Lecture Number 02 Unit 1: Quantum Theory of Collisions

Transcript

1 Seect/Speca Topcs n Theoy of Atomc Cosons and Spectoscopy Lectue Numbe Unt : Quantum Theoy of Cosons dθ P. C. Deshmuh Depatment of Physcs Indan Insttute of Technoogy Madas Chenna 636 PCD STTACS Unt Quantum Theoy of Cosons Pmay Refeence: Quantum Theoy of Cosons (Chaptes-,,3,4 by Chaes J Joachan (Noth-Hoand Pubshng Co.

2 Incdent beam of Monoenegetc patces Detecto Taget A + B A + B (eastc scatteng A + B A* + B (neastc scatteng A + B A* + B* * denotes new ntena state A + B C + D eactve scatteng - eaangement when codng patces ae composte objects. channe : possbe mode of fagmentaton pathway PCD STTACS Unt Quantum Theoy of Cosons

3 A not too ntense, no too wea. Incdent beam of Monoenegetc patces S Taget B Detecto n B S effectve coss-secton aea of the ncdent beam : numbe of taget patces B that ntecept the ncdent beam NA : Numbe of patces A eachng the taget pe unt tme Φ Fux of A w..t. taget B numbe of patces A x-ng pe unt tme pe unt aea noma to ncdent beam A ( T ( L PCD STTACS Unt Quantum Theoy of Cosons 3

4 Incdent beam of Monoenegetc patces A Detecto N A NInt P : : Numbe of patces A eachng the taget pe unt tme : Numbe of patces A whch nteact wth the taget pe unt tme. facton of N A P N N T Pobabty that an ncdent patce nteacts wth the taget and theeby gets emoved fom the ncdent fux by scatteng P tot, pehaps, fo thn taget A Int PCD STTACS Unt Quantum Theoy of Cosons 4

5 N P N T n B Int A n B Φ A Int tot A B Scatteng coss secton ( T ( L ncdent fux Φ N S A, A : numbe of taget patces B that ntecept the ncdent beam How s N Int eated to the taget patces B? NInt α What shoud be the dmensons of the popotonaty? ( L P N σ tot N σ Φ n Φ n N effectve taget aea that nteacts wth the ncdent beam and scattes t α Φ n Int A B A tendency of patces A & B to nteact PCD STTACS Unt Quantum Theoy of Cosons 5 B A

6 coss-secton numbe of events pe unt tme pe unt scattee fux of the ncdent patces w... t the taget x L P N A σ tot Φ AnB Scatteng coss secton dω L effectve taget aea that nteacts wth the ncdent beam and scattes t tendency of patces A & B to nteact PCD STTACS Unt Quantum Theoy of Cosons 6

7 effectve taget aea that nteacts wth the ncdent beam and scattes t Scatteng coss secton σ tot P N Φ n A coss-secton B A dθ ban cm 4 Mega-ban Mb cm numbe of events pe unt tme pe unt scattee fux of the ncdent patces w... t the taget ; 8 PCD STTACS Unt Quantum Theoy of Cosons 7

8 V( + ψ (, t m t t ( ψ (, ê θ eˆ sod ang e ê δ S ϕ δω θ δω eˆ eˆ eˆ θ ϕ PCD STTACS Unt Quantum Theoy of Cosons 8

9 ψ ( ˆ f Ω ( ; Ae + e ψ ( e e e nc cosθ ˆ e ρμ wth ρ & μ cosθ ˆz + m ˆ m, m e c Y ( j ( ρ a P cos θ j ( ρ ρμ ( μ e a P j ( ρ a? PCD STTACS Unt Quantum Theoy of Cosons 9

10 ρμ ( μ e a P j ( ρ ρμ e P ( μ dμ a P( μ P ( μ dμ j ( ρ a δ' j( ρ + a' j' ( ρ ' ' ' Othogonaty of the Legende poynomas + ρμ e P( μ dμ a j( ρ + PCD STTACS Unt Quantum Theoy of Cosons

11 + ρμ e P( μ dμ a j( ρ ρμ ρμ e P μ dμ P μ e dμ Intega of a poduct + ρμ + e ρμ ' e of two functons P( μ P ( μ dμ ρ ρ + P P st nd functon ρμ P( μ e P( μ e e P ( μ dμ ρ ρ ( μ ( μ P( μ ρ ( ( ρ ρ ρ + ( μ P( μ ρ ' ( μ ρμ P e dμ + + ρμ e e ' ρμ e P( μ dμ P ( μ e dμ ρ ρ PCD STTACS Unt Quantum Theoy of Cosons P ' d dμ

12 ρ ρ + + ρμ e e ρμ e P μ dμ P μ e dμ ρ ρ ρ ρ + ρμ e e e P ( μ dμ ρ + ρμ we had: e P( μ dμ a j( ρ + a + a j + ( ρ j e ρ ( ρ ρ ρ e ρ π e e e ρ ρ ' e Ο ( ρ gnoabe as ρ π ( π e ( PCD STTACS Unt Quantum Theoy of Cosons

13 a a a + j ( ρ ρ π e e e ρ ρ π π ρ ρ e e e e j( ρ + ρ π π π ρ ρ e e e e j( ρ e + ρ π ( π e ( ( π π π ρ ρ sn ρ e e a j( ρ + ρ ρ e π π e e PCD STTACS Unt Quantum Theoy of Cosons 3

14 π π ρ ρ e e a j( ρ + ρ π sn ρ a j( ρ + ρ π sn ρ Now, j ( ρ ρ ρ a + ρμ ( μ e a P j ρμ ( ρ We got a fom ρ, but t s ( μ e + P j( ρ vad fo a ρ snce a fn( ρ. cosθ ( θ e.. e + P cos j( PCD STTACS Unt Quantum Theoy of Cosons 4

15 cosθ X ( ( cos θ θ ( ˆ, ˆ e + P j Spheca hamoncs addton theoem (Unt, STAP X sde 94 Fo Z a pont on Z' axs, ϑ β; ϕ α θ' α β Z ϑ θ' Y ' e ˆ βα, uˆ z w..t. 4π + (X,Y,Z ' Y ˆ ϑ, ϕ vˆ ' ˆ θ', ϕ' vˆ ( ˆˆ ( ˆ ( ˆ m * m P u v Y v Y u ϑ ˆ ˆ 4 * ˆ ( ˆ e π j Y m Ym e m PCD STTACS Unt Quantum Theoy of Cosons m ( eˆ, ˆ z 5

16 ψ ( ˆ f Ω ( ; Ae + e ψ ψ ψ nc nc nc (; (+ P (cos θ ( (+ P (cos θ π π e e ( (+ P (cos θ e π sn( π π e π π + + e e e e PCD STTACS Unt Quantum Theoy of Cosons 6

17 ψ ψ ψ nc ψ nc ( ˆ f Ω ( ; Ae + e ( (+ P (cos θ nc ( (+ P (cos θ e π ( π e ( e e π π + + e e e e (+ P(cos θ e P(cos θ ( e ψ nc (+ P(cos θ e P( cos θ e PCD STTACS Unt Quantum Theoy of Cosons 7

18 e (+ P(cos θ e P( cos θ e z What w be the esut of scatteng by a potenta? ψ Tot ( condton: ( + δ ( + δ c(+ P(cos θ e P( cos θ e δ fo potentas that fa : phase shft of the pata wave faste than the Couomb potenta,.e. faste than PCD STTACS PCD Unt STfTACS Quantum Unt Quantum Theoy of Cosons Theoy of Cosons th as. 8

19 e (+ P(cos θ e P( cos θ e z ψ ψ Tot ( ˆ f Ω ( ; Ae + e ( δ ( + δ ( + δ c(+ P(cos θ e P( cos θ e Pease efe to detas fom : PCD STAP Unt 6 Pobng the Atom Lectue n: & /8 & /9 & /3 PCD STTACS Unt Quantum Theoy of Cosons 9 : phase shft of the pata wave th

20 ψ Tot ψ + Tot ψ Tot ( ( t, ( t, ( + δ ( + δ c (+ P(cos θ e P( cos θ e choce of nomazaton c Pease efe to detas fom : PCD STAP Unt 6 Pobng the Atom depends on the bounday condtons c Lectue n: & /8 & /9 & /3 e ± δ outgong wave bounday condtons ngong wave bounday condtons PCD STTACS Unt Quantum Theoy of Cosons

21 ˆ eˆ z ( ˆ ( ; f ψ + Ω A e e + c δ e gves: f ( Ω ˆ f e P δ ( (, θ + (cos θ Faxen-Hotzma s fomasm Each th tem gves the contbuton of the th pata wave to the scatteng amptude.? [ L] scatteng amptude Refeence: Quantum Theoy of Cosons by Chaes J Joachan Noth-Hoand Pubshng Co. // Secton 3. // see Eq.3.7, page 49 PCD STfTACS Unt Quantum Theoy of Cosons

22 ψ e ψ e + Tot Tot ( t, ( ωt + z+ ( t, ( ωt + z + + e e + + ( ωt ( + ωt c δ ( e descbes 'cosons' δ ( ( + e P (cos θ δ ( c e descbes 'photoonzaton' z e e ψ f e (+ P( cos θ + e P (cos θ δ δ ( Pease efe to detas fom : PCD STAP Unt 6 Pobng the Atom Lectue n: & /8 & /9 & /3 PCD STTACS PCD Unt STfTACS Quantum Unt Quantum Theoy of Cosons Theoy of Cosons

23 ψ e + Tot ( t, ( ωt + z + e + ( ωt Θ : opeato fo TIME REVERSAL SYMMETRY + e P (cos θ δ ( Θ[ ] ˆ eˆ ψ Tot (, t coson ( ωt e + z+ + Pease efe to detas fom : PCD STAP Unt 6 Pobng the Atom e z ( ωt + + δ ( e P (co PCD STTACS Unt Quantum Theoy of Cosons 3 photoonzaton + Lectue n: & /8 & /9 & /3 s θ

24 c e δ ( 'cosons' ( ˆ ( ; f ψ + Ω A e e + f e P δ ( (, θ + (cos θ Contbutons of Faxen-Hotzma s fomasm the pata waves to the scatteng amptude. QUESTIONS? Wte to: pcd@physcs.tm.ac.n Next cass: OPTICAL THEOREM Refeence: Quantum Theoy of Cosons by Chaes J Joachan Noth-Hoand Pubshng Co. // Secton 3. // see Eq.3.7, page 49 PCD STTACS PCD Unt STfTACS Quantum Unt Quantum Theoy of Cosons Theoy of Cosons 4

25 INTRODUCTORY ectue about ths couse on Seect/Speca Topcs fom Theoy of Atomc Cosons and Spectoscopy P. C. Deshmuh Depatment of Physcs Indan Insttute of Technoogy Madas Chenna 636 Unt Lectue Numbe 3 Quantum Theoy of Cosons OPTICAL dθ THEOREM PCD STTACS Unt Quantum Theoy of Cosons 5

26 hν ' + A * U.Fano & A.R.P.Rau: Theoy of Atomc Cosons & Specta PHOTOIONIZATION & eecton on scatteng have same fna state, but dffeent nta states. Pease efe to detas fom : PCD STAP Unt 6 Pobng the Atom Lectue n: & /8 & /9 & /3 PCD STTACS Unt Quantum Theoy of Cosons 6

27 ψ e + Tot ( t, ( ωt + z + e + ( ωt Θ : opeato fo TIME REVERSAL SYMMETRY + e P (cos θ δ ( Θ[ ] ˆ eˆ ψ Tot (, t coson ( ωt e + z+ + Pease efe to detas fom : PCD STAP Unt 6 Pobng the Atom e z ( ωt + + photoonzaton + δ ( e P (co Lectue n: & /8 & /9 & /3 s θ PCD STTACS Unt Quantum Theoy of Cosons 7

28 ( ˆ ( ; f ψ + Ω A e e + ncdent ncdent j m * Re ψ ( ψ ( m * * ψ ψ ψ ψ ncdent { } * z Re e ˆ + e ˆ +e ˆ z x y z j A e A e j S j S δ eˆ z A( vδ S A( δ f( ˆ Ω L scatteng amptude m x y z δ z j( A A v δ z m v e δ t ˆz cuent though aea δ S Pobabty cuent densty vecto δ Sδ z δv δ z δv δ S A δt δt 8 PCD STTACS Unt Quantum Theoy of Cosons

