1 Quantum Mechanics. Wave Mechanics. Physical Quantities of The System Are Given By Square Matrices. Classical mechanics Position time t>0

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1 Quantum chancs Lctur nots b Assaf Tal NR s a sm-classcal phnomnon On th on hand, w trat th lctromagntc flds n mattr as classcal, whch s justfd at th long wavlngths ncountrd On th othr hand, th basc unt w'll b dalng wth - th nuclar spn - s strctl a quantum mchancal (Q) ntt Although n crtan cass w can tak a classcal lmt n whch our sstm of spns can b dscrbd b classcal quatons, for th most part NR s bst thought of n quantum mchancal trms, and cannot b pland wll b classcal analogs Luckl, Q s farl bngn n NR snc t dals wth smpl sstms (mostl spn-/) W rvw hr som of th basc prncpls whch wll b of us to us n planng NR phnomna n subsqunt chaptrs Knmatcs: Frst, I wll plan how to dscrb a Q sstm W wll start wth a sngl spn n a magntc fld Instad of takng th "usual" vw adoptd b man Q ttbooks whch talk about wav functons and vctors, w wll mmdatl start wth th dnst matr formalsm, whch s n fact a gnralaton of thos concpts (and, paradocall, asr to undrstand!) Dnamcs: I wll show ou how to solv an problm n quantum mchancs That s, gvn th stat of th sstm at tm t= and th lctromagntc fld t's n, I'll plan how to solv - at last n thor - for ts stat at an tm t> Our phlosoph wll not b to "undrstand" quantum mchancs, whch s rathr ambtous, but to la down th basc ruls b whch w can carr out manngful calculatons In phscs-spak w'd sa w hav man photons pr unt volum Poston at tm t= Dnst matr at tm t= Nwton's nd Law ost books bgn b talkng about "wav mchancs" usng vctors to dscrb a pur quantum sstm, and gnral to statstcal nsmbls usng th dnst matr formulaton W wll crcumvnt that stp compltl to smplf th dscusson, whch mans w wll b sang "just bcaus" at a fw spots whr w wll lack th ncssar foundatons to plan thm W wll conclud b gnralng our notons to mor than on spn Wav chancs Classcal mchancs Poston tm t> Quantum mchancs (wav formulaton) Wavfuncton at tm t= Schrodngr's Equaton Quantum mchancs (dnst matr) Louvll's Equaton Dnst matr Wav mchancs Phscal Quantts of Th Sstm Ar Gvn B Squar atrcs Phscal obsrvabl quantts of a sstm such as nrg, angular momntum and magntc momnt ar gvn b matrcs Th smplst sstm n Q s possbl an solatd spn-/ Som atomc nucl hav nonro angular momntum It turns out that th magntud of th angular momntum onl appars 4 m kg n "stps" of h /, whr h 66 s sc Planck's constant So w hav spn-/ partcls (havng ntrnsc angular momntum h/), spn- at Wavfuncton at tm t> Dnst matr at tm t>

2 partcls (havng ntrnsc angular momntum h), spn-/ partcls (havng ntrnsc angular momntum h/) and so forth Som ampls from natur nclud: "Partcl" n p Spn (radh/t) Elctron / Proton ( H) / Nutron / Dutrum ( H) Carbon 6 6 ( C) Carbon 7 6 / ( C) Lthum ( 7 L) 4 / For a spn-/ partcl - such as th lctron, or proton - th spn angular-momntum componnts ar gvn b: S, S, S For a spn- partcl such as dutrum, th angular momntum componnts ar matrcs:, S S S In gnral, for a spn m/, th spn oprators wll b (m+)(m+) matrcs Th Sstm's Enrg s Calld Th Hamltonan It Is A atr W know from classcal lctromagntsm that th nrg of a magntc momnt m s an trnal fld B s gvn b E mb It s hghst whn m and B ar ant-paralll and lowst whn th ar paralll Th corrspondng quantum mchancal obsrvabl s calld th Hamlatonan, and s obtand b swappng m out for th quantum mchancal magntaton obsrvabls: H B B B B Substtutng th full forms for th s S and addng up all thr matrcs, w obtan: * B B H B B If th magntc fld s tm dpndnt, so s H Th "Quantum" n Quantum chancs A fundamntal proprt of man quantum sstms s that man of thr obsrvabls (phscal quantts) ar quantd Ths mans that f w tak a pur, wll solatd sstm, and masur that quantt, w wll alwas gt an answr that blongs to a dscrt st of valus Ths st concds wth th possbl gnvalus of th matr W rmnd that radr that () an mm matr has up to m dstnct gnvalus, and () a dagonal matr's gnvalus ar smpl ts ntrs on th dagonal For ampl, th -componnt of th angular momntum for a spn-/ partcl, S has two gnvalus, Ths mans that whn You wll gt to practc gnvalus a bt mor n th tutoral, but to rfrsh our mmor, a vctor v s an gnvalu of a matr s v=(som numbr)v A smpl ampl s th rotaton matr R () about th -as It s clar that,, s an gnvctor wth gnvalu, snc t s colnar wth th -as: R

