INTRODUCTION TO QUANTUM MECHANICS

Transcription

1 A. La Rsa Lctur Nts INTRODUCTION TO QUANTUM MECHANICS PART- I Th TRANSITION fr CLASSICAL t QUANTUM PHYSICS CHAPTER Th ORIGINS f QUANTUM PHYSICS. BLACKBODY RADIATION..A Th Kirchff Law and th cncpt f blackbdy radiatr..a.a Macr-scal hat-transfr cncpts Hat transfr, radiatin, absrbd radiatin, blackbdy radiatr...a.b Kirchhff s law Cnnctin btwn th issivity and absrptivity f radiatin by a bdy...a.c Cavity s aprtur as an appriatin t a blackbdy radiatr...b Radiatin in a cavity..b.a Eissin f radiatin by an acclratd charg Eissin f radiatin by an scillating charg..b.b Light Scattring Absrptin and r-issin f nrgy Fractin f th incidnt nrgy scattrd by th charg Scattring crss sctin..b.c Radiatin daping Rat at which th scillatr lss nrgy..c Radiatin and thral quilibriu..c.a Light intnsity spctral dnsity I at quilibriu..c.b Classical calculatin f th at s avrag nrgy W. Th Bltzann s distributin Th quipartitin thr Th ultravilt catastrph..c.c Light nrgy dnsity U at quilibriu. Th cncpt f lctragntic ds..c.d Th ultravilt catastrph..d Th birthday f Quantu Physics: Planck s Hypthsis t calculat th at s avrag nrgy W

2 Quantu physics viw t calculat avrag nrgy. Oscillatr s nrgy quantizatin.. PARTICLE-LIKE PROPERTIES f RADIATION..A Prcsss invlving th absrptin r scattring f radiatin by particls..a.a Phtlctric ffct. Einstin s hypthsis f th quantizd radiatin nrgy..a.b Cptn ffct. Scattring f a cplt quantu f radiatin in dfinit dirctin...a.c Pair prductin..b Prcsss whr intracting chargd particls prduc radiatin..b.a Brs-strahlung..B.b Pair annihilatin. WAVE-LIKE PROPERTIES f PARTICLES..A Th d Brgli Hypthsis..B Eprintal cnfiratin f th d Brgli hypthsis. Elctrn Diffractin. WAVE-PARTICLE DUALITY Rl f th sallnss f Planck s cnstant. Statistical intrprtatin Rfrncs R. Eisbrg and R. Rsnick, Quantu Physics, nd Editin, Wily, 985. Chaptrs and Richard Fynan, Th Fynan Lcturs n Physics, Vlu I, Chaptr and Chaptr Sctin -, -

3 CHAPTER THE ORIGINS OF QUANTUM PHYSICS By th nd f th 9 th cntury, classical physics appard t b n slid fundatins. Th univrs was cnciv as cntaining attr cnsisting f particls, bying Nwtn s law f tin; and radiatin wavs bying Mawll s quatin f lctragntis. Einstin s thry f spcial rlativity gnralizd classical physics t includ th rgin f high vlcitis. S ky prints, hwvr, culd nt b plaind by classical physics, Blackbdy Radiatin Phtlctric Effct Cptn Effct which frcd th intrductin f radical nw cncpts: Quantizatin f physical quantitis Particl prprtis f radiatin Wav prprtis f particls In th prvius chaptr w rviwd th strngth f classical physics lctragntis and rlativity. In this chaptr w ai t failiariz with th tpics that frcd a transitin fr classical t quantu cncpts.. BLACKBODY RADIATION On Dcbr, 9, at a ting f th Gran Physical Scity, Ma Planck prsntd his papr On th Thry f th Enrgy Distributin Law f th Nral Spctru. This dat is cnsidrd t b th birthday f quantu physics. H brk ranks with th thn quit accptd law f quiparticin nrgy that statd that th avrag nrgy f, fr apl, a harnic scillatr was indpndnt f its natural frquncy. H instad pstulatd a frquncy-dpndnt avrag nrgy and, vry rvlutinary, that

4 th nrgy culd hav nly discrt quantizd valus. Planck prsntd his papr whil trying t plain th bsrvd prprtis f thral radiatin. In this sctin w will rviw that dvlpnt, as t gain s grasp f why th discrtnss f a physical prprty lad t a radical diffrnt bhavir f attr. Th prbl that Plank had at hand was rlatd t thral radiatin. Thral radiatin rfrs t th issin f lctragntic wavs by any bjct at a givn tpratur abslut tpratur T. Mattr in a cndnsd stat slid r liquid its a cntinuus spctru f radiatin. But vn at tpraturs as high as a fw thusand f dgrs Klvin, st f th radiatin is invisibl t us as thy li in th infrard part f th lctragntic spctru. Radiatin nrgy K 5 K K Frquncy f Hz Fig. Spctral thral radianc f a bdy shwing th strng dpndnc n th bdy s tpraturs. With incrasing tpratur th bdy its r radiatin whil th frquncy f th st intns radiatin s dashd lins als incrass. Th rlatinship btwn th tpratur f a bdy and th frquncy spctru allws stiating th tpratur f a ht bdy, such as a star, by bsrving th radiatin it its. Th aild fr f th spctru dpnds swhat upn th cpsitin f th bdy. HOWEVER, thr ist a crtain typ f bdy cnfiguratin that it thral spctra f a univrsal charactr. Ths ar calld blackbdis thy fficintly absrb all radiatin incidnt upn th. Indpndnt f th ails f thir cpsitin, it is fund that all blackbdis at th sa tpratur it thral radiatin with th sa spctru

