Notes follow and parts taken from sources in Bibliography

Transcription

1 PYS 8 ots follow d prts t fro sourcs i Biliogrphy Two Prticl Systs Wh thr r two prticls i syst, th wv fuctio will dpd o th coordits of oth prticls d w writ it s (r,r,t). Th iltoi is V r, r t, whr th suscripts rfr to th idtitis of th two prticls. Th orlitio itgrl is writt r r, t d r d r, If V is idpdt of ti, w c writ s t / r, r, t r r i, d istd loo for (r, r ), th solutio to th ti idpdt Schrödigr qutio. V If th coid wv fuctio (r, r ) c writt s th product of two sigl-prticl wv fuctios (r ) (r ) (whr th lttrs idtify th prticls), choic pops up. I clssicl physics, w c clrly idtify ch prticl (v if thy r oth lctros, for pl) t ll tis. I qutu chics (d i rlity), w c t. lctros r idistiguishl prticls, ig tht (r ) (r ) d (r ) (r ) do t just loo li thr is o distictio tw th two stts. I this cs, how c w writ th stts? W writ th s qul coitio of our choics. For th two prticl cs, w hv two clr choics: r r A r r r, r It turs out tht th choic of plus or ius sig is ot up to us: th two choics r ssocitd with two diffrt ids of prticl. Thos prticls with itgr spis (photos, pios, d othr forc-crryig prticls which r ll collctivly ow s osos) will dscrid y th plus sig, d prticls with hlf-itgr spis (lctros d

2 PYS 8 utrios, s wll s protos, utros, d th qurs ig th up, collctivly ow s frios) will us th ius sig. If our two prticls r lctros, th, w c writ r r A r r r, r This dos t loo prticulrly sigifict yt, ut otic wht hpps if th two lctros try to occupy th s plc (ig r = r ). Th wv fuctio dispprs! This is th origi of th Puli clusio pricipl. To littl or prcis, th lctros would lso hv to hv th s spi orittio. If th two spis r i opposit dirctios, oth lctros c i th s spc. This is th rso toic rgy lvl dscrid y, l, (such s th groud stt of th hliu to) c ccoodt two lctros. A chg oprtor P c dfid such tht P( (r ) (r )) = (r ) (r ). Oprtig o stt twic with this P hs to rtur th iitil stt. This s P =, so th igvlus of P hv to. Agi, osos will hv th + igvlu d frios. This oprtor couts with th iltoi d is thrfor cosrvd prticls do t strt i sytric stt d th ov ito tisytric o. Th rquirt of sytry or tisytry trslts ito ffctiv forc o idticl prticls wh coprd to th forcs tht clssicl (d thrfor distiguishl) prticls would fl i th s situtio. Frios fl ffctiv forc of rpulsio d osos fl ffctiv forc of ttrctio. Followig your oo, w c dostrt this forc y looig t th for of th wv fuctio i ll thr css d clcultig th rsultt pcttio vlus of sprtio. Th thr wv fuctios (for distiguishl prticls, osos (+), d frios(-)) r D,,, d w us th to clcult th sprtio s Th rsult is siplst for th D wv fuctio, whr w gt

3 PYS 8 d d d siilrly < > = < >. Th cross tr loos li d d lvig us with d Th cs of idistiguishl prticls loos or coplictd cus of th tr trs i th wv fuctio. Th rsult is d d d d d d d d * * * * d th s for < > y idistiguishility. Th cross tr is ow

4 PYS 8 4 * * * * d d d d d d d d dfiig d * Th y rsult is tht, for distiguishl prticls, w gt d whr idistiguishl prticls giv us otic tht th diffrc is tht osos gt dditiol tr qul to <>, ig thir distc lss th wht would pctd for idticl prticls, whil th dditiol tr for frios is + <>, pushig th furthr wy fro ch othr. Th sigificc of this lst tr ivolvs th ovrlp itgrl <>. If two idticl prticls r wll sprtd, this tr is ro d thy ct li distiguishl prticls. This rstors littl it of clssicl ituitio if two lctros r vry fr prt, it s ss tht w c distiguish th y thir loctio lo. Wh th prticls r r o othr, though, th wv fuctios will hv so ovrlp d thy will ffctivly ttrct (osos) or rpl (frios) ch othr. To this lst sttt or prcis, w d to iclud th spi prt of ch wv fuctio. As tiod rlir, two lctros c hv th s sptil wv fuctio if thir spis r i opposit dirctios. Thrfor, rthr th syig th sptil wv fuctio of two lctros ds to

