or the harmonic oscillator s Hamiltonian, ˆ 1 m , (4.2) invite a similar treatment of momentum and coordinate. K. Likharev

Transcription

1 Chper 4. Br-ke Formlism The obecive of his chper is discssion of Dirc s br-ke formlism of qnm mechnics which no onl overcomes some inconveniences of wve mechnics b lso llows nrl descripion of sch inernl properies of pricles s heir spin. In he corse of discssion of he formlism I will give severl simple emples of is se leving more involved pplicions for he following chpers. 4.. Moivion We hve seen h wve mechnics gives mn resls of primr impornce. Moreover i is fll (or mosl) sfficien for mn pplicions for emple for solid se elecronics nd device phsics. However in he corse of or srve we hve filed severl grievnces bo his pproch. Le me briefl smmrie hese complins: (i) Wve mechnics is focsed on he spil dependence of wvefncions. On he oher hnd or emps o nle he emporl evolion of qnm ssems wihin his pproch (beond he rivil ime behvior of he eigenfncions described b Eq. (.6)) rn ino echnicl difficlies. For emple we cold derive Eq. (.59) describing ime dnmics of he mesble se or Eq. (.85) describing qnm oscillions in copled wells onl for he simples poenil profiles hogh i is iniivel cler h hese simple resls shold be common for ll problem of his kind. Deriving he eqions of sch processes for rbirr poenil profiles is possible sing perrbion heories (o be reviewed in Chper 6) b h in he wve mechnics lngge he wold reqire ver blk formls. (ii) The sme is re concerning oher isses h re concepll ddressble wihin wve mechnics e.g. he Fenmn ph inegrl pproch descripion of copling o environmen ec. ddressing hem in wve mechnics wold led o formls so blk h I hd (wisel :-) posponed hem nil we hve go more compc formlism on hnd. (iii) In he discssion of severl ke problems (for emple he hrmonic oscillor nd sphericll-smmeric poenils) we hve rn ino rher compliced eigenfncions coeising wih simple energ specr - h infer some simple bckgrond phsics. I is ver imporn o ge his phsics reveled. (iv) In he wve mechnics posles formled in ec.. qnm mechnicl operors of he coordine nd momenm re reed ver neqll see Eqs. (.6b). However some ke epressions e.g. for he fndmenl eigenfncion of free-pricle p r ep i (4.) or he hrmonic oscillor s Hmilonin m H m invie similr remen of momenm nd coordine. p r (4.) K. Likhrev

2 However he sronges moivion for more generl formlism comes from wve mechnics concepl incpbili o describe elemenr pricles spins nd oher inernl qnm degrees of freedom sch s qrk flvors or lepon nmbers. In his cone le s review he bsic fcs on spin (which is ver represenive nd eperimenll he mos ccessible of ll inernl qnm nmbers) o ndersnd wh more generl formlism shold eplin - s minimm. Figre shows he concepl scheme of he simples spin-reveling eperimen firs crried o b O. ern nd W. Gerlch in 9. collimed bem of elecrons is pssed hrogh gp beween poles of srong mgne where he mgneic field B whose orienion is ken for is in Fig. is non-niform so h boh B nd db /d re no eql o ero. s resl he bem splis ino wo prs of eql inensi. collimor N mgne W = 5% elecron sorce B B W = 5% pricle deecors Fig. 4.. The simples ern- Gerlch eperimen. This simples eperimen cn be semi-qniivel eplined on clssicl hogh somewh phenomenologicl gronds b ssming h ech elecron hs n inrinsic permnen mgneic dipole momen m. Indeed clssicl elecrodnmics ells s h he poenil energ U of mgneic dipole in n eernl mgneic field is eql o (-m B) so h he force cing on he pricle F U m B (4.3) hs nonvnishing vericl componen B F m B m. (4.4) Hence if we frher posle he eisence of wo possible discree vles of m = his eplins he ern-gerlch effec qliivel s resl of he inciden elecrons hving rndom sign b similr mgnide of m. qniive eplnion of he bem spliing ngle reqires he mgnide of o be eql (or close) o he so-clled Bohr mgneon 3 e 3 J B.974. (4.5) m e T s we will see below his vle cnno be eplined b n inernl moion of he elecron s is roion bo is. Bohr mgneon To m knowledge he concep of spin s n inernl roion of pricle ws firs sggesed b R. Kronig hen -er-old sden in Jnr 95 few monhs before wo oher sdens G. Uhlenbeck nd. Godsmi - o whom he ide is sll ribed. The concep ws hen cceped nd developed qniivel b W. Pli. ee e.g. EM ec. 5.4 in priclr Eq. (5.). 3 convenien mnemonic rle is h i is close o K/T. In he Gssin nis B e/m e c Chper 4 Pge of 4

3 Mch more impornl his semi-clssicl lngge cnno eplin he resls of he following se of mli-sge ern-gerlch eperimens shown in Fig. - even qliivel. In he firs of he eperimens he elecron bem is firs pssed hrogh mgneic field oriened (ogeher wih is grdien) long is s s in Fig.. Then one of he wo resling bems is bsorbed (or oherwise removed from he sep) while he oher one is pssed hrogh similr b -oriened field. The eperimen shows h his bem is spli gin ino wo componens of eql inensi. clssicl eplnion of his eperimen wold reqire ver nnrl sggesion h he iniil elecrons hd rndom b discree componens of he mgneic momen simlneosl in wo direcions nd. However even his ssmpion cnno eplin he resls of he hree-sge ern-gerlch eperimen shown on he middle pnel of Fig.. Here he previos wo-se sep is complemened wih one more bsorber nd one more mgne now wih he -orienion gin. Compleel coneriniivel i gin gives wo bems of eql inensi s if we hve no e filered o he elecrons wih m corresponding o he lower bem in he firs -sge. G () % G () 5% 5% bsorber G () G () % G () G () % % % G () 5% 5% Fig. 4.. Three mli-sge ern-gerlch eperimens. Boes G ( ) denoe mgnes similr o one shown in Fig. wih he is oriened in he indiced direcion. The onl w o sve he clssicl eplnion here is o s h mbe elecrons somehow inerc wih he mgneic field so h he -polried (non-bsorbed) bem becomes sponneosl depolried gin somewhere beween mgneic sges. B n hope for sch eplnion is rined b he conrol eperimen shown on he boom pnel of Fig. whose resls indice h no sch depolriion hppens. We will see below h ll hese (nd mn more) resls find nrl eplnion in he mri mechnics pioneered b W. Heisenberg M. Born nd P. Jordn in 95. However he mri formlism is inconvenien for he solion of mos problems discssed in Chpers -3 nd for ime i ws eclipsed b chrödinger s wve mechnics which hd been p forwrd s few monhs ler. However ver soon P.. M. Dirc inrodced more generl br-ke formlism which provides generliion of boh pproches nd proves heir eqivlence. Le me describe i. Chper 4 Pge 3 of 4