5 It turns ut that, whil th invariability f th spctru fr blackbdy t blackbdy can b plaind basd n thrdynaic argunts s Kirchff s law blw, which stats that th rati f th issiv pwr P t th absrptin cfficint a is th sa fr all bdis at th sa tpratur. th spcific fr f th spctru tnding t zr at vry lw and vry high frquncis can nt b plaind in trs f classical physics argunts. Planck was abl t slv this prbl n th basis f quantu argunts. Sctin. dscribs in r ail ths tw aspcts...a Th Kirchff Law and th Cncpt f Blackbdy Radiatr..A.a Macr-scal hat-transfr cncpts Hat transfr Whil hat transfr Q by cnductin and cnvctin rquirs a atrial diu, radiatin ds nt. Thral Radiatin Rfrs t th issin f lctragntic wavs by any bjct at a givn tpratur abslut tpratur T. Eprintally it was fund in 879 that th rat at which a syst its radiatin is prprtinal t th furth pwr f its abslut tpratur Q t ~ T bjct Sinc th rat f radiativ hat flw is als prprtinal t th ara f th bjct, th ittd pwr pr unit ara is typically prssd as, 5

6 R A Q t ittd T bjct Stfan-Bltzan cnstant = W/.K Eissivity f th surfac < < Absrbd Radiatin A syst absrbs radiatin prprtinal t th furth pwr f th abslut tpratur f its surrundings. R a absrbd T surrunding a indicats th rlativ ability f th surfac t absrb radiatin < a< Fr an bjct in quilibriu with its surrunding: Rittd R absrbd, which iplis a That is, Gd radiatrs ar gd absrbrs a Hat flw via radiatin fr an bjct f ara A t its surrundings Q A T bjct T surrundings 5 t..a.b Kirchhff s law Th issiv pwr P f a givn atrial is dfind as fllws s als Fig..A fr idntifying th variabls bing usd P k d d = ittd pwr pr unit ara f th bdy, cpsd f lctragntic radiatin f frquncy btwn and +d, and dirctd int a slid angl d abut th dirctin k 6

7 T k d=sin dd slid angl k = k = /c whr c is th spd f light. k = / Fig..A Radiatin f wav-vctr arund k ittd by a surfac at tpratur T. G. R. Kirchhff prvd in 859 by using gnral thrdynaic argunts that, at any givn lctragntic wavlngth, th rati f th issiv pwr P t th absrptin cfficint a is th sa fr all bdis at th sa tpratur P a P a P a tc Blackbdy Radiatr A prfct radiatr wuld hav an absrptin 7 Cfficint a= Such a syst is calld a blackbdy radiatr Nt: Th blackbdy na f riginats fr th fact that black lking bjcts ar in fact bjcts that d nt rflct any f th incidnt light. Black clr is in fact an absnc f clr. BUT in th cntt f th quantu physics that w ar dvlping hr, a blackbdy radiatr ay nt appar black at all. In fact, a blackbdy its vry fficintly all clrs. 6..A.c Cavity s aprtur as an appriatin t a blackbdy radiatr A blackbdy radiatr is an idalizatin; hwvr, a sall rific in a cavity clsly appriats n. 7

8 Fig..B Blackbdy radiatr. Any radiatin incidnt fr th utsid and passing thrugh th hl will alst cpltly b absrbd in ultipl rflctins insid th cavity; that is, th hl has an ffctiv absrptin cfficint cls t unity a= Sinc th cavity is in thral quilibriu, th radiatin gtting insid thrugh th sall pning and that scaping fr th sall pning will b qual =. W infr that th spctral charactristics f th radiatin ittd by th aprtur shuld hav thn th charactristics f a blackbdy radiatr...b Radiatin in a cavity Our nt bjctiv is t charactriz th bath f lctragntic radiatin istnt insid a blackbdy radiatr cavity. W wuld lik t knw, Hw uch light nrgy f angular frquncy acrss th lctragntic spctru ds ist insid th cavity at quilibriu cnditins at a givn tpratur T? 8 Tw apprachs fr answring this qustin, ach arriving t th sa gnral rsult, ar wrth ntining: APPROACH -: Th first fllws a Fynan s dscriptin vry rich in physics cntnt that dls an at as an lctrical harnic scillatr. Th quilibriu insid th cavity is btaind by a balanc btwn: 8

9 th rat f nrgy absrbd by th at dld as an lctrical harnic scillatr; i.. a charg attachd t a spring fr th light nrgy istnt in th cavity, and th rat at which th at lss nrgy this lss dld as dw W, whr W is th avrag nrgy f th at at a givn ti. Rflcting walls I At absrbs nrgy fr th incidnt radiatin q, dw At dld as an harnic scillatr W : Aplitud f scillatin Rat at which th at f avrag nrgy W lss nrgy Fig.. At dld as an lctrical harnic scillatr in quilibriu with a bath f lctragntic radiatin insid a b f prfctly rflcting walls. If it wr nt fr th incidnt radiatin, th at s nrgy W wuld dcay pnntially with ti. A prpr aunt f incidnt radiatin I allws W t rain cnstant. Th bjctiv in this sctin is t find ut such prpr aunt f radiatin; that is, s far w d nt knw th I that wuld prit quilibriu insid th b. T calculat rat f nrgy absrbd by th at fr th incidnt radiatin, w will appal t th cncpt bhind th prcss f scattring, in which th incidnt radiatin drivs th charg, and th lattr r-it it in all dirctins. This happns bcaus acclratd chargs it radiatin. Calculating th pwr <P> ittd by a chargd particl ittd will allw, thn, t calculat th pwr absrbd by th at fr th incidnt radiatin; thy ar th sa. Furthr, at quilibriu, <P> shuld b qual t W. 9

