Quantum Mechanics Lecture Notes, Slides of Quantum Mechanics

Download Quantum Mechanics Lecture Notes and more Slides Quantum Mechanics in PDF only on Docsity! Quantum Mechanics Lecture Notes OPTI 570, Fall semester 2020 Prof. Brian P. Anderson College of Optical Sciences University of Arizona Contents Section 1. - Wavefunction spaces, State spaces, Dirac notation Section 2. - Representations in state space Section 3. - Eigenvalue equations, observables Section 4. - Unitary operators Section 5. - {Ir>} and {Ip>} representations, R & P operators Section 6. - Postulates of Quantum Mechanics Section 7. - Evolution, Shrodinger & Heisenberg pictures, stationary states Section 8. - Free-particle wavepackets - example problem Section 9. - Quantum harmonic oscillator Section 10. - Energy eigenstates of the harmonic oscillator Section 11. - Harmonic oscillator: properties of the energy eigenstates Section 12. - Angular momentum in quantum mechanics Section 13. - Spin 1/2 Section 14. - Two-level systems Section 15. - Density operator Section 16. - Unitary operators and rotations (rotations not covered in class) Section 17. - Using rotation operators: j > 1/2 (not covered in class) Section 18. - Angular momentum and rotations (not covered in class) Section 19. - The hydrogen atom Section 20. - Tensor products Section 21. - Addition of angular momentum Section 22. - Stationary perturbation theory Section 23. - (removed) Section 24. - Fine and hyperfine structure of hydrogen Section 25. - Zeeman effect Section 26. - Time-dependent perturbation theory Section 27. - Optical Physics example problem Appendix B - Summary of topics, second half of semester Appendix FT - Fourier transforms in perturbation theory Appendix RF - Time-dependent reference frames Appendix OP - revised version of optical physics example problem of sec. 27 BASS Tyeeke loasls - weer reer dant terntrology 2 UrCey vegrems the goh ©) Roches Ulu a. & ® S > 2 ¥ = Suh x pele noranak it (ur, up = abr uh uy © 6y G an deSheagoned act a fe noadiwd te t o i 5 uo % & waste Me Fo on a s" 4 bay © Pon te dons >) The Ue gos : . ba erpandect vA one & oaks RO AS be) © Be, wu ie, hare ey ont, on8 bel | crebRrernbs Cy had att! que thes &e\ \ t Eu + |e Cangas He seh O} Crackoas gud gus avy zur v3 ee, Lt Vv = c. > . in the new sel a bass’ Ne > shore % ro longer Z unrart wads, Ww expend ath akeuah 9) ¥ : 4 2 ft és coo 4 (us 4, tut MS Louk alto be waiter t L &e gv+ 4 (ages) gta hermnme cy expansion coe FReients + EFS 2c, ui @® al y ur os Be, Ue * Cor ur’ = Be: (ara Cay, . 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Garsdizator te Goabruenn beet fo. (we wa) 2 J&e wT) Wyte bland th or Licg noe med» tal (as we &) wile’) = 6-7) Closure continuo, ort nannd Wears 7 fw tryy Roo4 Ww d Pr Caine yf kaw) (ig ty) hoes ) her) = Sd Cla) Wber) we 6G Ce Ce wy ce HE) Tess a a Sus $0 43 Anes bess Zui} Cond. beats 2 Maths ~ ” (ar us r* %% Cu, Wy S$ la-x') sons ui loager ds Se da we Wy = SGD « ct . a cs } de) + Ge elu, CF) Expense fa\s LY. Eegs coef : one Cu Ye 5 clad (a,b): (ar wir) He) favs. Cas fa) Clee} $7. (o,4) + (ae ey MLE) * 2 bee, oe J ; z a r Notes: squned C* é he (ee) . Vas (ete i o8 At a Syed Kee ! Aebned bn Spec ' CPeshdate 4°) WPS ee €& TL symbol represanbng olde Space - the setof all : L pessidle nacht sates of a Speck bow sacle Aw bet Te nok Bay a ar fleret way Te wast the a Cuncton of post ihe, i or rmomavbun, I 75 an alesinaed & seo! repmecertas overunthns Bs physically knovsuble alow a $98 (athens hoe) mn . Sealer peodiuche Do Stale space es, 1f> 6 & the “scalar pdt Bp 1P> by 14D" aegubotread by (1@, p>) © BSF me A aunaber ard ¥* » carmplene + (1, 18>) = (ies, 1@>) “nme ta tem = Cle, AMO # mB) = a, Herds) ea(rerih “ihe hmer nn Lena: (lererbeer,'9) 2 (rey iPr) +28 We) wee Norudizdsle > C42, LP?) ® cea epashye. i C42, 18s) 20 wae ener ie hos (ur THe w set A SHAE 4, PED, PD) Be the “More!!  by a we, “Dirac. Moreton Yowse 2 4a <el e B* Ss Fie evey bet 1€> stale space, thee 3% a correpondin clemert called a “brea” smecgeed by Sel) Heb man efron of he f * J dud space V 1@> 41> © E the saaker protuct (1e>,1P>) @ Syaboized by <@lP> =P <elb> F & comlex number Lmear Ricctoanal K: Raottakes a complex number col Gah, JH? EE iP > Bs x Ci) = canon Crreut) Laer Feudpet 3 Smut te Sanclsonal bneartty 4 Kr = ales a KUg>) +a. XU4,>) Dead Space: the Anak space Ze the sot of all Qndtondle dehaed assoctate A canplen numbor atl every Ibe E. Back Canchondd K of S* rs syatad by <KI, - <x! e 8” > bra rapresenk a Qinshonal, avt a slate. A bre Kl ache on IVD “to