29 ( ˆ ( ; f ψ + Ω A e e + ncdent j( A A v m ncdent A ψ e δ z * Pobabty densty ψψ ncdent Cuent densty: j( v m j S j Se ˆ ncdent ncdent δφ though δ δ aea δ S Densty of patces: patce pe unt voume;.e. patce x-sng unt aea n unt tme at veocty z v δ z δv v δs δs δt δt v δ z δ t veˆ z ncdent fux pe unt aea: δφ v PCD STTACS Unt Quantum Theoy of Cosons 9

30 ( ˆ ( ; f ψ + Ω A e e + scatteed pat f ( ˆ ψ + Ω ( ; A e scatteed pat j m * Re ψ ψ * ˆ ˆ f ( Ω Re f j A e Ω e ˆ +e ˆ +eˆ θ ϕ e m θ snθ ϕ f ( Ωˆ e Ο θ f ( Ωˆ e Ο snθ ϕ scatteed pat Ο gnoe w..t. Ο f ( Ω Re f j A e Ω e ˆ ( e m ˆ f ( Ω A eˆ m * ˆ ˆ PCD STTACS Unt Quantum Theoy of Cosons 3

31 ncdent fux pe unt aea: scatteed pat δφ A v j( ˆ f ( Ω A( eˆ m δ S δω Scatteed fux n the ada outwad decton though eementa aea δ S δω scatteed ˆ δ pat f j( δ Ω Φ Seˆ eˆ ˆ A Se m δ s f( ˆ Ω L scatteng amptude f ( Ωˆ L PCD STTACS Unt Quantum Theoy of Cosons 3

32 ( ˆ ( ; f ψ + Ω A e + e f( ˆ Ω L scatteng amptude ncdent fux pe unt aea: δφ A( v ˆ δ scatteed f j( δ Ω Φ Seˆ eˆ ˆ A e m δ Ω s Scatteed fux n the ada outwad decton s δφ? f ( Ωˆ δω δφ f ( Ωˆ : L PCD STTACS Unt Quantum Theoy of Cosons 3 dσ δσ m f ( Ωˆ dω δω δω scatteng x-sec pe unt sod ange Ths defnton dffeenta x-sec s ndependent of the nomazaton

33 ( ˆ ( ; f ψ + Ω A e e + Pobabty cuent densty vecto * * j ( ψ ( ψ ( ψ ( ψ ( m * ψ : tota Re ψ ( ψ ( m wave functon Rada * ˆ * f Ω A e e component + m of the j( eˆ Re ˆ e pobabty ( ˆ e ˆ + e ˆ +e ˆ f Ω A e e θ ϕ + cuent θ snθ ϕ densty vecto f( ˆ Ω L scatteng amptude * ˆ ˆ ˆ Re f e Ω f Ω e j e A e + e + m C.J.Joachan: Quantum Theoy of Cosons Eq.3.34, p 5 PCD STTACS Unt Quantum Theoy of Cosons 33

34 * ˆ ˆ ˆ Re f Ω e f Ω e j e A e + e + m ntefeence j eˆ j + j + j eˆ { } j e ˆ ncdent outgong ntefeence ˆ * ˆ Re f e f e Ω Ω A e + e m Ο gnoed w..t. Ο Rada component of the pobabty cuent densty vecto * ( ˆ ( ˆ Re f e f e Ω Ω A e ( + ( cosθ e m PCD STTACS Unt Quantum Theoy of Cosons 34

35 j e ntefeence ˆ Rada component of the pobabty cuent densty vecto ˆ * ˆ Re f e f e Ω Ω A e + e m * ( ˆ ( ˆ Re f e f e Ω Ω A e ( + ( cosθ e m Ο gnoed w..t. Ο ˆ ˆ f( Ω e f ( Ω e Re A ( + cosθ m ( cosθ * ( cosθ C.J.Joachan: Quantum Theoy of Cosons Eq.3.39, p 5 PCD STTACS Unt Quantum Theoy of Cosons 35

36 j eˆ ntefeence Rada component of the pobabty cuent densty vecto ( cosθ * ( cosθ ˆ ˆ f( Ω e f ( Ω e Re A + cosθ m Incdent enegy has some spead: spead n magntude of the wave vecto to + Δ ± ' e ( cosθ PCD STTACS Unt Quantum Theoy of Cosons ± ' ( cosθ e d' ± ( cosθ +Δ QUESTIONS? Wte to: pcd@physcs.tm.ac.n ( +Δ ( cosθ ( cosθ ± ± ± ' ( cosθ e e e d' ± +Δ ( cosθ +Δ numeato denomnato: Ο( Intefeence tem s of mpotance ony when cosθ θ 36

37 INTRODUCTORY ectue about ths couse on Seect/Speca Topcs fom Theoy of Atomc Cosons and Spectoscopy P. C. Deshmuh Depatment of Physcs Indan Insttute of Technoogy Madas Chenna 636 Unt Lectue Numbe 4 Quantum Theoy of Cosons OPTICAL dθ THEOREM. contnued PCD STTACS Unt Quantum Theoy of Cosons 37

38 j eˆ ntefeence Rada component of the pobabty cuent densty vecto ( cosθ * ( cosθ ˆ ˆ f( Ω e f ( Ω e Re A + cosθ m Incdent enegy has some spead: spead n magntude of the wave vecto to + Δ ± ' e ( cosθ ± ' ( cosθ e d' ± ( cosθ +Δ +Δ PCD STTACS Unt Quantum Theoy of Cosons ( +Δ ( cosθ ( cosθ ± ± ± ' ( cosθ e e e d' ± +Δ ( cosθ numeato denomnato: Ο( Intefeence tem s of mpotance ony when cosθ θ 38

39 j eˆ ntefeence ˆ ˆ f( Ω e f ( Ω e Re A + cosθ m ± ± ± ' ( cosθ e e e d' ± ( cosθ * ( cosθ ( cosθ ( cosθ +Δ +Δ +Δ ± ' ( cosθ ( cosθ e d', except nea θ fowad m scatteng C.J.Joachan: Quantum Theoy of Cosons Eq.3.4, p 5 numeato denomnato: Intefeence tem s of mpotance ony when θ consdeng the spead n magntude of the wave vecto fom to + Δ ony θ Ο( s mpotant wth egad to INTERFERENCE TERM PCD STTACS Unt Quantum Theoy of Cosons 39

40 δ S { j eˆ } ˆ jncdent + joutgong + jntefeence e δω j dω eˆ { } j ˆ ncdent + joutg ong + jntefeenc e dωe θ Ω ˆ j d e { } jncdent + joutgong + jntefeence dωe j( ds θ j ds + j ds + j ds ncdent outgong { } ntefeence dv j j ds ; j PCD STTACS Unt Quantum Theoy of Cosons 4 ˆ ρ t

41 j ds + j ds + j ds ncdent outgong ntefeence j ds + j ds outgong PCD STTACS Unt Quantum Theoy of Cosons ntefeence ˆ δ scatteed f ˆ eˆ ˆ outgong j δ Ω Φ Se A e m δ Ω s ˆ f ( Ω j ( ds A d m ˆ ˆ outgong e e Ω j ( ˆ outgong ds A f d m Ω Ω A m j ds d d j ds θ +Δθ π ntefeence θ ϕ θ sn θ θ ϕ Δ θ? ntefeence σ dσ dω sma : Δθ tota f ( Ω ˆ 4

42 j ds + j ds outgong ntefeence A ( σ + j ( e ˆ tota d ntefeence Ω m sma Δθ θ +Δθ π A ( σ + sn ntefeence ˆ tota d d j e m θ θ ϕ θ ϕ j eˆ ntefeence ˆ ˆ f( Ω e f ( Ω e Re A + cosθ m ( cosθ * ( cosθ C.J.Joachan: Quantum Theoy of Cosons Eq.3.39, p 5 A σ + m tota ˆ * ˆ π snθdθ Re A( + cos θ m θ θ +Δθ cosθ cosθ f( Ω e f ( Ω e PCD STTACS Unt Quantum Theoy of Cosons 4 NOTE : A( does not matte fo subsequent anayss

43 σ + tota ˆ ( ˆ f Ω e f Ω e π snθdθ Re + cos θ θ * θ +Δθ cosθ cosθ σ + tota θ π θ +Δθ * Re θ +Δθ f( e snθdθe cosθ + f ( e snθdθe + cosθ θ cosθ μ μ ( μ f e dμe snθdθ dμ + μ cosδθ σ + π Re tota μ * f ( e d e μ μ μ cosδθ PCD STTACS Unt Quantum Theoy of Cosons 43

44 μ μ * Re f( e μ σ + π dμe f ( e dμe tota + μ cosδ θ μ cosδθ Re ( e μ σ π f e f ( e + tota + μ μ μ * e cos ( e Δθ e f e + σ + π Re tota cosδθ * ( e e f e μ cosδ θ μ cosδθ ( cosδθ e f ( + σ + π Re tota ( cosδθ * e f ( PCD STTACS Unt Quantum Theoy of Cosons 44 μ

45 ± ± ± ' e e ( cosδθ d' howeve sma ( cos θ ( cos θ +Δ +Δ Δ Δ e Δθ ( cosδθ e f ( + σ + π Re tota ( cosδθ * e f ( ± cosδθ f ( + oscatoy tems + σ + π Re tota as * f ( oscatoy tems + PCD STTACS Unt Quantum Theoy of Cosons 45

46 σ + f f tota + * π Re ( ( σ + π Re Re f ( tota 4π σ + Re Re ( tota 4π σ + Im ( tota σ tota 4π ( f { } [ f ] [ Im f (] The tota scatteng x-sec s equa to 4π/ tmes the magnay pat of the fowad (compex scatteng amptude f( a+ b f( a b [ ] Re f( b [ f ] Im ( ( ea numbe OPTICAL THEOREM Boh-Pees-Pacze eaton PCD STTACS Unt Quantum Theoy of Cosons 46

47 4π σ [ Im f (] OPTICAL THEOREM tota ORIGINS: { } Boh-Pees-Pacze eaton dv j j ds ; j ρ t dθ ndependent of A( Shadow of the taget n the fowad decton esuts fom scatteng of the ncdent beam by the taget potenta. C.J.Joachan: Quantum Theoy of Cosons PCD STTACS Unt Quantum Theoy of Cosons Fg. 3.3, p 53 47

48 c Outgong wave ( ˆ ( ; ( f ψ + bounday Ω A e e + condton δ ( e descbes 'cosons' A ( : We have empoyed ths bounday condton, ncusve of an -dependent nomazaton. enegy dependent nomazaton of the ncdent wave that scaes the scatteed pat as we. OPTICAL THEOREM: ndependent of A( scatteng x-sec dσ Ths defnton pe unt sod ange f ( Ω ˆ s ndependent of dω dffeenta x-sec the nomazaton PCD STTACS Unt Quantum Theoy of Cosons 48

49 ( ˆ ( ; ( f ψ + Ω A e e + scatteng x-sec dσ Ths defnton pe unt sod ange f ( Ω ˆ s ndependent of dω dffeenta x-sec the nomazaton ψ + Tot ( π ( t, 3/ + ( ωt + (. ωt e δ ( A e + ( + e P (cos θ We empoyed mono-enegetc ncdent beam deazaton PCD STTACS Unt Quantum Theoy of Cosons 49

50 ψ + Tot ( π ( t, 3/ mono-enegetc / deazaton + ( ωt ( +. ωt e δ ( A e + ( + e P (cos θ d σ f(, Ω ˆ monoenegetc deazaton of dω ncdent beam popetes 3 Φ ncdent ( t, d Ae 3/ ( π 3 d Ae 3/ ( π whch s dσ f(, ˆ Ω dω + ( ωt + ( ω( t Reastc ncdent wave pacet PCD STTACS Unt Quantum Theoy of Cosons 5

51 3 Φ ncdent ( t, d Ae 3/ ( π ( π 3/ 3 d Ae + ( ωt + ( ω( t Reastc ncdent wave pacet A ( can be detemned f the wave-pacet s nown at t PCD STTACS Unt Quantum Theoy of Cosons 5

