt -5ê = constnt

17 IdelQuntumGses.nb 7 we find tht dibts for the nonreltivistic idel Fermi gs tke the fmilir form constnt S, ï pv 5ê = constnt The numericl nd nlyticl properties of Fermi functions re studied in detil within the notebook fermi.nb; here we summrie some of the results. These functions cn be represented by power series H-L k f = - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k n k= tht converges for ll >, n>. There is limiting vlue = nd useful downwrd recursion reltion f = ÅÅÅÅÅÅÅÅ f n@d For lrge, the symptotic expnsion known s Sommerfeld's lemm f bm D > ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ HbmLn i ÅÅÅÅÅÅÅÅ j + n Hn -L ÅÅÅÅÅÅÅ p G@n +D k 6 HbmL- + n Hn -L Hn -L Hn -L ÅÅÅÅÅÅÅÅÅÅÅÅÅ 7 p4 6 HbmL-4 + y { cn be derived by exploiting the fct tht the men occupncy for nerly degenerte systems hs shrp edge t the Fermi energy. Representtive Fermi functions re shown below for n =8 ÅÅÅÅ,, ÅÅÅÅ,, ÅÅÅÅ 5,, < with f ê lowest nd f highest. Fermi-Dirc functions fugcity, The temperture dependence of the chemicl potentil cn be obtined by numericl solution of the eqution f ã ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 4 è!!! p t-ê with the result plotted below.

18 8 IdelQuntumGses.nb mêhkbtfl Chemicl Potentil TêT F The chemicl potentil is defined s the Fermi energy t T =, vnishes t the Fermi temperture, nd is lrge nd negtive for high tempertures where the clssicl limit pplies. For low tempertures we cn pproximte the chemicl potentil s T ` T F ï m º i F j - p t ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k - p4 t 4 y ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 8 { The temperture nd energy dependence of the occuption probbility re illustrted below. For very low tempertures one finds tht sttes with < F re filled while sttes bove F re empty. As the temperture increses, the Fermi surfce becomes more diffuse s the popultion of energy levels bove F re populted t the expense of sttes below F. Occuption umber.8 n êhk B T F L TêT.4 F. The therml response of nerly degenerte Fermi gs with smll T ê T F is lrgely determined by orbitls in the immedite vicinity of the Fermi energy. Prticles lying deeper in the energy distribution cnnot bsorb smll mounts of energy becuse ll nerby orbitls re lredy occupied by other prticles nd re blocked by the exclusion principle. Only prticles in the pproximte energy rnge k B T cn prticipte strongly in therml processes. The shded region in the

19 IdelQuntumGses.nb 9 figure below illustrtes the frction the of system tht cn be considered thermodynmiclly ctive, which is rther smll frction t low tempertures. FD Occupncy for T=.5 T F êê n êm Thus, we might estimte the het cpcity for nerly degenerte Fermi gs using T ` T F ï C V ~ D@ F DH k B TLJÅÅÅÅÅ k B = ÅÅÅÅÅ 9 k T B ÅÅÅÅÅÅ Å T F where D@ F D is the density of sttes ner the Fermi surfce, the next fctor is the width of the ctive energy intervl, nd the finl fctor is the het cpcity per prticipting prticle. Therefore, we expect the het cpcity to be liner for low temperture nd to pproch the clssicl limit from below. A more rigorous nlysis bsed upon the lrge- expnsion of Fermi function gives the similr result T ` T F ï ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ U º ÅÅÅÅÅ F 5 + p t ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 4 - p4 t 4 Å 8 + ï C V ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ º ÅÅÅÅÅÅÅ p k B t - ÅÅÅÅÅÅÅÅÅÅÅÅÅ p4 t + except tht the coefficient ÅÅÅÅ 9 p is replced by ÅÅÅÅÅ. In fct, one does find tht the contribution mde by conduction electrons to the het cpcity of metls for T ` T F is liner in temperture nd tht this model provides good prediction for the slope. It is probbly worthwhile to evlute C V explicitly in order to prctice mnipultion of fermi functions. Using the chin rule, one obtins U = ÅÅÅÅÅ k B T f f ï C V = ÅÅÅÅÅ k B i j f 5ê@D k f + T i k j ÅÅÅÅÅÅÅÅ y T { Then using the recursion reltion f = f one soon finds C V = ÅÅÅÅÅ k B i j k f f + T ÅÅÅÅÅ i k j ÅÅÅÅÅÅÅÅ y T { V i j - f 5ê@D f y y ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k f { { V, i j ÅÅÅÅÅÅÅÅ k f y y f { { The isochoric temperture dependence of is obtined by dimensionl nlysis of the density eqution GB ÅÅÅÅÅ 5 F i k j k B T ÅÅÅÅÅÅÅÅÅÅÅÅÅ y F { ê f ã ï ÅÅÅÅÅÅ V Tê f Thus, constnt V, requires T ê f to remin constnt, such tht T ê f = constnt ï ÅÅÅÅÅ Tê f + T ê - f i k j ÅÅÅÅÅÅÅÅ y T { V, =