4 4.. es se vecors nd liner operors The bsic noion of he generl formlion of qnm mechnics is he qnm se of ssem. 4 To ge some g feeling of his noion if qnm se of pricle m be deqel described b wve mechnics his descripion is given b he corresponding wvefncion (r ). Noe however he se s sch is no mhemicl obec (sch s fncion) 5 nd cn pricipe in mhemicl formls onl s poiner e.g. he inde of fncion. On he oher hnd he wvefncion is no se b mhemicl obec ( comple fncion of spce nd ime) giving qniive descripion of he se - s s he rdis-vecor s fncion of ime is mhemicl obec describing he moion of clssicl pricle see Fig. 3. imilrl in he Dirc formlism cerin qnm se is described b eiher of wo mhemicl obecs clled he se vecors: he ke-vecor nd br-vecor. 6 One shold be cions wih he erm vecor here. Usl geomeric vecors re defined in he sl geomeric (s Ecliden) spce. In conrs br- nd ke-vecors re defined in bsrc Hilber spces of given ssem 7 nd despie cerin similriies wih he geomeric vecors re new mhemicl obecs so h we need new rles for hndling hem. The primr rles re essenill posles nd re sified onl he correc descripion/predicion of ll eperimenl observions heir corollries. While hese is generl consenss mong phsiciss wh he corollries re here re mn possible ws o crve from hem he bsic posle ses. Js s in ec.. I will no r oo hrd o be he nmber of he posles o he smlles possible minimm ring insed o keep heir phsicl mening rnspren. pricle in se mhemicl descripion clssicl mechnics : r( ) wve mechnics :eiher br - ke formlism :eiher ( r ) or Ψ or α ( r ) Fig Pricle s se nd is descripions. (i) Ke-vecors. Le s sr wih ke-vecors - someimes clled s kes for shor. Perhps he mos imporn proper of he vecors concerns heir liner sperposiion. Nmel if severl kevecors describe possible ses of qnm ssem hen n liner combinion (sperposiion) c (4.6) Liner sperposiion of ke-vecors 4 n enive reder cold noice m smggling erm ssem insed of pricle which ws sed in he previos chpers. Indeed he br-ke formlism llows he descripion of qnm ssems mch more comple hn single spinless pricle h is picl (hogh no he onl possible) sbec of wve mechnics. 5 s ws epressed nicel b. Peres one of pioneers of he qnm informion heor qnm phenomen do no occr in he Hilber spce he occr in lboror. 6 Terms br nd ke were sggesed o reflec he fc h pir nd m be considered s he se of prs of combinion (see Eq. () below) which reminds n epression in he sl ngle brckes. 7 The Hilber spce of given ssem is defined s he se of ll is possible se vecors. s shold be cler from his definiion i is no dvisble o spek bo Hilber spce of qnm ses. Chper 4 Pge 4 of 4

5 where c re n (possibl comple) c-nmbers lso describes possible se of he sme ssem. (One m s h vecor belongs o he sme Hilber spce s ll.) cll since ke-vecors re new mhemicl obecs he ec mening of he righ-hnd pr of Eq. (6) becomes cler onl fer we hve posled he following rles of smmion of hese vecors (4.7) ' ' nd heir mliplicion b c-nmbers: c c. (4.8) Noe h in he se of wve mechnics posles semens prllel o (7) nd (8) were nnecessr becse wvefncions re he sl (lbei comple) fncions of spce nd ime nd we know from he sl lgebr h sch relions re vlid. s eviden from Eq. (6) he comple coefficien c m be inerpreed s he weigh of se in he liner sperposiion. One imporn priclr cse is c = showing h se does no pricipe in he sperposiion. B he w he corresponding erm of sm (6) i.e. prodc Nll-se vecor (4.9) Liner sperposiion of br-vecors Inner br-ke prodc hs specil nme: he nll-se vecor. (I is imporn o void confsion beween he nll-se corresponding o vecor (9) nd he grond se of he ssem which is freqenl denoed b kevecor. In some sense he nll-se does no eis ll while he grond se does nd freqenl is he mos imporn qnm se of he ssem.) (ii) Br-vecors nd inner ( sclr ) prodcs. Br-vecors which obe he rles similr o Eqs. (7) nd (8) re no new independen obecs: if ke-vecor is known he corresponding brvecor describes he sme se. In oher words here is niqe dl correspondence beween nd 8 ver similr (hogh no idenicl) o h beween wvefncion nd is comple conge. The correspondence beween hese vecors is described b he following rle: if ke-vecor of liner sperposiion is described b Eq. (6) hen he corresponding br-vecor is c c. (4.) The mhemicl convenience of sing wo pes of vecors rher hn s one becomes cler from he noion of heir inner prodc (lso clled he shor brcke):. (4.) This is (generll comple) 9 sclr whose min proper is he lineri wih respec o n of is componen vecors. For emple if liner sperposiion is described b he ke-vecor (6) hen 8 Mhemicins like o s h he ke- nd br-vecors of he sme qnm ssem re defined in wo isomorphic Hilber spces. 9 This is one of he differences of br- nd ke-vecors from he sl (geomericl) vecors whose sclr prodc is lws rel sclr. Chper 4 Pge 5 of 4

6 while if Eq. () is re hen c (4.) c. (4.3) In plin English c-nmbers m be moved eiher ino or o of he inner prodcs. The second ke proper of he inner prodc is. (4.4) I is compible wih Eq. (); indeed he comple congion of boh prs of Eq. () gives: c c. (4.5) Finll one more rle: he inner prodc of he br- nd ke-vecors describing he sme se (clled he norm sqred) is rel nd non-negive. (4.6) In order o give he reder some feeling bo he mening of his rle: we will show below h if se m be described b wvefncion (r ) hen Inner prodc s comple conge e s norm sqred 3 d r. (4.7) Hence he role of he br-ke is ver similr o he comple congion of he wvefncion nd Eq. () emphsies his similri. (Noe h b convenion here is no congion sign in he br-pr of he inner prodc; is role is pled b he nglr brcke inversion.) (iii) Operors. One more ke noion of he Dirc formlism re qnm-mechnicl liner operors. Js s for he operors discssed in wve mechnics he fncion of n operor is he generion of one se from noher: if is possible ke of he ssem nd  is legiime operor hen he following combinion  (4.8) is lso ke-vecor describing possible se of he ssem i.e. ke-vecor in he sme Hilber spce s he iniil vecor. s follows from he decive liner he min rles governing he operors is heir lineri wih respec o boh n sperposiion of vecors: nd n sperposiion of operors: c c (4.9) c. (4.) c Chper 4 Pge 6 of 4

