lecture notes on qm11 - C h ap ter E lem en tso fF o rm a...

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Unformatted text preview: C h ap ter E lem en tso fF o rm a lS ca tter ing T h eo ry . S ca tter ingS ta te s In th ep rev iou schap ter ,w em o reo r lessph enom eno log ica llyd er iv ed th ebounda rycond i- t ion sfo rth esca tter ing sta tes .W enoww an tto in trodu ceam o re fo rm a ld en it ion .T h e t im e-d ep end en tS ch r od ing erequa t ion h i j _ ( t ) i = H j ( t ) i (.) w ith H = H + V ,and H = P m ha sth eso lu t ion j ( t ) i = e ? i h H t j () i : (.) A rea sonab lerequ irem en ttha t(.)isa sca tter ing sta tew illb eto requ iretha t long b e fo reanda fterth esca tter ingp rocess j ( t ) i b ehav es lik ea freew av epa ck et j ' a ( t ) i . M a th em a t ica llyw ecan fo rm u la teth isw ith th eno rm a s k j (+ ) a ( t ) i? j ' a ( t ) ik = k e ? i h H t j (+ ) a () i? e ? i h H t j ' a () ik t ! ? ?! (.) w h ereth e lim it t ! ? im p liestha t longb e fo reth esca tter ing j (+ ) a () i shou ldb ehav e a sa freew av e .T h e ind ex a d eno tesana rb itra ry in it ia ld istr ibu t iono fm om en ta in th e w av epa ck et . k e i h H t e ? i h H t j ' (+ ) a () i? e ? i h H t j ' a () ik = k j (+ ) a () i? e i h H t e ? h H t j ' a () ik t ! ? ?! : (.) T hu s ,w ecand en ea sca tter ing sta tea s j (+ ) a i = s ? lim t ! ? e i h H t e ? i h H t j ' a i = (+ ) j ' a i (.) w h erew ed en ed th eM llerop era to r (+ ) := s ? lim t ! ? W ( t )= s ? lim t ! ? e i h H t e ? i h H t : (.) Ino rd erto show th eex isten ceo fth eM llerop era to r ,w ehav eto show th econv erg en ce . L et t< t< and con s id er k [ W ( t ) ? W ( t ) ] ' a k = k Z t t d t dW ( t ) d t ' a k j Z t t d t k dW ( t ) d t ' a k j = j Z t t d t k e i h H t h ( H ? H ) e ? i h H t ' a k j = j Z t t d t h k Ve ? i h H t ' a k j = j Z t t d t h k V' a ( t ) k j : (.) T h eex isten ceo fth e in teg ra lm ean stha tth eno rm kk ha sto fa lloa t lea sta s j t j + " . W ea lready show ed tha ta freew av epa ck etd ecay sa s j t j = .I f V on lyex ists inan ite rang e ,on eon lyha sto in teg ra teov eran itereg ion ,and th en th e fa ll-o in t issuc ien t . T o show th is ,letu scon s id erth eno rm in (.)sepa ra te ly . k V' a ( ~ r ) k = Z dxV ( ~x ) j ' a ( ~x ;t ) j (.) W ea ssum etha t V issqua re in teg rab le ,i.e .,av ecto r in th eH ilb ertspa ce .I f,e .g ., V isloca land rea l,e .g ., e ? r =r ,th isiscerta in ly fu llled .( V squa re in teg rab lem ean s R dxV ( ~x ) < .)T h en (.)b ecom es Z dxV ( ~x ) j ' a ( ~x ;t ) j Z dx j V ( ~x ) jj ' a ( ~x ;t ) j Z dxV ( ~x ) c t = c t Z dxV ( x )= c t c (.) and thu s k V' a ( ~ r ) k c j t j = : (.) F o rth eo r ig ina lest im a te(.)th en fo llow s j Z t t d t h k V' a ( t ) k jj Z t t d t c h j t j = j j t j = ? j t j = j t ! ? ?! : (.) T hu s ,fo rsqua re- in teg rab lepo ten t ia ls ,th eM llerop era to rex ists .I f V _ V ( ~ r ) ,th en V r = + " ino rd ertob esqua re in teg rab le .Itcana ctua llyb eshow n (K up sch-Sandha s th eo rem )tha tth eM llerop era to rsex ist ifth epo ten t ia l j V ( ~ r ) j c r + " (.) fo r rR ,i.e .,fa llso...
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This note was uploaded on 12/12/2009 for the course PHYS 270 taught by Professor Wilfredlee during the Spring '09 term at École Normale Supérieure.

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lecture notes on qm11 - C h ap ter E lem en tso fF o rm a...

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