Lectue 0: Codesed matte systems Itoducg matte ts codesed state.! Ams: " Idstgushable patcles ad the quatum atue of matte: # Cosequeces # Revew of deal gas etopy # Femos ad Bosos " Quatum statstcs. # Occupato umbes, # Fem-Dac dstbuto; # Bose-Este dstbuto (ot fo examato). AA BB Apl 04 Lectue 0
Idstgushable patcles ad the etopy of a deal gas! Ideal gas: " Lectue 5 (p. 4) gave the followg esults fo a deal gas, (atoms of mass, m, cube of sde, a): a m g 4π h ( ε ) dε ε dε Aε dε Desty of states, at eegy ε 0 g A 3 β ( ε ) exp( βε ) 0 3 Patto fucto exp ( x ) dε " Evaluatg the tegal, substtutg fo β ad A, ad eplacg a 3 wth the volume V gves: mkt V πh 3 " The esult s coect fo a sgle atom (the ultmate low desty) 3 x d x β /kt costat Itegal π ½ /4 Apl 04 Lectue 0
Pefect gas of atoms! Themal popetes atoms: " Atoms do ot teact so eeges add, ε ε + ε + K gas ε A sum of sglepatcle eeges ad hece the patto fucto factoses K gas # ote: we have mplctly assumed that the atoms ae dstgushable (.e. classcal). gas K " The fee eegy, F, follows fom F - kt l() F gas kt l ad the etopy fom S - ( F/ T) V S gas ( V ) " We have a poblem: the etopy does ot scale popely wth V ad. Apl 04 Lectue 0 3 V 3 mkt + l πh 3 mkt k l( V ) + l πh mkt πh + 3 3!!
Coutg states - coectly " Etopy s a extesve vaable: # doublg the sze of the system (V ad ) should double S. Eq.!! " The fault les assumg the atoms ae dstgushable. " To appecate the souce of the poblem, cosde two atoms the gas,. Sum ove sgle-patcle eeges does ot scale wth V/. gas exp exp ( βε ) ( βε ) exp( βε ) + s ( s ) s ( β{ ε + ε }) s s, detcal tems fo each pa of States. eg. Fo, 4. K+ exp ( β{ ε + ε 4} ) K + exp β{ ε + ε } K ( )K 4 Apl 04 Lectue 0 4
Gas of dstgushable patcles " The states, s 4 ad 4, s ae dstgushable, quatum mechacally ad should oly be couted oce. " Wth atoms we ove-cout by a facto! " Fo dstgushable patcles a bette patto fucto fo the gas, tems of the sgle patcle patto fuctos s gas mkt V!! πh " Calculatg F - kt l() ad usg Stlg s appoxmato F gas kt V kt l ad the etopy s whch scales coectly wth V ad (c.f. p. 3) 3 { l( ) l( ) + } 3 mkt l πh + V 3 mkt 5 S gas k l + l + $$ πh Apl 04 Lectue 0 5
Idstgushablty ad the quatum atue of matte! Quatum popetes (key pots): " Agumet (above) shows macoscopc quattes (eg etopy) ae gve coectly whe the atoms ae teated as dstgushable (quatum patcles). " Eve dlute systems quatum effects ae mpotat. They ae eve moe evdet codesed systems (as we shall see). " ote that the etopy s ot, smply, tmes that fo a sgle atom; howeve, t scales coectly wth desty, /V.! ote o devato (detals): " Whe we wote gas /!, a appoxmato was made. Tems whee dffeet sgle patcle states ae occuped ae teated coectly by dvdg by!; howeve, states wth multple occupacy of the same sgle patcle state ae ude-couted. " I dlute systems (a gas) the appoxmato s a good oe sce the pobablty of multple occupacy s eglgble. It s less good some codesed matte system (eg. Boso gas at low tempeatues) " Geeally quatum effects ae moe evdet, ad mpotat, codesed-matte systems. Apl 04 Lectue 0 6
Othe quatum effects! Exchage symmety (See late, Quatum couse of Pat IB Physcs). " May-body wavefucto fo, detcal patcles, Ψ(x,x ): Ψ " We have dstgushable patcles so: Ψ " It follows that Ψ x, x ± Ψ x, x " Result s tue fo all patcles! Femos ad Bosos Symmetc Atsymmetc ( x, x ) d x d x ( x, x ) Ψ( x x ), Ψ Ψ Pobablty of of fdg: a patcle at at x ( dx ) a patcle at at x ( dx dx ) Apl 04 Lectue 0 7 Both cases gve: a patcle at at x ( dx ) a patcle at at x ( dx dx ) ( ) ( ) ( x, x ) +Ψ( x, x ) ( x, x ) Ψ( x, x ) Boso Femo