7 Hermiin conge operor Hermiin operor s definiion Long brcke Long brcke s comple conge These rles re evidenl similr o Eqs. (.53)-(.54) of wve mechnics. The bove rles impl h n operor cs on he ke-vecor on is righ; however combinion of he pe  is lso legiime nd presens new br-vecor. I is imporn h generll his vecor does no represen he sme se s ke-vecor (8); insed he br-vecor isomorphic o ke-vecor (8) is Â. (4.) This semen serves s he definiion of he Hermiin conge (or Hermiin doin )  of he iniil operor Â. For n imporn clss of operors clled he Hermiin operors he congion is inconseqenil i.e. for hem. (4.) (This eqli s well s n oher operor eqion below mens h hese operors c similrl on n br- or ke-vecor.) To proceed frher we need n ddiionl posle clled he ssociive iom of mliplicion: ino n legiime br-ke epression no inclding n eplici smmion we m inser or remove prenheses (s in he ordinr prodc of sclrs) mening s sl h he operion inside he prenheses is performed firs. The firs wo emples of his posle re given b Eqs. (9) nd () b he ssociive iom is more generl nd ss for emple: (4.3) This eqli serves s he definiion of he ls form clled he long brcke (evidenl lso sclr) wih n operor sndwiched beween br-vecor nd ke-vecor. This definiion when combined wih he definiion of he Hermiin conge nd Eq. (4) ields n imporn corollr: (4.4) which is mos freqenl rewrien s. (4.5) The ssociive iom lso enbles o redil eplore he following definiion of one more oer prodc of br- nd ke-vecors: If we consider c-nmbers s priclr pe of operors hen ccording o Eqs. () nd () for hem he Hermiin congion is eqivlen o he simple comple congion so h onl rel c-nmber m be considered s priclr cse of he Hermiin operor (). Here legiime mens hving cler sense in he br-ke formlism. ome emples of illegiime epressions:. Noe however h he ls wo epressions m be legiime if nd re ses of differen ssems i.e. if heir se vecors belong o differen Hilber spces. We will rn ino sch ensor prodcs of br- nd ke vecors (someimes denoed respecivel s nd ) in Chpers 6-8. Chper 4 Pge 7 of 4

8 . (4.6) In conrs o he inner prodc () which is sclr his mhemicl consrc is n operor. Indeed he ssociive iom llows s o remove prenheses in he following epression:. (4.7) B he ls br-ke is s sclr; hence he mhemicl obec (6) cing on ke-vecor (in his cse ) gives new ke-vecor which is he essence of operor s cion. Ver similrl (4.8) - gin picl operor s cion on br-vecor. Now le s perform he following clclion. We m se he prenheses inserion ino he brke eqli following from Eq. (4) o rnsform i o he following form: (4.9). (4.3) ince his eqion shold be vlid for n vecors nd is comprison wih Eq. (5) gives he following operor eqli. (4.3) This is he conge rle for oer prodcs; i reminds rle (4) for inner prodcs b involves he Hermiin (rher hn he sl comple) congion. The ssociive iom is lso vlid for he operor mliplicion : B B B B (4.3) showing h he cion of n operor prodc on se vecor is nohing more hn he seqenil cion of he opernds. However we hve o be rher crefl wih he operor prodcs; generll he do no comme: B B. This is wh he commor he operor defined s B B B is ver sefl opion. noher similr noion is he nicommor: (4.33) B B B. (4.34) Finll he br-ke formlism brodl ses wo specil operors: he nll operor defined b he following relions: Oer br-ke prodc Oer prodc s Hermiin conge Commor nicommor B noher poplr noion for he nicommor is ; i will no be sed in hese noes. Chper 4 Pge 8 of 4

9 Nll operor (4.35) Ideni operor for n rbirr se ; we m s h he nll operor kills n se rning i ino he nll-se. noher elemenr operor is he ideni operor which is lso defined b is cion (or rher incion :-) on n rbirr se vecor: I I. (4.36) Epnsion over bsis vecors Bsis vecors' orhonormli Epnsion coefficiens s inner prodcs 4.3. e bsis nd mri represenion While some operions in qnm mechnics m be crried o in he generl br-ke formlism olined bove mos clclions re done for specific qnm ssems h fere les one fll nd orhonorml se {} of ses freqenl clled bsis. These erms men h n se vecor of he ssem m be represened s niqe sm of he pe (6) or () over is bsis vecors: (4.37) (so h in priclr if is one of he bsis ses s hen = ) nd h ' '. (4.38) For he ssems h m be described b wve mechnics emples of he fll orhonorml bses re represened b n orhonorml se of eigenfncions clcled in he previos 3 chpers s he simples emple see Eq. (.76). De o he niqeness of epnsion (37) he fll se of coefficiens gives complee descripion of se (in fied bsis {}) s s he sl Cresin componens nd give complee descripion of sl geomeric 3D vecor (in fied reference frme). ill le me emphsie some differences beween he qnm-mechnicl br- nd ke-vecors nd he sl geomeric vecors: (i) bsis se m hve lrge or even infinie nmber of ses nd (ii) he epnsion coefficiens m be comple. Wih hese reservions in mind he nlog wih geomeric vecors m be pshed even frher. Le s inner-mlipl boh prs of he firs of Eqs. (37) b br-vecor nd hen rnsform he relion sing he lineri rles discssed in he previos secion nd Eq. (38): (4.39) ' ' Togeher wih Eq. (4) his mens h n of he epnsion coefficiens in Eq. (37) m be presened s n inner prodc: ' ' ; (4.4) hese relions re nlogs of eqliies = n of he sl vecor lgebr. Using hese imporn relions (which we will se on nmeros occsions) epnsions (37) m be rewrien s Chper 4 Pge 9 of 4

10 (4.4) comprison of hese relions wih Eq. (6) shows h he oer prodc defined s (4.4) Proecion operor is legiime liner operor. ch n operor cing on n se vecor of he pe (37) singles o s one of is componens for emple (4.43) i.e. kills ll componens of he liner sperposiion b one. In he geomeric nlog sch operor proecs he se vecor on is ( h ) direcion hence is nme he proecion operor. Probbl he mos imporn proper of he proecion operors clled he closre (or compleeness) relion immediel follows from Eq. (4): heir sm over he fll bsis is eqivlen o he ideni operor: I. (4.44) This mens in priclr h we m inser he lef-hnd pr of Eq. (44) ino n br-ke relion n plce he rick h we will se gin nd gin. Le s see how epnsions (37) rnsform ll he noions inrodced in he ls secion sring from he shor br-ke () (he inner prodc of wo se vecors):. (4.45) ' ' ' Besides he comple congion his epression is similr o he sclr prodc of he sl vecors. Now le s eplore he long br-ke (3): ' ' ' ' ' '. (4.46) Here he ls sep ses ver imporn noion of mri elemens of he operor defined s ' ' ' ' '. (4.47) s eviden from Eq. (46) he fll se of he mri elemens compleel chrceries he operor s s he fll se of epnsion coefficiens (4) fll chrceries qnm se. The erm mri mens firs of ll h i is convenien o presen he fll se of s sqre ble (mri) wih he liner dimension eql o he nmber of bsis ses of he ssem nder he considerion i.e. he sie of is Hilber spce. s wo simples emples ll mri elemens of he nll-operor defined b Eqs. (35) re evidenl eql o ero (in n bsis) nd hence i m be presened s mri of eros (he nll mri): (4.48) Closre relion Operor s mri elemens Nll mri Chper 4 Pge of 4