prdue a nuubers LX [Woe (+ Abou VIP> E39 SPLEE™ the cewmee Bact true. 2, given Kee 8 ot B not necessarily druethet MX EE  2 Pegje ters 3 RB = pe KP | © pedjectas on be” Bier = (EXE He> = TPXUIED = YD SD, an operfor Seal asallogy. : Me gzomedsicall a4. | Lot ao Tey FIe> i the octeagerel Qrejector ento “The Let pe> > proechon operators aan be used te obletn components a orteagonal bas 7% 2 Py = PPXPIYXH| = . “Prey chon onto A Subspace IE 1 in 1g? > orthonormal basrs spamming Nesubipace E, he a , aa Se! (b> Be = 4 1OXE | prajerbr anto & ost asl [PD = Linear combmativa 4. prajechsas onlo tle 14,7 bess vectors,  Lif where san bat aga kon ~ Debne acher of egeratar A on bra <¥} (corn) ws = gei(aip) = <elalys LA ee produces new pro kugs nen : t art eet : Linearity 3 A ach an Le (a,<61 <4) ae Ot8 + AL Se 1A Adjort Operator Let /$> 28, j¥'>> ALP> € E {\y > 2 <b | - bok lon e) yy ewe apy (nee bet corrsiedne Debwe adjoml apera by A’ ‘ <ylat z <4" | => ale>=lk'> em bias Hl A ( gh tosentets ecanning combatant e. Drep ) . * « Slated = ei aly » n LPS Leie'>™ ; Le jor= <ee'y by scaller prduct Abaihor e <i(eps <elt'o*® a. peat => “A m& Hermi bon" if A=At « ropgerhes of Hermite canjugedon Chr ang A) (nyt = & (aay = ovat (a+ay> ate et (any = ot pt Ciexbi yh = ie x yl © Rules of sthermidan caqiugahe “get the adjomt of an expression « | Replace : - uments by compler cogugales | - i> hw ME oe Zt bw ID ~ operators by ther Hormilion aagugl?s Reverse. : order of alll Gelors e TTpertnce mm QM ~- Mathematteel operations (Solvang prblers ) ~ Eigenwabue equachbns 5 mak x dragandlizaben - Observable pigs tcaQ quark hes relate bo Nerniknr qpersfols |  » Pojeuters aver Le enkra shite space Pug - lu, Ku, h 2 dincete, artlonorimadl. wacis u Cusualls ercourtered om 1 yoor Qua) ’ Fan < Sate Iweywyl oe & « (not pblen encoun fered mW | year Qua} * PReyectson ons & chonges nothing Cepceserbabrns 4 Hels and Bras Aseume 214: >3 deerele artrenonme bests. p> = ZC, \u> ™ now campletely speci bed by the set of aff cp <ujlP> Pe> — /duted) q <uelP> = ce ante Zlu>% i < Ugh? ? <s reprecedeadan, anak KHL (KH 4s, SPU, 41459, ) = Ce", cy, <3”, res ) —> THESE “Lersnonswrs are by Convenor, Nek peat te be fonalls “clerived » Can be generalized ta coxtrucus \wasis  “Lepresetabrs of Chemters In a Atecreke bests 21a >3, peetor An represenicd by Tle saadttk A — Ay Ae cee Aas Bray hee Rig . where Ax 2 <x us \ &\ us? => mahi ellament 7% Tow Colmes Contnuous baste (mater elemate we Alea’) = <w, Lalw > but Hee “Smakerx” Aves not he aeeete element Charclar to cence pads cea) => QM cyperabions become exercises m vecter, mati x mranipulabry ex: Gen A, WD delermne WY" > > AW) (fs; teoue’ Ladiah> kavon) Lee IW = Bel > ols <u, t{e'> = <ula\P~ Use closure = 2 <u; (Aa e> « Cu [A Prag 1¥> zs BdaialuykXu; (vd = 2hye 6 > C 1 > iy es 4 ' tol. wecder Cal. verb  Soulely i ie le>= é bi Lure, Woe Ze. \u.> Lelalv> wa nusaloer given by + Col oF ey Vibe Ba, a VS PA Aa, a Can also Shove LOLA — a few vecter lanetler- brad , + i t { \* lherans fran Lon] tales i (A Jy = {A.) fe Ae Herel brn ay . Ay C2 Ata < ced # * : ar, CONtnucrs basta! A land = A laa Ie At Hermite s Alay? APA Ay CHANGE 6F Pere esEN TATIONS Soom Arscake orHowoemal belt 2 ju. >d8 Te : - 2 14, 93 Delrwet bo the aunbers See Kur lbp Hhat “mabe up the trans lormat ion "Rade 8, Teens koowog a ket Geen PSs EB CuiP> Wey ee [5 t Kanon ee panshon coedhe ian ts | Wann: PDs B SHIPS TED oe [a8 ‘ TL sceck + Ltel> che ermme these 4 number $ edd 2-4  . BS AIPS = APES | hve Nw a vonplen 19> ean eymvecy Comerstate, eigen tet of A with ebperuafor A Dehwihe of eee eter ( emerke! we) : “Lemove alorgucd, Mm Ciqenvallue by actherng Te Consrantvos. hr Iy> rr pocmedirnell To bo vei =f = of Bishrnck DS domencin a & = non - degenerate eegenvelue + @aly aoe eoenket asso. wf Wr - a fold aegaerate eigenvalue * there ate agente’ that all have +e same ergewebXe. These eke make Ke +le Subspace ey &4 eo oa G7 terensimrel Subspace Re é. + Deterararag Ha eigenmlucs 3 A - Solve “Det [a HAL I =O Sole Cor at \ tentt valuss of H Aelemvad —Tate® oven anh “Determining He. ergenvectars - For atch A, salve fA-2x2 29,9 =0 ee reaise vette atsce wf Zigenvatas A See worked erange on he Callowomg pages  \beve_Exembie Svgpose had, & Vs yuwt B basis Sete S, eo supease Vaal He vets CD What we the mech Loren of Meacosy. elements: a : 2 Brgaveluses, Sraevectyy, ad Trenbrmabonrs a sete space het B spanned by lu, > ad lugs, wits <u luz >= i ié€, aay stebe WW> eZ Can he expanded berms of [uy ad by? operator A 1s dedwed to act an all re By and Ahab AW >= ZS lux + ue \u.