3 Th Stat Of An Ensmbl s Gvn B A Dnst atr A dnst matr, oftn dnotd, s just a squar matr that follows a fw ruls Frst, t s hrmtan, just lk obsrvabls: It s also postv smdfnt, manng that for an vctor, Fnall, t has a trac of unt: tr Th dnst matr has on fundamntal phscal proprt from whch ts ntr bhavor can b drvd: f th sstm s dscrbd b a dnst matr, thn th avrag valu of a masurd obsrvabl A s gvn b A tr A Our nt ordr of busnss wll b to gnral th stat vctor formulaton to handl statstcal nsmbls of partcls, such as th ~ 5 spn-/ partcls n a ml glass of watr Th Entrs of a Dagonal Dnst atr Rprsnt Probablts In th "convntonal" wa of tachng Q, on frst larns about wavfunctons and probablts It s mntond that a spn-/ partcl can b n thr an "up" or "down" stat, ach havng ts own nrg Usng ths, th followng ntrprtaton s thn gvn to th dnst matr: whn n a dagonal form, th lmnts along th dagonal rprsnts th probablt of th sstm of bng n th up or down stat, rspctvl: Pr Pr Ths s also consstnt wth th proprts of, naml tr()= (probablts sum up to ) and postv-smdfntnss (probablts ar ) Howvr, w wll not b takng ths rout and wll not b usng ths ntrprtaton cpt at on pont down th road, whn w dscuss th thrmal qulbrum stat of th sstm A Smpl Eampl: Spn-/ Partcl Lt's tak an ampl b lookng at a gnral matr for a spn-½ partcl:, whr th dffrnt lmnts can b compl Lt's carfull appl th condtons for to b a dnst matr Hrmtct mans * * * * * * W s that and, manng both quantts ar ral Lt's dnot thm b a d whr a, d ar both ral numbrs Furthrmor, lt's dnot wth b, c ral, such that b c a b c b c d Nt, w mpos th condton tr()=, from whch a+d=, so: a b c bc a Bfor mposng th fnal condton of postvsmdfntnss, lt's calculat th pctaton valu for th, and -componnts of th ntrnsc magntc momnt, S (=,,):

4 a bc tr b bc a a bc tr c bc a a bc tr a bc a Ths ar just nough quatons to solv for th coffcnts a, b & c: from whch or a b, c I S S S I S If H s tm ndpndnt, w can solv th Louvll formall: Ht / Ht / t UU / Th quantt Ut Ht s calld th propagator To s ths solvs th Louvll quaton, not that d U H U and dffrntat: d UU U U U U H H UU UU H t t H H, so t satsfs th Louvll quaton and s a soluton Eampl: Tm Evoluton of a agntc omnt In A Constant agntc Fld Lt's us all of th machnr w'v sn so far to calculat th tm voluton of a spn-/ magntc momnt, startng along th -as, and placd n a constant magntc fld along th -as I S Th Tm Evoluton Of Th Sstm Is Gvn B Th Louvll Equaton uch lk Nwton's nd law, F=ma, dctats th dnamcs of a classcal sstm, Louvll's quaton dctats th dnamcs of a quantum sstm: d, H Hr Ĥ s th hamltonan, and b dfnton, for an two matrcs, A, B AB BA B d at at For rgular functons and numbrs, a Ths dos not chang f w dscuss matrcs; th onl tm w nd to b carful whn handlng matrcs s whn w hav two dffrnt matrcs whch don t commut If w onl hav on matr, w can trat t as a numbr, so: d At At At A A Th last ln s tru bcaus, agan, w r dalng wth  and a functon of Â, whch commut (a matr commuts wth an functon of tslf)