52 Reastc ncdent wave pacet 3 Φ ncdent ( t, d Ae 3/ ( π 3 d Ae 3/ Goup veocty Patce veocty ( π dω d ω( v ( v m + ( ωt + ( ω( t E ω m m PCD STTACS Unt Quantum Theoy of Cosons 5

53 3 ( ( t ncdent ( t, d + ω Φ Ae 3/ can be ( π Eq.3.57 / p55 / Joachan s Quantum Coson Theoy detemned f the wave-pacet Reastc ncdent wave pacet 3 Φ ncdent (, d A e 3/ 3 A d Φ (, / ncdent e v ( π 3 ( π Each ndvdua wave ( π 3/ A e taves at the phase veocty φ + + PCD STTACS Unt Quantum Theoy of Cosons ( ω( t Eq.3.59 / p55 / Joachan s Quantum Coson Theoy ω( E( / /m m Eq.3.6 / p55 / Joachan s Quantum Coson Theoy A ( s nown at t wave-functon n the momentum (athe, wavevecto space nown at t Phase veocty s haf the goup veocty 53

54 3 ncdent ( t, d Φ Ae 3/ ( π Reastc ncdent wave pacet at t: 3 Φ ncdent (, d A e 3/ ( π ( ω( t + + naow spead Δ 3 A d Φ (, / ncdent e ( π 3 Nomazaton: d (, d A 3 3 Φ ncdent PCD STTACS Unt Quantum Theoy of Cosons 54 Let α 3 + ( t ncdent ( t, d A e e e ω Φ 3/ ( π 'spead/paced' n the egon Δ Δ A A e α

55 3 ncdent ( t, d Φ Ae 3/ ( π Reastc ncdent wave pacet at t: Let A A e α α 3 + ( t ncdent ( t, d A e e e ω Φ 3/ ( π 3 ( t, d A e 3/ π Φ ncdent Eq.3.65, 3.66 / p56 / Joachan s Quantum Coson Theoy ( ω( t + β ω α ( ( t+ ( PCD STTACS Unt Quantum Theoy of Cosons 55 β

56 β 3 Φ ncdent (, t d A( e 3/ ( π Unde what condtons s β ω α ( ( t+ ( Φ ncdent ( t, the agest? β e oscates n esponse to snce β β oscatng pats cance each othe's contbutons to Φ ncdent ( t, Fo Φ ncdent ( t, to be age, these oscatons must not happen β must not vay vey much wth espect to The equed condton s: β ( PCD STTACS Unt Quantum Theoy of Cosons 56

57 β condton fo Φ ncdent ( t, to be the agest -dmensona case 3-dmensona case Tme ogn: t snce ( β ω α ( ( t+ ( z ω( t + α( Eq.3.65, 3.66 / p56 / Joachan s Quantum Coson Theoy dω( α dβ d z t d + d d.e. ( dω( dα z t d d PCD STTACS Unt Quantum Theoy of Cosons 57 t ω( t α( t ( v t t + v ω( & α(

58 Comato d d: tansvese wdth of the wave pacet : b Longtudna wdth Δ a Schematc dagam of the chaactestc engths descbng the scatteng of a wave pacet by a potenta D d Scatteng egon Detecto D C.J.Joachan: Quantum Theoy of Cosons Fg. 3.4, p 56 d Δ Δ PCD STTACS Unt Quantum Theoy of Cosons 58

59 β 3 Φ ncdent (, t d A( e 3/ ( π Φ (, ncdent 3 t d A e 3/ π ω( ω( + ω... + ω( v ω( + v v +... β ω α ( ( t+ ( { ω( t+ α( } QUESTIONS? Wte to: pcd@physcs.tm.ac.n PCD STTACS Unt Quantum Theoy of Cosons 59

60 INTRODUCTORY ectue about ths couse on Seect/Speca Topcs fom Theoy of Atomc Cosons and Spectoscopy P. C. Deshmuh Depatment of Physcs Indan Insttute of Technoogy Madas Chenna 636 Unt Lectue Numbe 5 Quantum Theoy of Cosons dθ Dffeenta scatteng coss-secton dσ f(, ˆ Ω dω PCD STTACS Unt Quantum Theoy of Cosons 6

61 ( ˆ ( ; ( f ψ + Ω A e e + scatteng x-sec pe unt sod ange dffeenta x-sec ψ + Tot ( π ( t, 3/ dσ dω f ( Ω ˆ + ( ωt + (. ωt e δ ( A e + ( + e P (cos θ We empoyed mono-enegetc ncdent beam deazaton PCD STTACS Unt Quantum Theoy of Cosons 6

62 ψ + Tot ( π ( t, 3/ mono-enegetc / deazaton + ( ωt ( +. ωt e δ ( A e + ( + e P (cos θ d σ f(, Ω ˆ monoenegetc deazaton of dω ncdent beam popetes 3 Φ ncdent ( t, d Ae 3/ ( π 3 d Ae 3/ ( π whch s dσ f(, ˆ Ω dω + ( ωt + ( ω( t Reastc ncdent wave pacet PCD STTACS Unt Quantum Theoy of Cosons 6

63 3 Φ ncdent ( t, d Ae 3/ ( π β 3 Φ ncdent (, t d A( e 3/ ( π Unde what condtons s + ( ω( t PCD STTACS Unt Quantum Theoy of Cosons 63 β ω α ( ( t+ ( Φ ncdent ( t, the agest? β e oscates n esponse to snce β β oscatng pats cance each othe's contbutons to Φ ncdent ( t, Fo Φ ncdent ( t, to be age, these oscatons must not happen β must not vay vey much wth espect to β The equed condton s: A A e α (

64 β condton fo Φ ncdent ( t, to be the agest -dmensona case 3-dmensona case Tme ogn: t snce.e. ( β ω α ( ( t+ ( z ω( t + α( Eq.3.65, 3.66 / p56 / Joachan s Quantum Coson Theoy dω( α dβ d z t d + d d t ω( t α( t ( v t t + PCD STTACS Unt Quantum Theoy of Cosons 64 v ω( & α( ( dω( dα z t d d

65 β 3 Φ ncdent (, t d A( e 3/ ( π Φ (, ncdent 3 t d A e 3/ π ω( ω( + ω... + Can we ω( v negect hghe ode ω( + v v +... tems? v m ( ( v ω ω + ω( +... ω β ω α ( ( t+ ( { ω( t+ α( } v ω( m m m snce E ω ( m m ω( ω( + v +.. PCD STTACS Unt Quantum Theoy of Cosons 65

66 Φ (, ncdent ( π 3 3 t d A e 3/ π Φ ( t, ncdent ( π 3 d A e / Φ ( t, ncdent A d 3 ( 3/ e { ω( t+ α( } ω( ω( + v { + ω ( v } t t+ α { ( v } t + ω t+ α PCD STTACS Unt Quantum Theoy of Cosons 66

67 Φ ( t, ncdent π A 3 d ( 3/ { ( v t + ω ( t+ α ( } e α( α( + α... + α( α( [ ] wth ( α( Can we negect hghe ode tems? Φ ncdent ( t, 3 { ( v t + ω ( t+ α ( ( } d A e 3/ π 3 { ( v t + ω( t+ α ( + } d A( e 3/ π PCD STTACS Unt Quantum Theoy of Cosons 67

68 Φ ( t, ncdent π 3/ Φ (, t ncdent π 3 d A e 3 d ( 3/ A { ( v t + ω( t+ α ( + } e ( ( α α + { ( v t + ω ( t+ α ( } Φ ( t, ncdent π 3/ d 3 A( e { ( vt + ω( t} snce A A e α PCD STTACS Unt Quantum Theoy of Cosons 68

69 3 Φ ncdent ( t, d A( e 3/ π Φ ncdent ( t, ( vt e d e π { ( vt + ω( t} ω ( ( t t 3 ω ( t A e 3/ 3 + Φ ncdent (, d A( e 3/ π ( ( t t Φ ( ncdent e ω ncdent t t t Eq.3.79 / p57 / Joachan s Quantum Coson Theoy t ( + v( t t ;.e t ( v( t t ( t, Φ ( v, t PCD STTACS Unt Quantum Theoy of Cosons 69.

70 Φ ncdent ( t, d Ae 3 + ω( t Reastc 3/ π ncdent wave pacet ( 3 ( v t ω ω ( t t ( t A 3/ e d e e π Φ 3 + ncdent (, d A( e Φ 3/ ( π ( ω ( t t ncdent ncdent ( t, e Φ ( t v t t, t ( t ( + v( t ;.e ( v( t. t t t fee wave pacet centeed aound the pont at tme t w have same shape as a wave pacet centeed aound the pont + v ( t t at tme t PCD STTACS Unt Quantum Theoy of Cosons 7

71 ω( ω( + ω... + ω( v Can we negect hghe ode tems? α( α( + α... + α( α( [ ] Hghe ode tems gnoed Unde what condtons can we gnoe hghe ode tems? 7 PCD STTACS Unt Quantum Theoy of Cosons

72 dω ( ω( ω( d condton to gnoe hghe ode tems: d ω ( sma d m ( t m D Δ v ( t D v ( ( v ( ω ω ω( + v +.. E( ω( m m dω( d m m d ω( d m Phase veocty; haf the goup veocty PCD STTACS Unt Quantum Theoy of Cosons 7

73 m md ( Δ ( Δ.e. D eca : ( Δ ( Δ ( Δ ( Δ λ D ( Δ.e. λ D Δ In most expements: cm 3 cm Hence we can ndeed gnoe hghe ode tems. PCD STTACS Unt Quantum Theoy of Cosons 73

74 d Comato d D d: tansvese wdth of the wave pacet : : b Longtudna wdth Longtudna wdth a D Scatteng egon Detecto C.J.Joachan: Quantum Theoy of Cosons Fg. 3.4, p 56 a Δ a b: mpact paamete A patces descbed by same b as detaed shape of the wave pacet does not matte C.J.Joachan: Quantum Theoy of Cosons Fg. 3.5, p 58 PCD STTACS Unt Quantum Theoy of Cosons 74

75 : Longtudna wdth a Detaed shape of the wave pacet does not matte Δ a C.J.Joachan: Quantum Theoy of Cosons Fg. 3.5, p 58 PCD STTACS Unt Quantum Theoy of Cosons 75

76 Fee patce wave pacet 3 ncdent ( t, d Φ Ae 3/ E ω ( m m ( π ( π Fee patce wave pacet mpactng at 3 { } ( t b t, d Ae + ω Φ b 3/ ( ω( t + ( ( v ( ω ω ω( + v +.. b : mpact paamete C.J.Joachan: Quantum Theoy of Cosons Eq. 3.86, p 58 3 b + ω Φ ( t, d Ae e e b 3/ ( π ( π 3 b + ω + ( v Φ ( t, d Ae e e e b 3/ t t t PCD STTACS Unt Quantum Theoy of Cosons 76

77 3 b + ω + ( v Φ ( t, d Ae e e e b 3/ ( π t t mutpyng the ntegand by: { } { } + b v t b + vt e e e e + ω( t + ( b vt Φ ( t, e e e b ( π 3/ 3 d A e e + b v t PCD STTACS Unt Quantum Theoy of Cosons 77

78 + ω( t + ( b vt Φ ( t, e e e b ( π ( π 3/ 3 d A e e + ω( t + ( b vt Φ ( t, e e b 3/ + b v t 3 b t d + A e ( ( v { ( v } ( t b t t, e + + ω Φ χ b v t b ( 3 χ b v t d A( e 3/ ( π + b t ( ( v PCD STTACS Unt Quantum Theoy of Cosons Eq.3.88 / p.58 / Joachan s QCT 78

79 { ( v } ( t b t t, e + + ω Φ χ b v t b ( 3 χ b v t d A( e 3/ ( π + b t ( ( v Reca that: Nomazaton: d (, d A 3 3 Φ ncdent 3 ds χ( s PCD STTACS Unt Quantum Theoy of Cosons 79