> rol A \u,> « -vifElu> ; , f A ow Xe 14>, lun? reprasertabyn Ta A « <ulalu « <u, i(m lure ids Jus>) - Kaculure ode <u lu> < Ae Ae Lu, ta luer= <u| (-2%& 1u,>) Ag, = “Uz lalu>= <uz| Gate + ods i>) cm = iN% Au, = <UZLAlL> = -§% Lu, lu.> z= Oo a. [ia u|% ta the (ud, Ie> represenbalven. ile o => A ve termiban (othe exgenalues Aw He ono )  (2) Wheat are the etgervelues of A? Soe: Det la-»~T] = o be ~( ow ily, Oo-> (he -oy-w) -— %2 ew - Wy - #5 Car WeXn- 78) > W,--43 A+ 4B a a C2) What ave the eaenvectsrs ee At “Lenole Ve eksenvectrs as IW D> ond (>, Cosvespordnq ty a, 4 Ay Crepectuelys. Sree o> \¥ > e € wn oan We andad 4 te MLD Ma> vegesntawond of oe alu > + GQ Jud iv >= dtu > + 4, lu.d ee (ay Mena me tbe (4, >, \u.> representation, 3-G (4) Now seppase we wart te geht to ‘he 18>, > represerledhon Winet te the cbeoncharmabon macht S 7 Medex elements of Ss Sik = gar lS By. <ul >= 46 “Secginat cepreedade 3,2 <u lh >= 2 1% &,< Cu, >= V% Sie CUI De AE > 8 (2) Whek wn the adj of S ¢ ste [ V4 } x W% 1G “we chetk: Gaadaate Ssh G3 " \ \ og et [ie 8) % ye ANWR WG < hy, +4 ) WG _ h2/3 | \ O° “LE. ue 50" o | 3 3 Smee 3st “just Lens forens rete, one mepresentabren | en AJoans Gres back, Ss* > s's 2 4 WW \ ~ | fe) Whe ore the vectsr seprassrons Gor, I¥>a IR> m te IY, 2, > reeresentos Ban! (~ Shoda be oly tous, 5 weres EM. Use tho ~brons Gemaien maine ty soda. lara of = ches. 2 si duty > L new represen whe WS ae <HiY> Kato. E Buses I> Siduik>+ 3) dus¥S &(8)+ BUR) - Le IY>- stat > + 32 eu,\¥,> -H5( WR) + BUR) = E+E =e => if>- (5) ya Hee VY DIY representathron (SreutA be olovraus \ fee |% > Ih > = as We > Fat Cay, so du Was SH <ulY 8) > [‘e CH) + VElM5) \ (e) 19% (00%) + Hels) } LE (43 also Ghoulh have been alovinus | 3: These alse coud have been sotadnred mane guichly byt + IY. = S&S 14S. fm IS > ww ld, a> ira sig, basin ) 2 J%\f AB \ he °Y% YG © (| ard emorly be > (5 Trarcborn A mis He (W182 representerbon. A. st Au S Re ef, i% \ie Ye ills a wg Mle 0 NG WG 2 & f-é i (nw [: -u lz xe | \ ite fo} Nz 1 . - [% ° n(S: o O “ia ) TEAR eran Qeesuit 2 Whe an cosates ir expressed. rm He leash of ‘hs agpavectors tle eiaenabuas ASE On Avaganall and alle elopracts are. Berd, _ENb OF WokKed ewrnthle |  Urimes Opeearoe s CTE Comp. Se * Given en opambr A, Fag the prepert 4 » An operator AU te b> = ulbh>, ES - Ul > => <b>. Shiuurpee <v¥iky > Trang formate of beer vectors Leb Elve>3 be Aneeke lusts NE, [V7 we UV > Sui lvz >= Sj —> IN. 28 or-tlesa ermal’ Conodor IP 2 Eula Wz UPPD 2 & 2 i> ulW>= Be, lv;> uu ys Zc Ulv;> — 2a, > = I¥> => S)y. 8 ahe aA baew E " 5 Unitary sheen aborm akan carcaspandis “le & change of lezen uv UW» Sku. SLID = Bu ~> Solving EV equakms a» atng a bacts pe ditch agente tf Avaaenal goeawel analy robatiors reflectors, yaversnes frensh gas.  the share shay rs Grmahvas wets: MP >—s 1Y'd= ulP> ores: SPL ef =e <bjut agorainrs * Given epeely Ay te teaneGrn A % A wxl( be tle cooler wht - m the Hanglomed base 217 >3 - has the sare makri% clement as bk wm Stgmad BV 43 bares \G>- <vyplaivay ~ * Ae uaut Eiseal wes ak ~Ersenverts trans Gonna Hens Let AID = a AD | 1G >= ult > > ARi@>~ Cut I@S -ualute) ia ~ 2 UAE, >» aUibo=< a (fa > = Kie>-al&> —> etgavelucs of A BE Same as etgtrvalce ay A fh so ft >See KR a observatle, s vm OA iy & = ey, a Example of a woitary kA Uley) = (nr Sud => Ue) + ( 250 -sind uu ef wse { (rea? ~ wU Suppose hap the In Hae 3 Iw >3 bats , Debne an operator A which, wrltien as - Now, tranfora We tyasis vecbrs and A M>= wd > [me cane, (ae) PB tose e Sm © w,> " > wt < . V wy -en@ \ (250 S00 \ cos) |} Sin® \f @s -enB \ oO ° ( . 4 | wy }\ ev case o 4 + => A = ih (ua witout => UL ye ack Hered fang * ) SPR aot any ooseruable: 1 keds 1 > ark ine > Gra a aoe » E, woa[! ~>(0) o ' dw Re 2S vaptecetaken Ts (8 > temo | : bog U.  | | | 4-@ Nebon of SQ) ne 21g 23 cepresentatoan fet 4g) > <4 > > warebuchon Dr Lg>$ representa cre ae <4 (Sod 10> = <q-2i%> = ¥&G-2) ~ SCA) — Ararslahon lang 3 BASIS a 4G) oo Gg) BY > ~~ : > i > * stay }P> Actas & Pim 21573 re A téR wen se = Pe +p + Oe') Salk parameter a ~ 4? Kg 1362 1P> 2 8G) + de <gIRIeD + Oe) Nee = gre) => Wlgse) = fg) 4+ ek <4lP IPD equality ra cna BeEwe => KG JP IY De # bron Yared - Pig) BERS z 2+ - £5) ~ FAK |¥? => B assoc wv kA wm fle agers rep T ay Geneabeatn of, CTI-& result G, ~ 2 (es3 Sepresen foAonn . (con be Oden H BD Pe-rKV ia 3d ) | i ot BICDZ and los? representanons B +P eperaders. CT thee ‘Consrlar the state space e.. State- “Space equivalent et te acho. SALE ¥F ef ab php teal, realizable wayelinchoes puer the pes? ben