10 It is fr th last quality that that an analytical prssin will b btaind fr th rquird Light Intnsity Spctral Dnsity I that ust ist insid th cavity fr an quilibriu situatin t prvail. APPROACH-: Th scnd apprach is a bit r abstract; it cnsidrs th nrgy-intrchang quilibriu rachd btwn th lctragntic-ds insid th cavity and th cavity s walls. It invlvs th cunting f lctragntic ds. An prssin fr th Light Enrgy Dnsity U at quilibriu will b btaind. Th cnsistncy f ths tw viws will b vrifid thrugh th rlatinship I = c U whr c is th spd f light. It turns ut that th prcdur fllwd by ths tw apprachs lading t an prssin fr I and U rspctivly hav survivd th nw wav f quantu cncpts; that is, thy ar still cnsidrd valid. What has changd thn? What has changd draatically, du t th advnt f th quantu idas, is th spcific way t calculat th avrag-nrgy f an at in apprach- thd r th avrag nrgy f an lctragntic d apprach- thd. A aild dscriptin f ths quantu dificatins incurrd in th calculatin f I and U cnstituts th ssnc f Chaptr. In what fllws, w will ainly cncntrat n a aild accunt f th apprach- rfrrd abv. A dscriptin f th apprach- is givn in th appndi...b.a Eissin f radiatin by an acclratd charg Elctragntic radiatin is ittd by chargs in acclratd tin.

11 Acclratin a q r P E Elctric fild E prducd by an acclratd charg. At a pint P, E is rintd prpndicular t th lin f sight and its agnitud prprtinal t th cpnnt f th acclratin-vctr alng th dirctin prpndicular t th lin f sight. Fig.. An lctrn in an at dld as a charg attachd t th nd f a spring f apprpriat spring cnstant f natural frquncy. q sin E r, t a t r / c c r Rtardatin ti Elctric fild prducd by an acclratd charg q 9 Th Pynting vctr S S E cits agnitud S givs th aunt f nrgy pr squar tr pr unit ti that passs thrugh a surfac that is nral t th dirctin f prpagatin. Radiatin pwr P ittd by an acclratd charg a q r E P S [ da Intgral vr a sphr f radius r q sin c c r a' ] da Fig..5 Elctragntic pwr radiatd by an acclratd charg. q P 6 c a' whr a a t = a t - r/c

12 Eissin f radiatin by a charg undr harnic scillatin Cnsidr that a charg q scillats with angular frquncy j t t al { R} Cs t Cpl variabl Hr, is nt ncssarily th natural frquncy f th scillatr s Fig.8. Fr apl, th scillatr ay b bing drivn by a trnal surc f arbitrary frquncy. In what fllws w will b valuating ti avrags f physical quantitis lt s call it g vr n prid T / f scillatin: T g g t T Ntic, fr apl, that th ti avrag f Cs t vr n prid is ½. W als will b using indistinctly Cs t r. Fr th lattr it is assu that w will tak th ral part aftr th calculatins in rdr t btain a prpr classical physical intrprtatin. j t j t' Fr prssin w calculat an acclratin a' and a' /. Rplacing ths valus in prssin givs, Ttal avrag pwr ittd by a q P charg q undrging harnic c scillatins with aplitud, and angular frquncy..b.b Light Scattring Cnsidr a plan wav filling all th spac travling in a spcific dirctin, lt s say +X. Hr w highlight nly th tpral variatin:

13 t ERal { E j t } ECs t 5 ti variatin f th lctric fild f th incidnt radiatin. A travling wav carris nrgy. A quantificatin f this travlling nrgy is prssd by th crrspnding Pynting vctr Sc E ; its avrag valu, accrding t, is qual t, S ce 6 <S> is th avrag nrgy pr squar tr pr unit ti passing thrugh a surfac nral t th dirctin f prpagatin. E S ce E j t <S> is als calld th intnsity I E is th aplitud f th lctric fild f th incidnt wav Fig..6 Incidnt plan-wav radiatin f frquncy. Whn an at charactrizd by a rsnanc frquncy, is placd in a rgin whr thr is a bath f lctragntic radiatin, th j t radiatin s lctric fild E E will driv th at s charg q up and dwn; that is, it will acclrat th charg thus causing th at t r-it lctragntic radiatin. This prcss is calld scattring. That is, scattring is th prcss by which nrgy is absrbd by an at fr th incidnt radiatin 7 fild and r-ittd in all dirctins.

