QUANTUM MECHANICS I
QUANTUM MECHANICS I PHY 389K
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Date Created: 09/07/15
Perturbation Theory In section 1 we will gwc some descriptions of how and why appmxinlatiull methods are V H permrhntion theory are derived 11 Stationary Perturbation Theory and Its Underly ing Assumptions 111 Introduction Tlre deeerlpllrm or any physiml sysrerr is only epproxlmsre ll is Lrue of course elm ll ls h I Fer example 39 e 39 molecule only when llre vibrsllons ere rel mp violent And mhmollic forces are reglllsle u a similar win it ls possible no nd the energy levels u are hydrogen aLum ener lle g o prexsmele 11 Lake lulu aucounL the 3pm or the eleelrorr llre Pauli equerlor or Perrll Hamiltonian must be used no describe rhe eleccrerl The Dim o ls e relerlvlsele equsllen lel deserllres a spin 12 per39 le To rule inm scemmr rpm and elechn sl llre speed or llglrr Llle BetheSallie equation esp be emplpye Even when the elecrrorr ls traveling at speeds fa less than llral or light the electron mleracls the energy level calculsved from llre Pauli equaLiuu er Pauli llrrrrrrlrenlerr Tlrerelere if a pllysicisL wislres m calculele numbers llrrl erm be compared 31th experimemrl mhlei approximations are mevitable spurring thz Hamiltonian lrrro two pares H Hr H 11 1 1 CHAPTER 1 PER39I URBATIDN THEORY 1 39 Lhal 1139 V whoseeigw A I mien 39 H ix ux39llll lm in ihe syrem desu bed by ihe Tree Heinilmnien m In explain wlmL iL nienns ihnr the quanrnni physichi reins wrrh H and Ho are similar we lst consider rile rev ran 0139 ieelinn 1114 in ihe pr me el 11 mngnniin eld This is a situation in w in mM39 rims nor need nerrnrherinn ihenry 112 Magnetic Moment Operators The Himlilkmian or m interaction Free system in Lire mmlnl Hamilmninn Ila 39J39 of 111211 To conjeurure ihz Marniiron Hamiltonian 1 for Lbs inieraeiinn wiih nn exmnnl mngneiie eld H W rollew Hie usual procedure and Han wich are energy ul39 Um sysmnr hy replacing Lhe Llasmnl observables by their corresponding qnaninn uperuurs h ergy wieh n dipole magneeie nnnneni 1 in n magnetic eld 13 is given by E 7 27 E 12 r spinning nhi i n nng u n i in in general givnn hy 13 39 39 lhe nhjh n i ii a a semi or gyrernagneiie mlin is n pnrnmeeer which is spe i u inr ihe physical objer nr erbir wiLll orbital angular momeninrn l Lhen the magnetic momenL is given by 13 wirh g L4 m m h 1X3u The eleelmn with q 71502 10quot0 in n nenrly siruciurlesi relaiiviniie 39 39 39 quotnew n 5quot 7 due w QED wmeiiens urns 2 rr m 9 2 7 me gmer 1 a where m and 5 are ihe mm and spin or ihe electmll respeciwnly prawn has inirinsin euueune 141le EL megneiie mommh is giwn by rim 2 2 369 16 3 2m 1 mu VI 1 run LDAAI A where rt is the mass oithe proton Thus the Landiioetor or the proton is 17mm z3og n oi the proton which unlike the electron is not e neorly structureless particle A neutron has structure hut no net Charge its mngnotic moment is therefore only anomalous and is empirically determined to he e 3mm or 192 it 17 u t 2ms I J in quantum mechanics the megietic dipole moment operator it is obtained by replacing the Llaa slcal artgule memento by rhe angular momentum operators 1 2m run is where J ful ll the algebra or angular momentum 111216 The J nan he the orhital atignlnr momentum L the spin angular momentum Si or the sum of spin and orbital the iiingnetio dipole npcmwx is e 2m if llama inn 7 Z 25 1 9 w en 39 39 by the rotational motion oi the whole nluletule These magnetic moments result from the rototiou or the positively rhaxgcd nudei about the center of mass and the rotation of the negatively charged electron