80 Fee patce wave pacet nteactng wth the scattee at b : mpact paamete { ( v } ( t b t t, v b 3/ e + + ω Φ χ π b t 3 { } ( t b t, b 3/ d Ae + ω Φ π 3 b + ω Φ ( t, d Ae e e b 3/ π wave pacet fo the compete scatteng pobem 3 b ( t, d Ae ( e ω + + Ψ ψ b 3/ ( π Fee patce case PCD STTACS Unt Quantum Theoy of Cosons 8 t t

81 d Comato d D d: tansvese wdth of the wave pacet : : b Longtudna wdth Longtudna wdth a Δ a Scatteng egon Detecto C.J.Joachan: Quantum Theoy of Cosons Fg. 3.4, p 56 a D Δ D b: mpact paamete Hence the pacet does not oveap the taget when t s fa fom the taget C.J.Joachan: Quantum Theoy of Cosons Fg. 3.5, p 58 PCD STTACS Unt Quantum Theoy of Cosons 8

82 ( ˆ ( ; f ψ + Ω A e e + wave pacet fo the compete scatteng pobem 3 b ( t, d Ae ( e ω + + Ψ ψ b 3/ e ( π ( t + z ω ˆ eˆ z t In the next cass, we compete the poof that: dσ dω f(, Ω ˆ s appopate expesson even to descbe scatteng of the wave pacet. QUESTIONS? Wte to: pcd@physcs.tm.ac.n PCD STTACS Unt Quantum Theoy of Cosons 8

83 INTRODUCTORY ectue about ths couse on Seect/Speca Topcs fom Theoy of Atomc Cosons and Spectoscopy P. C. Deshmuh Depatment of Physcs Indan Insttute of Technoogy Madas Chenna 636 Unt Lectue Numbe 6 Quantum Theoy of Cosons PCD STTACS Unt Quantum Theoy of Cosons Dffeenta x-sec d (wave-pacets d Pata wave anayss σ Ω f(, Ω ˆ Refeence: Quantum Coson Theoy C.J.Joachan Chaptes 3 & 4 83

84 Fee patce wave pacet nteactng wth the scattee at b : mpact paamete 3 b + ω Φ ( t, d Ae e e b 3/ π wave pacet fo the compete scatteng pobem 3 b ( t, d Ae ( e ω + + Ψ ψ b 3/ ( π Snce the pacet does not oveap the taget when t s fa fom the taget, we may use the ( ˆ asymptotc fom: ( ; f ψ + Ω A e + e Ψ ˆ 3 b f ( Ω ( t ( t, ( t, d A( e e ω Φ + e 3/ ( π + b b ncdent wave pacet scatteed wave pacet t t PCD STTACS Unt Quantum Theoy of Cosons 84

85 3 b + ω Φ ( t, d Ae e e b 3/ π t Ψ ˆ 3 b f ( Ω ( t ( t, ( t, d A( e e ω Φ + e 3/ ( π + b b ncdent wave pacet scatteed wave pacet Ψ Φ + Ψ + 3 b ( t, ( t, d Ae f ˆ b b 3/ + b Ψ t ( t, Φ( t, b + b ( π ( ω( t e ( Ω Eef van Beveen t +? ( t, C.J.Joachan: Quantum Theoy of Cosons Eq. 3.86, p 58 PCD STTACS Unt Quantum Theoy of Cosons 85

86 + Ψ( t, Φ ( t, + + Ψ Φ + b + 3 b ( t, ( t, d Ae f ˆ b b 3/ + ˆ ( π ( π 3/ 3/ b f(, Ωˆ + Ψ( t, Φ ( t, + + b b e ω ( t e ( π 3 d ( ω( t e ( Ω ( ( v ω ω + ( ω( ˆ ( v ( ω t e e e e e t t ˆ ( v ω t 3 d Ae b f(, ˆ e e e e Ω Ae t f(, Ω ˆ f(, Ωˆ e f(, Ωˆ e ˆ b t Λ(, Ωˆ e v e e PCD STTACS Unt Quantum Theoy of Cosons 86 Λ(, Ωˆ Λ(, Ωˆ

87 + Ψ( t, Φ ( t, + + b ( π 3/ b f(, Ωˆ e ω ( t e 3 d Ae ( Λ(, Ωˆ Λ(, Ω ˆ + Λ(, Ωˆ (, ˆ ρ( ˆ ( Λ Ω + Ω ; ρ ( Ωˆ Δ + ρ( Ω ˆ Λ(, Ωˆ Ψ( t, Φ ( t, + b b ˆ b t Λ(, Ωˆ v e e ω ( t ˆ e e f( 3/, Ω + π ˆ ˆ ( ˆ 3 b v (, ( t Λ Ω ρ Ω d Ae e e e e e PCD STTACS Unt Quantum Theoy of Cosons 87

88 + Ψ( t, Φ ( t, + b b ω ( t ˆ e e f( 3/, Ω + π ˆ ˆ ( ˆ 3 b v (, ( t Λ Ω ρ Ω d Ae e e e e + Ψ( t, Φ ( t, + b b ω ( t (, ˆ ˆ e e Λ Ω b f(, 3/ Ω e e + π ˆ v ( ˆ 3 b t+ ρ Ω ( d Ae e PCD STTACS Unt Quantum Theoy of Cosons 88

89 + Ψ( t, Φ ( t, + b b ω ( t (, ˆ ˆ e e Λ Ω b f(, 3/ Ω e e ( + π ˆ v ( ˆ 3 b t+ ρ Ω ( d Ae e + Ψ( t, Φ ( t, + f(, Ωˆ b b { } ω ( t (, ˆ ˆ e e Λ Ω b f(, 3/ Ω e e ( + π ˆ v ( ˆ 3 b t+ ρ Ω ( d Ae e PCD STTACS Unt Quantum Theoy of Cosons 89

90 + Ψ( t, Φ ( t, + ω ( t (, ˆ e e b f 3/ Ω e ( + π ˆ v ( ˆ 3 t+ ρ Ω b ( d Ae shape of the wave pacet 3/ 3 + b vt we had: ( π d A e χ b vt + Ψ( t, Φ ( t, + b b b b { ω ( t } (, ˆ e b f ˆ + v ( ˆ Ω e χ t+ ρ Ω b 3/ ˆ ˆ 3 v t ρ b ( ( π d + Ω Ae χ ˆ v ( ˆ t+ ρ Ω b PCD STTACS Unt Quantum Theoy of Cosons 9

91 + Ψ( t, Φ ( t, + b b { ω( t } (, ˆ e b f ˆ + v ( ˆ Ω e χ t+ ρ Ω b Ψ Ω Ω + + scatteed pat ony ˆ (, (, ˆ t f χ ρ v b t b ( ˆ Pobabty of scatteng aong the decton ˆΩ P Ω d Ψ t f Ω d Ω + t b ˆ + scatteed ˆ (, (, ( ˆ ˆ b χ ρ v b f(, Ω d ( Ω + v t b snce v v ˆ ( ˆ ˆ ˆ χ ρ PCD STTACS Unt Quantum Theoy of Cosons 9

92 Pobabty of scatteng aong the decton ˆΩ P ( ˆ f(, ˆ Ω Ω d χ ρ( Ω ˆ + vt b b ( ˆ ( ˆ (, ˆ P Ω f Ω dz χ ρ( Ω ˆ + z b b ( ˆ z v t dσ ˆ ˆ dbp (, ( ˆ ˆ b Ω f Ω dz dbχ ρ Ω + z b dω Whoe s space ntega ρ( Ω ˆ + ˆ z b 3 ds χ( s dσ dω f(, Ω ˆ Appopate expesson even to descbe scatteng of the wave pacet. C.J.Joachan: Quantum Theoy of Cosons Eq. 3.7, p 6 PCD STTACS Unt Quantum Theoy of Cosons 9

93 Havng estabshed that dσ dω f(, Ω ˆ s an appopate expesson even to descbe scatteng of the wave pacet, we now poceed to study some mpotant and consequenta aspects of PARTIAL WAVE ANAYLSIS PCD STTACS Unt Quantum Theoy of Cosons 93

94 ψ ( ˆ f Ω ( ; Ae + e ψ ψ nc (; (+ P(cos θ π sn( π π e e ψ nc ( (+ P (cos θ nc (+ P(cos θ e P(cos θ ( e ψ nc (+ P(cos θ e P( cos θ e PCD STTACS Unt Quantum Theoy of Cosons 94

95 E > contnuum ( + μ R'' + R' R+ [ E V( ] R d d y ε( R ε( ;.e. yε( R ε( d ( + + V( E y ( + ε md m ( + + U( y (, + ε U ( mv ( M When m U ; M:constant and ε n the pesence of a scatteng taget potenta PCD STTACS Unt Quantum Theoy of Cosons 95

96 ( ( R (, y(, C ( j( C ( n (, " ange" + j ( : spheca Besse functons of the potenta n ( : spheca Neumann functons V π π sn cos j(, ; n(, π π sn cos ( ( y (, C ( C ( π π ( ( y(, C ( sn C ( cos C ( y(, π A( sn + δ ( tan δ ( C ( ( ( PCD STTACS Unt Quantum Theoy of Cosons 96

97 cosθ ( θ e + P cos j( ψ e ψ Tot + Tot ( t, ( ωt + z + e + ( ωt e e ˆ ˆ 4 * ˆ ( ˆ π j Y m Ym e m c ( + δ ( + δ c(+ P(cos θ e P( cos θ e Pease efe to detas fom : PCD STAP Unt 6 Pobng the Atom ( t + z ω ˆ δ ( e descbes 'cosons' + e P (cos θ eˆ δ ( Lectue n: & /8 & /9 & /3 z PCD STTACS Unt Quantum Theoy of Cosons 97

98 C ( y(, π A( sn + δ ( tan δ ( C ( π π ( ( y(, C ( sn C ( cos ( ( Lnea combnaton of Spheca Besse & Neumann We can aso wte the same as Lnea combnaton of spheca ngong waves & spheca outgong waves PCD STTACS Unt Quantum Theoy of Cosons 98

99 π sn + δ ( R(, A( e R(, A( e R(, y (, A ( y (, A ( A π π + δ( + δ( e π π + δ( + δ( e π π + δ( δ( e e e e e e π δ ( ( e e y (, e e e e δ ( + π PCD STTACS Unt Quantum Theoy of Cosons 99

100 A π δ ( ( e e y (, e e e e δ ( + π π π π π e e ( ( ; e ( e ( A ( δ ( ( e δ ( y (, e e e ( (, A δ ( y e e e Lnea combnaton of spheca ngong & spheca outgong waves A ( A ( δ ( ( e QUESTIONS? Wte to: pcd@physcs.tm.ac.n PCD STTACS Unt Quantum Theoy of Cosons

101 Seect/Speca Topcs n Theoy of Atomc Cosons and Spectoscopy P. C. Deshmuh Depatment of Physcs Indan Insttute of Technoogy Madas Chenna 636 Lectue Numbe 7 Unt : Quantum Theoy of Cosons OPTICAL THEOREM -Untaty of the Scatteng Opeato How many pata waves? Is thee an max? PCD STTACS Unt Quantum Theoy of Cosons Pmay Refeence: Quantum Mechancs Noneatvstc theoy by Landau & Lfshtz

102 e R(, y (, A ( (, A δ ( ( y e e e Lnea combnaton of spheca ngong & spheca outgong waves A PCD STTACS Unt Quantum Theoy of Cosons ( π sn + δ ( R(, A( ( ωt z ( ωt e δ ( ψ Tot ( t, e + ( + e P (cos θ e e A ( δ ( ( e π ; π ( e π ( π π + δ( + δ( e S ( ( e δ S Matx eement

103 e R(, y(, A ( natue of souton: m V( d d dr d ncudes couomb d dr ( + μ R + [ E V( ] R d d d d + R R( s a dr μ ( + R + [ E V( ] R d ( π π + δ( + δ( e Regadess of E,m s o ( + : R ( (any E y ( + 3

104 f ( θ ( + e P a ( θ δ ( (cos ( θ ( + (cos θ f ( θ f a P a ( e δ ( [ δ ] [ δ ] cos + sn a ( [ S ] : scatteng amptude ( a ( : pata wave amptude { [ δ ]} [ δ ] [ δ ] { } sn + sn cos ( ( { [ ]} + [ ] [ ] { } [ ] sn δ ( sn δ ( cos δ ( sn δ ( e a ( δ ( [ δ ] δ ( sn e f ( θ ( + P(cos θ PCD STTACS Unt Quantum Theoy of Cosons 4