foord note FE — Ff Y PGS 6 F hen there % & Cor vespordivg (pd < &, C wsih Avap Game Fane en convenience. ) Sealer Product Scalar product 3 ors, iry e F (ers, ery ) = fac Clery Vee) valegtelon ae aff space Scala potul % IP, 1P> 2 & (yes, >)» <eLe> Let whee SP. be retenbcal ¢ (a © Woke, “] lrof a wervabeny but based et ot Aobnthpn e, Saller paduch Bases OD Lunclvenad bases Gr Fy ¢ OY, Vp od§ 59> S(e-5) VA ROA VL Or 3 = lank) 2 exp Lt Oe J > altis ag we Can wag Hase so bese Gucters, Hey are nok alononk ay © ¥ BO), asscerts, & basis abte ig > v Vp,CF) 5 eesocraly a vasrs slate \a> —~ altinoudn He bases 215 >% 2 p>? Lan be used to kere any Wo ¢&, I>, I> € &, ;  + Stamented rele trong C beste elements y Ey CR IG > = 8l-G) Cpole > > SCpnB!D {, InXgh = £ Sotto, \eo Kea 7 A Celesure reledons ber conhauons bases } v (ps €E, ant the assocratect abies EF + rh>= far, Ig ><GlP> can ids repreurbeoe se PP>= Sa2p, Ipo<PIPD — | ctvore, Bled} or S1ps>% ) * dere SGI = Sah $7 Bees > see boved relahwa, 5 Ke lP> = Soee vities Per) pret Page. => <n 1¥> = 4) <p, lY> = lp) waar e Yd = Fourter fpanchren 3%) — Wate hinchron s Lops ba posi xen space - ¥O@) hn nopertum Space > wterpre lel as He cantruons set of components 2 ¥> as pajectest pa etter “tte ZIny3 er 3\p,>3 baets, * Smee tere Ba LI cacresponstence behsee Go and (> Wd 6 2 and betveen Po anal {Pe > Ve > Acop sabsecipt lo \, 7 | <A> = Bes Solb>= Fe) orien ere + <eje' ss SCr-r'y dp ip! >= s(e-p') closure : | (ge leXe} = [Bo Ie Xpl = A  Pore 210% Exabuate ech Ie e> = faecie><elRIY> > Cra © be ef e * Fp = lack Y Sao Pe exe Le Caxrey + Re Vp) . <o . _ = lene V% Sd’ . BR expe leet Pasheh Yip) " %, tee - (ark) Be \e et ¥6) 2 (fare steansiorne of >) Pas rt curb (ry x écle> a Vo ny jer 2 LOK Or, a <elPlyY> = ky KAY In B ahwmansionl spare ~ Sol lows Som Our Hosen: kon $ of ®, & cod Log wallena lug and rela ‘hors cps a He Bete 3 1e73 capresentahaas, LavBWo~ fae CO LER IO) .<clE&Aale> = «| - BLIYY = x<eIRPD-EZ GIL) 2 Ex&delo> - FEEL) ze ikdr|v> => xR de vk : wate ; . ce - L®, e;] =°0 CAI OIA aad Lp;,@;J-° ype 42,3 Lom unin gn) a *s : RELATIONS ce, 4 = ks: [er ¥y¥,%) S-& ° fe Pp are Hem than OL RAP >= SBLENC abn elm! i ‘ v, = ey eee 5) elovent je “ > Hermihan. Sane on Pe, Ge 842,33 ) “A ° Z menvectors of EF R,le> ~ cr, le> F lp >= p:. |p - thy os why we say ted le Drax Abt fuachon 7s an ergentucachon of le poses operates , ard fat a plane ware i an igen func hon ot Hag ewapren fur op ere tor, « Re Po ge Herm) bran, wt. ZD3 anak Fiprt vechrs Comets ba tong oo bases ow & c => g, 2 are Observakles © Siece a stele antguel Aabned by wpecilyrag a wordkircte bee, 4%, 2 > Cerganvalwes of & S92" cortegond bmg AA “te she ino, 49, zy Ba Gsaog, ° SF PB, Pad ts also a fdrdfoet> CSc. 0, ¥ eH: % alone. (ns an agervallue oe) Can not cooupletely Hers ee Guantues shake 3b Space, To bb, K would be a CSO by Hoel as woulh PR CT Tr -& Ir Ti Iv Postulates of Quantum Mechanics: At a fixed time f,, the state of a physical system is defined by specifying a ket |i )) belonging to the state space &. Note: Since & is a vector space, \ implies linearity of quantum mechanics. Every measurable physical quantity “A is described by an operator A acting in &3 this operator is an observable. Definition: A Hermitian operator A is an observable if the orthonormal system of eigenvectors of A forms a basis in &. In a finite-dimensional vector space this is always the case. The only possible result of the measurement of a physical quantity .A is one of the eigenvalues of the corresponding observable A. Note: Since A is hermitean a measurement of A is real. If the spectrum of A is discrete the results obtained by measuring A are quantized. (Discrete non-degenerate spectrum) When the physical quantity A is measured on a system in the normalized state |p), the probability P(a,) of obtaining the non-degenerate eigenvalue a, of the corresponding observable A is: 2 P(a,) =u, bv where ju) is the normalized eigenvector of A associated with the eigenvalue a,. Note: For the postulate 1V to be meaningful A must be an observable, since it must be possible to expand any state on the eigenvectors of A. IV implies that a measurement on |p) and el) yields identical results. Therefore a global phase factor has no physical significance. e-+4 ww. Peobabilibes of eblamang reselt a, (eval S} ots) Drerebe, non-degenerake ag etna of e-vala Plae (UIP > = <#/u,><u.