20 IdelQuntumGses.nb Therefore, i k j ÅÅÅÅÅÅÅÅ y T { V, = - ÅÅÅÅÅ ÅÅÅÅÅ T f f Finlly, substituting this result, we obtin C V ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = ÅÅÅÅÅÅÅÅ 5 k B 4 f f - ÅÅÅÅÅ 9 4 f f This result reduces to the clssicl limit for Ø or to the nerly degenerte result quoted bove for lrge or, equivlently, T ` T F. Although it is often useful to express forml results in terms of the reduced temperture, t=t ê T F = k B T ê F, we must lwys remember tht F depends upon density. The temperture dependencies of the principl thermodynmic functions re illustrted below nd re studied in more detil in the notebook fermi.nb. The internl energy is greter thn tht of clssicl idel gs, shown by the dshed line, becuse the Puli exclusion principle forces prticles into higher energy levels. Thus, the internl energy is ÅÅÅÅ 5 F t T Ø nd pproches the clssicl limit from bove. The corresponding degenercy pressure produced by the effective repulsion between identicl fermions is much lrger thn the kinetic pressure would be for distinguishble prticles. The het cpcity is smll t low tempertures becuse only the reltively smll number of prticles within bout k B T of the Fermi surfce cn prticipte in the thermodynmics. Similrly, the entropy is reduced t low temperture becuse nerly degenerte Fermi gs is highly ordered; permuttion symmetry strongly reduces the number of sttes vilble to fermions. UêH k BTFL CVêH k BL Internl Energy.5.5 TêT F Isochoric Het Cpcity TêT F FêH k BTFL SêH k BL Reduced Entropy 4 5 TêT F Reduced Free Energy 4 5 TêT F

21 IdelQuntumGses.nb Thermodynmics of nerly degenerte Bose gses The reltionship between chemicl potentil nd density for n idel Bose gs = ÅÅÅÅÅÅÅ - Exp@b D - requires more cre t low temperture becuse it is possible for the occupncy of the ground stte to become n pprecible frction of the totl number of prticles, but the continuous pproximtion to the density of sttes for uniform nonreltivistic system, D@D ê, gives no weight t ll to the ground stte. Under these conditions one cnnot simply replce the sum by n integrl becuse it would be impossible to ccount for ll the prticles. A simple solution to this problem is to seprte the sum into two contributions, = gs + exc, where the men number of prticles in the ground stte with energy is given by gs = g ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - where = Exp@bHm - LD, while the men number of prticles found in excited sttes is pproximted by the integrl exc = D@D êê n@, m, TD = ÅÅÅÅÅÅÅÅÅÅÅÅÅ gv 4 p J m ê ÅÅÅÅÅÅÅÅÅÅÅÅ ê Å ÅÅÅÅÅÅÅÅ - b - Ñ The requirement tht the number of prticles in excited sttes be positive limits the fugcity for Bose gses to the rnge nd the chemicl potentil to the rnge m. However, if pproches unity the ground-stte occupncy becomes mcroscopiclly lrge. The ccumultion of bosons in the ground stte is phenomenon, known s Bose-Einstein condenstion, tht hs profound consequences for the properties of n idel Bose gs. Expressing in terms of gs, gs = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ º - ÅÅÅÅÅÅÅÅ g ÅÅÅ g + gs gs where the intrinsic degenercy g is number of order unity, one finds tht is extremely close to unity whenever gs reches mcroscopic sie; in fct, is very close to unity even for s few s thousnd prticles in the ground stte nd is extremely close to unity if gs becomes nonnegligible frction of totl prticle number of order. Thus, the criticl temperture T c for Bose-Einstein condenstion is determined by the condition m@t c D ã t the specified density. Ordinrily one shifts the energy scle so tht Ø nd in the continuous pproximtion ignores the slight dependence of the ground-stte energy upon volume. It is useful to define fmily of Bose functions using g = ÅÅÅÅÅÅÅÅÅÅÅÅÅ G@nD x n- x ÅÅÅÅÅÅ - x - where the normlition fctor ws chosen to ensure tht g n Ø s Ø. Do not confuse the Bose function, g n, with the intrinsic degenercy of momentum sttes g; the nottionl similrity is unfortunte but trditionl. The detiled nlyticl nd numericl properties of Bose functions re studied in bose.nb nd here we summrie the slient results. The Bose functions increse monotoniclly with. For smll, power-series expnsion of the integrnd produces series representtion