clouds which move with the nurlet This rotntton or the l i own Lin The loetor g is a measure or the current distribution of the rotating mnlccuht and is deter mined empirically from the Zeeman splitting of the Energy levels eg ier the diatomic molmlle H2 g 0883 394 Mode J 183mm and Mmrdau 1 mquot Thurman in quantum mechanics two quantities with units of a magnetic dipole moment are often need ch Bollr mogneton M 2m 9272 1D Q JTeslu L11 Noelenr Bohl magneto no a 5049 r 10 quot7Tesla L12 The magnetic moment oi a proton or a nucleus is thue due to the mass ratio three orders of magnitude smaller than the magneth moments of Electronst Only when the letter are alreeot do these email nueleer momenta give noticeable eileets 4 CHAPTER 1 PERTURBATIDN THEORY is of thz same older of magnitude as that of u magnolic moments of Lb nuclgi Most moleculea in Lille ground state have no demonic angular momentum For that molecules me magnellc dipole moment i pruduced by the nuclei of me molecule or by the ruLaLimlxl motion of the entire molecule llan ls glwn by 1 MfgJAE 113 which in nfthl order ofnglliulde gmquot ly me nllclmr Bohr magnncon 1112 113 Quantum Rotatol in 3 Classical Magnetic Field If a quwlum mm wllll n mngnclic moment 2 and a Hulllillollian npemmr H H 5 14 for lhe simple mum ii placed ilan n nmgncllc eld the Hamiltonian npemur or llle quantum mum in lhe magnellc eld it ubhailml by conmpomlem lmm the classical incexacllun energy 12 ns Hxyrr i 114 This llamilmnian is no anger roaliulmlly imrlm which mmns that PLJ WJJE q r mlJAJIIBI 6 0 115 The lnzlgncllc eld I distinguishes a direcLiull und rstl39nyi MW isman nl Rpmn The In summary the Hamiltonian um we need be tonsillav rm L112 mmm m an nmmnl magmas eld ls l q N f gmBL 116 when the flue or quotnnpmnrbcdquot Hamiltonian is given by we also lu2n 7 2 H0 7 J m a 39 illlvrxl Lilm 39 39 39 ml l 1V2v H 118 Fa I H ilr 39 l1 39 39 adimccian LE me axbmxry dimcliull of me Ja cumpmmm nl Allgllhr momznlllm wlmsp clgcnwctols L7 13 one uses we dlmlluu cube magnum lield a mag The 115 Llulnllllm a II 7 17 33 l 21 an 13919 1 ADD u The eigenvccmrs n1 lliis Hamiltonian are already known Lo us They are the eigenvmnns mm M Lhe csca 11 found in sacrinn 1113 The eigenvalues ul the free Humilaunian 11 am 2 1H 120 The eigenvalues oi um exam Hamilkmiau Hlj in Elms are innuedinuely obtained by npplying H of 1 19 Lu ii 71 The eigenvalues of H depend new only on J like llne E but also 13 1 E Eu 1 1 15 Bin ml n 1 Huh eigellvodne E has a 2239 1fold degenern This imm w eanli 1 correspond 2 l eigenvccwrs 7 1393 in u zj 1 dimensional spuce RI The eigenvalues Eu or H are P n i n Minna 5 whidl merge ileo when the external lllagneLic eld is switched u 39 Lei when H l L 122 ELK l mil r i u u i 3 is plnood into a magnetic eld which blanks une symmetry 39 1M w I r II In U I J could not have been eigunvectnrs of H because for H oi 116 IHJI 6 n ln mini to obtain eigznvcclnrs of 116 we woulil ve n usn insmad or L1 1393 a basil eynmu jj v s lh of clgemecwls of the csco th39i J whey n 2 unit cctol in the dirwbion of m magnetic eld and plowed in ma same way as above Slug Inquot in place or L743 However 1 and Mn 0139 119 and 111 or 116 and 111 do not leviesem ulie general M each alher whlch in general is not ful lled nnmcly 1H 111 o For n gzlleml nnuuu syslembwl uia rolnwr in a magnetic eld wul the illLe ncLion Hamilmman 118 this is V u n m 01 any 11a f in LIIE general case the mace and nee Halllillollian do nDL commune 11110 36 o as is 11m use for 1me and molecules in n magnetic eld for wniul Hu would not be given in u funcLloxl mquot Bnl Hquot could stiLl