11 Ideni mri Mri elemen of n operor prodc Long brcke s mri prodc hor brcke s mri prodc while for he ideni operor Î defined b Eqs. (36) we redil ge i.e. is mri (clled he ideni mri) is digonl lso in n bsis: I ' I (4.49) ' ' '... I.... (4.5) The convenience of he mri lngge eends well beond he presenion of priclr operors. For emple le s se definiion (47) o clcle mri elemens for prodc of wo operors: B B. (4.5) ( ) " " Here we cn se Eq. (44) for he firs (b no he ls!) ime insering he ideni operor beween he wo operors nd hen epressing i vi sm of proecion operors: ( B) " B " IB " ' ' B " ' ' ' B '". (4.5) This resl corresponds o he sndrd row b colmn rle of clclion of n rbirr elemen of he mri prodc B... B B B B (4.53)... Hence he prodc of operors m be presened (in fied bsis!) b h of heir mrices (in he sme bsis). This is so convenien h he sme lngge is ofen sed o presen no onl he long brcke b even he simpler shor brcke:... ' ' (4.54) ' (4.55)... lhogh hese eqliies reqire he se of non-sqre mrices: rows of (comple-conge!) epnsion coefficiens for he presenion of br-vecors nd colmns of hese coefficiens for he presenion of ke-vecors. Wih h he mpping of ses nd operors on mrices becomes compleel generl. Now le s hve look he oer prodc operor (6). Is mri elemens re s Chper 4 Pge of 4

12 '. (4.56) These re elemens of ver specil sqre mri whose filling reqires he knowledge of s N sclrs (where N is he bsis se sie) rher hn N sclrs s for n rbirr operor. However simple generliion of sch oer prodc m presen n rbirr operor. Indeed le s inser wo ideni operors (44) wih differen smmion indices on boh sides of n operor: II ' ' ' (4.57) nd se he ssociive iom o rewrie his epression s ' '. (4.58) ' B he epression in he middle long brcke is s he mri elemen (47) so h we m wrie ' ' ' '. (4.59) ' The reder hs o gree h his forml which is nrl generliion of Eq. (44) is eremel elegn. lso noe he following prllel: if we consider he mri elemen definiion (47) s some sor of nlog of Eq. (4) hen Eq. (59) is similr nlog of he epnsion epressed b Eq. (37). The mri presenion is so convenien h i mkes sense o move i b one level lower from se vecor prodcs o bre se vecors resling from operor s cion pon given se. For emple le s se Eq. (59) o presen he ke-vecor (8) s ' ' ' ' ' ' '. (4.6) ccording o Eq. (4) he ls shor brcke is s so h ' ' ' ' ' (4.6) ' ' B epression in middle prenheses is s he coefficien of epnsion (37) of he resling kevecor (6) in he sme bsis so h ' ' '. (4.6) ' This resl corresponds o he sl rle of mliplicion of mri b colmn so h we m represen n ke-vecor b is colmn mri wih he operor cion looking like ' ' (4.63) bsolel similrl he operor cion on he br-vecor () represened b is row-mri is Operor s epression vi is mri elemens Chper 4 Pge of 4

13 Hermiin conge s mri elemens Operor s eigenses nd eigenvles Hermiin operor s eigenvles (4.64) ' ' B he w Eq. (64) nrll rises he following qesion: wh re he elemens of he mri in is righ-hnd pr or more ecl wh is he relion beween he mri elemens of n operor nd is Hermiin conge? The simples w o ge n nswer is o se Eq. (5) wih wo rbirr ses (s nd ) of he sme bsis in he role of nd. Togeher wih he orhonormli relion (38) his immediel gives 3 ' '. (4.65) Ths he mri of he Hermiin conge operor is he comple conged nd rnsposed mri of he iniil operor. This resl eposes ver clerl he essence of he Hermiin congion. I lso shows h for he Hermiin operors defined b Eq. () (4.66) ' ' i.e. n pir of heir mri elemens smmeric bo he min digonl shold be comple conge of ech oher. s corollr he min-digonl elemens hve o be rel: i.e. Im. (4.67) (Mri (5) evidenl sisfies Eq. (66) so h he ideni operor is Hermiin.) In order o fll pprecie he specil role pled b Hermiin operors in he qnm heor le s inrodce he ke noions of eigenses (described b heir eigenvecors nd ) nd eigenvles (c-nmbers) of n operor  defined b he eqion he hve o sisf: 4 Le s prove h eigenvles of n Hermiin operor re rel 5. (4.68) for... N (4.69) 3 For he ske of forml compcness below I will se he shorhnd noion in which he opernds of his eqli re s nd. I believe h i leves lile chnce for confsion becse he Hermiin congion sign m perin onl o n operor (or is mri) while he comple congion sign o sclr s mri elemen. 4 This eqion shold look fmilir o he reder see he sionr chrödinger eqion (.6) which ws he focs of or sdies in he firs hree chpers. We will see soon h h eqion is s priclr (coordine) represenion of Eq. (66) for he Hmilonin s he operor of energ. 5 The reciprocl semen is lso re: if ll eigenvles of n operor re rel i is Hermiin (in n bsis). This semen m be redil proved b ppling Eq. (93) below o he cse when kk = k kk wih k = k. Chper 4 Pge 3 of 4

14 while he eigenses corresponding o differen eigenvles re orhogonl: ' if. (4.7) ' The proof of boh semens is srprisingl simple. Le s inner-mlipl boh sides of Eq. (68) b br-vecor. In he righ-hnd pr of he resl he eigenvle s c-nmber m be ken o of he br-ke giving '. (4.7) This eqli shold hold for n pir of eigenses so h we m swp he indices in Eq. (7) nd comple-conge he resl: ' ' ' '. (4.7) Now sing Eqs. (4) nd (5) ogeher wih he Hermiin operor definiion () we m rnsform Eq. (7) o he following form: Hermiin operor s eigenvecors '. (4.73) ' ' brcing his eqion from Eq. (7) we ge ' '. (4.74) There re wo possibiliies o sisf his eqion. If indices nd re eql (denoe he sme eigense) hen he br-ke is he se s norm sqred nd cnno be eql o ero. Then he lef prenheses (wih = ) hve o be ero i.e. Eq. (69) is vlid. On he oher hnd if nd correspond o differen eigenvles of  he prenheses cnno eql ero (we hve s proved h ll re rel!) nd hence he se vecors indeed b nd shold be orhogonl e.g. Eq. (7) is vlid. s will be discssed below hese properies mke Hermiin operors sible for he descripion of phsicl observbles Chnge of bsis nd mri digonliion From he discssion of ls secion i m look h he mri lngge is fll similr o nd in mn insnces more convenien hn he generl br-ke formlism. In priclr Eqs. (5) (54) (55) show h n pr of n br-ke epression m be direcl mpped on he similr mri epression wih he onl sligh inconvenience of sing no onl colmns b lso rows (wih heir elemens comple-conged) for se vecor presenion. In his cone wh do we need he br-ke lngge ll? The nswer is h he elemens of he mrices depend on he priclr choice of he bsis se ver mch like he Cresin componens of sl vecor depend on he priclr choice of reference frme orienion (Fig. 4) nd ver freqenl i is convenien o se wo or more differen bsis ses for he sme ssem. Wih his moivion le s sd wh hppens if we chnge from one bsis {} o noher one {v} - boh fll nd orhonorml. Firs of ll le s prove h for ech sch pir of bses here eiss sch n operor U h firs Chper 4 Pge 4 of 4

