5- Scattering Stationary States

Transcription

1 Lctu 19 Pyscs Dpatmnt Yamou Unvsty 1163 Ibd Jodan Pys. 441: Nucla Pyscs 1 Pobablty Cunts D. Ndal Esadat ttp://ctaps.yu.du.jo/pyscs/couss/pys641/lc Scattng Statonay Stats Rfnc: Paagaps B and C Quantum Mcancs, Vol. Capt VIII C. Con-Tannoudj, B. Du and F. Laloë

2 Hamltonan of t Systm H w sall justfy t wav pact appoac. Lt s consd tat t scattng potntal V( ). T Hamltonan of t systm s: P 30 W P H ( + V ) s t patcl s ntc ngy. psnts,, t ducd of t studd systm (T ncdnt patcl and t scattng cnt). 3 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\ Dfnton of Scattng Statonay Stats

3 Statonay Stats Scödng s quaton dscbng t voluton of t patcl n t potntal V( ) s satsfs by solutons assocatd wt a wll-dfnd ngy E T gnal fom of ts solutons, calld statonay stats a: Ψ(, t ψ( ) ) ψ( 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\ ) E t W a t solutons of t tmndpndnt Scödng wav quaton,.. m + V ( ) ψ ( ) E ψ( ) An Infnt Numb of Solutons Dfnng: E and U( Eq. 3 s wttn n t fom: V ( ) + U( ) ψ( ) 0 ) Fo a gvn valu of (.. of E) an nfnt numb of wav functons satsfy q. 34. And ts s not an unusual stuaton n wav mcancs. (S dscusson Con-Tannoudj p 909) 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\

4 Statonay Scattng Stats (SSS) Wat w do s us t pyscs of t studd poblm n od to dtmn wc solutons a to b pt. H w av a 3D poblm wt an abtay 3D potntal. T pocdu s to us t popts of wav pacts n an ntutv way n od to spcfy t condtons tat must b mposd on t solutons of q. 34 f ty a to b usd n t dscpton of a scattng pocss. Solutons wc satsfy ts condtons a calld statonay scattng stats. 7 Asymptotc fom of Statonay Scattng Stats

5 Fst Constant As w sad wn w studd t puly wav pact appoac; at lag ngatv valus of t, t ncdnt patcl dos not fl t nflunc of t scattng potntal. T patcl s stat s psntd by a plan wav pact. Consquntly, a fst constant on t solutons s tat ty sould contan a plan wav tm,.. Wn t patcl bcoms und t nflunc of V( ) t stuctu of ts wav pact s pofoundly modfd and ts voluton (n tm) s complcatd. 9 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\ Wav Functon of t SSS Nvtlss at lag postv valus of t, t scattd patcl dos not fl any mo scattng potntal, t patcl s stat s psntd by t sum of two wav pacts: A tansmttd wav pact wc contnus n t dcton Oz and a scattd wav pact dfnd by a scattng angl θ T wav functon v dff ( ) s, n ts cas, a supposton of t plan wav and a scattd wav W sall not woy about t nomalzaton of t wav functons nvolvd. 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\ 10

6 Scattng Ampltud Scattng Ampltud Fom now on w follow a asonng smla to tat usd n t wav pact appoac. T wav functon v dff ( ) assocatd wt t statonay scattng stat s t soluton of q. 34 wos asymptotc bavo s of t fom: v dff ( ) T functon f(θ,φ) ampltud. + 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\ ~ f ( θ, φ) s calld t scattng 35 It s t only quantty n q. 35 wc dpnds on. V( ) 1

7 Pobablty Cunts Cunts T cunt assocatd wt a wav functon ψ s: 1 J R * ψ ψ 36 Fo t ncdnt wav functon z w av: 14 J 1 R z z 1 z R zˆ z 37 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\

8 Scattd Cunt T cunt assocatd wt t scattd statonay stat: 15 J 1 z * R + f z ( θ, φ) + f ( θ φ) f, 38 H w nd t componnts of t opato n spcal coodnats,.. 1 θ 1 sn θ φ, θ and φ 39 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\ Rgoous Calculatons T cunt assocatd wt t scattd componnt of t scattd statonay stat s: 1 * ( ) ( ) ( ) J R f θ, φ f θ φ f, 16 R R f f * * 1 ( θ, φ) f ( θ, φ) + ( θ, φ) f ( θ, φ) 1 f, ( θ φ) 40 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\

9 Angula Componnts 17 1 * 1 ( ) ( ) ( ) J R f θ, φ f θ φ f θ, θ R f * 1, θ ( θ φ) f ( θ, φ) 1 1 * R f ( θ, φ) ( θ, φ) 3 f θ 1 * 1 R f θ, φ sn θ φ ( ) ( ) ( ) J f θ φ f, φ * R f 3 sn θ, f ( θ, φ) ( θ φ) φ 41 4 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\ T Scattd Cunt s Radal At lag, 1/ 3 << 1/, wc mans tat t angula componnts of t scattd cunt (.. (J f ) and (J θ f ) φ a nglgbl compad to t adal componnt and t scattd cunt s pactcally adal. 18 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\

10 Intal Flux, dn T ncdnt flux, numb of patcls of t ncdnt bam wc coss a unt sufac ppndcula to Oz p unt tm) s popotonal to t flux of t vcto J acoss ts sufac,.. F C J C 43 Now t numb of patcls ttng t opnng of t dtcto placd at an angl θ p unt tm s popotonal to t flux of t vcto acoss t sufac of ts opnng,.. dn C J f. d S C J. dω f 44 f ( θ, φ) dω 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\ 19 Dffntal Coss Scton T dffntal coss scton s gvn by: dσ dω dn F dω f ( θ,φ) 45 0 Wc mans tat ts dffntal coss scton s smply t squa of t modulus of t scattng ampltud 'U1(UVKDLGDW3K\V&KDSWHU(OHPHQWVRI6FDWWHULQJ7KHRU\

11 Intfnc btwn t Incdnt and t Scattd Wavs S: Paagap B.d pags n Quantum Mcancs, Vol. Capt VIII C. Con-Tannoudj, B. Du and F. Laloë Nxt Lctu 5-6 Applcaton Cntal Potntal End of Lctu 19