lecture notes on qm10 - C h ap ter E lem en ta ryS ca tter...

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Unformatted text preview: C h ap ter E lem en ta ryS ca tter ingT h eo ry A sseen in th ep rev iou schap ter ,th ebound sta tep rob lem ischa ra cter ized th rough th e sta t iona ry ,no rm a lizab lesta tes in th eH ilb ertspa ce .S in cequan tumm echan ica lsta tes in p r in c ip lea lw ay shav etob eno rm a lized (b ecau seo fth esta t ist ica lin terp reta t iono fth e w av e fun ct ion s) ,sca tter ing sta tesshou ldb e non-sta t ion a ry ,no rm a lizab leso lu t ion so f th eS ch r od ing erequa t ion . A ne lem en ta ry ,con cep tua llyno tqu itesa t is fa cto ry ,how ev er ,inp ra ct ica lapp lica t ion s ex trem e ly su ccess fu lapp roa ch ,con s istso fd ropp ing th eno rm a liza t ioncond it ion .T h is a llow sto in trodu cesca tter ingsta tesa ssu itab lycho sen sta t ion a ry ,non-no rm a lizab le so lu t ion so fth eS ch rod ing erequa t ion .S ta t iona ry sta tesa re inp r in c ip lee ig en sta teso f th eH am ilton ian .I fth eya resuppo sed tob enon-no rm a lizab le ,th eyhav etob esp ec ia l so lu t ion sfrom th e con t inuou s sp ectrum o f H . . F reeM o t ion T h e fo rce free ,s ing lepa rt ic lem o t ion ( freem o t ion ) isg iv enbyaH am ilton ian ,o ften ca lled freeH am ilton ian , H = ~ P m : (.) P o ss ib lee ig enva lu eso f H a reth em om en tum e ig en sta tesd en edby ~ P j ' ~p i = ~ p j ' ~p i : (.) W ith (.)and (.)fo llow s H j ' ~p i = ~ P m j ' ~p i = E p j ' ~p i ; (.) th e"no rm "o ftho sesta tescanb echo sena s h ' ~p j ' ~p i = ( ~ p ? ~ p ) : (.) T h esp ec ic fo rm o f j ' ~p i isob ta in edw h en th eexp lic itrep resen ta t iono fth eop era to r ~ P in coo rd ina tespa ce isem p loy ed : h ~x j ~ P j ' ~p i = h i ~ r h ~x j ' ~p i = h i ~ r ' ~p ( ~x )= ~ p' ~p ( ~x ) ; (.) w h ichha sa sso lu t iono fth ed ieren t ia lequa t ion ' ~p ( ~x )= ( h ) = e i h ~p~x := h ~x j ~ p i ; (.) w h ich co rrespond stoap lan ew av e .T h eno rm isexp lic it lyg iv ena s h ' ~p j ' ~p i = ( h ) Z dxe ? i h ~p~x e i h ~p~x = ( ~ p ? ~ p ) ; (.) thu s ,th eno rm ina regu la rsen sedoesno tex ist . Ing en era l,th et im eevo lu t iono fso lu t ion so fth e freeS ch r od ingerequ a t ion isg iv en by ? h i d d t j ' ( t ) i = H j ' ( t ) i (.) w ith th eso lu t ion j ' ( t ) i = e ? i h H t j ' i : (.) W ith j ' ~p i a ssta rt ingv ecto rfo llow sw ith (.) j ' ~p ( t ) i = e ? i h E p t j ' ~p i : (.) T h is isa sta t iona ry so lu t ionw ith th etyp ica lt im ed ep end en ceg iv en inapha se fa cto r .It isobv iou s lyno tno rm a lizab le h ' ~p ( t ) j ' ~p ( t ) i = h ' ~p j ' ~p i = ( ~ p ? ~ p ) : (.) In sert ing (.)in to (.)g iv esth eexp lic itrep resen ta t ion ' ~p ( ~x ;t )= e ? i h E p t ' ~p ( ~x )= ( h ) = e i h ( ~p~x ? E p t ) ; (.) w h ich isju stth eD e-B rog liew av e .Incoo rd ina tespa ce ,th ephy s ica lin terp reta t iono f (.)canb eread ily stud ied .W ehav eap lan ew av e .P o s it ion sw ith th esam epha se ~ p~x ? E p t = con stan t (.) sp readw ith th epha sev e loc ity ~ v ph = d~x d t = E p p ^ p = p p m ^ p = p m...
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lecture notes on qm10 - C h ap ter E lem en ta ryS ca tter...

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