105 [ ] δ ( δ e sn f ( θ ( + P(cos θ dσ dω f * ( θ f ( θ ( [ δ ] δ ( sn e ( + P (cos θ + ' δ ' [ δ ] e sn ( ' ' P'(cos θ σ Tota ( + ( ' + sn [ δ' ( ] sn [ δ( ] π ' π [ δ' ( δ( ] e sn θdθp(cos θ P' (cos θ PCD STTACS Unt Quantum Theoy of Cosons 5

106 σ Tota ( + ( ' + sn [ δ' ( ] sn [ δ( ] π ' π [ δ' ( δ( ] e sn θdθp(cos θ P' (cos θ σ Tota π ( + ( ' + sn [ δ ( ] sn [ δ ( ] ' ' [ δ ' ] δ e δ ' + σ Tota 4π + ( sn [ δ ( ] PCD STTACS Unt Quantum Theoy of Cosons 6

107 σ Tota σ ( σ Tota 4π + ( sn [ δ ( ] 4π σ( + sn 4π σ ( ( + δ max n+ n, ±, ±, ± 3,... σ ( δ ( nπ mn [ δ ] π No contbuton to scatteng by that pata wave σ Tota σ ( usuay, max ~ a; not PCD STTACS Unt Quantum Theoy of Cosons 7

108 e ( t + z ω ˆ eˆ z δσ c Scatteng of cassca patces bb δ δϕ PLANE of the IMPACT PARAMETER C.J.Joachan: Quantum Theoy of Cosons Fg. 4., p 74 δb δσ b δ( cosθ δϕ c δ( cosθ δb δσ b snθδθ δϕ c { snθδθ} { } Y pane δσ c δσ bδϕ δb δσ c bb δ δϕ b δb δω snθ δθ dσ c b db dω snθ dθ PCD STTACS Unt Quantum Theoy of Cosons 8 X

109 e ( t + z ω ˆ eˆ z Scatteng of cassca patces PLANE of the IMPACT PARAMETER dσ c b db dω snθ dθ What woud be the angua momentum of a cassca C.J.Joachan: Quantum Theoy of Cosons Fg. b 4., p 74 ρ p b p max ap a fo b ~ a:" ange" patce at mpact paamete? a max max + max a PCD STTACS Unt Quantum Theoy of Cosons 9

110 pata waves: a a :" ange" of the potenta Often, just few pata waves suffce n the pata wave expanson Pease efe to detas fom : PCD STAP Unt, Lectue 5 Lectue n Speca futhe consdeatons: ( Resonances etc. ( V( fas off extemey sowy n the asymptotc egon. (3 Eecton coeatons. PCD STTACS Unt Quantum Theoy of Cosons

111 [ ] δ ( δ e sn f ( θ ( + P(cos θ ( θ ( + [ δ ] sn Im f P(cos θ fo evey, fo θ, cos( θ, P (cos θ σ Tota above Im f sn ( ( θ ( + [ δ ] 4π sde 7 : σ sn ( Tota + [ δ ] 4π Im f ( θ OPTICAL THEOREM

112 ˆn ˆ ' n Incdence decton Scatteng decton ψ ( nˆˆ.' n + e f nˆ, nˆ' e Any LINEAR COMBINATION of functons of the above fom fo dffeent dectons of ncdence w aso be a souton to the scatteng pocess. ( ˆ, ˆ' ( ˆ ˆˆ.' f n n F n n n e Ψ e dο F( nˆ dο + dο : eementa sod ange ( F( nˆ nˆˆ.' n e Ψ e dο+ f ( nˆ, nˆ' F( nˆ dο Ref.: Landau & Lfshtz, NR-QM 5, page 58 ˆn NOTE: ntegaton s ove dffeent dectons of ncdence PCD STTACS Unt Quantum Theoy of Cosons

113 ( F( nˆ nˆˆ.' n e Ψ e dο+ F( nˆ f ( nˆ, nˆ' dο en ˆˆ.' n Integaton s ove dffeent dectons of ncdence π π Ψ( snϑdϑ dφf n oscates apdy at age as ncdent decton ˆn changes hence detemned by nˆ ± nˆ' F nˆ e n ˆˆ.' n dο whee F nˆ F ± nˆ' ϑ φ e + ( ˆ ( ˆ ( ˆ, ˆ' F n f n n e nˆˆ.' n dο ϑ ˆn φ ˆ ' n PCD STTACS Unt Quantum Theoy of Cosons 3

114 π ϑ ϑ ˆn φ ˆ ' n π Ψ( π sn ϑdϑf( nˆ en ˆˆ.' n + f ( nˆ, nˆ' F( nˆ dο e π Ψ( snϑdϑ dφf n ϑ φ e + ( ˆ ( ˆ, ˆ' ( ˆ f n n cosϑ F( nˆ cosϑ e Ψ( π + f ( nˆ, nˆ' F( nˆ dο cosϑ nˆˆ.' n F n dο F( nˆ' F( nˆ' Ψ( π + π + f ( nˆ, nˆ' F( nˆ dο e e e e e e e e F( nˆ' F( nˆ' π Ψ( + f ( nˆ, nˆ' F( nˆ dο PCD STTACS Unt Quantum Theoy of Cosons 4

115 e e e F( nˆ' F( nˆ' π Ψ( + f ( nˆ, nˆ' F( nˆ dο e e e F( nˆ' F( nˆ' π Ψ( f ( nˆ, nˆ' F( nˆ d + Ο π doppng the facto π e e e F( nˆ' F( nˆ' Ψ( + f ( nˆ, nˆ' F( nˆ dο π ngong outgong Spheca wave spheca wave e e Ψ( F( nˆ' F( nˆ' f ( nˆ, nˆ' F( nˆ dο π PCD STTACS Unt Quantum Theoy of Cosons 5

116 e e Ψ( F( nˆ' F( nˆ' f ( nˆ, nˆ' F( nˆ d Ο π f n n F n d f F n ( ˆ, ˆ' ( ˆ Ο 4 π ( ˆ' f F( nˆ' f ( nˆ, nˆ' F( nˆ dο 4π defnton of the opeato f e e Ψ( F n' F n' 4 f F n' π ( ˆ ( ˆ π ( ˆ e e Ψ( F n' + f F n' ( ˆ ( ˆ PCD STTACS Unt Quantum Theoy of Cosons 6

117 e e Ψ( F n' + f F n' f F( nˆ' f ( nˆ, nˆ' F( nˆ dο 4π ( ˆ ( ˆ e Ψ( F n' S F n' e ( ˆ ( ˆ Scatteng Opeato (defnton S + f Ref.: Landau & Lfshtz, NR-QM 5, Eq.5.3, page 59 Hesenbeg (943 PCD STTACS Unt Quantum Theoy of Cosons 7

118 Scatteng Opeato (defnton e Ψ( F n' S F n' ngong outgong ( ˆ' ( ˆ' F n F n ( ˆ' F( nˆ' F n SS? e ( ˆ ( ˆ Consevaton of ngong and outgong fux S f F n ( ˆ ' F n dο 4 f ( nn ˆ, ˆ' ( ˆ π + f measue of ntensty of ngong wave measue of the ntensty of the outgong wave SS SS S : untay Ref.: Landau & Lfshtz, NR-QM 5, page 58 PCD STTACS Unt Quantum Theoy of Cosons 8

119 e Ψ( F n' S F n' e ( ˆ ( ˆ Scatteng Opeato (defnton f F( nˆ' f ( nn ˆ. ˆ' F( nˆ dο 4π SS + f f SS f + f + 4 f f SS + f f + 4 f f S S + f f SS f f f f SS Ref.: Landau & Lfshtz, NR-QM 5, page 58 PCD STTACS Unt Quantum Theoy of Cosons 9

120 f f f f ( f f F n ˆ ' f f F( nˆ' f F nˆ' f F nˆ' f f F( nˆ' ( ˆ' ( ˆ, ˆ' ( ˆ Integaton s ove f F n f n n F n dο 4π unpmed vaabes nd ndex Integaton s ove f F( nˆ' f * ( nˆ', nˆ" F( nˆ" dο" doube-pmed 4π vaabes st ndex * f ( nˆ, nˆ' F( nˆ dο f ( nˆ', nˆ" F( nˆ" d " 4π Ο 4π * f f ( nˆ', nˆ" F ( nˆ" dο" 4π PCD STTACS Unt Quantum Theoy of Cosons

121 * ( ˆ ˆ ( ˆ ( ˆ ˆ ( ˆ f n, n' F n dο f n', n" F n" dο " ( ˆ ˆ ( ˆ Ο * f f n ', n" F n" d " * ( ˆ ˆ ( ˆ ( ˆ ˆ F( nˆ f n, n' F n dο f n', n" " dο " * f f ( nˆ', nˆ" F( nˆ " dο" * ( ˆ, ˆ' ( ˆ ( ˆ', ˆ ( ˆ f n n F n dο f n n F n dο * f ( nˆ', nˆ" f F( nˆ" dο" PCD STTACS Unt Quantum Theoy of Cosons

122 * ( ˆ, ˆ' ( ˆ ( ˆ', ˆ ( ˆ f n n F n dο f n n F n dο * f ( nˆ', nˆ" { f F( nˆ" } dο" f F( nˆ' f ( nˆ, nˆ' F( nˆ dο 4π f F( nˆ" f ( nˆ, nˆ" F( nˆ dο 4π * ( ˆ, ˆ' ( ˆ', ˆ ( ˆ f n n f n n F n dο * f ( nˆ', nˆ" f ( nˆ, nˆ" F( nˆ dο dο" 4π * ( ˆ, ˆ' ( ˆ', ˆ ( ˆ f n n f n n F n dο π ( ˆ nˆ ( ˆ ˆ" ( ˆ ', ", " * f n f n n F n dοdο PCD STTACS Unt Quantum Theoy of Cosons

123 fo nˆ' * ( ˆ, ˆ' ( ˆ', ˆ ( ˆ f n n f n n F n dο * f ( nˆ', nˆ" f ( nˆ, nˆ" F( nˆ dοdο" π * ( ˆ, ˆ' ( ˆ', ˆ ( ˆ f n n f n n nˆ F n dο * f ( nˆ', nˆ" f ( nˆ, nˆ" dο" F ( nˆ d π Ο * * f ( nˆ, nˆ' f ( nˆ', nˆ f ( nˆ', nˆ" f ( nˆ, nˆ" dο" π f( nn ˆ, ˆ f * ( nn ˆ, ˆ f * ( nn ˆ, ˆ" f( nn ˆ, ˆ" dο" π Im ( ˆ, ˆ ( ˆ, ˆ" f n n f n n dο" π Im f ( nˆ, nˆ σ π Tota ( ˆ, ˆ" f n n 4π σ Im ( ˆ, ˆ Tota f nn S : untay dσ d Ο " optca theoem QUESTIONS? Wte to: pcd@physcs.tm.ac.n PCD STTACS Unt Quantum Theoy of Cosons 3

124 Seect/Speca Topcs n Theoy of Atomc Cosons and Spectoscopy P. C. Deshmuh Depatment of Physcs Indan Insttute of Technoogy Madas Chenna 636 Lectue Numbe 8 Unt : Quantum Theoy of Cosons RECIPROCITY THEOREM - fom Landau & Lfshtz NR-QM Phase-shft anayss - fom Joachan s Quantum Coson Theoy PCD STTACS Unt Quantum Theoy of Cosons 4

125 f F( nˆ' f ( nn ˆ. ˆ' F( nˆ dο 4π S + f Scatteng Opeato (defnton e Ψ( F n' S F n' e ( ˆ ( ˆ e e e e ωt ωt Ψ( t, F n' SF n' ( ω ( ˆ ( ˆ ( ω + t + t e e Ψ( t, F( nˆ' SF n' ( ˆ PCD STTACS Unt Quantum Theoy of Cosons 5

126 ( ω ( ω + t + t e e Ψ( t, F( nˆ' SF n' ( ˆ ( ω ( ω + + t t * e * e * Ψ ( t, F( nˆ' S F n' ( ˆ ( ω ( ω + t + t * e * e * Ψ (, t F ( nˆ' S F n' ( ˆ PCD STTACS Unt Quantum Theoy of Cosons 6