22 IdelQuntumGses.nb k n> ï g = ÅÅÅÅÅÅÅ g k n =@nd g = ÅÅÅÅÅÅÅÅ g n@d k= tht is convergent over the entire physicl rnge provided tht n >. Thus, Bose functions with n > re closely relted to the Riemnn et More cre is needed for n becuse the integrnd for = is singulr t x =. Although simple result with logrithmic divergence is obtined for n =, = -Log@ - D more generl methods re needed for rbitrry n. The derivtion of n symptotic expnsion due to Robinson (Phys. Rev. 8, 678 (95)) Ø ï g > H-ln L n- Hln L -md G@ -nd + m! m= is outlined in bose.nb. ote tht this expnsion cn be used for ll n becuse the singulrities for positive integers cncel to ll orders. Representtive Bose functions re shown below for n =8 ÅÅÅÅ,, ÅÅÅÅ,, ÅÅÅÅ 5,, < with g ê highest nd = lowest. Thus, we find tht g Ø D diverges for n, converges with finite slope for n > ÅÅÅÅ, while for n = ÅÅÅÅ the limiting vlue is finite even though the slope is infinite t =. Bose-Einstein functions.5 After this mthemticl interlude, we re now redy to determine the density dependence of the criticl temperture. The number of excited prticles in three-dimensionl nonreltivistic Bose gs cn be expressed in terms of Bose functions s n Q = exc l ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = g gv where l is the therml wvelength. Recogniing tht g =@ ÅÅÅÅ D is finite, the mximum number of prticles tht cn be plced in excited sttes is limited to exc ÅÅÅÅÅÅÅÅ gv ÅÅÅÅ B ÅÅÅÅÅ l F ÅÅÅÅ D º.68. Any dditionl prticles must be found in the ground-stte. Therefore, the temperture

23 IdelQuntumGses.nb p i T c = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ j y ÅÅÅÅ k gv { Dê ê Ñ ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ mk B º.5 i j y ÅÅÅÅÅÅÅÅÅÅÅÅ k gv { ê Ñ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ mk B t which the mximum number of prticles in excited sttes becomes equl to the totl number of prticles represents criticl temperture below which the ground-stte begins to receive mcroscopic occupncy. Below the criticl temperture the occuption of the ground stte becomes T T c ï gs ÅÅÅÅÅÅÅÅÅÅÅ = - J ÅÅÅÅÅÅÅ T ê T c wheres for higher tempertures the frction of the totl number of prticles found exctly t the ground-stte energy is negligible. Although ll prticles occupy the sme volume, below T c it is useful to describe the system in terms of coexistence between two phses, the norml phse consisting of prticles in excited sttes nd condensed phse consisting of prticles in the ground stte. For this system the condenstion occurs in momentum spce rther thn configurtion spce, so tht the two phses coexist in the sme volume but hve distinctly different properties. Most notbly, we will find tht the energy nd pressure contributed by the condensed phse re negligible. or does the highly ordered condensed phse contribute to entropy. Therefore, the thermodynmic functions re determined by the frction tht is in the norml phse. The ground-stte frction gs ê, illustrted below, serves s the order prmeter chrcteriing the phse trnsition. g s ê, e x c ê Ground-Stte Popultion gs ê exc ê TêT c The chemicl potentil nd fugcity obtined by numericl solution of the eqution t ê g ã B ÅÅÅÅÅ F for T > T c re shown below; both re constnt below the criticl temperture. Detils of the numericl solution of this eqution re provided in bose.nb.