he s spllerlcnlly symmetric Hamiltonian ic ulnll These systems will alm have Zeemnn splining however we eiguuvalus EN 39 on Q 39 39 39 quot HIIandHn do not commune one has Lu use a prowdlmz called pmulbalion Lhwry We now consider lliis gonoial case 139 CHAPTER L PHU URBATION THEORY 114 Perturbation Theory We haw two Hamiltonian In and H H 4 I Ebwuw for very special tasa such 15 the one dimensional hennanie 39 mm one has in addnion to ht energy qunnuun munth niher quenrnin numbels such es 3 in rhe case or the mman Then we have to sFJecl other npcrnihrs whim ingenhnr wirh Ha Drm n en whuee eigenvnlnee re known we rienme Hus sen ofoperatuis B unliecrively by B and rheir Av we denote byh 39 39 o ii 1111 V Then we hnyn rwo csu u39s nonsiecing of N opennure 11mm E BN which we ahhxeviam HnB L23 3 31 BM whim we nhhrevrnre 33 mm 1r Lliis ix not rhc case i ir nnn nr more or are 5 do urn cninrnure wirh H Lhen nun A 7 4 I 39 L I as vr 7M n wen er his we will discus heiew The m c has 11x and 124 can he mm as r rhey wer n a cousins of N2 npernuns 1155 and 115 respectivclyr eeee We heyn hhp following snua u The physical system desalile hy L23 has soluciune n b b 4 mid eigenvalues E5 5 h hm hm Lhal ere cnmplntv y known We nenh Ln nd the eigenneeunn my gt ham gt mid 39 E E l M at ihe p I 39 sysueni dzscribu hy 124 and expreee hlmm in henna ni rhe eohninne for rhe free eyeiernn Per 2 special case um 11 in 0 gt13 in my one can dluuse me gt Us eince Lhe eigenvmmis of u are myme r r u m m r v unknown Eigenwlhws E gnu and eigenvenwrs mm or H from the 1mm eigenvalues 39g nwczure mg ann The sihuuliun Lu whieh one npniiee perturbuion hheory ie ourerwiee very ninuinr Ln rhn 1 v A39 A 122 539 E or 39 39 w ch nee degeneme except for ihe my special um char B has unly um eigenveiue er ihe h N J u M vein Eb of the sysuem whh inwrnerian Hh eueh u in name rnnnuer all these energy leveh Em merge in Iv 5quot E 2 vi 2 E n ur innnirn runnher cl enenny Ie rhni fox zuw interaction Lu Hi inw rhe same energy urine by ivEmb Ema m quot o E3 125 em However 1 rherinerinn to we 122 Ln rnch 1 uon ebpundx a vector mm gt dl 39rmnt h m r13 i v1 under the condiLiun lhnL 11 in later remitrtan it me nx smh gtvgt 15quoth for H1 7 n v 1 911 ffquot nite or in nitc 126 or in nite knovvn eigenvcctars mg belonging to Lhc one known algIlwahm 55 0f 11 They span the nite or innnite dimellsinnal space ui slgenvecters ni 115 with the same eigenvalue E3 The nettnrhnnan 11 then splits the energy level into the eublevel Ert the degeneracy oi the eigenvalue 53 is remtvvedt often as is the case of the rotetur in the r s in I perturbation Hernilteninn H then breaks the symmetry thereby removing the y ln order for the sulmme I15 and L26 to make physienl sense the splitting inside a mnltiplet it between thc Equotl for it given wine of n end di emxiL velnes of 7 should he 39 39 39 h munn F IF 391 Ho The dcgener39 75 5 n This means the interacLion opamLm must be small in a ruin mathematical senstl gt Mammalian ltlt1E2E l Hawm reelsclyrerrnnlete themethemntieul linmns an the a enLurs Hu d perturbetien procedure converge the procedure to calculate llnlllbels penurbatiun ies somehow convex p H such that 1 25 mul L25 result and such that s is u dif cult mathematical problem Physicists use and hope that H hes been chosen right so thet the Before we dambe in the next ctien how the unknown eiganwdms E and the eigen vectnrs lbw gt at H am determined in terms or the known eigenvalus its and knmvn i n I r U f U 39 to do in the case that Hm D but HH st 1 In this case we could ll use b as labels for the ei envecwts of H but these eigellvecbuxs