127 e Ψ( F n' S F n' ( ω e ( ˆ ( ˆ ( ω + t + t e e Ψ( t, F( nˆ' SF n' ( ˆ Ogna functon tme evesed functon: + ( ωt ( + ωt * e * e Ψ (, t F ( nˆ' * * S F n' ( ˆ space pat of the tme-evesed functon: e + e ( ˆ' ( ˆ' * * * F n S F n PCD STTACS Unt Quantum Theoy of Cosons 7

128 space pat of the tme-evesed functon: e e + ( ˆ' ( ˆ' * * * F n S F n et S F n n n ( ˆ Φ( ˆ defnton of Φ( ˆ * * : ' ' ' ˆ' ( ˆ' * * * * F n S S F n ˆ' Φ( ˆ' * * F n S n ˆ' Φ( ˆ' * * F n S n Paty: ( ˆ' ( ˆ' * * F n PF n PS Φ ( nˆ ' Φ * S ( nˆ ' * F nˆ SΦ nˆ * snce S ( ' ( ' S PSP Φ ( ˆ' Φ( ˆ' * F n PS P n ( nˆ ' PCD STTACS Unt Quantum Theoy of Cosons 8

129 space pat of the tme-evesed functon: e + e e ( ˆ' ( ˆ' * * * F n S F n space pat of the tme-evesed functon: + PSP n Φ S F n e * * ( ˆ' ( ˆ' PSPΦ ( ˆ' Φ( ˆ' * F n PS P n ( nˆ ' * * S F nˆ nˆ nˆ ( ' Φ( ' defnton of Φ( ' space pat of the tme-evesed functon: e + PSPΦ n Φ n e ( ˆ' ( ˆ' PCD STTACS Unt Quantum Theoy of Cosons 9

130 space pat of the tme-evesed functon: e + PSPΦ n Φ n e ( ˆ' ( ˆ' e e + ( nˆ' Φ PSPΦ( nˆ' ogna functon: e e + F n S F n ( ˆ' ( ˆ' F nˆ' o Φ nˆ'.. matte ony of notaton... PSP S PCD STTACS Unt Quantum Theoy of Cosons 3

131 ˆn Intechangng ˆn & nˆ ' ˆ ' n ˆ ' n ˆn ˆ ' n Snn ( ˆ, ˆ' S( nˆ', nˆ ˆn Revesng the sgns of & ˆn ˆ ' n PCD STTACS Unt Quantum Theoy of Cosons 3

132 ntechange ncdence & scatteed dectons & evese sgns Snn ( ˆ, ˆ' S( nˆ', nˆ scatteng amptudes: f( nˆ, nˆ' f( nˆ', nˆ RECIPROCITY THEOREM The scatteng amptudes fo two scatteng pocesses whch ae tme-evesed pocesses of each othe ae the same. PCD STTACS Unt Quantum Theoy of Cosons 3

133 Snn ( ˆ, ˆ' S( nˆ', nˆ ˆn ˆ ' n Intechangng ˆn & nˆ ' ˆ ' n ˆn ˆ ' n ˆn Revesng the sgns of & ˆn ˆ ' n Tme-evesa ntechanges the nta and fna states, and eveses the decton of moton of patces n those states. PCD STTACS Unt Quantum Theoy of Cosons 33

134 hν ħω A A + e PHOTIOIONIZATION SCATTERING (A + * + e FINAL STATE: SAME PCD STTACS Unt Quantum Theoy of Cosons 34

135 hν ' + A * HALF-SCATTERING SCATTERING U.Fano & A.R.P.Rau: Theoy of Atomc Cosons & Specta Moton-Revesa PCD STTACS Unt Quantum Theoy of Cosons 35

136 Pata wave anayss σ Tota σ ( hν ' + 4π σ( + sn ~ max a δ nπ [ δ ] Consde s-wave scatteng Eectons just go though the taget! - no scatteng! A * Ramsaue- Townsend effect PCD STTACS Unt Quantum Theoy of Cosons Low enegy (~ev scatteng of eectons by ae gas atoms Xe, K, A Demonstaton of Ramsaue Townsend Effect n Xenon by Kuoch Am. J. Phys. 968 Vo.3, No.8 36

137 ψ nc (+ P(cos θ e P(cos θ ( e ψ Tot { } + δ c e + P θ e e P θ e δ ( ( (cos ( cos Phase shfts pay a centa oe n quantum coson physcs. PCD STTACS Unt Quantum Theoy of Cosons 37

138 ψ nc (+ P(cos θ e P(cos θ ( e ψ Tot { } + δ c e + P θ e e P θ e δ ( ( (cos ( cos Phase shfts ae caused by the scatteng potenta, so to study them we consde two dffeent scatteng potentas. PCD STTACS Unt Quantum Theoy of Cosons 38

139 R ( ε d d d d y ε( d d ( + + U( y (, ( + + U( y (, ( + + U( y (, U ( U ( U ( mv ( Fo two potentas mv ( mv ( Nomazaton π π y(, sn tanδ( cos + π π y(, sn tanδ( cos + PCD STTACS Unt Quantum Theoy of Cosons 39

140 d d d d ( + + U( y (, ( + + U( y (, y (, y (, Eq.A Eq.B y " y y " y U U y y Eq.A - Eq.B Wonsan of the two soutons (defnton: [ ] W y(,, y(, y( y, '(, y( y, '(, pme devatve wth espect to dw U U y y d y (, and y (, PCD STTACS Unt Quantum Theoy of Cosons 4 dw d ( U U y y

141 dw d ( U U y y ( ( b b b dw d U U y y d y U U y d d a a a b [ (,, (, ] b W y y y a U U yd a b a b [ (, '(, (, '(, ] y y y y y U U y d a [ (, '(, (, '(, ] y y y y y U U y d PCD STTACS Unt Quantum Theoy of Cosons 4

142 [ (, '(, (, '(, ] y y y y y U U y d [ (, '(, (, '(, ] y y y y y U U yd y + (, y + (, π π y(, sn tanδ( cos + π π y(, sn tanδ( cos + PCD STTACS Unt Quantum Theoy of Cosons 4

143 Evauaton n the asymptotc egon [ y (, '(, (, '(, ] y y y y U U yd π π y(, sn tanδ( cos + π π y(, sn tanδ( cos + st devatve w..t. [ y (, y '(, y( y, '(, ] π π y '(, cos tanδ( sn π π y '(, cos tan δ( sn + π π π π sn tan ( cos s n tan ( cos + δ δ + π π π cos tanδ ( sn π cos tanδ ( sn PCD STTACS Unt Quantum Theoy of Cosons 43

144 Evauaton n the asymptotc egon [ y (, '(, (, '(, ] y y y y U U yd [ y (, y '(, y( y, '(, ] π π π π sn tan ( cos s n tan ( cos + δ δ + π π π cos tanδ ( sn π cos tanδ ( sn [ y (, y '(, y( y, '(, ] π π π π sn + tanδ ( cos sn tanδ ( cos + π π π cos tanδ ( sn π cos tanδ ( sn PCD STTACS Unt Quantum Theoy of Cosons 44

145 Evauaton n the asymptotc egon [ y (, '(, (, '(, ] y y y y U U yd π π π sn cos sn tanδ π + tanδ ( cos π π tanδ( sn tanδ( cos π π π sn cos sn tanδ π + tanδ ( cos π π tanδ( sn tanδ( cos tanδ + tanδ ( ( PCD STTACS Unt Quantum Theoy of Cosons 45

146 Evauaton n the asymptotc egon [ y (, '(, (, '(, ] y y y y U U yd tanδ y ( U tanδ + U yd δ ( tanδ tan y (, U U y ( d, Nomazaton : V { } { V } V tan δ j( UR, ( d, R (, j(, tan δ ( n (, V when U ( (fee patce! tan δ j(, U R (, d PCD STTACS Unt Quantum Theoy of Cosons 46

147 V tan δ j( UR, ( d, { } { V } tan δ j(, U R (, d { R j } sn π ( V asymptotc behavo E > contnuum fo V (, π sn V R + δ V R (, sn π Examne the noda behavo PCD STTACS Unt Quantum Theoy of Cosons 47

148 π π (, sn + δ has nodes at + δ π π π (, sn has nodes at π n,,,3, 4,... V R n V R n π π (, sn π π π + n,,,3, 4,... V R + δ nπ + δ V R (, sn n δ ( (, ae pued/pushed by espect to those of (, dependng on V nodes of R V wth R δ ( o δ (. PCD STTACS Unt Quantum Theoy of Cosons 48

149 V( QUESTIONS? Wte to: V δ ( (, ae pushed/pued by espect to those of (, dependng on V nodes of R V wth R V δ ( o δ (. R V (, π nπ δ ( + Repusve Attactve Potenta Potenta V( Refeence: Joachan: Quantum Coson Theoy / page 8 PCD STTACS Unt Quantum Theoy of Cosons 49

150 Seect/Speca Topcs n Theoy of Atomc Cosons and Spectoscopy P. C. Deshmuh Depatment of Physcs Indan Insttute of Technoogy Madas Chenna 636 Lectue Numbe 9 Unt : Quantum Theoy of Cosons Moe on: Phase-shft anayss - fom Joachan s Quantum Coson Theoy PCD STTACS Unt Quantum Theoy of Cosons 5

151 Repusve Potenta V( V R V (, π nπ δ ( + Attactve Potenta δ ( (, ae pushed/pued by espect to those of (, dependng on V nodes of R V wth R V δ ( o δ (. Repusve Attactve Potenta Potenta V( Refeence: Joachan: Quantum Coson Theoy / page 8 PCD STTACS Unt Quantum Theoy of Cosons 5

152 δ ( tanδ tan y (, U U y ( d, tanδ ( tanδ ( ( U ( U ( δλ y(, y( d, δλ d U( λ, ( tan δ( y(, y( d, dλ λ cos V tan δ j( UR, ( d, ( dδ U( λ, y(, y( d, δ dλ λ U U( λ, U ( U( λ, λ : coupng stength paamete PCD STTACS Unt Quantum Theoy of Cosons 5

153 U U( λ, U ( U( λ, λ : coupng stength paamete cos ( dδ U( λ, y(, y( d, δ dλ λ V tan δ j( UR, ( d, dδ { } U( λ, cos δ y(, y( d, dλ λ dδ U( λ, {} y(, y(, d dλ λ PCD STTACS Unt Quantum Theoy of Cosons 53

154 dδ U( λ, y(, y(, d dλ λ U U( λ, λ : coupng stength paamete dδ U( λ, y(, y(, {...} d dλ + λ dδ [ ] U( λ, y (, d dλ λ U( λ, dδ ( f the sgn of then has λ λ does not change n the opposte sgn egon PCD STTACS Unt Quantum Theoy of Cosons 54

155 U( λ, dδ ( f the sgn of then has λ λ does not change n the opposte sgn egon δ ( htheto defned moduo π can now be defned as an absoute ange by settng δ ( fo U, and et δ ( evove contnuousy wth the conto paamete λ to get : tan δ j( UR, ( d, PCD STTACS Unt Quantum Theoy of Cosons 55

156 V( V a aa Stct fnte ange ( a R (, Aˆ ] exteo [ δ ] ( j( tan ( n( dr (, R (, and ae contnuous at a. d dr(, dr(, d d γ ( R(, R(, γ ( nteo exteo Aˆ Aˆ ' ' j a δ n a ( [ j( tan δ( n( ] a ' ' ( j( tan δ( n ( tan [ j ( a tan δ ( n ( a ] Pme: devatve w..t. Pme: devatve w..t. evauated at at Logathmc devatve of the ada wave functon at a PCD STTACS Unt Quantum Theoy of Cosons a a 56

157 Logathmc devatve of the ada wave functon at a γ ( tan δ ( V( tan δ ( tan ' ' j a δ n a [ j ( a tan δ ( n ( a ] ' j ( a γ j a ' n ( a γ n a a d a V ( d: neggbe even f not zeo ' j ( d γ j d ' n ( d γ n d Pme: devatve w..t. V ( a We sha consde ( V a PCD STTACS Unt Quantum Theoy of Cosons 57

158 ' j ( a γ j a tan δ ( ' n ( a γ n a ( a ' j ' j ( a j ( a j a q ( tan δ ( ' j ( a ' j ( a n ( a n a q ( tan δ ( γ ( / ' j a j a q ( γ ( defnton : dmensoness j ( a ( a ( a ' j ' j ( a j ( a j a q ( ' j ( a ' j ( a n ( a n a q ( ' q ( j PCD STTACS Unt Quantum Theoy of Cosons 58

159 tan δ ( ' j a ' n a ( a ' j j ( a j ( a q ( ' j ( a j ( a n ( a q ( tan δ ( ' j a ' n a ( a ' j j ( a j ( a q ( ' j ( a j ( a n ( a q ( tan δ ( ' j ( a q ( ' j ( a j ( a n ( a q ( ' n a tan δ ( { } ' ' j ( a j a q ' q n a j a j a n a PCD STTACS Unt Quantum Theoy of Cosons 59

160 tan δ ( z n ( z a z z z j ( z +... ( +!!! ( + 3! ( + 3( + 5 ( +!! (+ (!! ( z (!! z z! (! ( ( 3 ( + { } ' ' j ( a j ' a q ( q ( n ( a j a j a n ( a ow enegy ( z!! j( z n( z z!! z z + PCD STTACS Unt Quantum Theoy of Cosons 6?