24 4 IdelQuntumGses.nb mêhkbtcl Chemicl Potentil Fugcity 4 5 TêT c. 4 5 TêT c The grnd potentil cn be seprted into ground-stte nd excited-stte contributions using G ÅÅÅÅÅÅÅÅÅÅÅÅÅ k B T = g Log@ - D + D@D Log@ - -b D The second term cn be integrted by prts, whereby gv ÅÅÅÅÅÅÅÅÅÅÅÅÅ 4 p J ÅÅÅÅÅÅÅÅÅÅÅÅ m Ñ ê ê Log@ - -b D = - ÅÅÅÅÅÅÅÅÅÅÅÅÅ gv i 6 p j mk B T y Å k Ñ { Recogniing the coefficient from the eqution for in terms of, we find T T c ï ÅÅÅÅÅÅÅÅÅÅÅÅÅ G k B T = g Log@ - D - g 5ê@D ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ g ê bove the criticl temperture. ext, using the reltionship between nd gs, Ä É gs g = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ï Log@ - D = Log ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ g + gs ÇÅ g + gs ÖÑ ~-Log@ gsd x ê x ÅÅÅÅÅÅ - x - we find tht for lrge the first term is negligible in comprison with the second nd my be omitted except perhps for very smll systems. Hence, in this pproximtion the energy, pressure, nd entropy contributed by the ground stte re negligible compred with the contributions of excited sttes. Therefore, the eqution of stte becomes T T c ï ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ pv Å k B T = g 5ê@D g for tempertures bove the phse trnsition. Below the phse trnsition we require = such tht the contribution of the norml phse reduces to T T c ï ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ pv Å k B T = J ÅÅÅÅÅÅÅ T ÅÅÅÅ 5 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ D T ÅÅÅÅ D while the condensed phse is neglected becuse Log@ gs D `. Therefore, the pressure on n isotherm is ctully independent of density for T T c becuse T c H ê VL ê. Thus, upon evlution of the numericl fctors, the trnsition line T = T c ï p c J V 5ê c ÅÅÅÅÅÅÅ = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ p ÅÅÅÅ 5 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ D mg ê ÅÅÅÅÅÅ ÅÅÅÅ D5ê Ñ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ mg ê

25 IdelQuntumGses.nb 5 reltes the criticl pressure to the criticl density. otice tht the eqution for the trnsition line, p c H ê V c L 5ê, hs the sme form s isentropes for n idel gs; indeed, the entropy reches its minimum vlue nd is constnt on the trnsition line. The figure below shows isotherms for nonreltivistic idel Bose gs, where both condensed nd norml phses coexist under the dshed trnsition line. Isotherms for idel Bose systems 8 6 p V In the coexistence region wvepckets with dimensions chrcteried by the therml wvelength overlp sufficiently strongly for quntum correltions to strongly enhnce the popultion of the ground stte. As the temperture increses this phse trnsition requires incresing density (decresing V for fixed ) to compenste for the decresing therml wvelength. Recogniing tht the density of the norml phse in the coexistence region depends only upon temperture, T T c ï exc ÅÅÅÅÅÅÅÅÅÅÅÅÅ V = ÅÅÅÅÅÅ V J ÅÅÅÅÅÅÅ T ê T c = g i k j mk B T y ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅ p Ñ { ê B ÅÅÅÅÅ F we find tht the energy density nd pressure for the norml phse remin constnt during isotherml compression. Therefore, the primry effect of isotherml compression is to push more prticles into the condensed phse for which the energy density nd pressure re negligible. The temperture dependencies of the principl thermodynmic functions,derived in the notebook bose.nb, re tbulted below ssuming tht =. T T c T T c exc I ÅÅÅÅÅ T T c M ê U exc k B T ÅÅÅÅ C V exc k B 5 ÅÅÅÅ ÅÅÅÅ D ÅÅÅÅÅÅÅÅÅÅÅÅ k D B T 5 ÅÅÅÅÅÅÅÅ ÅÅÅÅ D 5 ÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ k D B I ÅÅÅÅÅ 5 g 5ê@D ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ g g 5ê@D ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 4 g - ÅÅÅÅ 9 g ê@d ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 4 g M S exc k 5 B ÅÅÅÅ k D B I ÅÅÅÅ 5 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ g ê - ln F - exc k B 5 ÅÅÅÅÅÅÅÅÅÅÅÅ D ÅÅÅÅ D B T I g 5ê@D ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ g + ln M g 5ê@D These functions re plotted in reduced form below using t =T ê T c.