15 Bsis rnsform Unir operor s definiion v U (4.75) nd second U U U U I. (4.76) (De o he ls proper 6 U is clled nir operor nd Eq. (75) nir rnsformion.) ' α ' ' ' Fig Trnsformion of componens of D vecor reference frme roion. Unir operor of bsis rnsform Conge nir rnsform operor ver simple proof of boh semens m be chieved b consrcion. Indeed le s ke - n eviden generliion of Eq. (44). Then ' U v ' (4.77) ' ' ' ' ' U v v v (4.78) so h Eq. (75) hs been proved. Now ppling Eq. (3) o ech erm of sm (77) we ge so h ' ' ' ' ' U ' v (4.79) U U v v v v v v. (4.8) ' ' B ccording o he closre relion (44) he ls epression is s he ideni operor q.e.d. 7 (The proof of he second eqli in Eq. (76) is bsolel similr.) s b-prodc of or proof we hve lso go noher imporn epression (79). I implies in ' ' ' Reciprocl bsis rnsform priclr h while ccording o Eq. (77) operor U performs he rnsform from he old bsis o he new bsis v is Hermiin doin U performs he reciprocl nir rnsform: U v. (4.8) ' ' ' 6 n lernive w o epress Eq. (76) is o wrie U U b I will r o void his lngge. 7 Qod er demonsrndm (L.) wh needed o be proved. Chper 4 Pge 5 of 4

16 Now le s see how do he mri elemens of he nir rnsform operors look like. Generll s ws sed bove operor s elemens depend on he bsis we clcle hem in so we shold be crefl - iniill. For emple le s clcle he elemens in bsis {}: U ' in U ' vk k ' v '. (4.8) k Now performing similr clclion in bsis {v} we ge U ' in v v U v ' v vk k v ' v '. (4.83) k rprisingl he resl is he sme! This is of corse re for he Hermiin conge of he nir rnsform operor s well: U ' in U v. (4.84) ' in v These epressions m be sed firs of ll o rewrie Eq. (75) in more direc form. ppling he firs of Eqs. (4) o se v of he new bsis we ge imilrl he reciprocl rnsform is v v ' ' ' U '. (4.85) v v ' ' U ' v. (4.86) These eqions re ver convenien for pplicions; we will se hem lred ler in his secion. Ne we m se Eqs. (83) (84) o epress he effec of he nir rnsform on epnsion coefficiens (37) of vecors of n rbirr se. In he old bsis {} he re given b Eq. (4). imilrl in he new bsis {v} in v v. (4.87) gin insering he ideni operor in he form of closre (44) wih inernl inde nd hen sing Eq. (84) we ge in v v ' ' v ' ' U ' ' U ' ' ' ' ' ' The reciprocl rnsform is (of corse) performed b mri elemens of operor U : in U ' ' in v ' in. (4.88). (4.89) Boh srcrll nd philosophicll hese epressions re similr o he rnsformion of componens of sl vecor coordine frme roion. For emple in wo dimensions (Fig. 4): Bsis rnsforms: mri form Chper 4 Pge 6 of 4

17 Mri elemens rnsforms Operor/ mri rce ' cos sin. sin cos (4.9) ' (In his nlog he eqli o of he deerminn of he roion mri in Eq. (9) corresponds o he nir proper (76) of he nir rnsform operors.) Plese p enion here: while he rnsform (75) from he old bsis {} o he new bsis {v} is performed b he nir operor he chnge (88) of se vecors componens his rnsformion reqires is Hermiin conge. cll his is lso nrl from he poin of view of he geomeric nlog of he nir rnsform (Fig. 4): if he new reference frme { } is obined b conerclockwise roion of he old frme { } b some ngle for he observer roing wih he frme vecor (which is iself nchnged) roes clockwise. De o he nlog beween epressions (88) nd (89) on one hnd nd or old friend Eq. (6) on he oher hnd i is emping o skip indices in or new resls b wriing in v U U. (4.9) in ince mri elemens of U nd U do no depend on bsis sch lngge is no oo bd; sill he smbolic Eq. (9) shold no be confsed wih genine (bsis-independen) br-ke eqliies. Now le s se he sme rick of ideni operor inserion repeed wice o find he rnsformion rle for mri elemens of n rbirr operor: ' v v v ' v k k in k' k' v ' U k kk' in U ; (4.9) k'' k k' k k' bsolel similrl we cn ge ' in U k kk' in vu k''. (4.93) k k' In he spiri of Eq. (9) we m presen hese resls smbolicll s well in compc br-ke form: in U U U U. (4.94) in v s sni check le s ppl his resl o he ideni operor: I in v in in in in v in v U IU U U I in (4.95) in - s i shold be. One more invrin of he bsis chnge is he rce of n operor defined s he sm of he digonl erms of is mri in cerin bsis: Tr Tr. (4.96) The (es) proof of his fc sing he relions we hve lred discssed is lef for reder s eercise. o fr I hve implied h boh se bses {} nd {v} re known nd he nrl qesion is where does his informion comes from in qnm mechnics of cl phsicl ssems. To ge pril nswer o his qesion le s rern o Eq. (68) h defines eigenses nd eigenvles of n Chper 4 Pge 7 of 4

18 operor. Le s ssme h he eigenses of cerin operor  form fll nd orhonorml se nd find he mri elemens of he operor in he bsis of hese ses. For h i is sfficien o innermlipl boh sides of Eq. (68) wrien for inde b he br-vecor of n rbirr se of he sme se:. (4.97) ' ' ' The lef-hnd pr is s he mri elemen we re looking for while he righ hnd pr is s. s resl we see h he mri is digonl wih he digonl consising of eigenvles:. (4.98) ' In priclr in he eigense bsis (b no necessril in n rbirr bsis!) mens he sme s. Ths he mos imporn problem of finding he eigenvles nd eigenses of n operor is eqivlen o he digonliion of is mri 8 i.e. finding he bsis in which he corresponding operor cqires he digonl form (98); hen he digonl elemens re he eigenvles nd he bsis iself is he desirble se of eigenses. Le s modif he bove clclion b inner-mlipling Eq. (68) b br-vecor of differen bsis s he one denoed {} in which we know he mri elemens. The mliplicion gives k k '. (4.99) In he lef-hnd pr we cn (s sl :-) inser he ideni operor beween he operor nd he kevecor nd hen se he closre relion (44) while in he righ-hnd pr we cn move he eigenvle o of he br-ke nd hen inser smmion over new inde compensing i wih he proper Kronecker del smbol: k k' k' k' k '. (4.) Moving o he sign of smmion over k nd sing definiion (47) of he mri elemens we ge k ' kk ' kk ' k ' k ' kk'. (4.) B he se of sch eqliies for ll N possible vles of inde k is s ssem of liner homogeneos eqions for nknown c-nmbers k. B ccording o Eqs. (8)-(84) hese nmbers re nohing else hn he mri elemens U k of nir mri providing he reqired rnsformion from he iniil bsis {} o he bsis {} h digonlies mri. The ssem m be presened in he mri form: U... U (4.) Mri elemens in eigense bsis Operor digonliion 8 Noe h epression mri digonliion is common nd convenien b dngeros rgon. ( mri is s mri n ordered se of c-nmbers nd cnno be digonlied.) I is OK o se his rgon if o remember clerl wh i cll mens see he definiion bove. Chper 4 Pge 8 of 4