14 E E, j t [ j ] j t Fig..7 Incidnt light is absrbd and r-ittd scattrd by an at in all dirctins. Lt s calculat th nrgy absrbd hnc r-ittd by th charg Actually th nrgy ittd by th acclratd charg has alrady bn calculatd in prssin abv, cpt that w nd t find ut th aplitud f scillatin ; th lattr will dpnd n th lctric fild aplitud E, as wll as th frquncy f th incidnt radiatin. In thr wrds, lt s calculat th rlatinship btwn E,, and. Incidnt radiatin Scattrd r-ittd light q P c q, At dl as an scillatr f natural rsnanc frquncy. Th spring cnstant is chsn accrding t k Fig..8 At dld as an scillatr f natural frquncy. Th ability f th scillatr t absrb nrgy fr th incidnt radiatin dpnds n. T find, lt s dl th at as a dapd harnic scillatr. Accrdingly th quatin f tin fr th charg q is givn by,

15 d d j t k q E d Hr th tr accunts fr th prsnc f a dissipatin nrgy surc, which, in ur cas, can b idntifid in th lss f nrgy du t th lctragntic radiatin by th acclratd charg. A statinary slutin f 8 is givn by, whr j j t [ ] 9 8 q / E / Aplitud f scillatin as a functin f frquncy and tan Eprssin indicats that th aplitud f scillatin and hnc th acclratin f th charg dpnds n th incidnt radiatin s frquncy. Lt s prcd nw t calculat th ttal pwr radiat by th acclratd charg undr th influnc f an lctric fild f aplitud E and frquncy. Rplacing th valu f givn in fr int th prssin fr th radiatin pwr givn in, n btains, Rarranging trs, P q c E q q / P. c 8 q P ce c 5

16 Eprssin givs th ttal avrag nrgy ittd by th charg q whn subjctd t a harnic lctric fild givn in prssin 5 f aplitud E and frquncy. Th cncpt f scattring crss sctin If w cnsidrd a hypthtical crss sctin f ara intrscting th incidnt radiatin, th aunt f nrgy pr scnd hitting that ara wuld b P [ ce ] I S ce crss sctin f ara <S> is calld th light intnsity I Fig..9 Pictrial rprsntatin f scattring crss sctin. On can us th analgy f an affctiv ara bing intrcptd by th incidnt radiatin t dfin hw ffctivly th radiatin is absrbd and scattrd i.. r-ittd by an at. In ffct, cparing prssins and, th ttal pwr scattrd by an at is nurically qual t th nrgy pr scnd incidnt n a surfac f crss-sctin ara, scattring P scattring [ ce ] whr scattring scattring 8 q c has units f ara. scattring 6

17 scattring Fig.. Sktch f th at s scattring crss sctin. Crtainly, vry phtn absrbd by th at will b scattrd ut. But what prssin is suggsting is that phtns f frquncy quit diffrnt than will hav littl chanc t b absrbd and rittd. Th clsr is t, th highr th chancs t b absrbd and subsquntly b r-ittd...b.c Elctragntic Radiatin Daping What is th valu f? W addrss hr th fact that d th tr in Eq. 8 th tr in th diffrntial quatin that taks int accunt th nrgy dissipatin, shuld b cpatibl with prssin that givs th lctragntic nrgy dissipatd by th scillatr. W shuld rquir thn that ths tw prssins b cnsistnt with ach thr. Indd, n n hand, th pwr dissipat by a scillatr is givn by d [frc]vlcity = [ ] d = [ j ] j = =. j j t Hr w us th prssin fr t givn in 9 [ ]. Th avrag valu f th dissipatd pwr will b, 7

18 /. On th thr hand, accrding t, th ittd lctragntic pwr is, q P c Th last tw prssins shuld b qual. q / c This allws t idntify c q 6 c q c. Rarranging trs, lctragntic radiatin daping cnstant 5 Fr practical purpss, hwvr givn th vry narrw bandwih f th crss sctin shwn in Fig. abv, will b typically nd up bing valuatd at =,i.. th narrw bandwih f tll us that st f th physics happns arund =. Rat at which th scillatr lss nrgy A r aild dscriptin f this sctin is givn in th supplntary Appndi- f this chaptr. Lt W W t b th avrag nrgy f an scillating 6 charg at a givn ti t. If th scillating charg is lft aln t scillat, its aplitud f vibratin will di ut prgrssivly as th scillatr lss its chanical nrgy by itting lctragntic radiatin. Th rat at which th scillating charg lss nrgy is givn by, dw W 7 with its crrspnding slutin 8

19 W t t W 8 As an apl, an at that has a rsnanc frquncy crrspnding t = 6 n, wuld hav a daping cnstant f ~ 8 s -. That is, th radiatin will ffctivly dy ut aftr ~ -8 s r aftr ~ 7 scillatins...c Radiatin and thral quilibriu Lt s cnsidr an at nclsd in a b ad f irrr walls which cntains lctragntic radiatin. Radiatin r-ittd by th at rains insid th b undrging ultipl buncs n th irrr walls. Lt s furthr assu that th tpratur f th whl syst is T. Scattrd r-ittd light q Incidnt radiatin At dld as an scillatr B at tpratur T Fig.. Schatic rprsntatin f an at as an harnic scillatr that radiats nrgy. Th ats absrb nrgy fr th lctragntic radiatin istnt insid th b th lattr assud t b ad f prfctly rflcting walls. Hw t ak th tpratur T intrvn in an prssin lik that givs th pwr scattrd by an at in th fr P scattring q? c It is plausibl t assu that th quilibriu tpratur shuld crrspnd t prpr valu f th aplitud f th lctric fild, E, sinc th highr th valu f E, th highr th charg s 9