{> 2 <¥/R 1% x C poyectr anh ~tle ket IWS Drcvete, Ata ercrite spe 4 e-valy Pla,) = 2 lu, “is|* ((Su0s over ail assoc, Aeawaalt Stebes + : 2 <b ue Kuk te> - <0i (2 lai kalt \ w> 2 <b lB YD penjedty ant sakpace E, in All cases, Fra.) = Ze leh JP 2 | get sun off ath probabs (freq Commonly calhed $ha “collapse” postales Mate maheatly relafes prdjectre ty He meaguremart process, “ee bette degenerate anel ron-cleger evel spectra. Besudkng atate ones ioe property poralizedl + SE. =D Lidamental staterert of how quontun Sastems evolve ra Home. Note - We ase not iRnow tp wihe tt yw debe pacha lehe => There ar no, places ~~ he peghyates ware kom of aa operat js specth hp or =p Pashilures af e Vncteperclar of represesteaker  Ldwon Keaton wiles Postulate TL slates het every mreacuraele guankhy must cerresperd To an observable, but does not spect fy hows “to zonctract Vte vbservaule, Fer gueaktres mm Cardestan coord alts, an addi kena Ssommedieaher mle” car be usec “We conctruch operates. Whey weeded ee! re = er mm Car teston Space, Bat, BB ABB Ceomnatehen rales), Eutus @-= & ZB but need Be. create ar ‘operatar symmetrre mm BP cc RG lament) > ERR + BA) cece we BR BER cee = BA LR 22 ~ at Classrcad Energy = fea Vir He dn + VE) RL P Nov Alaags the case. See CTMR- Sb-f { “Dent need “ta waacty about “hoy Se ous class  TThtetpredaten evdo-c Mean value of an observable A om te clite Iy> : —= nolaten <A Dy er just CAY if content» leas. —> Mean value +thatd wold be obbamed Ma lange of measurements en den Keally prepared guanhins Systems fie tack ta state I> J —> Abo called expectaher value, wat Te oot necessarily a sesetble meacurenet result (re, <A D> not Ae. an etgenvelus of AK y Seve d 3 A ~<A. = Ba Par+ Sa. <P lumi ] ay pres, | measuring Nea kegngnhe case zB + 2 <P lan pun un iPro = 2 <P La fugu Pd Samy = <Plal?> CNP = CVAD oe RMS dartaben + AA# Jen? -<ay not ar & number Operator ? Co mapactiole Observable s be related phrysteal quankhes an be srmultnresusly eternametl “i AB ae compatible, EAael<o =D AB share a set of etgenvectars Hoact pan E 2 for ane, (G., by), A le de Est emu ltaneous meagcrement of At BR always ghyes Cesult s Ga, b, ) - (© LABIA, set of e-vectrs Fe not sshareck , Sen not eeboe er-ve® pars (a,b, ) ant Fehake dam @ amale- siote 6-9 Duperpas! hon Praceige, “Plvystend. ct Zxample let A [ud = Balu, (i woop & &) sta tonary stales, ake grerey, atgions fates Nor naw Conner (Pld (luds iu,>) (Bae & (14> ~ pu,>) netmabieed 2 7 orHunaad ? Le )B>~ 4 (ker da, lue> ~ + Lu, lu, > - Lug lugr) ZO “ IB >, n> Spar He subspace cmsiciing of all bef Het car be created tam ID, [UZ >, Bet, [HYD wor steahonary sheles, Let 1@> = €, TA > + elt > Lnoresali ad sok fef?t Kegf?=!) Wht a le evloabtlst, ted an Aa reasucmed will gue &, | PY = (Lulerl= fe, uate Kate > 1° = fe [Cayo + leah (du, ye ol® + 2-@e $C Cok Cu RMU LY. leh Pile) + lel Pe) | + 2 -Re $ Cr LUN hay (YR NX | ONT ' yalerlerence erm Prob Hal a mase rere on IMD sll give Ala ceo E_~ a Ce Ore Ormplitvoles , aot proba’: bes tm ~Homse lives, MO as net a slabahead omixtwe but a eraperpasinns  Evotunion Ovcenmat CT Mt = dome Ee (ae OR Pee ee cs “The atutihits delermine Hla eudaten wa Rae of a slate IRCLYD Sect ast mmporte MW Quantum Mechanics, TRE Sees dvager Equatan THe What allows sue calenlakon, Onve we know ree>, after specifams ant hay YY we tan cateulafe dhe “bee abapendvnce of an expectation value, of the me depevtence of probable outtones of mneasuremuce, The evolubroa operetber Ul, 4,) allows one way emulate the evoluhor of a shabe Com IHC, S to wba : Pears = alee y 1b ep “Ting sechon ATscusses prope bes of ul4t,). Geneene Proteenges udty, ty = AL ~ . he [ues busy | = Hed} [ ute, eked PLD | Cees => me Ulbb,) = HEED KGS) * - ul&t, d= ft - ET ha) uty eat! Rh, » U(eeds (44) + ule) (reverse evoluden ) » be uN hary Coustevatis Sus reas ~it (4-4,3 ’ as {Sr ¥ 5 ~ 18, (ttn SE WIGS Bled, UBAUEAr C7 Ve H mdep of hme => ultr,)- ©  7-4 “Tw. approaches + OS We could expard Wl > = lad wm Aeres oF Ye energy etgensiale laaais (Te, the caborary J fes\, tee la,> = 2 4 lu? = \WiO>: &B ay et lu > 3 Is CHP conde ro toe Yes, since ERHt eo, lg LAD condcat to Hae? a t Gonerallly vot , Since FELD RATAN? Fo ( assume A has no exghrcte Swe. Sagedonce ) Tefermme <A? US» Lasts <vlala t¥as>d 2 Sak TO Ca ialyyd wn Fee the cpecea? cope abe . lupe = lap>> Cook goer rae 3, Ht Suen becomes ile; -By Ye/y S Ay A, e* a 5, 3 < 2 2B; Pag adack ig ? bea - redierdiad,  7-D (25 Pe word agptoacd, Te perhaps more racer hl, Us ng dle hee Aroluchon opetber- tee) = @ (¥) > = Ulgos 1¥lo> - ulsed las . ots lay Lilt New we ask ibe. ques han, does (kLd> Vernal. Mw On exqens ticle of At C we brow hy ack os an Qrqeystile >} , since, we narasured the quank