Perturbation Theory and Algebraic Equations with Examples | PHYS 6124, Study notes of Physics

Download Perturbation Theory and Algebraic Equations with Examples | PHYS 6124 and more Study notes Physics in PDF only on Docsity! _Pectucbatoon Theory Somakimes cannot RofI@ @ pooblem Cuaot bey , bet oan) ele. tt appcovs ty a time ‘ shia. t Appcorimaseley, in @ Umut of smal C axge paramokec (2), Often solutoon obtained Loe extcan Small Vara & works rather well Loc wodecoke &, Example: We Exn4 =O, O64 Can sobve oxachhy 2 ya ck aNavee Comacga, ef (e1<2 7 Teuncetad socteg _QNL Naky Qood approximatcon of the Ask Tosukt, Tor 3% CLOUGH need 4 dace ~ leleoos Q ate eee OY Bar AR Ant ee AS en re small: Catmost wear cpmecaoneg boumetar ys \ Atgelcate Equatlons toa How do we obtain an approximate anewar na Sy seamatic way ? Expansion method Repeesent Soliton as a Setiegin &8 WCE) = Wet ERATE KAS TK. Hey ox Bes 2 aera YOER- 4 RO (2gavn). 2 2 Crt OK te Vag ye SOCK 4 SC a4 = = (WP SEM KH KE LEK H gt) HC EMOt Byte) 4 = = (KS 4S 4 22m TK) COTTA SO. Ae > Pr -L eo => Ye et4 St, Qe sO => MLS * Cry ZAM gtk, =O => Xa = may Coty? ya ad —™ ~actdtes dette < cotnerdes Wath the expanse of the React colbson “p to, 0 (<*) Reautec perha'docion expansions AS Bo dolukton of tue par tustoa A Ce oy poob Lenn PRCORONRE Uo Solution ot Ae cenportucba dh Ce~o) gob Ler (245 x4 20) Exowmp Sa 2 eee xX +4 =O Pap) Exack goleton: ya aN 4k, = ts Cn4es%e Ze 227 2k wm wc th (eee e ete) = [AT e-2ehs - at Mie BS gd?a 4 => S- 8% = yrhy-4=0 Da © A 4 ory 4 erat, tocye Senabh Wo derms balancing Pay \ OS = £ > ex er-l =o d & a a A by 4 cena Darge Moke. Domdmant balantear: %-t =O => *=44 < WoOdingulec root & Nop tat toca Ruletg Coun wie days assume that the xoanslon X(E) cs. to Enbeqrar Powers of e ? Exompder: U-@>x*- 2x +4 =0 Foc GO + KR -Oet (a Cee no -S Kea l a ite Deganer acy — sign of tcoubte I MEY = A+ EWE Kates Qn=e)Cat Bx 4 Sea ceey te) +s = G- edt 2EXp ao) = 2-2EK Hts SHS AKOURZEK His HZ-22Kp rm +h HO Se ane ai ao Lthose 44 =4 vorveckly E's -4 42%,- 2%, = 0 ~ cannot be Sabiafesd tar Ay ons tee of X44 rong oxpan sion Y Loole. at The pansion of the arackroltis,, ko con whet Werk won,’ y te. a “(Ee = 2tN 4-4 (rey Aa oO iagtag te TTee zy \-e +ere..  Use <ascahing xe ttby tr Beye +dyy- 2A+by\ 44 = AA a re Gree ory x (1-858? y* ~ 2 28y 4S = -~e2-228y +Q-e)8%? =O can appoxionaka Le with £ ty dong dorminant bakanoe: gry 8ay)- 2 - E(B Ye - 8% -=0 a) SaaS => B= 8 => EY HE mK a =O 4 “ No BALANCE! coche emolR ox ad Cs % bee => EY -2y ol =o -p Yao # 0 * a. & o Large. Sirah srnalh \ Bo BALANCE! | “ Mea => §=e% = yao Bo. ~4 =O * * Lorge Small ange. Doosnan’ dolanag 2 y ~l=O0 = y= #4d > R= Aaety = Ab Ee en Appsoperake. Mx ponsion t xCe\= 44 exe BY, + OK bes Eyecose 2 Bd Ky. ve by aquaking powets o aie Tkevarine Mathod Cigh+ Lxo aM ELON by sokvina tha Bq ation ibecariu the. £ixS <0 oguivalent to y% = AON 4% EOD Tkerate 2x4, = F Od? Fexcad point of cherakion = Toeay ic acslitton ol £Oe\=0  . Ad, Convergence of Aac dons Culd See again Un stability theory) 4 Can we Lind the Pixel point by therakeng ? * No Khe Veet ty of x S Xn = xP Bye x Hu ag a Bar = FOR = FOCHE) = FOF FOALS OE) / nh 2 Saag ODE, = FOS Say ee FON 8 Trecaktions, ComJerae iQ (elo <4 Note: Chalee of FOS ts nok unique, have. to choose Carefults ho en®uce Comerar nee, Notes Not every expansion is good, Look for txpan sions thet COnVERER etl | > good okotee of yy te erthear | ra Excmp he: H- 8) - 2x44 =0, Rancike as exta x® = A =? Nnay = Ab 2x, Stacking with x, =4¢ Example: ye =e 4 = eo “ Ee y / NE g y Kay > NN vo Xp BS 2% =O 4 A se eg Yg= C27 = & Mas feta C22 = C484 = yeti D Kyar= Ont/ey~ Wns -%y = OnCve) Maw Oacves «Bn Chal ved) oxkcarnely alow by Convacaing ~