20 aplitud f vibratin, th gratr tpratur t b assciatd with th at i.. th aplitud f th scillatr shuld incras with tpratur. If ur assuptin wr crrct, hw t find thn th prpr valu f E crrspnding t a givn tpratur T? Aiing t find a prpr answr lt s utlin s cnsidratins: - If an atic scillatr had n charg, it wuld scillat frvr. It wuld hav an avrag nrgy W cpatibl with th tpratur in th b; that is W W T. In thr wrds, it wuld scillat frvr with an aplitud sttld by th tpratur T in th b. ½ k = ½ kt r ½ = ½ kt. - But ur atic scillatr is chargd. If it wr lft aln, its aplitud f vibratin wuld di ut prgrssivly, as th scillatr lss its nrgy by itting lctragntic radiatin. - If ur atic chargd scillatr wr in physical cntact with thr ats, nrgy in th prpr aunt will b supplid by thir utual cllisins as t kp th sa tpratur ang thslvs. Hr, hwvr, w will cnsidr that th nrgy is supplid via lctragntic intractin: th at draws nrgy fr th radiatin istnt in th cavity t cpnsat th nrgy bing lst by radiatin acclratd chargd particls it radiatin. Whn this cpnsatin f nrgy atchs, thn w ar at an quilibriu situatin, which inhrntly shuld ccur at a givn tpratur th lattr sttling th charg s aplitud f scillatin at a crrspnding valu. W will rtak th subjct f tpratur dpndnc f in Sctin..c.b blw...c.a Light intnsity spctral dnsity I at quilibriu T fraliz th quilibriu situatin w hav t kp in ind that radiatin f diffrnt frquncis ight b prsnt in th cavity. It is cnvnint, thn, t intrduc th cncpt f spctral dnsity I: I = light intnsity spctral dnsity I d = light intnsity f frquncy within a rang, +d insid th b

21 = cntributin t th avrag lctragntic nrgy pr squar tr pr unit ti passing thrugh a surfac nral t th dirctin f prpagatin fr radiatin cpnnts f frquncy within a rang d [ Intnsity ] J / s J Units f [I ] = = [ ] / s Bfr stablishing th cnditin f quilibriu, lt s ak tw prtinnt bsrvatins: Accrding t and and kping in ind that diffrnt frquncis can b prsnt in th cavity th ttal pwr ittd by th at is givn by I w d. But, in a scattring prcss, vry phtn incidnt is r-radiatd by th at f natural frquncy. i.. phtn-in phtn-ut. Thrfr th prssin abv can b intrprtd as th rat at which nrgy is incidnt and capturd by th at. On th thr hand, this sa issin f pwr i.. I w d, can b sn fr th prspctiv f a syst lsing nrgy du t a daping prcss d[ W ] [ W ] charactrizd by a daping cnstant. Als, th rquirnt f cpatibility btwn i pwr r-radiatd by th at, and ii a sipl daping harnic d d scillatr dl j t k q E, lad t q prssin 5. But sinc all th c c dynaics ccurs at, that is W~ fr, w can dw us W, with th intrprtatin that W is th ttal nrgy f th at. Fralizatin f th thral quilibriu cnditin:

22 Hw uch light intnsity spctral dnsity I thr ust b insid th b at tpratur T fr, th nrgy absrbd by th scillatr fr th radiatin bath pr unit ti dw I w d 9 t b qual t th nrgy r-radiatd by th scillatr pr unit ti dw W Avrag nrgy f th scillatr at tpratur T B at tpratur T q Fig.. At f natural frquncy in a bath f lctragntic radiatin f spctral dnsity I. Lt s valuat th intgral that appars in 9. Sinc th prssin fr paks at thn tnding th intgral dwn t ds nt caus any significant chang this is dn just t facilitat th calculatin I d I d 8 q whr scattring c

23 Fr th sa rasn that nly th valus f vry cls t will significantly cntribut t th intgral w can pictur in ur ind that I d I d Fig.. Sktch f th at s scattring crss sctin and th spctral dnsity light intnsity prsnt in sid th cavity Accrdingly th fllwing appriatins can b cnsidrd apprpriat, I I [ [ ] ] All ths appriatins lad t I d I d 8 q I c d

24 d c q I 8 d c q I / using a a a d arctan / c q I c q I c q I d I At quilibriu w shuld hav, d w I dw = W dw, which lads t W c q I W q c I Using prssin 5 fr th valu f c q c valuatd at n btains, W c q c q c I, r W c I, W c I Avrag nrgy f th scillatr I is th light spctral dnsity at =. Hr is th natural frquncy f th scillatr w wr fcusing in. Had w usd an scillatr f a diffrnt natural frquncy, lt s say

25 , w wuld hav btaind a siilar prssin but with instad f. Hnc, in gnral, I W c Avrag nrgy f th scillatr at tpratur T Rquird light intnsity spctral dnsity I insid th b at tpratur T in rdr t aintain quilibriu. Ntic J Units f [I d] = [Intnsity] = s [ Intnsity ] J / s J Units f [I ] = = [ ] / s It is wrth t highlight that, Eprssin has raind undisputd. That is, it is still cnsidrd crrct vn whn th nw quantu chanics cncpts ar intrducd. It is in th calculatin f th avrag nrgy W whr th classical and quantu apprachs fundantally divrg...c.b Classical calculatin f th at s avrag nrgy W. In classical statistical chanics thr ists a vry gnral rsult s calld quipartitin thr, which stats that th an valu f a quadratic tr in th nrgy is qual t ½ k B T. Hr k B is th Bltzann s cnstant and T is th abslut tpratur. Th Bltzann distributin Th quipartitin thr can b btaind fr th Bltzann s prbability distributin fr a sall syst A in quilibriu with a hug rsrvir at tpratur T. Th Bltzann distributin stats 5