19 nd he sl condiion of is consisenc Chrcerisic eqion for finding eigenvles (4.3) Pli mrices pls he role of he chrcerisic eqion of he ssem. This eqion hs N roos ; plgging ech of hem bck ino ssem () we cn se i o find N mri elemens U k (k = N) corresponding o his priclr eigenvle. However since eqions (3) re homogeneos he llow finding U k onl o consn mliplier. In order o ensre heir normliion i.e. he nir chrcer of mri U we m se he condiion h ll eigenvecors re normlied (s s he bsis vecors re): U (4.4) k k for ech. This normliion complees he digonliion. 9 Now ( ls!) I cn give he reder some emples. s simple b ver imporn cse le s digonlie he operors described (in cerin -fncion bsis {}) b he so-clled Pli mrices i σ σ σ. (4.5) i Thogh inrodced b phsicis wih specific prpose o describe elecron s spin hese mrices hve generl mhemicl significnce becse ogeher wih he ideni mri I he provide fll linerl-independen bsis - mening h n rbirr mri m be presened s I σ σ σ (4.6) wih niqe se of 4 coefficiens. Le s sr wih digonliing mri. For i he chrcerisic eqion (3) is evidenl k k k (4.7) nd hs wo roos = ±. (gin he nmbering is rbirr!) The reder m redil check h he eigenvles of mrices nd re similr. However he eigenvecors of he operors corresponding o ll hese mrices re differen. To find hem for le s plg is firs eigenvle = + bck ino eqions () wrien for his priclr cse:. (4.8) 9 possible sligh complicion here re degenere cses when chrcerisic eqion gives cerin eql eigenvles corresponding o differen eigenvecors. In his cse he reqiremen of he ml orhogonli of hese ses shold be ddiionll enforced. Chper 4 Pge 9 of 4

20 The eqions re compible (of corse becse he sed eigenvle = + sisfies he chrcerisic eqion) nd n of hem gives i. e. U. (4.9) U Wih h he normliion condiion (4) ields U U. (4.) lhogh he normliion is insensiive o he simlneos mliplicion of U nd U b he sme phse fcor ep{i} wih n rel i is convenien o keep he coefficiens rel for emple king = i.e. o ge U U. (4.) Performing n bsolel similr clclion for he second chrcerisic vle = - we ge U = -U nd we m choose he common phse o ge U U (4.) so h he whole nir mri for digonliion of he operor corresponding o is U U (4.3) For wh follows i will be convenien o hve his resl epressed in he ke-relion form see Eqs. (85)-(86): U U U U (4.4) U U U U (4.5) These resls re lred sfficien o ndersnd he ern-gerlch eperimens described in ec. - wih wo ddiionl posles. The firs of hem is h pricle s inercion wih eernl mgneic field m be described b he following vecor operor of he dipole mgneic momen: m (4.6) where he coefficien specific for ever pricle pe is clled he gromgneic rio nd Ŝ is he vecor operor of spin. For he so-clled spin-½ pricles (inclding he elecron) his operor m be represened in he so-clled -bsis b he following 3D vecor of he Pli mrices (5): Unir mri digonliing Mgneic momen operor Noe h hogh his priclr nir mri is Hermiin his is no re for n rbirr choice of phses. This is he ke poin in he elecron s spin descripion developed b W. Pli in For n elecron wih is negive chrge q = -e he gromgneic rio is negive: e = -g e e/m e where g e is he dimensionless g-fcor. De o qnm elecrodnmics effecs he fcor is slighl higher hn : g e = ( + / + ).3934 where e /4 c E H /m e c /37 is he fine srcre consn. (The origin of is nme will be cler from he discssion in ec. 6.3.) Chper 4 Pge of 4

21 pin-½ mri Qnm mesremen pole n n in n in n n σ n σ n σ (4.7) nd n re he sl Cresin ni vecors in 3D spce. (In he qnm-mechnics sense he re s c-nmbers or rher c-vecors.) The -bsis in which Eq. (77) is vlid is defined s n orhonorml bsis of wo ses freqenl denoed n in which he -componen of he vecor operor of spin is digonl wih eigenvles +/ nd -/. Noe h we do no ndersnd wh ecl hese ses re 3 b loosel ssocie hem wih cerin inernl roion of he elecron bo -is wih eiher posiive or negive nglr momenm componen. However n emp o se sch clssicl inerpreion for qniive predicions rns ino fndmenl difficlies see ec. 5.7 below. The second new posle describes he generl relion beween he br-ke formlism nd eperimen. 4 Nmel in qnm mechnics ech rel observble is represened b Hermiin operor nd resl of is mesremen in qnm se described b liner sperposiion of he eigenses of he operor wih (4.8) m be onl one of corresponding eigenvles. 5 If se (8) nd ll eigenses re normlied o ni hen he probbili of ocome is 6 W (4.9) (4.) This relion is evidenl generliion of Eq. (.) in wve mechnics. s sni check le s ssme h he se of eigenses is fll nd clcle he sm of ll he probbiliies: W I. (4.) Now rerning o he ern-gerlch eperimen concepll he descripion of he firs (oriened) eperimen shown in Fig. is he hrdes for s becse he sisicl ensemble describing he npolried elecron bem is inp is mied ( incoheren ) nd cnno be described b pre 3 If o hink bo i word ndersnd picll mens h we cn eplin new more comple noion in erms of hose discssed erlier nd considered known. In or emple we cnno epress he spin ses b some wvefncion (r) or n oher mhemicl noion discssed erlier. The br-ke formlism hs been invened ecl o enble mhemicl nlsis of sch new qnm ses. 4 Here gin s like in ec.. he semen implies he bsrc (mhemicl) noion of idel eperimens posponing he discssion of rel (phsicl) mesremens nil ec s reminder in he end of ec. 3 we hve lred proved h sch eigenses corresponding o differen re orhogonl. If n of hese vles is degenere i.e. corresponds o severl differen eigenses he shold be lso seleced orhogonl in order for Eq. (8) o be vlid. 6 This ke relion in priclr eplins he mos common erm for he (generll comple) coefficiens he probbili mplides. Chper 4 Pge of 4