5 PYS 8 tisytric, it is or ccurt to sy tht th totl wv fuctio (spc ultiplid y spi) of two lctros ust tisytric. Th olcul is pl of this. Plcig th lctros tw oth protos (i vrg ss, of cours, sic w still do t hv prfctly dfid positios d ot du to th ucrtity pricipl) would lowr th Coulo rgy d th to or rgticlly stl. This s sytric sptil wv fuctio for th lctros, which c oly hpp if th spi stt is tisytric. W gt th tisytric spi stt y coiig th two spis i such wy tht w gt totl spi of ro (spi siglt stt). W would writ this stt s ( > - >) (igorig th orlitio fctor d ). W could hv gott ro totl spi y usig th sytric coitio ( > + >), ut rr tht our gol ws tisytric fuctio ot prticulr spi vlu. Th liu Ato Th Schrödigr qutio for th lctros i hliu to c writt s 4 r 4 r 4 r r cpt for th lst tr (d th fctor of i th third d fifth trs) this loos vry uch li th qutio w would gt if w hd two hydrog tos r ch othr. Bcus th rgy scls s Z for lctro r uclus of chrg Z, w would pct th lctro i ch of ths wird hydrog-li tos to hv groud stt rgy of.6 V * = V, givig totl for th pir of 8.8 V. If w cotiu to igor th lctro-lctro itrctio, our groud stt wv fuctio would sipl product of two hydrog wv fuctios: r r / r, r r r 8 Sic th wv fuctio hr is oviously sytric wh r d r r itrchgd, th tisytric prt would hv to cotid i th spi wv fuctios: w gi pct th siglt pirig w sw rlir ( > - >) ultiplid y costt. Igorig th lctro-lctro tr is too uch of pproitio, s it hpps. W prdictd totl rgy of 9 V for th pir of lctros, which is sigifictly diffrt fro th tru rgy cssry to rov oth lctros fro hliu to (foud pritlly to out 79 V). Th first ioitio rgy is littl ovr 4 V, ig tht our origil stit of th idig rgy of o of ths lctros s 54 V ws oticly fr fro rlity. This s tht th fifth tr, which w glctd d which rprsts th lctrosttic rpulsio tw th two lctros, is cosidrl. It s 5

6 PYS 8 worth tioig, though, tht th scod ioitio rgy dos gr with our 54 V stit. This is rsol cus ftr th first lctro hs lrdy rovd, th riig lctro ss (lost ctly) th s pottil s it would fro doulychrgd hydrog uclus. Although th tru groud stt is rstrictd to th tisytric spi (= siglt) stt, if w lv o lctro i th groud stt d ov th othr to citd stt, w could hv ithr siglt stts or sytric spi (triplt i this cs) stts. Sytric spi stts of hliu r ow s orthohliu whil th siglt stts r clld prhliu. Prhliu rgis will highr th orthohliu rgis cus th tisytric spi stt is pird with sytric sptil wv fuctio ig th lctros r closr togthr o vrg d th rpulsiv itrctio (which is positiv rgy tr) will lrgr. Th Priodic Tl Th dditio of spi s w c fit twic s y lctros i giv shll or sushll. For pl, loo t th priodic tl low (fro 6

