Lecture 5 Hhlht Phy 40 e are ow o to coder the tattcal mechac of quatum ytem. I partcular we hall tudy the macrocopc properte of a collecto of may (N ~ 0 detcal ad dtuhable Fermo ad Boo wth overlapp wavefucto. e wll tudy a umber of ytem whoe macrocopc thermodyamc behavor domated by quatum mechac, clud: Electro a old, ad upercoductvty Lqud 4 He ad uperfludty Photo a box (black body radato 4 Ultra-cold atom a optcal lattce (Boe-Ete codeato The ytem we wll coder wll be at a fte temperature T. Temperature a meaure of the averae ketc eery of the partcle the ytem. May of the partcle wll occupy quatum eery tate above the roud tate. Becaue the umber of partcle N o lare, there are may mcrocopc cofurato of the partcle that are cotet wth a fxed partcle umber (N ad total eery (E. The fudametal aumpto of tattcal mechac that the ytem explore all the poble mcrocopc tate that have the ame eery, wth equal lkelhood. Th the cocept of erodcty emboded the erodc hypothe of tattcal mechac. Coder the example of partcle weakly teract a oe-dmeoal q π fte quare well. The partcle have le-partcle eeeere of E q =. ma If the total eery of the three partcle π π E = E A + EB + EC = ( q A + qb + qc = 4, what are the poble ma ma mcrocopc cofurato cotet wth th total eery? The awer deped o what kd of partcle we are talk about. For the cae of completely dtuhable Newtoa partcle, we lve uder the fcto that each partcle ha a uque label, ad there are 0 poble tate: Dtuhable: (9,9,9; (,,5; (,5,; (5,,; (5,7,; (5,,7; (7,5,; (7,,5; (,5,7; (,7,5, where the trplet repreet the quatum umber (q A, q B, q C. For the cae of dtuhable Fermo, we caot have multple occupato of the ame quatum tate. I addto we caot dtuh the partcle oce ther wavefucto overlap, o fact there oly oe tate poble that whch the partcle occupy dtct tate: 5, 7 ad (wthout pecfy whch partcle whch tate! e eed a better / ew otato to decrbe th tuato. e hould mply pecfy the occupato umber of each tate a follow: 5 =, 7 =, =, = 0 for all ot equal to 5, 7 or. The cae of dtuhable Boo mlar, except that we do ot have to atfy the Paul excluo prcple. I th cae there are three dtct quatum tate of the dtuhable partcle: Cofurato : 9 =, = 0 for all ot equal to 9. Cofurato : =, 5 =, = 0 for all ot equal to ad 5. Cofurato : 5 =, 7 =, =, = 0 for all ot equal to 5, 7 or.
To eeralze th proce to a lare umber of partcle N, coder the follow exerce. Coder a eeral ytem that ha a fte umber of dcrete boud tate, wth eere labeled a E, where ru from to fty. Each tate ha deeeracy. [Recall that the uperturbed hydroe atom wth o p the lt of quatum umber,, m, ad the deeeracy of the tate equal to. Th mea for example that for = 00 there are 00 = 0 4 dtct lt of quatum umber,, m all wth the ame.6 ev eery:! The meae that deeeracy quatum ytem eerally row 00 very quckly wth crea ee-eery.] e have the job of dtrbut the N partcle to thee tate, ubject to two cotrat: the total umber of partcle fxed at N ( = N, ad the total eery fxed at E ( E = E. How ca we pobly do th? The approach to calculate all of the poble mcrocopc cofurato of the partcle dtrbuted to the avalable tate (at fxed N ad E ad the fd the cofurato that mot lkely to occur, aum erodcty. Eetally we mut calculate the relatve probablty of fd every poble mcrocopc cofurato, ad eek to maxmze that probablty. Th tate, ad may other that dffer from t oly lhtly, wll domate the thermodyamc properte of the ytem. Frt we wll calculate the umber of way that partcle ca be dtrbuted to tate of eery E ad deeeracy. Th wll be called P. Next we wll calculate the total umber of arraemet for a etre et of occupato umber,,, 4,, Th wll be the tattcal weht of the arraemet (,,, 4,, : (,,..., P = P P P P4 P = Th weht wll be proportoal to the probablty of fd th partcular dtrbuto of occupato umber. {Note that f = 0, there oly oe way to make that happe, o the correpod P =.} The ext tep to maxmze by vary all of the occupato umber value ubject to the umber ad total eery cotrat. e wll the do thermodyamc wth the mot probable mcrocopc cofurato. Coder cae: Dtuhable clacal partcle Idtuhable detcal Fermo Idtuhable detcal Boo Dtuhable clacal partcle: Th ometh of a fcto the ee that each partcle ha a uque detty ad we ca keep track of t locato ad eery wth arbtrary preco. Start wth the roud tate ( =, eery E wth deeeracy. How may way are there to put dtuhable partcle th eery level? The awer ;