22 ( coheren ) sperposiion of he pe (6) h hve been he sbec of or sdies so fr. (We will discss he mied ensembles in Chper 7.) However i is iniivel cler h is resls nd in priclr Eq. (6) re compible wih he descripion of is wo op bems s ses of elecrons in pre ses nd respecivel. The bsorber following h firs sge (Fig. ) s kes ll spin-down elecrons o of he picre prodcing n op bem of polried elecrons in pre se. For sch bem probbiliies () re W = nd W =. This is cerinl compible wih he resl of he conrol eperimen shown on he boom pnel of Fig. : he repeed G () sge does no spli sch bem keeping he probbiliies he sme. Now le s discss he doble ern-gerlch eperimen shown on he op pnel of Fig.. For h le s presen he -polried bem in noher bsis of wo ses (I will denoe hem s nd ) in which b definiion he mri of operor Ŝ is digonl. B his is ecl he se we clled in he mri digonliion problem solved bove. On he oher hnd ses nd re ecl wh we clled in h problem becse in his bsis mrices nd hence re digonl. Hence in pplicion o he elecron spin problem we m rewrie Eqs. (4)-(5) s (4.) (4.3) Crrenl for s he firs of Eqs. (3) is mos imporn becse i shows h he qnm se of elecrons enering he G () sge m be presened s coheren sperposiion of elecrons wih = +/ nd = -/. Noice h he bems hve eql probbili mplide modli so h ccording o Eq. () he spli bems nd hve eql inensiies in ccordnce wih eperimen. (The mins sign before he second ke-vecor is of no conseqence here hogh i m hve n impc on ocome of oher eperimens for emple if he nd bems re brogh ogeher gin.) Now le s discss he mos mserios (from he clssicl poin of view) mli-sge G eperimen shown on he middle pnel of Fig.. fer he second bsorber hs ken o ll elecrons in s he se he remining elecrons in se re pssed o he finl G () sge. B ccording o he firs of Eqs. () his se m be presened s (coheren) liner sperposiion of he nd ses wih eql mplides. The sge sepres hese wo ses ino sepre bems wih eql probbiliies W = W = ½ o find n elecron in ech of hem hs eplining he eperimenl resls. To conclde or discssion of he mlisge ern-gerlch eperimen le me noe h hogh i cnno be eplined in erms of wve mechnics (which operes wih sclr de Broglie wves) i hs n nlog in clssicl heories of vecor fields sch s he clssicl elecrodnmics. Le plne elecromgneic wve propge perpendiclr o he plne of drwing in Fig. 5 nd pss hrogh liner polrier. imilrl o he iniil G () sges (inclding he following bsorbers) shown in Fig. he polrier prodces wve linerl polried in one direcion he vericl direcion in Fig. 5. Is elecric field vecor hs no horionl componen s m be reveled b wve s fll bsorpion in perpendiclr polrier 3. However le s pss he wve hrogh polrier firs. In his cse he op wve does cqire horionl componen s cn be gin reveled b pssing i hrogh polrier 3. If ngles beween polriion direcion nd nd beween nd 3 re boh eql /4 ech polrier redces he wve mplide b fcor of nd hence inensi b fcor of ecl Relion beween eigenvecors of operors nd Chper 4 Pge of 4

23 like in he mlisge G eperimen wih polrier pling he role of he G () sge. The onl difference is h he necessr ngle is /4 rher hn b / for he ern-gerlch eperimen. In qnm elecrodnmics (see Chper 9 below) which confirms he clssicl predicions for his eperimen his difference is eplined b h beween he ineger spin of he elecromgneic field qn phoons nd he hlf-ineger spin of elecrons. 3 Fig Ligh polriion seqence similr o he 3-sge ern-gerlch eperimen shown on he middle pnel of Fig.. Epecion vle s long brcke 4.5. Observbles: Epecion vles nd ncerinies fer his priclr (nd hopefll ver inspiring) emple le s discss he generl relion beween he Dirc formlism nd eperimen in more deil. The epecion vle of n observble over n sisicl ensemble (no necessril coheren) m be lws clcled sing he generl rle (.37). For he priclr cse of coheren sperposiion (8) we cn combine h definiion wih Eq. () nd he second of Eqs. (8) nd hen se Eqs. (59) nd (98) o wrie W. (4.4) Now sing he compleeness relion (44) wice wih indices nd we rrive ver simple nd imporn forml 7 '. (4.5) This is cler nlog of he wve-mechnics forml (.3) nd s we will see in he ne chper m be sed o derive i. hge dvnge of Eq. (5) is h i does no eplicil involve he eigenvecor se of he corresponding operor nd llows he clclion o be performed in n convenien bsis. 8 For emple le s consider n rbirr se of spin-½ nd clcle he epecion vles of is componens. The clclions re esies in he -bsis becse we know he operors of he componens in h bsis see Eq. (7). Represening he ke- nd br-vecors of or se s liner sperposiions of vecors of he bsis ses nd. (4.6) ' ' 7 This eqli revels he fll be of Dirc s noion. Indeed iniill he qnm-mechnicl brckes s reminded he nglr brckes sed for sisicl verging. Now we see h in his priclr (b mos imporn) cse he nglr brckes of hese wo pes m be indeed eql o ech oher! 8 Noe h Eq. () m be rewrien in he form similr o Eq. (5): W where is he operor (4) of proecion pon he h eigense. Chper 4 Pge 3 of 4

24 Chper 4 Pge 4 of 4 nd plgging hese epressions o Eq. (5) wrien for observble we ge. (4.7) Now here re wo eqivlen ws (boh ver simple :-) o clcle he br-kes in his epression. The firs one is o represen ech of hem in he mri form in he -bsis in which br- nd ke-vecors of ses nd re respecivel mri-rows ( ) nd ( ) or he similr mricolmns. noher (perhps more elegn) w is o se he generl Eq. (59) for he -bsis o wrie i. (4.8) For or priclr clclion we m plg he ls of hese epressions ino Eq. (7) nd o se he orhonormli condiions (9):. (4.9) Boh clclions give (of corse) he sme resl:. (4.3) This priclr resl migh be lso obined sing Eq. () for probbiliies W = nd W = : W W. (4.3) The forml w (7) bsed on sing Eq. (5) hs however n dvnge of being pplicble wiho n chnge o finding he observbles whose operors re no digonl in he -bsis s well. In priclr bsolel similr clclions give (4.3) i (4.33) imilrl we cn epress vi he sme coefficiens nd he r.m.s. flcions of ll spin componens. For emple le s hve good look he spin se. ccording o Eq. (6) in his se = nd = so h Eqs. (3)-(33) ield:. (4.34) Now le s se he sme Eq. (5) o clcle he spin componen ncerinies. ccording o Eqs. (5) nd (7) operors of spin componen sqred re eql o (/) Î so h he generl Eq. (.33) ields pin-½ componen operors