7 PYS 8 otic tht th top row cotis oly hydrog d hliu. I hydrog s lowst rgy stt, it will hv o lctro i th = stt. For this lctro, if =, l = d l =. O th opposit sid of th tl w hv hliu, which hs two lctros. Thy c oth hv =, l = d l = (ig thy r oth i th groud stt) if thy hv diffrt vlus of s (thy c t hv diffrt vlus of s sic thy r oth lctros d ll lctros hv s = ½). Th lctros i hliu s groud stt will th hv th s qutu urs for vrythig cpt s, d o will hv s = +½ whil th othr hs s = -½. This s hliu hs o or roo for lctros i th = lvl d w c th sy tht its outr shll is filld. This s it trly ulily tht hliu will try to coi with y othr lt sic hvig filld shll is th gol of y to. Th t row strts with lithiu which hs th filld = shll d o lctro i th = shll. This shll hs sushlls with l = d l =. I thos sushlls, l c ithr (l = ) or,, or + (l = ). Tht givs 4 possil coitios of l d l, which would tht th scod shll would filld y th ti w gt to lt ur 6. owvr, th fct tht w ow hv th spi gtic qutu ur s s w c doul th vill spc for lctros, d th scod shll dos t fill up util w rch lt ur (o, othr lt tht grlly rfuss to coi with ythig ls cus of its filld shll). Ths stts will td to fill up th = stts for th = stts cus th lctro will spd or ti clos to th uclus (whr its rgy is lowr) i = stts. I lowr rows of th priodic tl, this is iportt ough for th lctros to strt fillig i shlls t highr vlus with = rthr th coplt th op shlls with highr vlus of. This is th rso why two rows of th priodic tl r st off y thslvs t th otto. For pl, Lthu (lt 57) is holdig plc for itslf d ll of th othr lts out to Luttiu (lt 7). To discuss th spcific lctro cofigurtios, you d to ow tht chists will grlly rfr to stts y vlus of d i odd fort: th vlu is giv s ur, ut th vlu is writt s lttr, whr th / lttr corrspodc is = s, = p, = d, = f, cotiuig lphticlly for fw or lttrs. Th lctro cofigurtio for hydrog s groud stt (=, =) would th writt s s. For hliu, whr oth lctros fit i this stt du to th diffrc i spi, w would writ th cofigurtio s s, ig thr r lctros i th =, = stt. Th t lt, Lithiu, could writt s s s, ut sic w ow th = shll is full, w could writ this i trs of ol gs + othr stuff. W ow s =, so w could writ th cofigurtio for Lithiu s [] s. Usig this ottio, w gt th dt low for lts 57-7, showig tht th =6 shll is filld whil thr r still vccis i th =4 d =5 lvls. Lthu [X] 5 d 6 s Criu [X] 4 f 5 d 6 s Prsodyiu [X] 4 f 6 s odyiu [X] 4 f 4 6 s Prothiu [X] 4 f 5 6 s 7

8 PYS 8 Sriu [X] 4 f 6 6 s uropiu [X] 4 f 7 6 s Gdoliiu [X] 4 f 7 5 d 6 s Triu [X] 4 f 9 6 s Dysprosiu [X] 4 f 6 s oliu [X] 4 f 6 s riu [X] 4 f 6 s Thuliu [X] 4 f 6 s Yttriu [X] 4 f 4 6 s Luttiu [X] 4 f 4 5 d 6 s To dtri th wy stts fill i grl, w c us ud s ruls. Ths thr ruls stt tht ) th iu vlu of totl spi S is prfrrd ) for giv vlu of S, th iu vlu of L (totl oritl gulr otu) is prfrrd, d ) wh sushll with giv vlu of (, ) is hlf full or lss th lowst rgy lvl hs J= L-S. If it s or th hlf full, th lowst rgy lvl hs J=L+S. Solids lctros i solid hv vry diffrtly th th lctros o isoltd to. A to s outrost lctros (ow s vlc lctros) r oud to th solid s whol rthr th to idividul to. Th siplst odl is to trt th lctros s ig oud i -D ifiit squr wll. Th pottil is ro isid th solid d ifiit outsid of it. If it s rctgulr solid, w c sprt th Schrödigr qutio ito thr pics: d X d Y d Z X y Y d d y d Z for totl rgy = + y +. Th ssocitd wv urs r rltd i th usul wy: /, tc. Solutios to ths qutios will lir coitio of si d cosi trs. Th oudry coditios (= t =, y=, =, =L, y = L y, d = L ) liit th cosi trs d plc coditios o th wv urs to sur tht th si trs r ro t ths oudris. It would giv us L y L y y L whr ch is oro positiv itgr. Th orlid wv fuctio is th y L 8 L L y y si si y si L Ly L 8