by, corsasgaral ren te 5 Checks - af to, Wey > = fay ure Al > = a la, > . S, ML> slats gar m ar eigenstate 4 A, “For £0 AUS © ARS 1POYD = AU) 1&5 ison equal to 14> 4 CH wedd we WF IES slays Won ergantabe ef lad) Only if TA, 21 Cornea + it TaU3 <0, then { AWS = Uleed & lady : Olga? lay> o Ay Wey Laud =O andy e& Taye dso, Stove Land to ww queral, ~fhen he. susiers Aes aot Slag Mm Sa Ligerwaks, OF A, ~  2-© Se ulvek lergage do ue use ty charecerive Hes Stiuaion? Aw of 4a G\ouwrng ere cacrecti . \Ways> VS tho chake ot shoe Sgsteon al £70, 4 : ad. f ri _ « sy % % the oith, wh sa, Sig en as b©o. © (eLES> ow the choke ah tba Sige ak alg tay het. : Wit, > is the ote ol Ha <p haan ack dare +c and sererall, Abes ig ret ai $ as He slot, “ot Su Syaten ; et ant Lyne se the quackun Reb evolves pw Ame lmmedtesh abler to measurement, he syle 7 an eroens boot of A, bat 4g He Svrolves If it ig Me longer Man - taonhet, of A Cageeral \ S. However) -He spn with SHI he characterised ee , \Pus> as (vag as HH sheya wrdapendont of ime” €4 ny measurements are. rade, whrda is molred by seymg Wo stage Matopendect af Are } + “Snect Kees“ or “wane Badly alk a measurenant at £20 mélam Woes ~ Mat fy we . Uld,o) = 2 S [ — THAR GO, Bee > Alwar> =(A- cA )itio> s AWh>d = ide A lyey> - nla - ide 2 dy Ae; ls> J t ‘ tegenstake, start of Tle evolulron ok ft Bd bg firm faz> > AINW>< aa —- ae (4 ZAG; Ruy ) => AIWKY> F aylar a long ag “Hin Second cheren reeds oral, ot as loag aa Ya progedan of NW onto 1a, > doermnrales Hoa decerprien of hl?  “groucd stays” Seed, but “parser mpves, | 78 Taterprerhalsan Sed vs evelving Coordicahe soslem euh pre. wh pie. Sch. Pre yes, Pre Drsserutea: Os, Pe, Ny oe R, Ty Os (oot wt, 31%, 41.53 (ip se % 31,23 i% Vn ? Be ead igul alta? Shh) Skike evclves 19, 1S to? _® be + 14 a gy b> i Coord ahh, System i Ovo hvedn Anahogy Aasceireg soln of a RErBOA GA A coovirg dred Sc pre Less pee + you are Slarding onthe Girard. + you are alse on dem A greundk wpves” but “petses Aoestt move” Zhe wey , wm the > She moasurale py srcall queakhy B Masperctant of daccrigkba> bof ane Ae sectpkon may | aster dhen te oller Alger dng en we pre blew ersan cets fs A to B ? g e pt Saat Avo at  Llane t 4 ae b eRnbers oredoane t ghipreal aaend Py HA, (24,6) | > mahal tome Aaponvlrce, Whe om OAsrerend mecharien Altes, pl, ). Hetrerburs pire Eq: 2) ponent mm A Ruy = -AVER) ee Gly lok the a ak Concvead 4's » moafoon Compre wf Chrenlas tam: t J ANCES ™ 2 LQ = <u) > Cioerwenetion Pieced - Tree depetene m states anol aperators » Use hl wher Here RB Ree Leprdedef? ALN a hee, wickepenctint pet 4 H. yo Es bet Hoe Hiv vie), . Te Be. pthwe Chime datep rent Der : rhe D>~ uy, (4,4) [¥els> s -th b-4) 4 ik due 2 WU, ~P@ UbLI IM YD = hte veehr mw benches prehue  Ping [Heed >= WlslO> mh Sz, ik Aue (> eit U, Sele >= EDU PORE MED = Ve Me> Ve UO = bo SS YO . SE, Talerauteon picfuns, . Evoluhen duet aca perhela ton! anhs H ° Tale : - boordinale sgatern evolves m kwe. ‘ rhe evolves if ye fo  “Probabilih, current ' i bards fh x ThAd~ be bie)" Yoo - ce) x ne jal® T_ clrssreed velocity , pesiie = Yoreael* correspruds te moka af purtcte coming laa - 2 Momerchon represerla.Sen Voy 2 deo PH wm 2. & ~ pith, | lex ° =) 400 = |e é 28 [he oo (AK fae) © ~ SCP) plane veut A post bn. space a=> § Crenehrvn M momentum pace | Wav enaceg Ts (aespteed alaks } tye Conver Vix ds AGE Aud 20 be xt Vax eG =) (A - bw yx iy 26K Feds a fae Ai) e —p FT of Alw), tanked at pr ke Note ¢ WD Ts wor a Soluba af “he Bray eve eg. Cor Ne V, —P Net a sTateuaey smire, Wavep ackels are Not stahonary = spread w/ Ame Cese Lusotn » yy ree space , “52 plane waves as a basis fer dob Curchors), Wave packets ase plptral, ape mimate solubors (fe slowly vargey. Aled) +  vik) bre te. S _aibaile, ID__sqiere well l 1s, UGRD < 5 © OLKLa ae loeutee re. Genend solr's to sé. i wile Vane Ylxd = Aci + pel les Ae BeKenw o é\seunore Boundary condi bons |! € =o lad <0 Nac malized €, — E, = wx), $C) Gite bE) Londons euey vere Cle) corbruous evryubere extent where Nik) = fae => Asa =>!) BIA om kx => b- ne fa, nom keger a Je e/a) (rete, negahve tategers are technical nN mbger, 2] Ok, but jut megly an ever please hele, Saw State as Tle poss Ave n shies Ap AS ema*  Memectarn,_represeta ivan ip) = asf ark = 2 Ja Zi Nak © Jeo (100 OM* 4, (nf = gq) lr (o-) + Fis) = = CP*/ ox, (PAAR) (-1 “Ele ee \ |  Positon and momentum ranslakion operate rs -U4,P/§ Derwe : Six) 2 € x bape & fe Tle) Fe Qr what do dese dol how ate eeu used SO) Exawine Khe commutator C&,§ (2,4 L&, ee Swe Leslee = OC, cesih-o | PB Ral] -o, CS, 