5 To solv ths problm, w wll ask ourslvs four qustons n ths ordr: What s our dnst matr at tm t=? What s our Hamltonan? What s our propagator, U(t)? 4 What s our dnst matr as a functon of tm? (obtand b computng U U ) Our ntal dnst matr s obtand b puttng =(,,) n our prsson for : I S Th constant magntc fld s Our Hamltonan s wth H B B B S B B radkh For a proton 4576 mt n a B = Tsla magntc fld, 7 H Th propagator s as to calculat bcaus th Hamltonan s dagonal For an dagonal matr, A, matr multplcaton wth tslf s vr smpl:!! so for th matr ponntal Â, A A A I A!! A A A! A A! A Addng up th matrcs, w gt A A A!! A A A A!! W s that, along th dagonal, w gt Talor A A pansons of and, so A A A Ths rsult s not tru n gnral For a nondagonal matr, n gnral A A A A A A A A Usng ths, w can mmdatl wrt down th propagator: A A A A A A A N N A A A A A A N U t t/ Ht / t/ Fnall, wth all of ths n plac, w can comput th form of th dnst matr as a functon of tm: W can us ths proprt b pandng th matr ponntal wth a Talor panson I rmnd ou that Talor panson for s

6 t t UU t/ t/ t/ t/ t/ t/ t/ t/ t/ t/ t t Ths looks smpl, but w d lk to also undrstand t b rcastng t n th gnral form t I ts Not that, smpl b nspcton, t t t t I t cos sn t t I cos t sn t I cos t S sn t S From ths w can mmdatl rad off th componnts of th magntc momnt as a functon of tm: cos sn t t t t t W can also wrt ths usng matr notaton to mak t a bt clarr: t t t cos t sn t t sn t cos t LH R t Ths dscrbs a crcular moton of n th plan about th as along whch th trnal constant fld B ponts, whch s th -as Th rotatonal moton s lft handd, manng th sns of drcton s obtand b puttng our lft hand along B, curlng t and notng th drcton n whch our fngrs curl agntc omnts Prcss About An Etrnal Fld Thr was nothng spcal about th drctons chosn n th prvous scton W could go back and solv for a gnral ntal dnst matr, but our phscal ntuton should tll us alrad that t shouldn't mattr how w start out: f B s constant n spac, th magntaton wll just prcss around t (not th lft handd sns of rotaton): W won't go back and actuall solv th tdous Louvll quaton wth an arbtrar fld and ntal condton, although - asd from trml long-wndd algbra - t s qut possbl to do so Snc w know (t), w can also mmdatl wrt down th dnst matr as a functon of tm, usng th formula t I t S A Spn-/ Sstm Can B Undrstood Classcall: Bloch's Equatons Th abov quantum mchancal drvaton has a compltl classcal analogu To s ths, w bgn b provng Ehrnfst's Thorm, whch stats that th tm voluton of th pctaton of valu an B

7 obsrvabl, A tr A accordng to:, volvs n tm d A AH, da Th proof s obtand smpl b applng th dfnton of a drvatv and usng th fact th trac s lnar n th drvatv: d A d tr A d da tr A tr H, da tr A tr HA HA da tr tr tr Now, th frst trm on th last ln can b changd tr AH snc th trac s cclc: to tr ABC tr BCA tr CAB Usng th fact that A tr A for an oprator Â, w can wrt ths as d A AH HA da whch s prcsl quvalnt to th clam mad Wth Ehrnfst's thorm, w can tak ach of th magntaton oprators and wrt down: d, H whr th plct tm drvatv of th oprators s ro snc th ar tm ndpndnt 4 Rmmbr what th Hamltonan of th sstm looks lk: H B Substtutng n Ehrnfst's thorm, w gt: d, B B B, B, B, B,, I lav t to th radr to vrf that,,, from whch (along wth th fact that [A,B]=-[B,A] and [A,A]= for an A,B) follows that d d d B B B B B B Lookng closl at ths thr quatons, w s th can b succnctl wrttn usng vctor notaton as: d B Th sam quaton can b drvd compltl classcall as follows: A mcroscopc (pont-lk) magntc momnt m n a magntc fld B wll b affctd n two was: t wll fl a torqu (whch s th tm drvatv of th angular momntum): dl mb 4 Som obsrvabls can b tm dpndnt, but that s a topc w wll not touch upon n ths cours