9 PYS 8 givig rgis of y y L Ly L Bcus of th rstrictios o ch copot of ( L =, tc.), w c sy tht ch stt occupis volu of L L L y V Th two spi stts llow two lctros to sty withi volu this si (otic tht th volu is ot i rgulr spc; it is volu i rciprocl spc or spc). Sic ch of our copots, y, d is positiv, w will oly usig /8 of spc. W thrfor hv wht is ow s th positiv octt of sphr with rdius F. Th suscript F stds for Fri. If w igi th solid to t (or ritrrily r) solut ro, th oudry i spc tw filld stts (lowr i rgy) d pty, vill stts (highr i rgy) is clld th Fri surfc. If w wt to fid th rdius F of th sphr tiod ov tht rs th Fri surfc, w c st th volu of th positiv octt of th sphr qul to th volu dd to coti th fr lctros. Tht givs us 4 q F 8 V lvig us with F / q V whr th q rprsts th ur of fr lctros pr to (rr tht y of th to s lctros r still tightly oud to th uclus). Th Fri rgy ssocitd with th lctro gs ov is F F / Th -spc volu of o octt of thi sphricl shll would 9

10 PYS 8 dv 4 8 d th ur of stts i th shll would (for th spi) * dv / ( spc volu dd for stt, clcultd for to /V): d Wightig ch rgy with V d /, w s tht th rgy i giv shll is V d d W fid th totl rgy y itgrtig this fro to th dg of th Fri surfc, F : q 5 / / F 5 V 4 F V tot d V Th prssur rtd hr is foud fro d/dv, which w c writ s d P dv tot This prssur is ow s th dgrcy prssur, d it prvts th collps of solid ojcts. otic tht this is ot lctrosttic rpulsio of lctros, cus w hv o lctro-lctro itrctios i this odl. It s siply th fct tht frios c t occupy th s spc. Th rgy of th lctros r th Fri surfc (sotis rfrrd to s ig th top of th Fri s of lctros) is vry lrg coprd to thrl rgis t roo tprtur. Wht this s is tht th lctros i solid (v vry hot solid K, for pl) r lost ll i th lowst rgy stts vill to th. Oly tiy frctio, of ordr B T/ F, whr B is Bolt s costt, will foud ov th Fri surfc. I othr words, s fr s th fr lctro gs i solid is cocrd, its hvior is out th s r solut ro s it is t K. rgy Bds V tot

11 PYS 8 If w wt to td our odl of solid to or ccurtly rflct rlity, w d to loo t priodic pottil. A lctro ovig through lttic will coutr th s situtio ovr d ovr gi. W r t quit rdy to th odl so rlistic tht w r usig th Coulo pottil, so w strt with wht is ow s Dirc co. This is just rgulrly spcd st of dlt fuctio pottils (rrirs, ot wlls w do t wt oud stts hr). Th dfiitio of priodic pottil is o tht rpts ftr prticulr distc. This s V(+) = V() for so vlu of. I our cs, would clrly th spcig of th dlt fuctios. Usig Bloch s thor, th hvior of th wv fuctio i priodic pottil is ik for so costt K. Th proof of this, outlid i your oo, is rsoly sipl. First, dfi displct oprtor (D) tht ovs fuctio fro f() to f(+). This oprtor D hs to cout with th iltoi (p / is clrly idpdt of positio, d if th pottil rgy fuctio is uchgd s +, th iltoi itslf is lso), which llows us to choos siultous igfuctios of D d. Th igvlus of D will dotd y, so D =. Oprtig with D o will giv us D Sic our is i grl copl, it c writt s ik for so vlu of K. It is coo to dl with th dgs of th solid y iposig priodic oudry coditios. This s tht w trt th solid li it wrps roud i -D, istd of li of pottils, w hv circl of th. I -D, istd of pl rry of pottils, it would th surfc of dout with pottils ll ovr it. Sic th pottil vtully wrps roud, thr is so vlu (strooiclly lrg for croscopic solid) for which Applyig Bloch s thor to this, w gt This plcs rstrictio o K: i K K,,,...