8lxy3 CR, 8 AS HY ys the (~ %/pY SOx, e " x, 8 (x) a tes, LR, SG) ~ £868) - $GD% > BOY ~ SSK = x, 80x) Ka = Slay % + x, 8%.) = Say (K + 2) Exanrne achson of Stu) om 1e> &8U) bs) = Bah KD = 3ln,) (Rix? + 418?) = 3a CH? © IK) Sa ee wD (a4, )( SE) n>) > 3éx,3 n> je an etqencecda of KX wi e-vall (x4%,) is _ a w ty = \SG, > = \xta5>! LEE 7  Sle IK? = REG? > Sl) +onstabe dhe erapnstates of X ‘oy an aronk Ky is Slx,3 usilery § . . ix, O/e Nes - sce PR Verntisan & ani lary, - ant Ux, b/® > Stings = SG) = but Yes & just Sl-x,) > Sle) = $x) 0% RQUWALERTLY Six) - Sten) Smee Slr = \K+%, Lx StS = £x\5C%) <x+4,\ = ths also ruples ~Hhot L4-%) = S418, \ Now bot oat He positon representa kan, ath arbi beary cafe \Y, BOS = Lxi¥> thou te dasertbe Ula, where OC = CxlO> = Ce (Sexy 1 > oS 4, (Y> = Blew) = Rix) im “yak a copy of Bx), sintled Wy aor" amount Yo + Wer = Slav? 3 ™ posrtia  i RVI De es LAILARY = LYS S & SGI IYY We Wer already Gund (on p. wP-3) Hoa VE la y= SR +%) = EKO = SPS ar Stalk +a IHD 2 <8 eK IY> . ance Slt,) is usilery = HIRIYD © A LOIYD = %, + %e Even witha wvokma the pothen representation, woe con soo that S(x,y Bill anG pesrhvs expedtahian values, | “erly, # 1B = SEI, i thes Bind) con be obbmed Cam Dey | | boy replacing ahh wstancea of arth Y-% a xy BG-w Fle.) moe ah) a fo X 7K . Aw analysers of T Ce) = & proceeds with ~he One meteds 24 os clone wonlh Se». Stet, DE, Fp rsd. CA TAT Mg = RUA FR) 2 ps Tp) => Tin) = Tasl(P+~) > BF/ (4p, \p> = (prep dlp? => Fopripr >= lperp?  After — splurw Parr A > we will sel up He olubten via appre OS, ond we wth solve she aroblem sing agerocch (2). BeGre omy shah, Ehren Ces b's ive wrt Qn LR 9S using Jhesrem om peels ty demonsttale nes ublthy, anc to emphasize that we do not need IYLeSY order “HR eernme La7 @). Aeproach (39+ \Ne shart cord ~He 2qua hens of one Ran as LEY ord <B>, USN 2qua Wen on Qme b-24 of et Chm: AKKW= EET, WIY + CHS Ee KBP = LC TBAT) © Flay ~s need CRA] = (& Mad + [K, ves] > 09,8): 8 2 th B/m ak LA] - 18, Phe Te C8, vee) » [8% 1%, s -th hy > Ley = KBD sme Poe vo Tuis is threest’s bLoprem, #iby - KE as alee gren mm CT cat, 4h D-34 and 3S,  Our provlem Mvalues a Lee parkele, cad hus ANG) _ ae ° (sur debaer ie. ° feee partele’ case ‘, 7 2&4 => <b> = const. >- {2> = 8D Me SS We are aiten hak ab bate, dé = > > <P) = valied Gr alk & => the centr of 46,4) unl stay fixed od € i rmeomartiun Space, > EE = Phe as LOW + Pty + Conch Ubsmeg eo aren railed conde tron CK . Ko, 5 % sigle te _ che kerning. i LES lye Po (t-b) + Ke This 1s He togwer we wh abla hen wae o THhrouds tLe are. teyalved Laleh 3 bons, her solving Par A  F -10 Fedyen “te Peer A Ragin, we ase Ladheseg for a sloke > suche Yea KdUbS* %, ABMS pe and Yea ig & Qqusian gay + heQh-urdt “of a. Approach we'll start wrth a verg single Cand pacarrect } Gest ques 12>, and “than try To refre | the guess lnaced an what we” brow adoout it - ets Guess + @ lee = <elay = ne Pfea® where 4 ib a normallizalion conghert 2, - - -P/a2 book at Wirt rls MGs phe” —» this & a Gaussian with b hdl f -wietth of a. (a Gr, 2, Evalue <AY: A -* <ede {se forte) p ? (pe) ol fae ~P Yat O srace. this 1 re is on odd Lanetian Ya sie va bared. => oesnt sahely loodi Kan <é >) = Po 3, Who alooud £x> : KKY = Cap OGE) (HAVE GAS 2 nck (ao o fleet Q Phar t { ve Ze : tute ck (-ge) Sap pote <O [cad bec) => ales goeedt salioby LEU) = Ke  Bus Se, @, (2,4) * yo 2 wl Cm) (e-) oP Te i erry ne = oy Peek = we ie fs we already mete, i | Ko Tde Xce. Gorwg Litas, for's God “: Swace we ae Gee te chsese ae 0 vert. constant prase actor, Bax 4 be reel and posite, Sace tat wont affect . measuraiole quon Phe s, Need - le | ae 19,6, 6 )\ = = yt -@ py => |\s " de e ePo Ye ft bt welp-P/a Sy daw dela eur cae bene (adu @ x nia \ dus @ Ne? EWP gee separa page REF -2 va course wots, ate  S-i¢ Les autcely check “to see 4 ce = Ry & Lé>« vf j ee Pe Vee L jar F we (pted fa due Fela p> Aw + By a " (ade > lau*Pe > « ot oye [« {ae ue” + p {dae | Nes Ce =O Vr = u a We e, <é> Pe o> works be 2e> Br we lenoea ie st nok quod br 2x> Now we need to create aartler stake, > oblamed hy usras he post Kon transkehten operator Sly); aS = SOR) VY Te origi seem whakve. to (eed get w &, ty b Fourer transform, tun to replace ah mciancas of me tthe AK, Tis does work, but hs ackrally voy dasy to do Hie re. qurted opera Ben A Yhe mamenhent, rep resenbadtans @, 6) = eo Orb ¥ lb) where 6 zp ™ Sa Whe procter buna cepresealahin, —_—_ Sly) TL TO “ty =| é, Co, 4) = ye ees o fe-Pe Vee ‘ i epee = sthadls ib ~ escentrally po werk, ‘  Be -\6 “Te Oneck | ln calcula hong bY, hy