26 that th prbability that th syst S b fund in a stat f nrgy E is prprtinal t P E E / k T B ; that is, E / k T B E / k B T C prbability t find th syst A in a stat f nrgy E Rsrvir at tpratur T E Enrgy chang Sall syst A PE Bltzann distributin Enrgy E Fig. Lft: A syst intracting with a thral rsrvir. Right: Bltzann s distributin t find th syst in a stat f nrgy E. Th valus f E culd g fr t infinity th rsrvir bing in charg f kping th tpratur cnstant; but, as th prssin abv indicats, th stats f lwr nrgy hav a highr prbability. Sinc fr a givn nrgy thr ay b svral stats charactrizd by th sa nrgy, it is usual t dfin, g E de nubr f stats with nrgy E, within an intrval de, thus giving E / k T C B g E de prbability t find th syst A in a stat f nrgy btwn E and E+ de, which suggsts t rathr idntify a prbability-dnsity P E dfind as fllws E / k T P E de C B g E de th prbability t find th 5 syst in a stat f nrgy btwn E and E+ de, 6

27 with C bing a cnstant t b rind. Sinc th prbabilitis addd vr all th pssibl stats shuld b E'/ k T qual t, w ust rquir, C B g E' de', which givs, C = E' / k T B g E' de' 6 A slf cnsistnt prssin fr P E is thrfr givn by, E / kbt g E de P E de E'/ kbt 7 g E' de' Ntic in th dninatr w ar using a duy variabl E '. Fr prssin 7 w can frally calculat th avrag nrgy f th syst, E / kbt E g E de E E'/ k T 8 B g E' de' Th Equipartitin Thr It turns ut, vry ftn th nrgy f th syst ay cntain a quadratic tr. Cnsidr fr apl p E k..., and w wuld lik t calculat, fr apl, th avrag valu f th kintic nrgy tr p aln. As w knw, bing th syst in p cntact with a hat rsrvir, th valu f is stis high, stis it is lw bcaus it gains r lss nrgy fr th hat rsrvir; w wuld lik t knw what wuld b its avrag valu p. 7

28 8 / / B B dp dp p p T k T k p p 9 Lt s call k B T In trs f prssin 9 bcs, dp dp β dp dp p p p p p p β β β β dp dp β p p β β ln dp β p p β Dfining th variabl p β u, ln ln du β du β p u u ln ln du β p u Tr indpndnt f ln β p T k p B Had w chsn any thr quadratic tr f th nrgy w wuld hav btaind th sa rsult. This is th quipartitin thr. It

29 stats that th an valu f ach indpndnt quadratic tr in th nrgy is qual t k B T. Th ultravilt catastrph Using this rsult in prssin I W, th c diffrnt dgrs f frd lt s assu thr ar f diffrnt quadratic trs in th prssin fr th ttal nrgy invlvd in th calculatin f W wuld lad t a finit nubr f f k B T. Thus, I f k BT c. Sinc th valu f is accidntal w wuld btain a siilar prssin if w wr t chs an at f diffrnt rsnanc frquncy, w hav th fllwing rsult, I f kbt classical prdictin 6 c I Classical prdictin Eprintal rsults Frquncy Fig.. Th srius discrpancy btwn th printal rsults and th thrtical prdictin is calld th ultravilt catastrph...d Th birthday f Quantu Physics: Planck s Hypthsis t calculat th at s avrag nrgy W T bring th thrtical prdictin clsr t th printal rsults Planck cnsidrd th pssibility f a vilatin f th law f quipartitin f nrgy dscribd abv. Rathr than using prssin, th starting prssin wuld b prssin 9

30 I W, with th avrag nrgy f th scillatr W n c t hav a cnstant valu as th quipartitin thr prdictd but rathr bing a functin f th frquncy, W, with th fllwing rquirnts, W and W Fr th statistical calculatin f W, Planck did nt qustin th classical Bltzann s Statistics dscribd in th sctin abv; that 5 wuld still b cnsidrd valid. Planck ralizd that h culd btain th dsird bhavir prssd in if, rathr than trating th nrgy f th scillatr as a cntinuus variabl, th nrgy stats f th scillatr shuld tak nly discrt valus:,,,, 6 th nrgy stps wuld b diffrnt fr ach frquncy = 7 whr th spcific dpndnc f in trs f t b rind Incidnt radiatin q Planck pstulatd that th nrgy f th scillatr is quantizd

31 Fig..5 An at rciving radiatin f frquncy, can b citd nly by discrt valus f nrgy,,,, Accrding t Planck, in th classical intgral prssin 7 E / kbt E g E de W classical E E'/ kbt, n wuld hav t rplac: g E' de' gede [ gede givs th nubr f stats with nrgy E within an intrval de ] de de thus btaining, W Planckl whr n E n E n n E E E n = n ; n=,, n n / k / k B B T T 8 A graphic illustratin can hlp undrstand why this hypthsis culd indd wrk: First w shw hw classical physics valuat th avrag nrgy. E classical E [ P E ] de = ara undr th curv EP E PE E PE Bltzann distributin <E>= =Ara Classic calculatin: cntinuu additin intgral Enrgy E Enrgy E

32 Fig..6 Schatic rprsntatins t calculat th avrag nrgy f th scillatr undr a classical physics apprach. Using Plank s hypthsis E E P E Planck Cas: Lw frquncy valus f Fr this cas, Planck assud shuld hav a sall valu fr th rasns plaind in Fig..7. sall valu 9 n n n PE E PE Bltzann distributin Quantu calculatin discrt additin Enrgy E Enrgy E Fig..7 Schatic rprsntatins t calculat th avrag nrgy f th scillatr assuing th scillatr can adit nly discrt valus f nrgy, fr th cas whr th sparatin btwn cntiguus nrgy lvls bing f rlativly lw valu. Indd, cparing Fig..6 and Fig.7 n ntics that if n is sall thn th valu f EnP En will b vry cls t th classical valu. It is indd dsirabl that Planck s rsults agr with th classic rsults at lw frquncis, sinc th classical prdictins and th printal rsults agr wll at lw frquncis s Fig.. abv. Cas: High frquncy valus f Fr this cas, Planck assud, Planck assud shuld hav a biggr valu