of H are not cigmvenwrs of B and the label I in lztt gt would only be nn approximate quantum number dn ncd by the limit of 1251 This is what is orten dune in Samuel39ng theory or the L39onLinumls eigeuvnlnes E or I where l are the asymptadc momente with igcn l H evul Hg with discrete spectra if is better it one does an operator A on it system of operators AhAn V A esco m vaneoan nu not use lEtb gt but tries in nd M whirh together with H iorms e ILA u 127 and which also hns the property that Hem o 125 One then obtains A new system of b is vectors er the internction free system E n Ho A a 014le Is PERTURBATION THEORY basis vecnnu i331 E iu Z lEI iibMEMlEf nu 129 i These basis Vecle lE mn am like me rfb eigenvecwxs nl 11 Willi tigenvalue 53 I s m n I from the known man by paiiumniiun lIIBnlyi This means fur the case um HE ll in 129 to line previous case 123 and l 24 which can be LreaLed like n nmndngunmuc perturbation since ills degeneru labels 1 are llul n ecledl The nansroxmaiinn ma rix animal dppends upon Ill piopurzies or the operators B and A and mu uprln H and n An example oi ilns will be disl39lmed in dam in chamel39 1X The csnn Hm is HWLQ 135453 llan wlisre IT is aha mbiml angular quotlumenmm nnd IS me spin ninl Hn is n function of ulic 39 39 h fHTM Hi I l product basis Vecwrs I111 its I Inns Wilma lass Ivan wlme 5 null 701 si we the am Hamiltonian H of seclinn IX 391 ranmins an 135 coupling HHiE 132 wing f may I my mi be a lnnciiun or upemlurs innl commute win L and sh an my where Q are lihe components of ihe pwiliull opernlor q the exact linmilmnisn H will uni nommucn with Li and El Hi L 0 ms Hs 134 lloweyen jlle exam Hsmilmninn will commute with the Lou mlgulur mnmcnlllm operator s 11 Id 0 135 Sr it explime lmni 132 sillre s l l E 3 511 Lquot 5 135 and 11D commutes with JP Therefan HmJ IL 51 a Hadl 137 12 SIA I39IONARYPEKI39UREATION SERIES 9 as well as HJ JL s e 111 135 are c gtLo s which differ only u zhs openth H aml Hquot The Llgenvzcmrs lam 31139le 139 at 137 m ubLade fmm the dim product vectors 131 by cf v212 lb jjulslZlEM lvlavsvsales 140 1m when 113551jj3 we die 01th Gurduxl uoel ricnts obtained in sechinn v2 There Is a apecinl case not covered above which rennin special cunsldcuzion This is lhe case leu lHB as o lmri one cannot nd an npnrnua A or it is oh pmnucnl m nd an upnrnlnl A which has the plopmy 127 and 128 and mgcther with an I Imus a cscn ln uhls case am has no apply alluhlmr pmceduu uuuwy unllerl dzgencmte perturbation lhmryquot 12 Stationary Perturbation Series h 39 39 39 5 39 39 or H in mm of the energy eigenvetlnl s nnd clgomlues umquot wuhnuu loss of generality we msmcu uur uLLenHmI to systems for which nnly two opmcuu axe leqllived m form a unmle sea at commlung nhsm39ablcs The gellurulization to systems for which fume 39 n I The farmldas we derive can be used for llle cwu Lypcs of problem lislled below Case a A llelmicinn opernux 8 Dr to an or operators 13 3 EN is known which satis es 3 HQ BHlo Ml 151901 39 A 39 gt the eigenvalue h lakes only one value In case n we assume uuu Lhe free Humlhonlnn unperturbed Ilamiltulliuu Hr has hucn solved and the basis vechms IEZJJ satisfying Hamill Film 142 MM blEl h 143 A known Usually lu eigenvalues 0111n lepmslld only nu uh quantum number n hul ml an I nihhnugh uhuc m exceulluus For Inter rammime the eigenvarlols 53 b are normalle lulluws EMlbw mm 144 10 CHAPTER 1 PHRTUREATION THEORY For ease u we seek eigenvenmrs EM 7 e of me quotexactquot Hnmilmmaxl H HEM gt EmoleJ gti ms elm 51 isry 5 st gtgt IE 39i b for H A u L45 is F r 39 I i ii i L unmiH 39 l39l 39 Ens Em as shown in