12 PYS 8 Bcus of th priodicity of th pottil, th solutio will priodic ovr th s itrvl. For th Dirc co fuctio show i your oo, w c writ V j j I tw th spis, w wt to solv th V= Schrödigr qutio, which is d d d d whr solvd y Asi Bcos for O priod (or uit cll) ovr to th lft w hv i K Asi Bcos for Cotiuity of t th oudry lvs us with th qutio B i K Asi Bcos d th discotiuity i th first drivtiv (s s rlir for th dlt fuctio pottil) givs A i K Solvig th t-to-lst qutio givs Acos Bsi B i K Asi cos d coiig th prvious two qutios yilds i K K i cos i K si si cos B

13 PYS 8 d th W th followig sustitutios cos K cos si d th w could rwrit th prvious qutio s cos K f cos si Th vril rprsts th strgth of th dlt fuctio i disiolss uits. otic tht, for so vlus of, f() will grtr th or lss th. Th cos(k) tr is cofid to th rgio to +, so thr r so vlus of tht r foridd. A rgy corrspodig to tht vlu of (i.., tht vlu of ) is ot llowd. Ths r gps i th rgy spctru. Also, th llowd vlus of rgy (whr f() <) r discrt, ut th spcig tw djct llowd rgy stts (typiclly fw V dividd y th ur of prticls i th solid, which will ~ ) is so ridiculously sll tht th stts ight s wll cotiuous. W c th divid rgis ito llowd d foridd ds. For v ur of fr lctros pr to q, th (q/)-th d will filld d th t d ov it will pty. If q is odd, th (q+)/-th d will hlf-filld. This hs iportt cosqucs for th coductivity of th solid. If d is filld, thr is rgy gp tw th highst currtly occupid stt d th lowst (ut highr) uoccupid stt. A currt c t flow sily sic ddig sll out of rgy to th lctro is t possil. Oly vry lrg pottil diffrc will llow th lctros to jup fro th filld lowr d to th pty highr d. This tril would isultor. O th othr hd, if th d is hlf full, thr r plty of stts (strtig ~ - V ov th highst prviously filld stt) vill to ccpt lctros. This w tril will good coductor. If th gp tw th vlc (lowr) d coductio (uppr) ds is vry sll, th tril is ow s sicoductor d so lctros will prootd fro th vlc d to th coductio d t tprturs ov solut ro. Wh th lctros ov to th coductio d (llowig th to crry so currt), thy lv hid hols. W c thi of th hols s positiv chrgs fr to ov through th tril, lthough thir ovt i th dirctio of th lctric fild is rlly du to th ovt of lctros i th opposit dirctio.