25 (4.35) I I (4.35b) I. (4.35c) While Eqs. (34) nd (35) re compible wih he clssicl noion of he spin being definiel in he se his correspondence shold no be oversreched o he inerpreion of his se s cerin () orienion of elecron s mgneic momen m becse sch clssicl picre cnno eplin Eqs. (35b) nd (35c). The bes (b sill imprecise!) clssicl imge I cn offer is he mgneic momen m oriened on he verge in he -direcion b sill hving - nd -componens srongl wobbling bo heir ero verge vles. I is srighforwrd o verif h in he -polried nd -polried ses he siion is similr wih he corresponding chnge of indices. Ths in neiher se m ll 3 componens of he spin hve ec vles. Le me show h his is no s n occsionl fc b reflecs he mos profond proper of qnm mechnics he ncerin relions. Consider observbles nd B h m be mesred in he sme qnm se. There re wo possibiliies here. If operors corresponding o he observbles comme B (4.36) hen ll he mri elemens of he commor in n orhogonl bsis (in priclr in he bsis of eigenses of operor ) re lso ero. From here we ge B B B. (4.37) ' In he firs br-ke of he middle epression le s c b operor  on he br-vecor while in he second one on he ke-vecor. ccording o Eq. (68) sch cion rns operors ino he corresponding eigenvles so h we ge ' B ' ' B ' ' B '. (4.38) This mens h if eigenses of operor  re non-degenere (i.e. if ) he mri of operor B hs o be digonl in bsis i.e. he eigense ses of operors  nd B coincide. ch pirs of observbles h shre heir eigenses re clled compible. For emple in wve mechnics of pricle momenm (.6) nd he kineic energ (.7) re compible shring eigenfncions (.9). Now we see h his is no occsionl becse ech Cresin componen of he kineic energ is proporionl o he sqre of he corresponding componen of he momenm nd n operor commes wih n rbirr power of iself: n '. (4.39) n n n Chper 4 Pge 5 of 4

26 Now wh if operors  nd B do no comme? Then he following generl ncerin relion is vlid: 9 B B. (4.4) The proof of Eq. (4) m be divided ino wo seps he firs of which proves he so-clled chwr ineqli: 3. (4.4) The proof m be sred b sing posle (6) - h he norm of n legiime se of he ssem cnno be negive. Le s ppl his posle o he se wih he following ke-vecor: (4.4) Generl ncerin relion chwr ineqli where nd re possible non-nll ses of he ssem so h he denominor in Eq. (4) is no eql o ero. For his cse Eq. (6) gives. (4.43) Opening he prenheses we ge. (4.44) fer he cncellion of one inner prodc in he nmeror nd denominor of he ls erm i cncels wih he rd (or 3 rd ) erm proving he chwr ineqli (4). Now le s ppl his ineqli o ses  ~ nd B~ (4.45) where in boh relions is he sme (b oherwise rbirr) possible se of he ssem nd he deviions operors re defined similrl o observble deviions (see ec..) for emple ~. (4.46) Wih his sbsiion nd king ino ccon h he observble operors  nd B re Hermiin Eq. (4) ields ~ ~ ~ ~ B B. (4.47) 9 Noe h boh sides of Eq. (4) re se-specific; he ncerin relion semen is h his ineqli shold be vlid for n possible qnm se of he ssem. 3 This ineqli is he qnm-mechnicl nlog of he sl vecor lgebr resl. Chper 4 Pge 6 of 4

27 Commion relion for spin-/ componen operors ince se is rbirr we m se Eq. (5) o rewrie his relion s n operor ineqli: ~ ~ B B. (4.48) cll his is lred n ncerin relion even beer (sronger) hn is sndrd form (4); moreover i is more convenien in some cses. In order o proceed o Eq. (4) we need cople more seps. Firs le s noice h he operor prodc in Eq. (48) m be recs s ~ ~ ~ ~ i ~ ~ B B C where C i B. (4.49) n nicommor of Hermiin operors inclding h in Eq. (49) is Hermiin operor nd is eigenvles re prel rel so h is epecion vle (in n se) is lso prel rel. On he oher hnd he commor pr of Eq. (49) is s C ~ i ~ B i B B ib B i B B i B. (4.5) econd ccording o Eqs. (5) nd (65) he Hermiin conge of n prodc of Hermiin operors  nd B is s he prodc of swpped operors. Using he fc we m wrie C i B i B ( ) i ( B ) ib ib i B C (4.5) so h operor Ĉ is lso Hermiin i.e. is eigenvles re lso rel nd hs is verge is prel rel s well. s resl he sqre of he verge of he operor prodc (49) m be presened s ~ ~ ~ ~ B B C. (4.5) ince he firs erm in he righ-hnd pr of his eqli cnno be negive B ~ ~ i B C (4.53) nd we cn conine Eq. (48) s ~ ~ B B B (4.54) hs proving Eq. (4). For he priclr cse of operors nd p (or similr pir of operors for noher Cresin coordine) we cn redil combine Eq. (4) wih Eq. (.4b) nd o prove he originl Heisenberg s ncerin relion (.3). For he spin-/ operors defined b Eq. (7) i is srighforwrd (nd highl recommended o he reder) o show h i (4.55) wih similr relions for oher pirs of indices ken in he correc order (from o o o ec.). s resl he ncerin relions (4) for spin-/ pricles nobl inclding elecrons re Chper 4 Pge 7 of 4

28 ec. (4.56) In priclr in he se he righ-hnd pr of his relion eqls (/) nd neiher of he ncerinies cn eql ero. s reminder or direc clclion erlier in his secion hs shown h ech of hese ncerinies is eql o / i.e. heir prodc eqls o he lowes vle llowed b he ncerin relion (56). In his spec he spin-polried ses re similr o he Gssin wve pckes sdied in ec... Uncerin relions for spin-/ componens 4.6. Qnm dnmics: Three picres o fr in his chper I shied w from he discssion of ssem dnmics impling h he br- nd ke-vecors of he ssem re heir snpshos cerin insn. Now we re sfficienl prepred o emine heir ime dependence. One of he mos beifl feres of qnm mechnics is h he ime evolion m be described sing eiher of hree lernive picres giving ecl he sme finl resls for epecion vles of ll observbles. From he sndpoin of or wve mechnics eperience he chrödinger picre is he mos nrl. In his picre he operors corresponding o ime-independen observbles (e.g. o he Hmilonin fncion H of n isoled ssem) re lso consn while he br- nd ke-vecors of he qnm se of he ssem evolve in ime s ) ( ) ( ) ( ) ( ) ( ) (4.57) ( where ( ) is he ime-evolion operor which obes he following differenil eqion: i H (4.58) where Ĥ is he Hmilonin operor of he ssem (h is lws Hermiin H H ) nd he do mens he differeniion is over rgmen b no. While his eqion is ver nrl replcemen of he wve-mechnicl eqion (.5) nd is lso freqenl clled he chrödinger eqion 3 i sill shold be considered s new more generl posle which finds is finl sificion (s i is sl in phsics) in he greemen beween is corollries wih eperimen - more ecl in hving no single credible conrdicion wih eperimen. ring he discssion of Eqs. (57)-(58) le s firs consider he cse of ssem described b ime-independen Hmilonin whose eigenses n nd eigenvles E n obe Eq. (68) 3 H E (4.59) n nd hence re lso ime-independen. (imilrl o he wvefncions n defined b Eq. (.6) n re clled he sionr ses of he ssem.) Le s se Eqs. (57)-(59) o clcle he lw of ime evolion of he epnsion coefficiens n defined b Eq. (8) in he sionr se bsis: n n chrödinger eqion of operor evolion 3 Moreover we will be ble o derive Eq. (.5) from Eq. (54) see ec Here I inenionll se inde n rher hn o emphsie he specil role pled b he sionr eigenses n in qnm dnmics. Chper 4 Pge 8 of 4