L25 For case in we seek eigenvcvmrs inquot gt at ilie quotvisaquot Hmnilmniau H Hit gt Elm h 147 elm saiisry IE gt4 113 for 11 a 148 If I commuted win some other cparawr uie eigulvectnm u gt wnnld be labeled by additional qunnium mime which are m39nitlcd here because iliey are ich of lumen em fox our diecussiuu Hemusc for iliis Easel ilie energy eigenvalues of In axe nondegenexale L l 1 Ex V with the Sam mllle 0f the quaninm number 14 Since nus is a special case of e we will Ilnl discus KI any Euthzr 21 MINE b gt mg Myquot Hiwwh gt 149 when 13 All E E The matrix alumni on line le rhmld side or L49 is simpli ed h I I I f l39 39 39 39 evniuniesl using the 5 The rm lluilrix elemeni39 a rithiand side a L45 39 liennil you n the and 142 We wins nblnin for 149 bzhuE ublEnnb e 524 ME A gt ltE nhlllilznm 1150 i 39 gt n r a n y Am by mud nm ihe value E3 Fur HIE same mnsnn me cigmvuclnni lEmm gt and E b are 39 i ll cerLaiu way also close to each other This in paniculm quotmums what for n u lllmJlxelmeul mammal b e 53 Mag 1 l Thamfow we can divide I an by iliis I A l hoary e L EblHilEnw in my EmI gt 1n Ami 151 p EM mil fr v i m 39 A is llie eller shin mm the mpmurbed energy value 2 or llie iniemeiinn rm smem m 1392 STATIONARY PERTURBATION SERIES 11 the allexgy values EM hi the system with interaetiun it is given by the matrix element of the quotsmn 1quot interaction Hamiltonian 11 which needs to be calculated 1mg 39 llm F 39 l Hl ftlh Vht r i 439 on wu n w c one can divide 150 by EM 7 2754 obtaining Exit thi Ent b gt e E EM or n e n l 52 ExhblEmhb gt 39 39 we expand the quotextantquot eigenvectai lam gt of II in term at the Eigenvecml s mall at Hg The eigenvectms iE b 01 the 03cc H v ml rm a complete axis system of the spnnn H nfstate teetois Thus every west 1 s H can be expnnded in tmns of this svslemt basis 2 lain wait we 153 quotW t 4 39 W 39 M M HBiBN1 H H mm be expanded as in 153 mm gt 2 lb tb39xa n b lEntb gt 2 wwgtltt22nul tt t gt 154 nW n The second equality halds because 5quot h lEmt gt ll or N e it L55 si e MIME and map are alganvmtms oithc same svswm of hermitian operatoris B 39 39 The we in 39 39 M I venture 53 with the same value as on the Mt hand side appear and we can write l 54 as Enttb gt1 Z lE n anE rlEanb gt Eib E il39lEntttb gt 2 Win FINEXublEntib u n39fn l 56 For a g39ven value of n the terms with n n and n39 e n in 155 are or quite different magnitude since the matrix elements E liEaM gt2 ELI 73 t 157 airtight It ys Eff bimh n for n n L58 3 i i 39 iEthfH 39 lamb of 11 plus a m nne sum of terms of much lnallel39 lnagn39 n bLain the traditional nude 39 0 arm we add and snhnnct mg to the rst term in 156 and insert 152 in the second teim the nite sum over n39 of 155 Then 155 heeumes 7 t n t n n Eto lh HllEn bf with gt With EniMEnttJI gt llllini39l T WWW 159 12 CHAPTER I PEKTURBATION THEORY Exec for m m mm of 0m ugh hand side we coelliciellls or the aigcnvccmvs 1E3 1 and 115141 are n11 ofsmall magniLude due to 157 and 133 Fm rm H H III39 39 39 39 theory they 111 energy 39 F of H rt speclively But the unknan quantities Em nnd Emma appear on both sides of 11w mun mm 11nd these equations nnn be solved for muse unknown quantities We du unis by successively inseniing 119 Mininmd sides mm L e rig i7 and aid in carrying uni its itembiv procedure in 15 ilnponnnL u keep 111 mind lImL Hi is 1 small perturbation n 1 1 g 1 39Lk a Il mh u 39 39l I r39 39 we aim over 1139 cocmcieum umpared wiih zhn rs term E39b Tquot 39 39 39 39 quot MM denoted by 252 and WWW mspcctivaly is ubtained by mung HI u in 1151 and 159 m a 53 E2 150 iE1 H Enib 1521 161 Since the Min an