14 PYS 8 Th sicoductor c cotitd (dopd) with lt fro ighorig colu i th priodic tl. This will ithr dd o lctro or o hol for vry to of dopt icorportd ito th lttic. Th coductivity c ltrd to suit prticulr pplictio y doig this. I crystl of Si (4 vlc lctros), th dditio of Atioy, Phosphorus, or Arsic (ch with 5 vlc lctros) will dd spr lctro to th tril. Addig tr lctros, with thir gtiv chrg, s -typ sicoductor. If w put tos of Glliu, Idiu, tc. ( vlc lctros) i th Si lttic, w r ffctivly losig lctro t ch stp, which is quivlt to ddig positivly-chrgd hol. For tht rso, w cll ths p-typ sicoductors. A ordiry diod is d t juctio tw p-typ d -typ sicoductors. Sttisticl Physics Th hvior of lrg urs (~ ) of prticls hs lwys trtd proilisticlly. v for th discovry of qutu chics, it ws cssry du to th ipossiility of dirctly ddrssig th otios of croscopic collctios of prticls. Aog th y cocpts is tht of thrl quiliriu. Wh syst is i thrl quiliriu, ll stts with th s rgy r qully prol outcos for th syst. Th tprtur of syst (i thrl quiliriu) is th sur of tht syst s rgy. Th fudtl diffrc tw th clssicl d qutu pprochs to sttisticl physics is th distiguishility of prticls, first tiod t th giig of this group of ots i coctio with two-prticl systs. As outlid i your oo, if w strt with sll ur () of distiguishl prticls i -D squr wll with totl syst rgy qul to 6 uits (ig th su of th squrs of th vlus for ll thr prticls A + B + C = 6), thr r ultipl choics of A, B, d C tht will stisfy ths rstrictios. For pl, o possiility is A = B = C =. W could lso hv two s = d th third = 9, ut thr r (clrly) thr wys for this to hpp. Two = d o =5 c lso hpp thr wys. Filly, thr r si coitios with o =5, o =7, d o =7. If w oly d to worry out th ur of prticls i ch stt (th occuptio ur), w c siplify this y just listig th occuptio ur for ch. If ll thr hv th s vlu of (= i our first pl), w hv = d = for ll othr vlus of. Thr is oly o wy this c occur, ut th lst situtio i th prvious prgrph (which hs 5 = 7 = 7 = ) c hpp si wys. It is thrfor si tis or lily to occur th th = rrgt. Th qustio of fidig th proility for rdoly-chos prticl of th thr to hv rgy is foud y looig t ll th possil vlus of th occuptio urs (th s) d how oft ch cos up. Th rgy 5 oly pprs i th thr wy d si wy cofigurtios. Sic thr r totl of wys th vrious itgr vlus of A, B, d C c coi to giv A + B + C = 6, w fid th proility of gttig 5 y th followig: thr r thr wys tht (,, 5) will wor sic th 5 could t ithr of th thr positios. Tht givs us / chc. owvr, rdoly picig o of th 4

15 PYS 8 thr prticls rducs tht to / / = / chc. I th (5, 7, 7) stup, th si wys thos thr urs c rrgd would giv 6/. Agi rliig tht w oly hv / chc of picig th prticl tht is i stt 5 rducs tht to / 6/ = /. Thos r th oly two wys 5 c co up if our totl rgy is to 6 uits, so w c sy th chc of rdoly picig prticl d fidig it to hv rgy 5 is / + / = /. (Rr tht ths prticls r distiguishl). If w hv idistiguishl frios, ll occuptio urs ust ithr or. Oly th (5, 7, 7) cs is possil hr, d thr is oly o rlitio of it (ot si). Tht s th chc of gttig 5 fro rdo frio i this syst (whr totl rgy = 6) is /. For idistiguishl osos, w c still hv (,,), (5,, ), (,, 9), d (5, 7, 7) ut w c hv oly o pl of ch. ch rrgt th hs proility of /4, d fidig th o with = 5 is gi / shot for th scod d fourth rrgts (thr is of cours o chc of fidig prticl with = 5 i ithr th first or third rrgt). Our totl proility is th /4 / + /4 / = /6. W could or grl d loo t th cs for totl prticls whr th occuptio ur for stt with rgy is (d of cours ( ) = ). W would li to ow th ur of wys w c rrg ths prticls for prticulr choic of vlus. It will ttr whthr th prticls r frios, osos, or distiguishl (sotis clld Bolt) prticls. W strt with th ur of wys to pic fro th totl of. This is wll-stlishd proility thory, d you c vrify for yourslf tht thr r!!! possiilitis. If thr r th d possil sigl-prticl stts with rgy, th fctor ov is ultiplid y th wy thos prticls c rrgd, which is d Followig through for ch diffrt d, w gt totl ur of cofigurtios qul to Q,,! d! (p i id tht, for th prvious pl of thr prticls with totl rgy = 6 uits, ll d = sic thr ws o dgrcy i th sigl-prticl stts). For frios, w gt sothig siplr du to th uch or liitd choic of occuptio urs ( or ): 5

