1) They represent a continuum of energies (there is no energy quantization). where all values of p are allowed so there is a continuum of energies.

Transcription

1 Unbound Stats OK, u untl now, w a dalt solly wt stats tat ar bound nsd a otntal wll. [Wll, ct for our tratnt of t fr artcl and w want to tat n nd r.] W want to now consdr wat ans f t artcl s unbound. Rbr wat ts ans t artcl ay a on or no turnng onts. In ts cas, wat ar t allowd nrgs? Wll, wat quantzd t allowd nrgs n all t cass so far? It was t boundary condtons. For t unbound artcl, tr ar no boundary condtons, and so (n classcal trnology) tr ar no standng was, and nc no dscrt alus of t allowd nrgs all nrgs ar allowd. s n t cas for t fr artcl, unbound stats a sral caractrstcs dffrnt fro bound stats: ) Ty rrsnt a contnuu of nrgs (tr s no nrgy quantzaton). T fr artcl al of ts s: ) H E π wr all alus of ar allowd so tr s a contnuu of nrgs. ) Ty ar not noralzabl (altoug lnar cobnatons of t ar). gan, loong at t fr artcl wafuncton and tryng to noralz: π π ( ) d d and so s not noralzabl. Of cours, f w add a wgtng functon of nto t ntgral, t can b ad to b noralzabl.

2 3) Tr ar dgnracs n nrgy (sral stats ay a t sa nrgy). For t fr artcl, bot sa nrgy: E. and ( ) π π a t OK, all nrgs ar allowd, all ostons ar allowd wat tn ar w loong for? T classcal analogy s tat of dscrbng a collson: w a a artcl tat s ong n so ntal drcton wt an ntal ontu, t ntracts wt sotng (anotr artcl, a gratatonal fld tc.), and tn contnus off n so (otr) drcton wt so nw ontu. V(r) out n W wll aroac t robl as nowng t otntal, and sng t robablty tat t ncong stat wll scattr nto a artcular outgong stat. In gnral, yscsts do t rrs ty asur a robablty to scattr nto an outgong stat gn an ncong stat, and tn try to surs t otntal. Now, rbr tat wn w wr drng t Scrödngr s quaton w dfnd t robablty currnt dnsty as: (, t) Ψ ) Ψ Ψ ( Ψ Ψ ( Ψ ) Ψ) ) Ψ but, snc t ontu orator s Hrtan, ) (, t) Ψ Ψ I[ Ψ Ψ] and w dtrnd t contnuty quaton as:

3 d dt dρ dt ( Ψ Ψ) dv ( Ψ ( Ψ) ( Ψ ) Ψ) V V V S V dv d dv ( Ψ ( Ψ) ( Ψ ) Ψ) dv Ts just says tat t robablty to fnd a artcl wtn a olu V ars wt t dndng on t flow of robablty () nto and out of t surfac tat bounds t olu. Now, n a stady-stat stuaton, ρ wouldn t cang wt t and so: S d For scattrng n on dnson, d, t I Ψ Ψ, and on can loo at t d currnt dnsty n four arts, gong n fro t lft, gong out to t lft, gong n fro t rgt and gong out to t rgt: n Rgt n out out Rgt So tat for a stady stat stuaton: S d Rgt Rgt ( ) n n out out Now, lt s narrow t layng fld down a lttl bt, by only consdrng ncong artcls fro on drcton, say fro t lft so tat n Rgt. Tn, n out Rgt out

4 W wll also ta tat t otntal at /- nfnty gos to a constant (n ost cass, w wll ta t constant on bot sds to b zro, altoug for now, w wll consdr t or gnral cas): rfl n V() way fro t non-constant otntal, t wafunctons for t ncong ( n ), rflctd ( rfl ) and ttd ( ) was wll loo l lan was: n rfl Tn t currnt dnsts ar gn by: n rfl d ( t), I I[ ] (, t) (, t) d Notc tat s just t grou locty so tat t currnt dnsty s rlatd to t locty ts t altud squard, or, to ut t a natr way, t locty ts t robablty dnsty. Now, lt s dfn t rflcton and sson coffcnts as: R T rfl n n

5 and, fro t contnuty quaton, t s tral to sow tat T R. Now, fro our dfntons of T and R, and our currnt dnsts w a: R T rfl n n gan, ry oftn, as n scattrng of a ba of artcls fro a targt, (otntal s t sa far away fro t targt n bot drctons). Lt s now consdr so als: Scattrng fro a Dlta-Functon Potntal In t cas of unbound artcls scattrng fro a Dlta Functon otntal, w ran t solutons to t TISE for t cas wr t nrgy s gratr tan zro. To t lft of t otntal (V ) and: d d d E d E, E Ts s just t fr artcl wa quaton and ylds solutons:, < Ts solutons don t blow u (l t ral onnts dd for bound stats) and so w can t dro tr tr. Slarly, for t rgt sd of t otntal, D, > Now, f w consdr an ncong artcl (or ba of artcls) fro t lft sd, tn w can quat wt t ncong wa, wt t rflctd wa, and wt t

6 ttd wa. D would b t cas for an ncong wa fro t rgt, and w tn st t qual to zro: >, Now, contnuty of t wafuncton at gs us and t rstrcton on t dscontnuty of t frst drat: ε ε U gs us: U U U U, Ts, wt gs us: and So tat our sson ad rflcton coffcnts bco:

7 R ( ) ( E U ) T ( ) ( U E) Sral tngs to not r: ) R T dos n fact ) Wt ncrasng nrgy, R gos to zro and T gos to on. 3) s t strngt of t dlta functon otntal bcos larg (wt E fd) R gos to on and T gos to zro. 4) T wafunctons tat w usd ar not noralzabl, so w a to us a wa act as w a any ts tald about. Tn, ts sson and rflcton coffcnts sould b consdrd as aroat robablts n t cnty of E. Wat about a dlta functon barrr nstad of t wll? To a t barrr fro t wll all you a to do s to cang t sgn of U fro ngat to ost, but loo at t sson and rflcton coffcnts ty only dnd on U, so ty do not cang. So, t artcl s just as lly to ass troug t (nfnt) barrr as t s to ass or t wll. Quantu cancs s full of suc surrss! Scattrng fro a St Potntal onsdr an ncong artcl (fro t lft) scattrng fro a st otntal of gt V : rfl n E > V V wr, ( ) E E V,

8 T goal r, as n all scattrng robls, s to fnd t altuds, and. To do ts w agan loy our boundary condtons tat t wafunctons and tr frst drats ust atc (accordng to t otntal at t boundary). In ts cas, t otntal s fnt, so bot t wafuncton and t frst drat ust b qual: d d d d and Rgt Rgt Puttng ts two togtr: and So tat ouflcton and sson coffcnts bco: 4 4 T R

9 Not agan tat: ) R T dos n fact ) Wt ncrasng nrgy, aroacs and R gos to zro and T gos to on. 3) s t gt of t st otntal aroacs E, R gos to on and T gos to zro. Not tat as t nrgy gos blow t st gt, tat n toug tr s ntraton nto tat rgon, tat tr s % rflcton.