shidens wth Ve wroner bus repre sextaban | he exh en rhs tern will cancel its daughex conugale eoths Witlar dle LEY retegradl = mre otsa, arth. term has no cffed an £8), Lets oblam <KY using Sele muenactim represen 3 a UX, -lp-po =v - (-py C85 2 x - eo” Ah le-Po Wea? lx d eth Cp hea? u * he a a Prk oben (-2 iy - na) \6 oie of wY he FR a C a tk qi |4e ef PN /ae (= iv, ~ (ee \ |e | OX, “a € eee ee 7 Pot a _ (4 fe pes eed a et rn - (e-Pe Sit =o (oa& 5 n° % {ap @ 7 * y Hh & jane 7 z a %o oe CVG YE A, — =? @, CP, bY Ves all of he prepertres we wart, So Lt i rn Vn wiXep sR - Cee “pe [tat Peed We é 2} \ LP -12 Obtamma | ¢ BAMA Oe) bey Founer deonsbom of 4p,4.) bes, bs) = t Pip Var av € tere 4 : . + « fo la ip Wi - iP ¥o ~ (Po Vta® | pe e Lt. e Po) fea et up ted YY, de ot ~ oto? a ( e é fp Vf 202 _ tt 6 lp rPesle-%a/n Lpy bee : 2 j4ee e oe Xead/q 6 lp-poV tak 2 te ge , _blePo EK} . me { ee K o Ce-Po Vat Wk we @PV/Go 1. ‘eles dus a0 /Ga eM UR Ok Ufo ae oto dh mi e gia (de EE gt Se b = tha mS 2 Ma uP le-%e VE 6 2bu 2 1s © ( aus ° 2bu ) 2 Ue ilk _ (ut -tou + &® firm e 2 {ane > . ipfee Wa ke, 6 he gterth noone (| oa ie é due v oe < a ~UP¥le — p,xk x Nay = oe & & e on 4 SOF og ¥lxye,) = - tfo%e hi Upote ee ae \ar/, . | (a\e ea 2 Coe Seer | %/ / : RMS DK ke ~ (6% VA } ae e° 2 ewe where wee Ma, Ae (&) Ve. \" Crean Po,  Pq , vs r Comparing Wax, LY aka 4 2 opens we Can Mow See Hut ~ XSawe tok d = Ser) (ae (0 \ Coos -spece rep® ) . ~ la klar warsks, he anda hm week St T are, agetried — d posthus space 8 the sow ag Mm oro martian Space wm order to hag the Fourteen dens Grms CarreSpondy en @-17, Ge wk. this ts not Surprising ~ PY = SQ SFIS den Habs de LSGQO TORY 1GY onl Pets > <p |S GQ terlea> eT hrs pert of Hee too Repressres 16 he seme We also see how He usdts of He mornen hunt - Space ond. pos fron - space wave lrackuns we relaled > Wwe Kla Bebre mona an to Paar B, Bs Look af one mere, aspect, to ts problem : the Shaler clewiakers of % ae ard hod Hey relate te Hetsenkerg’s unceramk prmcsde 4 Delailen. AM = [Za*9 tay y* Well wont Ax anh AB. re te EK ot BIB al tre > Ned <x*? wt <b?  BY - 20 These ase shaightterward te cableulate + ED 2 [as cen ae Bab we Cem ddw => daar ax /nu “> Ke Wu + %5 ec yt aon XW {au (wuee) & ub yt c ktw | { au: ee Zui, fd ae oe fae™ | eo ey See Ve /2, ad = Sir ew je + Zz Nt \ wis fF yn } = we xe ep (ARS = CRED - LRM “ Wa! -x°* co Wh So dhe shaker doraten o A adusmiren Tes wehe pendent * the” verrable offsel, Aude scuact 4 Briley we Cain Named ately (a . 2 z (ab eos as  X23 Easier approach use the theisenbera ercture —> obbam <K>CO petbheut cadealabn, Yljt) o Spey was 4 7 This methad wnvelves a Seb of a Crome of referee suck “Mak AY skays constant ™~ bree y Nauk Le ob servaldes tebe on wend — Qorms, wast: <UL = £¥4d\ x AY WY “Teg 75 often seen as ~Khe mare “intudve! Apervack , and 1 he methed enytrtoned by Skrjdmager + wance. the “s') on XY to Morente our ‘usual’ pesibe. ppereter, we have peen usiea Le all along) Bat HOD = UW, IGG + $e, <R7W) = KHQIL UE & AGES [HOD 2 <4G5 1x) \¥ G&S whore &, EY = UE & YK, OH) nat Clee) < en Pee lank Lr a fee parbele, > dalerome Syl, van gel <E?S, le thts really ana easier | Th tan be. Also, EH you know ” % "for a giv scenarte (ithe. the Gree parkole prallems, ~Wen pu Can age) Hes procectiue ane sas Pead, Shake 4 withoud Wnaraq th erabidiete the me-deperdence of Ue elake~rbself ray Rowe work ute a da CRerat “stake, es —p The Wersen'o rehuce tan be a relpbd Cadadtabusad ‘ool !  Ene We'll see how I hos moetrod werks ‘ou cabicalals, YS eeplicctly using UO, 4.) “a Tauck frst, weil 42k Xa an easier USO . Asing &4, 2 Mw CT compllencett Gx , we Carr ane BE 5 =O ey P, = tong ae TH ‘ C bar the Gree partrede ) ond HO 2 tq Py (ky Note that these ave eQuabrans Gos ae operakars , vot We expectaivan values, dyerg radio pendent \ arch since LA AY +o Gr tte Gree particle, Py = 6 (sez lotfom of QBIL, anck eqs 4¢ S mw er Cereglament Gogg 5 => ds - & ak %, (eS = 8 fon > RLY LB ete) + Mule) A Sace Hl = Ww) UL Grd Uld,, by) * A Nn na Yd = Y e 1K = LR Gt + Xl rt Thn vs tle Uererber positen gpanctar. Noke hak dry Ts on dxplrertly dae higeactint — aperatar,  &-25 Ce fo Ko > <RIWIs Pye) + % me Ge our evaugle poblem 1 We'th C08 cob ahh LES LS 4 One Laat hime, Sis lyme cbtamrmn Xy CO with te Ouralueon operaker ' us mq se nromntun representa hon, Sy) = UTE) Xe Ul) tp" -& Vawk 5 o ip? Lets fon $ Ber any Sree parkele ardolem. Be aoe a ow an a Gack & new Loew Cor oa UN 4g “Ay etoame ual ab does te a test furetren Bas ~ use he omuntue remresetaden MO GG = oP / ng [e's S «)