16 PYS 8 Q,, d!! d! I th cs of osos, sic w c put s y of ths (idistiguishl) prticls i stt s w wt, w d to fid th ur of wys th prticls c put ito d possil stts. Your oo drivs this fctor to d d!! d! for fil prssio of Q,, d! d!! Sic w hv lrdy s tht thr is o prfrc for o prticulr stt ovr othr with th s rgy, th iportt thig is to fid out which o hs th ost possil rlitios. Our costrits r o th prticl ur d totl rgy: To solv this prol, your oo wors through th us of Lgrg ultiplirs. This is usful tchiqu i clssicl chics, d you c fid good pltio of it t Th rsult i your oo lso rlis o th us of th Stirlig pproitio for th vlu of! wh is lrg. It sys l! l wh Wh w us ths thticl tools, w fid tht th ost prol occuptio urs r for frios d, for osos, w gt d 6

17 PYS 8 7 d To figur out wht d, your oo is th cs of idl gs (prticls tht do t itrct with ch othr). W lrdy ow th rgis of prticls i this situtio:,,, y y L L L Th ur of stts i ch /8 of sphricl shll is th dgrcy d d it is d V V d d / 4 8 For our clssicl (d thrfor distiguishl) prticls, our coditio tht th su of ll occuptio urs (writt hr s itgrl) = totl ur of prticls. W th gt / / / V V d V Th costrit o totl rgy cos / 4 / V d V Usig th wll-ow rsult tht = / B T, w fid tht = /( B T) whr T is th solut tprtur d B is Bolt s costt. Th fil vril is rltd to sothig ow s th chicl pottil. This qutity (dotd y ) is wht drivs diffusio, d it is th drivtiv of th fr rgy with rspct to prticl ur. Th forul w d is T T B All of this wor c surid if w writ th prssio for th ost prol ur of prticls i stt with rgy dfid s

18 PYS 8 / T B for distiguishl prticls, / T B for frios, d / T B for osos. Ths distriutio fuctios r lso ssocitd with th s Mwll- Bolt, Fri-Dirc, d Bos-isti, rspctivly. I th cs of low tprturs (or out wht ctly low is ltr), foud i th liit T, w would gt () = for ll rgis low th Fri rgy ( F ) d () = for ll rgis ov it. I othr words, s th tprtur flls, th lowst rgy stts fill up util, t T=, ll th stts r filld ctly i ordr of rgy d thr is cutoff wh ch prticl is i th lowst stt it c fid. This rgy F lso ust th vlu of (T=). Plottig () vs. t ro tprtur will giv us stp fuctio. It will qul for F d ov tht. At highr tprturs, thigs chg slightly. Frios with rgis r ~ ( F B T) c upd up (through thrl itrctios) to vill (pty) highr rgy stts. Ths stts r withi out B T ov F. I th first plot low, w hv chos = 4 d T =. (pproitly ro). As you c s, th fuctio is sstilly stp d () gos to ro wh th rgy quls th chicl pottil (d vry prticl hs plc) rgy 8

19 PYS 8 If w up th tprtur up highr (to T=.) d focus o th rgio r th Fri rgy F, w gt rgy Th width of th rgio tht is diffrt fro th low-tprtur cs is out B T. O pplictio of ths tchiqus lds us to th forul for th lcody spctru. W hv to lrdy ow tht, for photos, d =/c. Additiolly, w hv oly two spi stts (polritios) for sslss prticls, v though th photo hs spi. Filly, thr is o cosrvtio lw for photo ur. Thy c crtd or dstroyd t will, s log s rgy is cosrvd. This is prssd i thticl trs s. Th rgy dsity ssocitd with this distriutio is c / B T Th rgy dsity is ()d i th frqucy rg d, d this would lso / V. Th Tiplr & Llwlly oo plors othr rsults du to th diffrt sttistics oyd y prticls of diffrt spis, icludig suprfluidity d th hvior of lctros i tls. Ti Idpdt Prturtio Thory For sipl pottil, w c sily solv th ti idpdt Schrödigr qutio (writt low with suprscripts of ro idictig this sipl cs) whr th wv fuctios r orthoorl, s usul. If w ow loo t w pottil, closly rltd to th old o ut just slightly diffrt, w would pct our igvlus (rgis) d igfuctios (wv fuctios) to vry clos to wht w hv ov th w os will slightly prturd fro th origil. Th pl i your oo cosidrs th cs whr th origil pottil is tht of th ifiit squr wll d th prturd pottil dds sll up to th wll. 9

