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L4.4 QET CST L4| DRiVeW RARMoNic OscillateR (pHo) 1| Calculus o¢ variations avd Evuler—lacrance EYVATION We start by The AcTiow: te (4) | $lam,¢mJ= J L (4,49, t) de. ty We calculate tre variation with Respect to Zand §° t te fat fa -\ (2) $s=(sl(og)de= f ( 22 Sg +t] SG) de. t, ty \ 9 aa We do wot vary time, thus S49 = (£4) = < G 4) =? t L \dt “7 2 2 t2 G) fee sgde- | th Logae = f ae d (se) ay) i, 22 # i, 94 Now we jutecaate by parts tHe last Expaession: te [ot \ t2 e 1 9L j 2 dls) =(45 $4) -— J SteoF t, 22 \og Tt ty Cr — <= ~— _ {ae ¢\% _ ff J (aL) po de mat “hy jp dt \agl ~ RoUNJARY TERM 11.2 We fwd tae Evleg- ly crance EQUATION TOGETHER With A bouwdary covdititiow: \ —— oO ~~ / \t. d 9k _ ak And (aL) salt)! =o . (Eom ) at ah OL \az@) A, The classical DHO The laceanGiaw for tHe classical DAlvew HARMONIC oscillator IS (4) LW) = ga -F wey Pe) + FE), where wt) 1s A time-dependent frequency asd F(t) 1s ad ExtERWVAL force. The equation of motion (6) Reduces to: @) G+ wrtayga- lt. v w) 4) : Ce) Owe CAN Intaoduce tne foace Yewsity J (4) = ——— | Thus Wt g 4) + w*(t)4) = T(t). This is aw inhomoGewEous Secowd oRdeR dif fEREwTIAL Equation with VAaiable coefficients. Solving THIS EqvAtion iv GeweRAL is impossible. Foe simplicity ove cad Assumé time-independent fRreyvewcy , Thus (10) Ftp +e Let) = TK). This EQUATION caw wow casily be solved to ceive: 11.4 be more conveniént te woRk with dewsities: ti) =< FH / TREREfoRE | , P= we ™ | bie = 5> ze > hd) =e Lp*4Lu*4* —THgce), (1+) 2 Awd tne Heisenbere equations ( 16) become : an Po-w'4+TGt pwd Fe (49) We wote that tHe exreawal force JC) sears A classica( poRce, i.e. it has wor been gvantized: Notice also that tHe Hamiltow ial (42) wow haS aw Explicit rime dépendewce awd thus is not necessarily Con SERVEd.- Te THe ewerRGr of tue DHO \¢ not conseaved 1+ could lead +0 paaricles paoduction. As vsuval wé choose rue alcebaaic WAY OF ZYAMTIZA-— TION Vig CREATION 47d Anwihi la TIOW OPERATORS: nw acest (Fit) ++ piu) , wW v T _ 2 h— / ° a ) abH=sfe (gry + pa) 2 \ w 7? 3 ray =1. The wveese ts : A ToGetHeR with [ ACt) at rn GSfr) ys = 4]- . (22) fo ( ates ate) z \ )- PH) =i 4 [4.5 Lee vs substirote (24) awd (22) iw (18): e . \ w | gta) wf ate ag jatd Q> iva ( )-- Ge \* 777 . | / At_ 7 ee 4 NN a-h&=iwat tiwa-irnf2 57 . w + => Ay n ° At . a Aat+QA=atWHa' -wWwa ° ' a (3) |@Q@)= -iwatt) + JH), V2 au) [dtwmet wate) - Taq, (2w 4 (ast _solvé tHe Homoceneu VATION 4)=0: 4 - > a a -twt (2s) att) s—iwatt) | | alt)=Ae (26) At(t) = imate) Aatwy=8 ei" | a a Aa Aa Next we will vary me coefficients A=A(t) avd B= B(E), thus Wt wt (23) aca) = A(t) e7 Alt) = Awe’? -~iwhae ; a a i wot —— a z - . + (ay) ||&* (4) = BCE) e° ata) = bwe!?? 4 iweae? . We wow substitute (2%) Awd (28) back iv (23) gh (eu): L1.4 Ww “ a aa A a aA OK a -2(- at+ataraat—ga+ at Bt ate GS) wo 2 | zo tata — La at]=Aat- Gat =4 A ft - A nw \ A A (33) klo= w [atwaws+et) — 22 (atw+aa). 2] Jaw /] The scattering problem ("iw" avs “out seaces) SECTOR § SEeroR 2 SEctoR 3 |in> K Bo. | out> fo) Int€RACTIoW Ss T = <0 t o