33 big valu 5 PE E PE Bltzann distributin Quantu calculatin discrt additin Enrgy E Enrgy E Fig..8 Schatic rprsntatins t calculat th avrag nrgy f th scillatr assuing th scillatr can adit nly discrt valus f nrgy, fr th cas whr th sparatin btwn cntiguus nrgy lvls bing f rlativly larg valu. Ntic in this cas, th valu f Planck n E EnP En dpictd in Fig..7 will b vry diffrnt fr th n dpictd in Fig..8 th lattr clsr t th classical rsult. n In fact, EnP En tnds t zr as th stp is chsn largr. This rsult is dsirabl, sinc th printal rsult indicat that E tnds t zr at high frquncis. It appars thn that th Planck s dl 8 culd wrk. Nw, which functin culd b chsn such that is sall at lw, and larg at larg? An bvius sipl chic is t assu a linar rlatinship, = 5 whr is a cnstant f prprtinality t b rind. Rplacing 5 in 8 n btains,

34 Lt W W Planckl Planckl kt E kt n n n E n n n E E n n n n Sinc d[ lnu ] /d = /udu/d W Planckl W Planckl k T If w dfin n W Planckl n / k T B B / k T n k T n n n ln[ n ], thn... This sris is qual t /- = = k T k T k T ln ln [ [ n n n/ k T B B n/ k T n n n ] k T ln[ n ] k T kt 5 ] Ntic: W Planckl kt

35 W Planckl With this rsult, prssin bcs I c 5 W Planck I Eprintal rsults Fig..9 Planck s dl quantizatin f th scillatr nrgy fits wll th printal rsults. Particl s and wav s nrgy quantizatin Histrically. Planck initially 9 pstulatd nly that th nrgy f th scillating particl lctrns in th walls f th blackbdy is quantizd. Th lctragntic nrgy, nc radiatd, wuld sprad as a cntinuus. It was nt until latr that Plank accptd that th scillating lctragntic wavs wr thslvs quantizd. Th lattr hypthsis was intrducd by Einstin 95 in th cntt f plaining th phtlctric ffct, which was crrbratd latr by Millikan 9...B.g Light Enrgy Dnsity U at quilibriu. Th Cncpt f Elctragntic Mds s Cplnt Chaptr- nts wbsit 5

36 . PARTICLE-LIKE PROPERTIES f RADIATION..A Prcsss invlving th absrptin r scattring f radiatin by particls..a.a Phtlctric ffct Einstin s hypthsis f th quantizd radiatin nrgy Einstin s intrprtatin: In th wav pictur, th intnsity f radiatin is givn by th avrag f th Pynting vctr S, I S ce whr E is th lctric fild f th wav In th particl pictur, th intnsity is writtn as I N h whr N is th nubr f phtns pr unit ti crssing a unit ara prpndicular t th dirctin f prpagatin Einstin suggstd that E culd b intrprtd as a asur f th avrag nubr f phtns pr unit vlu. Th radiatin granularity cncpt lads t cnsidr th light intnsity as a statistical variabl A light surc its phtns randly in all dirctins Th act nubr f phtns crssing an ara fluctuats arund an avrag nubr in ti and spac..a.b Th Cptn Effct Scattring f a cplt quantu f radiatin in dfinit dirctin. Cptn plaind th rsults n th assuptins that ach lctrn scattrs a cplt quantu f radiatin phtn; and th quanta f radiatin ar rcivd fr dfinit dirctins and ar scattrd in dfinit dirctins. WAVE-LIKE PROPERTIES f PARTICLES..A Th Luis d Brgli Hypthsis 6

37 Th Cptn s Effct print had prvidd strng vidnc n th particl natur f radiatin. Th typical sytry prssd by Natur in any physical prcsss fr apl, chargs in tin prduc agntis and agnts in tin prduc currnts ay hav ld Luis d Brgli t prps th wav natur f particls. In 9, Luis d Brgli, using argunts basd n th prprtis f wav packts, prpsd that, th wav aspct f attr ar rlatd t its particl aspct in actly th sa quantitativ way displayd by radiatin. h / p whr is th wavlngth f th wav 66 assciatd with th particl s tin; and E / h th ttal nrgy E is rlatd t th frquncy f th wav assciatd with its tin. Hr h is th Planck s cnstant; h =6.6 - Js. Sinc in th cas f radiatin ass w alrady knw that c, nly n f th rlatinships abv is ndd t btain bth and fr its nrgy and ntu s particl prprtis. In cntrast, fr a particl f ass, n nds bth rlatinships givn abv t btain th crrspnding and. Ordr f agnitud calculatin f d Brgli Cas: = - Kg, v= /s p = v = - Kg /s =h/p = This valu is inaccssibl t any asurnt currntly pssibl Cas: = lctrn s ass = 9. - Kg, q =.6-9 C If th lctrn acclratd thrugh a ptntial diffrnc f V vlts its kintic nrgy will b rind by q V =p / /, which givs p q V Fr V =5Vlts: p q V /.56 8 / 7