20 PYS 8 Th w iltoi will stisfy (otic tht this is trivil to writ dow, ut it y vry hrd to ctully solv dirctly tht s why w r ivstigtig prturtio thory). W writ th w iltoi s th coitio of th origil cs ( ) d so w (prturig) pottil : ' If th cofficit is st to ro, w rcovr th origil ifiit squr wll; if it s o, w gt th w prol. W pct to s sooth trsitio tw th two prols s gos fro ro to o. W us th fctor to pd our pctd solutios ( d ) i powr sris i For oth th rgis d th wv fuctios, w c loo t th suprscript s providig th ordr of th corrctio to th origil prol. is th first-ordr corrctio to th rgy, is th scod ordr corrctio, tc., d siilrly for. W put ths psios ito our prssio. W put ths psios ito our prssio = d gt ' d th group powrs of to gt ' ' ow w loo t th trs t ch ordr. Th roth ordr tr is th old rsult: To first ordr, w gt '

21 PYS 8 Ad t scod ordr, w fid ' W could oviously cotiu this to whtvr ordr w flt cssry. ow, s Griffiths dos, w ov to th full prol y sttig =. iig th first-ordr cs, w t th dot product of tht qutio d < to gt ' Th first tr o th lft ccls th first tr o th right, sic w c oprt with to th lft o < to gt <. Th scod tr o th right is just du to th orthoorlity of th wv fuctios. W r lft with ' W gt our first corrctio to th rgy y fidig th pcttio vlu of th prturtio usig th uprturd wv fuctios. W lso d to corrct th wv fuctios thslvs. W t th first ordr prssio d gi group th ordrs i to gt ' Th thigs o th right sid r ow w hv our origil wv fuctios, th prturd rgis w just foud, d th prturd iltoi which w should ow (otic tht w do t yt hv th solutios to th prturd iltoi, just its for). Bcus our uprturd wv fuctios r coplt st, w c solv this y writig s psio i trs of : ow w put this psio ito d gt ( ) c '

22 PYS 8 ) ( ' c W th sdwich this fro th lft with to gt ) ( ' c For th cs wh, th ir product o th lft will ro d w gt th rlir rsult ' If istd, w c writ ) ( ' c W us this to solv for th cofficits c fro th othr ow qutitis. Thy r ) ( ' c lvig us with ' W will hv prol hr if th rgis for th two diffrt stts d hpp to th s (i.., r dgrt). Our cofficit will udfid if th doitor gos to ro, d i tht cs w hv to switch to dgrt prturtio thory. If w wt to go to scod ordr (frqutly cssry), w gi sdwich fro th lft with (this ti o th scod-ordr trs of our psio) d gt '

23 PYS 8 Tchiqus siilr to thos usd rlir ld to siilr rsults: th first trs o ch sid ccl, d th scod tr o th right dispprs du to th orthoorlity of th wv fuctios. W th gt ' which w c writ s ' ' c ( ) ' ' ' ch tr i this su c thought of s th product of th proility plituds to ov fro stt to itrdit stt c to th origil stt. Th su rflcts th fct tht w r cosidrig ll possil itrdit stts. Biliogrphy Qutu Mchics, d ditio Dvid J. Griffiths Modr Physics 5 th ditio, Tiplr & Llwlly Physics 6 th ditio, Cutll & Johso Fudtls of Physics 7 th ditio, llidy, Rsic, & Wlr Qutu Physics of Atos, Molculs, Solids, ucli, d Prticls d ditio, isrg & Rsic

24 PYS 8 Astrooy Tody, Chisso & McMill Discovrig Astrooy, Shwl, Rois & Jffrys UIVRS, Frd & Kuf Th Physics of Digostic Igig, Dowstt, Ky, d Johsto Qutu Mchics, F. Mdl Mthticl Mthods for Physicists, Arf Qutu Mchics: Foudtios Ad Applictios, Dold Gry Swso 4