161 D D + D +!!!! D D ( +!!(!! ( ( ( D ! (! D +!! ( + (! (!!! PCD STTACS Unt Quantum Theoy of Cosons 6 D ( D

162 tan δ ( z a z n( z + z D z + z j z { } ' ' j ( a j ' a q ( q ( n ( a j a j a n ( a ow enegy D +!! D!! +? D z ; +!! ' ' j z n z z z!! ( + { } z + z ; z +!! z ' ' j z n z!!( + + PCD STTACS Unt Quantum Theoy of Cosons 6 z

163 tan δ ( z a ' ' j z n z z and n( z + z D z + z j z { } ' ' j ( a j a q ' q n a j a j a n a z ; z D z tan δ ( ow enegy + { } tan δ ow enegy D ( + + z q z z D D q ( + z + z + z z ow { } enegy q D+ D+ D ( + z z D q ( + + z D+ D+ z tan δ ( ow enegy PCD STTACS Unt Quantum Theoy of Cosons 63 D? q ( + ( + DD + q ( + z

164 ow enegy + q ( z tan δ ( DD ( + + q ( + ' V j( a / j( a γ ( q ( defnton γ ( γ ( ' j ( a V γ ( V γ j ( a PCD STTACS Unt Quantum Theoy of Cosons 64 ow enegy (? ρ ρ a ( a j ( ρ ρ a ρ ( a j ' ( ρ ρ D+ D + ρ D+ D+ γ V tan δ ( a ( ( a a a q ( ow enegy + aγ ( z DD aγ ( aγ (

165 tan δ ( tan δ ( ow enegy ow enegy tan δ ( + aγ ( z DD aγ ( ow enegy + aγ ( z D D + + aγ ( + + a aˆ γ ( D D + + aˆ γ ( + z a ˆ γ ( m γ ( ow enegy tan δ ( + a ( f a ˆ γ + ' zeo enegy esonance ' PCD STTACS Unt Quantum Theoy of Cosons 65

166 ow enegy tan δ ( + ( a [ aˆ γ ] D ( + + a ˆ γ D D + D +!!!! fo fo RECALL: ow enegy whee ˆ γ γ ( ( θ ( + (cos θ f ( θ f a P a ( e δ ( [ S ] ( : scatteng amptude a ( : pata wave amptude ow enegy Fo sma δ(, δ( tan δ( + PCD STTACS Unt Quantum Theoy of Cosons 66

167 S ( ( e δ S matx eement S ( cos δ + sn δ + δ fo sma ow enegy ( + + snce δ + S c δ Pata wave amptude Contbuton to pata wave cosssecton PCD STTACS Unt Quantum Theoy of Cosons a ( a ( [ ] ( + S ( c c 4 Fas apdy fo sma, except fo scatteng ength especay usefu to descbe ow enegy s-wave ony scatteng 67

168 Scatteng ength defnton Pata wave amptude α m tanδ Low enegy s wave scatteng ( snδ tanδ tanδ m a( m m m [ S ] Dmenson: L δ ( e cos δ + sn δ a ( ( δ ( cos δ + sn m a ( m m m m a ( m ( δ( δ( δ( δ( cos sn + sn cos ( δ( δ( δ( sn + sn cos ( sn δ( + snδ( cosδ( δ : sma α PCD STTACS Unt Quantum Theoy of Cosons 68

169 m a ( m tanδ ( α ( θ ( + (cos θ f ( θ f a P a ( e δ ( [ S ] ( a ( : Low enegy s-wave ony scatteng θ α θ α f a f : scatteng amptude pata wave amptude σ f θ dω 4πα tota P (cos θ PCD STTACS Unt Quantum Theoy of Cosons 69

170 ' j ( a γ j a tan δ ( fo a. ' n ( a γ n a ' j ( a γ ( j ( a n ( a γ n a tan δ ( fo ' fo sn z cos z j( z ; n( z z z ' cos z sn z ' sn z cos z j ( z ; n ( z + z z z z z a PCD STTACS Unt Quantum Theoy of Cosons 7

171 tan δ ( ' ' j ( a γ ( j ( a n ( a γ n a sn z cos z j( z ; n( z z z cos z sn z sn z cos z z z z z ' ' j ( z ; n ( z + γ ( / j a j a ' q ( / ' ' j a j a j ( a j ( a q( tan δ ( ' ' j( a / j( a n ( a n ( a q( tan δ ( ' ' j ( a q( j a n ( a q ( j a ' ' j n ( a ( a PCD STTACS Unt Quantum Theoy of Cosons 7

172 tan δ ( δ tan ( ' ' j ( a q( j a n ( a q ( j a ' ' j n ( a ( a [ ] ' j a q n ( a q ( j a ' ' j n ( a ( a sn z cos z j( z ; n( z z z cos z sn z sn z cos z z z z z ' ' j ( z ; n ( z + [ q ( ] cos a sn a tan δ ( a ( a sn a cos a cos a sn a + q ( { cot( a } a ( a a ( a PCD STTACS Unt Quantum Theoy of Cosons 7

173 [ q ( ] cos a sn a tan δ ( a ( a sn a cos a cos a sn a + q ( { cot( a } a ( a a ( a δ ( a ( a ( tan ( acos a sn a 4 θ θ cosθ +...! 4! 3 5 θ θ sn θ θ ! 5! [ q( ] ( a sn a + cos a q ( ( a cos a sn a { cot ( a } 3 5 θ θ θcos θ θ +...! 4! θ θ θ θ θ 3! 5! 4 6 sn θ θ 3 θ θcosθ sn θ θ +.. θ +.. θ! 3! 6 3 θ θ! sn + cos θ θ θ θ PCD STTACS Unt Quantum Theoy of Cosons 73

174 δ ( tan ( acos a sn a [ q( ] ( a sn a + cos a q ( ( a cos a sn a { cot ( a } θ θ 3 θ θcosθ sn θ θ +.. θ +.. θ! 3! 6 3 θ θ θsnθ + cos θ θ ! 3 ( a [ q( ] 3 ( a ( a tan δ ( 3 + q( { cot( a } θ θ ( a ( a cot θ.. θ 3 45 a.. 3 a ( a 3 ( a [ q( ] 3 ( a tan δ ( q ( 3 QUESTIONS? Wte to: pcd@physcs.tm.ac.n tan ( ( a [ q ( ] δ q a 3 + PCD STTACS Unt Quantum Theoy of Cosons 74

175 Seect/Speca Topcs n Theoy of Atomc Cosons and Spectoscopy PCD STTACS Unt Quantum Theoy of Cosons P. C. Deshmuh Depatment of Physcs Indan Insttute of Technoogy Madas Chenna 636 Lectue Numbe Unt : Quantum Theoy of Cosons aˆ γ + esonant condton th n the pata wave zeo enegy esonance 75

176 a ( ow enegy tan δ ( tan δ ( + a [ ] ( + S ( c ow enegy c + a aˆ γ ( D D + + aˆ γ ( + S ( cos δ + sn δ + δ fo sma δ ow enegy S( + ( c + snce δ + Phase shft tends to zeo (moduo π a ( 4 Fas apdy fo sma, except fo PCD STTACS Unt Quantum Theoy of Cosons 76

177 tan δ ( ow enegy + a aˆ γ ( D D + + aˆ γ ( + tan ( ( f aˆ γ + ( a [ q ( ] δ q a 3 + ' zeo enegy esonance ' PCD STTACS Unt Quantum Theoy of Cosons 77

178 tan ( ( a [ q( ] q ( a δ 3 cos sn j ( z sn z ; j ( z z z ' z z z wth q ( / j a j a ' γ ( 4 θ θ cosθ +...! 4! 3 5 θ θ sn θ θ ! 5! 4 snθ θ θ +... θ 3! 5! 4 z z... z j z + +Ο z 3! 5! ' z z z z j ( z z! 4! z 3! 5! PCD STTACS Unt Quantum Theoy of Cosons 4 3 cosθ θ θ +... θ θ! 4! 3 snθ θ θ +... θ θ 3! 5! z + z 6 4! 5! e.. j ( z z z ' 3 78

179 ( a PCD STTACS Unt Quantum Theoy of Cosons [ q( ] q ( a δ 3 tan z 4 j ( z +Ο( z 6 j ( z z z z a ' 3 a a q ( q ( wth q ( / j a j a ' γ ( ( a +Ο 6 γ ( ( a ( a ( a γ ( a +Ο 6 ( 4 ( a ( a ( a 4 ( 4 ( a 3 79

180 ( a [ q( ] q ( a δ 3 tan wth q ( / j a j a ' γ ( q ( ( a γ ( a +Ο 6 ( a ( a 4 ( 4 ( a γ ( a ( a ( a 4 q ( +Ο 3 γ ( a ( a ( a 4 PCD STTACS Unt Quantum Theoy of Cosons q ( 3 γ ( a ( a 8

181 [ q ( ] q ( a δ 3 tan ( a... fo ( a 3 γ ( a [ q ] 3 q( a 3 q( ( a [ ] γ gnong q a γ q a 3 ( γ ( a ( a γ ( a 3 3 q( ( a γ ( a 3 q ( a 3 q ( a ( a weae tems PCD STTACS Unt Quantum Theoy of Cosons 8

182 [ q ( ] q ( a δ 3 tan ( a... fo [ ] γ gnong q a γ 3 q ( a 3 q ( a ( a weae tems γ ( a tan δ ( ( a... fo 3 q( ( a γ ( a PCD STTACS Unt Quantum Theoy of Cosons 8

183 γ ( a tan δ ( ( a... fo 3 q( ( a γ ( a tan ( a... fo δ 3 q( ( a j ( a / j ( a 3 a ' ; q( γ ˆ ( γ q whee: ˆ γ m γ ( fo ( a tan δ ( ( a... fo ( a 3 3 ( a ˆ γ a PCD STTACS Unt Quantum Theoy of Cosons 83

184 tan δ ( ( a... fo ( a 3 3 ( a ˆ γ a tan δ ( ( a... fo + ˆ γ a ˆ γ a tan δ ( a... fo [ + ˆ γ a ] PCD STTACS Unt Quantum Theoy of Cosons 84

185 ow + enegy ( a aˆ γ tan δ (.. fo DD + + aˆ γ + Both cases: ow + enegy a aˆ γ tan δ (.. fo DD + + aˆ γ + ˆ γ a tan δ ( a... fo [ + ˆ γ a ] what f: aˆ γ + esonant condton th n the pata wave We sha fst consde such esonant condtons fo The case w be consdee d ate. PCD STTACS Unt Quantum Theoy of Cosons. 85

186 fst, Consde the next tem n the ow enegy appoxmaton and compae ts mpotance wth that of the consequence of the esonant condton: z z z j ( z +... ( +!!! ( + 3! ( + 3( + 5 th ( aˆ γ + esonant condton n the pata wave z j ( z +Ο z!! ( + ( + z ( Coectons : Ο z + PCD STTACS Unt Quantum Theoy of Cosons 86