states. S€ct2 2 Is THE oUutGOlwG REGION with |our> strates. SEctoR 2 18 WhERE tHE IWitial QUANTUM SYStEM |In> IWwtERACTS with tee ExteRnAl force Tt) foR a period OF time Of FST. le I$ simple To write ree solutions oF (31) awe (32) IW SECTORS 1 and 3: a a —twt (4) & (t) = Ain © SectoR 4: op PR $40, Tlt)=0 , a + twt j (3s) at (t)=aij, € L418 a _ . t (36) At = dow e Sector 3: . . fr t2T, Tit)=0 (sz) atith= Ase @ ~~ lw SecroR2 WE caw only account for tHE AVERAGEd IWF luEemCe na a oF THe exterRwAl force TCe) on the opeRATORS Aw 4A at i OnE Gets A Iw such a WAY th CT and Oba, Le. x . 7 T A c lwt 4 (38) Aout = Aw + J TIT e@ dt = Biw +To1, 2W 6 a+ + i we =—t (39) Qour = Ajn -—1t- f Tice dz at +55" y2~W 0 ¥ Where Jo awd Jo are tHe AVERAGED forces OVER TRE WwterAction terval o = (Aw +Jot) |Ow> = Tol Oiy>. (46) Therefore, |O0w> is av EiGenstat€ oF Qout with Aw cicenvalve Jo, We Remembeg tHe followwe definition: @ Eicenstates oc tre Awwih; lation operator with Now-2ERO Eigenvalues pre called coherent states, A Ln (43) Aw [Ow >= O Ava aw | Oout >=-To | Oour > t (4) |Qour[Oour7?=0 avs Aone [Ow > = Jo |Ow> - AL a+ Applyine TRE CREATION OfERATRS Ayy Av Clout 0” THE Cope- spondivG Ground states lOw> avd [Oour>, we caw boi ld two disgerent but complete ornthondRmAl sets oF Excited Stakes: A papyn (43) | Inw> = — ( w Ow > \™! ne IN 4 [ap \un_, (s0) [None > = — ( Aour ) |Ooue > yu! IN this case TRE Actiow oF tee Hamiltonian 15 A i 4-\ (1) HUt) (Nw =w(n+7) (Nin > , foe ted, (2) | H(t) | Noue> =wln+l) |Your, fe b2T. This means that INin > ARE EicenstAtes OF tHE Hamiltonian foR téo0, while | Vouep> ARE Ei GEn- L444 aA states of H foe t2T. For tis Reason it is NATURAL to Wwteepret |Niy> as describing iwitial paaticles £0 d (vn eT e€at iwteeaction with T(t) for t2T What we have fouwd 15 that [Ow > Is wo lowGer tHe lowest-EweRGy SrAt€e oR t2T. This 1s because tHe ENERGY UF THE SyStEm chawGeEs dve To THE Ex tERwAl force J(t). ity . u 4" w Relation between “Iw” avd “our” states Both SEtS O€ states { Inw>J awa {[noue>d SEPARA- tely form complere basis in tre Hilvert sence. This SuGGests that owe caw Expand THE INCOMING GROUMA SEALE |Ow> 1 teams oe cre basis vectors !Noug>, 6-2. lOw> = ZF. Ay | Nowe > ($3) n=O Wheae tHE THE Constant coefficients Ny_satisey tne following REccuRENcE RELATION: 64) Nye = 22 A, = n Yn+t let us prove tuis. We staet by considering tHE following Vey O= CNout | An | Ow > ——— (oe) 14.43 The constant No caw be computed by tue worma lization condition < Owl Ow > =t 1 Le, oo 4= <0n | Ow 7 = O Nx ZKout| 2 Ay \Nout> k=O n=0 eee eee eee” <0! 0m > ee — ¥ oe z =2. > Ny Nu < Koue | Nout > = > No An =7_ [Au] Kzo n=o ee n=o n=o Sun i nN Tz = use(ss) = D | 22 | \al*- [p|* > | o| =o fn: ! ! ! nzo ni Is. |2 1Fol? = N\* ee” => é =4} 31° (56) | No =@ 27° This @esolt fivally Allows us to Expaess |Ow> In beens O€ THE OUtGoinG states such as (see Ey. (63) ): £155}? n (sz) | l0w>= € 2 2 | Nour > =5 n=o n! ———~— fem (50) { {4 \"nln »S \Qeoue) V0 out {u! Jo Qt 1 15,}27 ($8) \Ow> = é 2 | Oout > It follows that tre iwitjal VACUUM StAtE 1S A SupERpoSitioW of excited states for t27T, Pagticularly, tne probabi- L1.14 lixy pe detecting tre oscillatoR w AN Excited State n ls measured by Lu —|3.1* {Jo\?™ (53> | |An|” = @ n! Tg we think of A StAtE with aw occupation number AS dJ€éscRibi icles ¢ € Inte 5 odu- ction OF particles dvé to THE presencE OF AN ExtERAl force Jt). 