38 p.8 / s h / p 6.6 J s /.8 / s.7. This valu is f th sa rdr f agnitud f th X- rays wavlngth...b Eprintal cnfiratin f th d Brgli hypthsis. Elctrn Diffractin In 897 J. J. Thpsn discvrd th lctrn stablishing th ratin q /. and was awardd th Nbl Priz in 96. Th d Brgli wavlngth d Brgli assciatd t an lctrn falls in th rdr f Angstrs whn acclratd at drat vltags ~ 5 Vlts, as shwn in th apl abv, which happns t b th intr-atic spacing in a crystallin slid. In 96 Elsassr suggstd that th wav natur f particls culd b tstd by vrifying whthr r nt lctrns f th prpr nrgy ar diffractd by a crystallin slid. This ida was cnfird by Davissn and Grr in th Unitd Stats and J. J. Thpsn s sn G. P. Thpsn in Sctland 97. Elctrn gun Ni crystal Dtctr Schatic diagra f th Davissn-Grr print d D=.5 Å 8

39 d D=.5 Å A B Fig..9 Eprintal stup fr dnstrating th wav charactr f lctrns If th incidnts lctrns wr intrprtd as wavs f wavlngth, rflcting fr th Brag s pans siilar t th -rays, thn thy culd giv ris t intrfrnc phnna. In particular, th cnditin fr btaining cnstructiv intrfrnc s figur abv is givn by lngth f path ABC = n r, d Cs / = n n=,,... Fr th figur abv, d=d Sin /. Thus th cnditin f cnstructiv intrfrnc bcs D Sin / Cs / = n, r D Sin = n n=,, Fr an acclrating vltag V = 5 vlts, C d Brgli = h/p=.66 Å as calculatd abv. Accrding t prssin 6 it wuld b pctd t bsrv a aiu f intrfrnc at, Sin = D = dgrs thrtical prdictin. Th figur blw shws a schatic f th printal rsult btaind by Davissn and Grr. Th pak signal f ctd lctrns arund = 5 supprts th d Brgli hypthsis, sinc it can b plaind as cnstructiv intrfrnc f wavs 9

40 scattrd by th pridic arrangnt f th ats in th crystal I At V =5 Vlts Fig.. Eprintal rsults btaind by Davissn and Grr. Cllctr currnt I as a functin f th ctr s angl fr a fid valu f th acclrating vltag Th lctrn intrfrnc is nt btwn wavs assciatd with n lctrn, and wavs assciatd with anthr. Instad, it is th intrfrnc btwn diffrnt parts f th wav assciatd with a singl lctrn. Th abv can b dnstratd by using an lctrn gun f such lw intnsity that n lctrn at a ti incidnt th crystal and by shwing that th pattrn f scattrd lctrns rains th sa.. WAVE-PARTICLE DUALITY Nt nly lctrns but all atrials, charg r unchargd, shw wavlik charactristics: Estrann, Strn, and Frish Mlcular bas f hydrgn and atic bas f hliu fr a lithiu flurid crystal. Th difficultis f ths prints li in th lw intnsity f th lcular bas attainabl. Fri. Marshall and Zinn

41 Us th abundant slw nutrns availabl fr nuclar ractrs. X-ray analysis is basd n th phtn-lctrn intractin; nutrns rathr intract with th nuclus. Th particl aspct is phasizd whn studying thir issin and ctin Th wav aspct is phasizd whn studying thir tin Th sall valu f h ipds th bsrvatin f wav-particl duality in th acrscpic wrld. Sinc h / p, a zr valu fr h wuld iply an absnc f diffractin ffcts. On th thr hand, th rlativly larg ntu f acrscpic bjcts cpard t h als aks t sall t b ctd. Th dpr undrstanding f th link btwn th wav dl and th particl dl is prvidd by a prbabilistic intrprtatin f th wav-particl duality: In th cas f radiatin, it was Einstin wh unitd th wav and particl thris usd in th planatin f th phtlctric ffct. Ma Brn any yars aftr applid latr siilar argunts t unit th wav and particl thris f attr. Einstin, hwvr, did nt accpt it. This lattr aspct is r invlvd and bcs th subjct f th nt chaptr.

42 Ovrlk th fllwing APPENDIX Th lctragntic daping cnstant Fr cparisn, lt s cnsidr first th cas f a dissipativ frc prprtinal t th vlcity F dis d, Th Eq. f tin is d k F t, which can b prssd in trs f nrgy balanc as fllws, d d k P t t d Fr j t, th avrag dissipating pwr is givn by, d d d d d Pdis Fdis * d d * Pdis Th avrag ttal chanical nrgy f th scillatr varis as, d d k t t P Fr a charg q scillating accrding t jt, th avrag lctragntic nrgy dissipatd is givn by, d d k P t t q c

43 Fr th sak f cparisn f th ffctiv dissipatin nrgy btwn th chanical dissipatin and lctragntic dissipatin cass, th fllwing dfinitin is typically intrducd, lctragntic Thus lctragntic q c q 6 c q c / c d d 5 k Pt t lctragntic Nt: Whn a trnal surc is prsnt and aks th charg scillat at w hav t us d d k P t t lctragntic which is ssntially, W t W t t lctragntic P Ntic this tr is sallr than W, and sinc is als uch sallr than th aplitud at rsnanc, thrfr th lss f nrgy utsid th rsnanc frquncy is cparativly vry sall. R. Eisbrg and R. Rsnick, Quantu Physics, nd Editin, Wily, 985. S Chaptr Fynan Lcturs, Vl - I, Chaptr