Excluso pcple! Paul pcple fo Femos " Cosde, sgle-patcle states φ ad φ s ; fo example fom a -D well. " Fo, o-teactg patcles, we ca costuct a two-body wavefucto, wth appopate exchage symmety, usg the sglepatcles states. " Fo example: # A boso Ψ boso # A femo Ψ femo " ote, they have the coect exchage symmety.! What f ad s ae the same? " o poblem fo Ψ boso. " Howeve, fo a femo ( x x ) φ ( x ) φ ( x ) + φ ( x ) φ ( ), s s x ( x x ) φ ( x ) φ ( x ) φ ( x ) φ ( ), s s x ( ) ) 0 Ψfemo x x φ ( x ) φ ( x) φ ( x) φ ( x " Two femos caot be the same state. " The Paul Excluso pcple., Apl 04 Lectue 0 8
Femos ad Bosos! Exchage ad agula mometum " Thee s a dect coecto betwee exchage symmety ad tsc agula mometum a cosequece of elatvstc quatum mechacs. " Bosos have tegal sp, 0,, # Photo, π-meso, 4 He, " Femos have half-tegal sp, /, 3/, # Electo, poto, euto, 3 He,! Quatum statstcs: " Cosde the gas of o-teactg patcles wth sgle-patcle states eeges ε < ε < ε 3 < ε < " The total eegy s the sum of the sgle-patcle eeges E (, K K) ε + ε + L ε L, " We have two cases: # Femos: 0, Fem-Dac statstcs # Bosos: 0,,.. Bose-Este statstcs " Subject ( both cases) to: Occupato umbes Total umbe of of patcles Apl 04 Lectue 0 9
Fem-Dac ad Bose-Este dstbutos! Exctatos at T>0, >0, fo 3-D 3 D pefect gas. " We wll deduce the dstbuto fom the fee eegy, F, fo whch we eed the eegy ad the etopy. " Label states as the fgue: the states have eegy ε degeeacy g ad s occuped by patcles. " Eegy of state : ε " Etopy of state : k l Ω " Etopy of whole gas Eq. A F U TS [ ε kt l Ω ( )] Apl 04 Lectue 0 0 See late S kt l g kt l( Ω ( ))
Chemcal potetal! Mmse F to deteme the. " Equlbum occus at the mmum of F, (as usual). Howeve, we caot smply set F/ 0, sce Σ s fxed. " Cosde the followg eaagemet: k 0 fo all k except ad j, whe - j F F F + j 0 j - F F - µ ( T ) + ote µ ca oly be a fucto of T as the agumet must be depedet of ad j.! Asde o the chemcal potetal " µ s a fucto of state, whch ases whe a system ca exchage patcles wth a esevo F µ ( T ) T, V " Equlbum # I A: T a T b # I B: T a T b ad µ a µ b j AA BB Chemcal potetal Apl 04 Lectue 0
FD ad BE dstbutos " Recall eq. A (p.0) F U TS [ ε kt l Ω ( )] Dffeetatg gves F l Ω ( ) ε µ l Ω kt ( ) ( ε µ ) kt! Ω( ) fo femos ad bosos! Fem-Dac statstcs " Oe patcle (at most) ay state " umbe of aagemets of femos g states, of eegy ε. o. of aagemets, Ω( ) g! /!(g - )! (We have selected states to get femo fom the g avalable). " Stlg s appoxmato gves l Ω ( ) g l g l ( g ) l( g ) ad l Ω ( ) g l + l( g ) l, Eq. A Eq. B Apl 04 Lectue 0
Fem dstbuto fucto " Usg Eq. B ε µ ( T ) g l kt g exp ( ε µ ) g g exp exp ( kt ) (( ε µ ) kt ) (( ε µ ) kt ) + Fem dstbuto p F ( ε ) g ( ε ) ( ε ) ( ε µ ( T )) exp( kt ) + " We wll dscove the cosequeces of ths dstbuto fucto the ext lectue " ote the occupato of state does ot, geeal, follow a Boltzma dstbuto " Becomes lke a Boltzma dstbuto at hgh tempeatues, whe (ε µ) >> kt. Apl 04 Lectue 0 3
Bose dstbuto (ot fo examato)! Bose-Este statstcs " o lmt o the umbe of patcles pe state. " umbe of aagemets of bosos g states, of eegy ε. # Image the patcles ad g - pattos aaged a le. 6 g 7 # ( + g -) objects to aage ( + g -)! of these,! (g -)! ae the same sce pemutg patcles o pattos makes o dffeece. o. of aagemets, Ω( )( + g -)! /!(g - )! Ω Ω Ω ( ) ( + g ) l( + g ) ( g ) l( g ) ( ) l( + g ) l ( ) + g l l l + g Apl 04 Lectue 0 4
Bose dstbuto, cotued.. (ot fo examato) " Usg Eq. B, as befoe ε µ ( T ) + g l kt g + exp ( ε µ ) kt ( ) Bose dstbuto p B ( ε ) g ( ε ) ( ε ) ( ε µ ( T )) exp( kt ) " Aga the dstbuto s ot, geeal, lke a Boltzma dstbuto, though at hgh tempeatues, ad whe ε >>µ the dstbuto teds to a Boltzma fom. " Do ot be deceved by the appaet vsual smlaty wth the Fem-Dac dstbuto. The subtle dffeece leads to pofoudly dffeet physcal pheomea. Apl 04 Lectue 0 5