N P N N! =, where the bomal coeffcet =. The!( N! bomal coeffcet are becaue we ca dtuh each partcle ad there are may dtct way to chooe a ubet of all the partcle N, wthout reard to the order whch they are choe. The partcle ca each be put to ay of poble tate, hece the factor of. he cotruct P there a mlar factor, except that there are ow oly N N partcle to tart wth. Hece P =, ad o o. he we cotruct the relatve tattcal weht, the reult ha a lot of cacellato: N! ( N! ( N! Dt (,,...,...!( N!!( N!!( N! Dt (,,..., N! =. The b P a product.! Idtuhable detcal Fermo: I th cae we do ot have the problem of choo partcle out of N ce they are all completely detcal ad there o eed to eumerate how uch choce ca be made there oly oe way. Itead we are ow cocered wth eforc the Paul excluo prcple. I th cae t mea that mut be le tha or equal to, but ever reater. If le tha we have the freedom to dtrbute the partcle may dfferet way. I fact there are way to put the partcle to the avalable tate. Note that f =, th reduce to a factor of ce there oly oe way to dtrbute oe of the detcal partcle to each avalable quatum tate. Smlarly f = 0there oly oe way to accomplh that, o P =. Fally, f > there would be a volato of the Paul excluo prcple. Th create the factoral of a eatve umber the deomator from (! From the properte of the Gamma fucto, whch the eeralzato of the factoral fucto to the complex plae, we kow that the factoral of eatve teer evaluate to fty, hece the correpod P zero, ad the correpod tattcal weht alo zero. Th a trct eforcemet of the Paul excluo prcple! Now the tattcal weht ; =! Fermo (,,...,,... =!(! Idtuhable detcal Boo: From the treatmet Grffth, oe fd the reult for the tattcal weht : ( +! Boo (,,...,!(! =
The ext tep to maxmze,,...,,... by vary all of the occupato ( umber, ubject to the umber ad total eery cotrat: = N ad E = E. e wll clude the cotrat u the method of Larae multpler. Th method allow oe to perform a cotraed maxmzato. e wll form a ew fucto to maxmze, amely; G(,,...,,..., α, β = (,,...,,... + α N + β E E The lat two term are add zero, dreed by the Larae multpler. Thee two term wll modfy the radet of the fucto the hh-dmeoal pace paed by the value. To maxmze th fucto we mut eforce thee codto: = 0 ad = = 0. α β The form of G already atfe the lat two codto. e foud the tattcal weht of the arraemet (,,, 4,, : = (,,...,,... P, = for three dfferet type of tattc. Th weht proportoal to the probablty of fd th partcular dtrbuto of occupato umber. Dtuhable clacal partcle Dt (,,..., N! ( =! Idtuhable detcal Fermo! Fermo (,,..., =!(! ( Idtuhable detcal Boo ( +! Boo (,,..., ( =!(! Becaue of the product appear Eq. (-(, t eaer to maxmze the loarthm of, rather tha telf. Th wll yeld the ame reult ce ad l have maxma at the ame value of ther arumet. Tak the atural lo of a product equvalet to the um of the atural lo (e. l(xxxx = l(xx + l (yy, hece l( = = = l (. The ewly defed G for dtuhable partcle ow : G Dt (,,...,,..., α, β = l N! + α N + β E E =! = + + + l N! l l! α N β E E = ( 4
To take the dervatve of G wth repect to a partcular t we mut ow decde what to do wth the loarthm of!. Oe approach to employ Strl approxmato: l x! x l x x, ood for x >> (t alo work for x = 0. th th approxmato, G Dt become: GDt (,,...,,..., α, β l N! + l l + + α N + β E = Tak the dervatve of G wth repect to ome partcular (called t the lecture ad ett t equal to zero (to fd the maxmum, yeld; ( α +βe = e Dtuhable partcle For the other cae oe et = Idetcal Fermo + ( α + βe e + = Idetcal Boo + ( α + βe e hat are the Larae multpler α ad β? They are determed by the umber ( E ad eery cotrat = N ad E = E. The challee to determe the eere ad deeerace of all of the le-partcle tate of the ytem - th the hardet part of quatum tattcal mechac. Calculat the total eery of a deal a, whch a relatvely eay cae, Grffth (pp. 9-40 fd that β = / k B T, where T the abolute temperature of the a. The other parameter α re-defed term of the chemcal potetal µ a α µβ. The chemcal potetal a meaure of how much eery requred to chae the partcle umber of the ytem from N to N +. The three dtrbuto fucto ca ow be wrtte a: ( E µ / kbt = e Dtuhable partcle = Idetcal Fermo + ( E µ / kbt e + = Idetcal Boo + ( E µ / kbt e e ca ow do tattcal mechac. 5