7. Gaeens functions and mAtRix Elements o¢ tre DHO The classical equatioy (10) fow tee DHO, 2 b* gu) +w* ge) = TU), dt? can be solved by (64) 4H = 9, 4)+ J teyved edt’ 0° where Jy (t) =ASinwt +B coswt 1s tae solution To tHE homo- Genéovs EQUATION At Tt)co. The Green's fuvction G (tt!) defines THE RESPONSE OF THE SyStEM To THE Wwf lu- Encé o¢ tue eExteewal force T(t). Lt satisfies tre Follow ing IwhomoGenEous £4UATION y Glt,t!) + w* Gt) = § Ce-29. ot? Eg vation (62) Gives AwotheR interpretation oF Glt;t!). 14.16 LOw|R4£0)| Ow > = A = w +2 < Owl 4 | Ow > =#»s e~_-~_—- —+~_~ 2 ——_-_— a {7}. + 4 Y Vv i (63 < Ow |A(t40)] Ow>d = 2, [ec ( oR- tua) f Ay (4a) x Sector 3: t>T => H(t) =W ( Bene Aout +=) =? als) + W < Ow | Ch + I" 1) (Gw +Jo1) 1Ow > = _ 2 eY&’+wilye lata — | Aw 2> SN 4+ 20m! 44, Tol Ow> + IW IN 7 v v C lo 2 Ww tw —— —— vo o) a + <0w| Jo* Aw |Ow> + < Ow | t* To 1 |Ow> as S rp) oO vo = feos sw 7,4) => \ ‘ 2/] n (63b) {Ow | H(t27) 1Ow> = w [Itol*+ +) = (420) +wlrol2 As Expected tre fiwal AVERAGEd EWERGy OF THE SyStEM IS moRe thaw thE witial owe. This suggests that praricles Miche have beew Produced ASLER tHE INtERACTION WithJ- ThE coWstawt |Jo|2 18 THE SguRREd AVERAGEd foRcE From (38) awd (335) OME caw weite L1. 14 bw (4, —te Jt, dt, Cr 544 Tete). + Iml* = T* 5, = + J ZW 0 ory let us calculate tue averaged wumber of paaticles (bu): A A a (at dw, #40 Gs) N®=atHam=4 " — _. | Qoue Gout, t2T SeéctR 4: a A A (64a) i Z i = = = <0; | (at, +3*4) (dy +3,1)l0w><= | Fol =0 There fore, um less THe AVERAGES S4uhRe foRCE IS 2ER0, there js A wow- 2ERo wumber of pARticles ON AVERAGE |v The final REGion, AFtER the wteRActiow (65a) Bi Sector 4: = < Cour lace Qour ]0,)>=0. VS (oe) Iw both cases we dowor have Am ivteRpREtATIoW OF THE Result, bécavuse tunis mixed matrix Element js NOt AW obseevable guantity. L1.43 Aa A -—twt _ n a -t t (ww) | ALESTI = Ague @ = (Au +7) en, Thus one caw weRite 4, (€27) such as nat ¥4) piwt y x =—twt a5) leer) = EWE + Chow + BF) Jaw We will weed also tne ExpressSin$S for tae averaGed LoRces : Fiwally (66) caw be compure d iw Sector 3 : nN - A tot A r aR (66) KOw| 4(t2TOw>= (Seu) | com dt +52 lo,.2€ + LO wl Ay +JolOwre wt zlfao yt Te" eivt + To ewe) = USE (#6) = =! zi ( Ja/ priwts iy piwt t t y,! piwe! ny prwe -— ove IVTT ED +r aoc Cc TRE 2wW NE 5 vow 2 T , \ ( -t! - (t-t! __! yer ( pie lee) be 89) 5 cary ; J c Cc ALW on \ T +o° _— 41. ( gmfwe-en Tag dt’ = J G,,, (e97e dt’ J e - 7 Ww 0 -o (Exa) where Gpo, (t,t) = Lsm[w(t-t)J OCE-t9 1s tHe Retarded Green's fuwcrio. Combining (66a) awd (664) owe caw weite (64). L140 lee us calculate THE w-oue"" MATRIX ELEMENts OF THE Position OPERATOR. Sector q: Oout 4 (420)| Ow >= A At twt A -bwt aw — = _ 20,..1at 10 (a) ett = out | Aw Te iy —— out am tw dud Vw. — . () twt =f Oour| Ow> => < Oout | 4(t20)| Ow > -_ 5? eve XK Ooue | Om > afaw Sector 3: ( HOMEWORK 4): Z Oout | 4 (t2T) | Ow > ¢ - eet < Dour | Ow>D aw HOMEWORK 2: LOw | 4 (t’) 4(t") lOw> =? , Is Sectors 1 ar 3. < Oourl 2(t') $e") [Din > | LO out | Ow > LITERATURE | V. Murhanov, S. Wwirtzni, lurroductiow to QUAWTUM EFfECES WW GRAVITY, CUP, QooF, | tanining hepi. bsu, qe/rtn/ncriyities/sougces/Lect MF Trev. p dé (Sannei au) 