LII righlIland side at 151 and 159 that involv En or 15 gt are small we obtain the rsvurdvr nppmxinmlions l0 1 and 112 1 gt by snbsiiznung Emil H 39quot V39anndh 1n rospccnivniy Deming we rsirunier nppmximnnmn in cncxgy by 153 151 become Ewwiiannw EEVMEanu 39 a 1 391 i n rm 151 39 L 39 39 1 1 1 15111 this becvmes E2 5 162 12112 a 1 13211Hisz b E A3 163 To 11m unler the energy shin A3 of the degenerate energy level E33 ui 11m Free in 1 milcouian equnin Lhe matrix elmnmn of nhc perturbation Hamiltonian H bemeu thP iinps niuhcd known eigensiaies 3 Dnnoiing the rstrm39dur ap oximminn for energy eigenvacbors by 1E gt and using 160 and 11ii in 1151 we obtain 1112 rst oxdsr appmxinmiun in the exact oigenveclors 1mm 1m E39nthEmJ 7 11112 2 laun W quotWg 1641 m E 7E H quot L K 39 me no D 39 39 RM II R Linn 1141 me above equation 1mg We Form mm 11111 z in nb b quot 31 m 1 w 53 n1 1 2 STATIONARY PERTUEBATIDN SERIES 13 l V V 39 Em and M1th 51 And 159 14 and 15 m as follows l 66 l EMIH 53 7 z Eff E b bH E h A Equotb HE b a 13 MHHE MbHF bHJE b Emmm r HELEN Mawwvg WWWTL IWV 157 Cumin mg m this fashile n is pusxible Lo nhlnin higllex older apyruximaliuns39 fur the 39 I H exact Eigenvalue and eigmvccmrs onus exam HamilLullml l h m Em h h However since lhe mam cleman of H is small we sea ham 155 and 167 and he dsvlmons from unhonumlality m small HT 5 39 l V Insmad it is conventional w choose Llle normalization of the Emhb gt such that EquotmbEgh gt l ms T also xes the relative phase belwesn the exact eigenwrtnr EMJ gt and the me eigenvectox mm with the who he 168m nm39malizauun the WA undnnlmlu 9mm sqhhlions 151 and 159 bewme respectively 1 E2 ltEquotblHlEnab gt 169 lamb gt mg 2 WWW mm m Ens n If she hemluh procedure to solving 151 and 150 is no make sense Lhe mm slush venmr l5 b gt must nnl differ much mm my Mum 159 we Lhm conclude that the rulluwhh uhmllnoh will have m be uL llnd HELVMHJEHM b gt l ltlt IEquot E for all n 7 171 um mum I H l l gt 39 L70 converges and Inwal alder solutions nn attuallv approximations to the exan solulion anl39cvnr ch r ls 39 f a large dm 0 problems 0 which the parlurhauon apmelmMInn does 39ludillg the speclm of moms and mnlsculcs whcrc 11 is given by the Coulomb n The inmacdon Hamiltonian H can describe m incarnation wth h wank 39 L x 39 the lllngneLlL39 eld due no the nuclear muL ll Pl39lY 359K Homework 4 v 1quot I I 39 1rnwn xaud s ncul39mn wilh angular mumanij jm y 1 Considering only rotational dcgmcs of fmwom whal is the space of physical states of the Amman Jin terms of lhc product basis vccmrs of mm andj39mmn c 39 391 fima 392 2 The electmu wiui spin in the hydrogen mom is h combinminn of its oxbiial moxion and its spin motioui The crbixal mclion is descrihm by Lhc orbilzl spam 54 wih the basis vecms 1mm whurc R HuinJmi 774mm LZHLm a I11IU inlm LJIIJJuhmnlm 2 i R Rydberg constam and Ho 2L 3 is the hydmgen Hmnillunian r r j Lu act The oval Ihmikonian including spinnihii inlzmctinn is given isy 1 11 s H wih H 7 Ln where G is n coupling 005mmquot a Are the dime pmde basis Vemors IIX IIII Inn of 5H 9 R39i39 cigcnvwlurs of In and H b inhe answer39 v nigcnveums 39 nnmmi m 39 5 nn nn 3 Consider ms combiimliml uriwo rulalurs wiuu V and 1quot Inn with the dim producl hnns mm gr m39 034quot quotMal1 yami mm Vi rn z ivy s A M vi le Jamili Js n Which oflhese venom are eigenvecwrs nu N N527 h ann 39 39 39 L 39 Check vhcthar Au each other r and 0 the vectors found in a c Calcch lhc aciion of J E J u on the eigenvccmrs nr d ninniimhxwiihcr W39 39 39vimquot39i39imin 1 an 21 Semi 392 of the texibook
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