8 A nuclus has an ntrnsc angular momntum S and proportonal momnt =S Its drvatv s S d d ds mb Th soluton of th Bloch quatons, whch w wll not pursu hr, prdcts - ou gussd t - that th momnt m wll prcss around an constant magntc fld B Th Intal Stat Of A Sstm s Gvn B Boltmann's Dstrbuton W now know how to solv for th tm voluton of a sstm gvn ts ntal stat But what should that ntal stat b? Wll, that s a quston that should b answrd b ou: what phscal stat s our sstm n ntall? Howvr, on ntal condton prsnts tslf rpatdl: thrmal qulbrum I ll gv ou th answr for how should look lk and thn w ll dscuss ts manng:, Z tr Z H H TE kt kt Lt s assum our fld s along th -as Th Hamltonan s: H B S B Bcaus t s dagonal, w can mmdatl wrt down: H kt kt kt H kt kt kt Z tr W wll b workng at room tmpratur, and can thus smplf, b notng Ths mans th trm n th ponntal s vr small Gong back to our Talor panson, for small w can appromat qut wll, whnc and H kt kt kt kt kt I S kt kt kt kt Z kt kt Smpl! Dvdng th two, TE I S kt I 4kT TE Snc s now n th form I S, w can mmdatl wrt down th -componnt of th magntaton 5 : q 4kT Ths tlls us somthng that s phscall ntutv: whn B s statc and along, th spns algn along B bcaus thr s an nrgtc prfrnc for thm to pont along th fld Howvr, th amount of ths algnmnt s qut small bcaus of th thrmal ffcts that dsprs th magntaton Th & componnts of th magntaton ar ro, whch can also b confrmd b calculatng S 6 8 J kt 4 J 5 W could hav quall drvd th thrmal qulbrum componnts of th magntaton b frst calculatng tr and thn takng th tracs

9 tr, tr, and ths s bcaus thr s no nrgtc prfrnc for th spns to pont prpndcular to th man statc fld Ths calculaton could hav bn rpatd wth an spn and ldd a smlar rsult For a spn S (S=/,, /,, ), th qulbrum magntaton s q SS kt What happns f w hav N spns? If th do not ntract wth ach othr, th total magntc momnt of th nsmbl wll (on phscal grounds) just b multpld b a factor of N: q N BSS kt (t s wrttn n a slghtl dffrnt wa I took and wrot t plctl as B ) Th Idntt Elmnt Is Oftn Omttd From Th Dnst atr Rmmbr that for a sngl spn-/, w saw that th most gnral dnst matr has th form I S In othr words, t looks lk ths: I Now, whatvr our propagator s, UU U IU U U Th part wth th dntt matr stas th sam: U IU IUU I bcaus th propagator s untar (so UU I ) Ths mans t s borng Not onl dos t not chang wth tm for whatvr ntracton w hav! but t also usuall dos not vn contrbut to man obsrvabls For an obsrvabl Â, f I thn 6 tr A tr A A an obsrvabls satsf tr A chck that S S S You r fr to tr tr tr, and that for a spn n an trnal constant magntc fld, tr H as wll Ths s wh man authors smpl omt t W wll do that as wll and smpl not wrt t out On consqunc of ths s that th dnst matr at thrmal qulbrum assums a smpl form: TE S Product Oprators W'v sn that for a sngl spn-/, I S Ths mans w can wrt t as a lnar sum of th form: aiai ai ai whr w'v dfnd Ŝ I, and whr 7 I j S j (j=,,) Ths s lk sang that th st of oprators I, I,, I I forms a bass for th spac of dnst matrcs of spn-/ partcls: vr dnst matr can b wrttn as a lnar sum of ths four oprators Now, S I S S S tr tr tr tr tr 6 Of cours ths s not strctl a dnst matr bcaus ts trac s not on Howvr, ths rasonng holds for an part of that s proportonal to I, lk th I trm 7 I m dvdng b to mak all bass lmnts dmnsonlss (t would b wrd to hav on lmnt of th bass hav no unts, whl anothr hav unts of angular momntum)

10 Th dmand tr mans tr tr nsn n tr S n n n Ths s of cours nothng nw: t mans w gt th I trm Tm Evoluton of Product Oprators Th ntrstng thng about ths formulaton s th wa t maks us thnk about propagaton Gvn a propagator U, an ntal dnst matr volvs as: UU Pluggng n our dnst matr prsson, w gt U a nin U n ai auiu n n So, n ths compltl quvalnt pctur, th coffcnts ar tm ndpndnt and th bass vctors thmslvs volv ovr tm If w know what th propagator dos to ach of th propagators, w can mmdatl wrt th soluton down Ths wa of thnkng wll bcom trml usful whn trng to undrstand basc prmnts down th ln Lt s do an mportant ampl For a spn-/ sstm, w want to calculat UI U n undr th nflunc of a gnral and constant magntc fld B I ll show ou that ou alrad know th answr W v sad that (t) prcsss about a constant magntc fld B: LH t R t n n LH whr Rn s a lft handd rotaton matr about an as dfnd b th unt vctor n, whch ponts along th drcton of th constant fld B Th angular vloct of th prcsson s B Ths mans that th dnst matr changs from at tm t=, to I S I I I R t I Now, anothr wa to rprsnt th dot product s usng matrcs: ab ab ab ab b a a ab T ab b Th transpos T has a nc proprt 8, b whch t swtchs th ordr of th matrcs (just lk for th hrmtan conjugat): T T T AB B A Ths s tru rgardlss of th ss of th matrcs, and th don t vn hav to b squar So, for our multplcaton, R LH t n R LH t n T R LH t n I 8 Ths s asl provd Th transpos of A has lmnts A T j A For th product AB, T j j AB AB A B N j N k k jk k k jk k T I I N T T T T B A B A B A k kj j

11 In othr words, th bass matrcs I,, I I thmslvs prform a prcsson around th magntc fld For ampl, f our fld s along th -as, thn our Hamltonan s and so B B H B I Ht U t t ti U t I U ti ti I I cos t I t sn whr n th last ln w just tratd I as a vctor along th -as and appld a lft handd rotaton to t wth an angl ultpl Spn Sstms ultpl Spn Sstms Ar Dscrbd B Outr (Kronckr) Products B now w'v sn how to dscrb th stat of a sstm and calculat ts tm voluton f w know ts Hamltonan, usng Louvll's quaton (at last n thor, or for th smpl sstm of a sngl spn- /) Ths can b tndd to multpl spn sstms b usng th kronckr product of ths spn spacs Th ruls ar smpl and ar lstd blow for two spn-/ sstms; th can b gnrald n a farl straghtforward mannr: An oprator A of sstm (), a A a t a a s now transformd nto a a AI a a a a a a a a a a whr I s th dntt oprator of sstm two An oprator A of sstm (), a A a s now transformd nto a a a a I A a a a a a a a a a a For ampl, th -componnt of th angular momntum for spn # s S S I Anothr ampl: th -componnt of angular momntum for spn # s: S I S Ths das can b tndd to an numbr of spns, b takng succssv kronckr products For

12 ampl, for thr spn-/s, th -componnt of th nd spn s ths form a bass for all dnst matr of th combnd sstm: S I S I nm, S S nm n m Ths maks sns n trms of numbr of constants: s now a 44 matr havng 6 compl lmnts, or ral numbrs Hrmtct cuts ths down to 6, prcsl th numbr of coffcnts n th panson abov (not th rqurmnt tr()= fs, lavng us wth 5 ral coffcnts) 4 That s trul a frghtnng matr! Down th road, w r gong to com up wth othr was of dalng wth multpl spn sstms that avods mostl dalng wth drct matr multplcaton Product Oprators For ultpl Spns Can ths b tndd to two spn-/ oprators asl Thr ar a total of 6 componnts of angular momntum: S,, I S I S I I S, I S, I S W can also form kronckr products that do not corrspond to an phscal obsrvabl: S S, S S S S S S, S S S S S S, S S S S Along wth th trval dntt oprator, I I, w gt 6 combnatons S n Sm whr, as bfor, n,m=,,, and Ŝ I, S S, S S, S S W won't prov t mathmatcall, but