187 fst, γ V ' j ( a V γ ( j ( a a ( ( a a a q ( defnton V γ ( γ ( q ( q ( Coectons : z a a γ ( Ο aγ ( ( + z γ V Next ode modfcatons: ( ( +Ο a +Ο a ( ; q( a aγ ( PCD STTACS Unt Quantum Theoy of Cosons 87

188 fst, eca: tan δ ( ow enegy + ( a ( + q ( DD + q ( + Use next ode tem: q ( +Ο ( a aγ ( tan δ ( ( a +Ο + ( + ( a DD + +Ο ( a + ( + ow enegy th aˆ γ ( + esonant condton n the pata wave q +Ο ( ( + ( a ( + PCD STTACS Unt Quantum Theoy of Cosons 88

189 tan δ ( ow enegy tan δ ( ( a +Ο + DD ow enegy ( + ( a +Ο + ( + ( a DD + +Ο ( a + ( + + a ( a Ο + + ( + tan δ ( ow enegy ( a +Ο DD ( + + ( a tan δ ( ow enegy ( + DD + ( a ow enegy tan δ ( a PCD STTACS Unt Quantum Theoy of Cosons 89

190 ow enegy tan δ ( a S e ( δ ( δ ( δ δ cos + sn + fo sma δ ow enegy ( + snce δ S d ( θ ( + (cos θ f ( θ f a P a ( e δ ( [ S ] ( : scatteng amptude a ( : pata wave amptude a ( ( d + d PCD STTACS Unt Quantum Theoy of Cosons 9

191 ( θ ( + (cos θ f ( θ f a P a ( e δ ( [ S ] : scatteng amptude ( a ( : pata wave amptude a ( ( d + d fo : fo, a d What s the contbuton of ths tem to the scatteng amptude? + a ( P (cos θ 3 d cos θ β cos θ PCD STTACS Unt Quantum Theoy of Cosons 9

192 ( θ ( + (cos θ f ( θ f a P a ( e δ ( [ S ] : scatteng amptude ( a ( : pata wave amptude We have, fo : + a ( P (cos θ βcosθ We have, fo : + a ( P (cos θ α scatteng amptude f θ α + β θ cos when aˆ γ ( + esonant condton n the pata wave fo. PCD STTACS Unt Quantum Theoy of Cosons 9

193 ( θ ( + (cos θ f ( θ f a P a ( a ( e δ ( [ S ] : scatteng amptude ( a ( : pata wave amptude ( d + d f, a ( d f, a cos scatteng amptude f θ α + β θ Thus the utty of s-waves scatteng ength fomasm. when aˆ γ ( + esonant condton n the pata wave fo. PCD STTACS Unt Quantum Theoy of Cosons 93

194 ow + enegy ( a aˆ γ tan δ (.. fo DD + + aˆ γ Above, + f/when: aˆ γ + th esonant condton n the pata wave we fst consdeed esonant condtons fo. NOW, we consde the case fo. Fo, aˆ γ + PCD STTACS Unt Quantum Theoy of Cosons 94

195 [ q ( ] q ( a tan δ ( a... fo 3 +Ο 3 q ( γ ( a [ ] q ( a ( a 4 3 q( a 3 q( ( a ( +Ο 3 γ ( a [ ] γ gnong q a ( a ( a 4 4 ( a +Ο( a γ ( a 3 3 q( ( a γ ( a γ 3 q ( a 3 q ( a ( a weae tems PCD STTACS Unt Quantum Theoy of Cosons 95

196 [ q ( ] q ( a tan δ ( a... fo 3 [ q ( ] γ eadng a γ 3 q ( a 3 q ( a ( a tems ˆ γ a tan δ ( ( a... fo ˆ γa 3 ( ˆ γa q( ( a ˆ γ m γ ( PCD STTACS Unt Quantum Theoy of Cosons 96

197 ˆ γ a tan δ ( ( a... fo ˆ γa 3 ( ˆ γa q( ( a PCD STTACS Unt Quantum Theoy of Cosons ( a ( a 4 ' j( a / j( a +Ο 3 ; q( γ ˆ ( γa q when a ˆ γ (non-esonant, we had gnoed tan δ ( ( a ( a 4 γ Fo, when aˆ + we consde next ode tem n a ˆ γ a 4 ( a +Ο( a ˆ 3 γa 3( ˆ γa a ˆ γ a esonant pat 4 97

198 tan δ ( ( a γ 4 PCD STTACS Unt Quantum Theoy of Cosons Fo, when aˆ + we esonant pat consde next ode tem n a ˆ γ a ( a +Ο( a ˆ 3 γa 3( ˆ γa a ˆ γ a tan δ ( ( a +Ο( a 3 aˆ γ ( 3( ( + ( 4 tan δ ( ( a ( 3 3 Ο( a 3 { } 3( a 98

199 γ Fo, aˆ + esonant pat consdeng the next ode tem n a 4 tan ( δ 3 a tanδ m a( m α defnton : scatteng ength mα as, scatteng ength dveges as tan ( δ bows up π tan δ ( ± when δ ( ± PCD STTACS Unt Quantum Theoy of Cosons 99

200 sn θ π Just beow θ, both sn θ and cosθ ae postve.. π π π Just above θ, sn θ > and cosθ <. 3 π cosθ tan θ eveses + to π 3π 5π sgn at θ,,,... θ n adans PCD STTACS Unt Quantum Theoy of Cosons

201 3π π π tan δ ( 3 ( a tanθ π tan δ (, and δ ( moduo π tan θ eveses + to π 3π 5π sgn at θ,,,... ow enegy 7 tan θ θ + θ + θ + θ +... tan δ ( non esonant π δ ( moduo π 3π Fo, aγ ˆ esonant pat Zeo enegy esonance ( π + π θ PCD STTACS Unt Quantum Theoy of Cosons

202 as, δ ( π moduo π ( θ ( + (cos θ f ( θ f a P a ( e δ ( [ S ] : scatteng amptude ( a ( : pata wave amptude a a S ( cosδ + snδ δ π ( δ π cosπ + snπ ( PCD STTACS Unt Quantum Theoy of Cosons

203 σ tota cosπ + snπ a( δ π ( θ ( + (cos θ f ( θ f a P a ( Fo aˆ γ + e δ ( ( [ S ] ( a ( : f dω dω ( θ esonant pat Zeo enegy esonance x sec bows up as.e. as as E π 3π 5π δ,,,... pata wave amptude : scatteng amptude 4π f ( θ Zeo enegy esonance QUESTIONS? Wte to: pcd@physcs.tm.ac.n PCD STTACS Unt Quantum Theoy of Cosons 3

204 Seect/Speca Topcs n Theoy of Atomc Cosons and Spectoscopy P. C. Deshmuh Depatment of Physcs Indan Insttute of Technoogy Madas Chenna 636 Lectue Numbe Unt : Quantum Theoy of Cosons Levnson s theoem 949 Numbe of bound states of an attactve potenta Scatteng phase shfts PCD STTACS Unt Quantum Theoy of Cosons 4

205 γ Fo, aˆ + esonant pat consdeng the next ode tem n a 4 tan ( δ 3 a tanδ m a( m α defnton : scatteng ength mα as, scatteng ength dveges as tan ( δ bows up π tan δ ( ± when δ ( ± PCD STTACS Unt Quantum Theoy of Cosons 5

206 3π π π tan δ ( 3 ( a tanθ π tan δ (, and δ ( moduo π ( π Fo aˆ γ + tan θ eveses + to π 3π 5π sgn at θ,,,... ow enegy 7 tan θ θ + θ + θ + θ +... tan δ ( non esonant π δ ( moduo 3π esonant pat Zeo enegy esonance ( π + π θ PCD STTACS Unt Quantum Theoy of Cosons 6

207 as, δ ( π Fo when aˆ + esonant pat ( θ ( + (cos θ f ( θ f a P a ( e δ ( γ [ S ] moduo π Zeo enegy esonance : scatteng amptude ( a ( : pata wave amptude a a ( S ( cosδ + snδ δ π cosπ snπ + δ π ( PCD STTACS Unt Quantum Theoy of Cosons 7

208 σ tota cosπ + snπ a( δ π ( θ ( + (cos θ f ( θ f a P a ( Fo aˆ γ + e δ ( ( [ S ] ( a ( : f dω dω ( θ esonant pat Zeo enegy esonance x sec bows up as.e. as as E π 3π 5π δ,,,... pata wave amptude : scatteng amptude 4π f ( θ Zeo enegy esonance PCD STTACS Unt Quantum Theoy of Cosons 8

209 LEVINSON s THEOREM zeo of δ ( : δ ( efeence Kg. Danse Vdensab. Sasab. Mat. Fys. Medd.5 9 ( fo : δ ( n π o δ n + π f thee s a (esonant zeo enegy esonance bows π σtota ( when λ a Ua up π δ( δ π haf-bound state bound state souton at zeo enegy. n... fo PCD STTACS Unt Quantum Theoy of Cosons 9

210 Squae we attactve potenta U ( β m m U ( V ( a y R ( ε( ε U β fo < a fo > d ( + + V( E y ( + ε md m d d m + { U (} + E yε ( a me γ dscete bound state d d + { U (} γ yε ( PCD STTACS Unt Quantum Theoy of Cosons

211 Bound states of the SPHERICAL we attactve potenta U ( yε( R ε( a d E γ + { U } γ y β d d d d PCD STTACS Unt Quantum Theoy of Cosons dscete bound state ( ( ε d U β fo < a fo > a + β γ y ( fo ( sn ε < a yε A β γ y ( > yε( B γ fo ε Contnuty at β γ e γ a tan a tan θ popetes... β γ γ

212 DISCRETE BOUND STATES of a SPHERICAL squae we attactve potenta U ( a β m U ( V ( ξ tanξ η E γ dscete bound state contnuty at tan ( a β γ a β γ ξ a β γ & η aγ Bound state dscete enegy eves ae gven by the ntesecton of the cuves descbed by these two equatons. ξ + η a β γ + a γ a β U a γ PCD STTACS Unt Quantum Theoy of Cosons

213 U ( β a tanθ tan a β γ tanθ 3 β γ γ dscete bound state Dscete bound states < a 3π π π yε ( Asn β γ > y ( B ε e γ PCD STTACS Unt Quantum Theoy of Cosons 3 π ππ 3π π 5π 3π dscete bound states π 3π tanξ < aβ < 7π 4 θ aβ dscete bound states ξ η ξ a β γ & η aγ Etc. 3

214 U ( a β V( x V x a a PCD STTACS Unt Quantum Theoy of Cosons 4

215 a V V( x a I II III V V x d + V( x ψ( x Eψ( x mdx d + V ψ( x Eψ( x mdx I d ψ( x Eψ( x E > mdx II d + V ψ( x Eψ( x mdx III ψ x Fe + De De } I } β x βx βx ψ ( x Asnαx+ Bcosαx } III II ψ x Ce + Ge Ce β x βx βx me α + ( E mv β + V > E (bound states PCD STTACS Unt Quantum Theoy of Cosons 5

216 a V V( x a I II III V V V α + β + > E me mv ( E (bound states x } ψ ( x} I De β ψ ( x Asnαx+ Bcosαx x a: ψ ( x} III x x Ce β Asnαa+ Bcosαa Ce β a Aαcosαa Bαsn αa x a: Asnαa+ Bcosαa De β a Aαcosαa+ Bαsn αa βde β a β a PCD STTACS Unt Quantum Theoy of Cosons 6

217 V V( x a: Asnαa+ Bcosαa Ce β a Aαcosαa Bαsn αa βce β a a a I II III V V me α + ( E mv β + V > E (bound x a: Asnαa+ Bcosαa De x β a Aαcosαa+ Bαsn αa βde Asnαa ( C D e β a Aαcosαa β( D C e β a B cosαa ( C+ D e β a Bαsnαa β( C+ D e β a β a PCD STTACS Unt Quantum Theoy of Cosons 7

218 a α + V V( x a me ( E mv β + V > E (bound states x Asnαa ( C D e β a Aαcosαa β( D C e β a B cosαa ( C+ D e β a Bαsnαa β( C+ D e β a A and C D whence Bcosαa Ce β a αbsnαa Cβe β a α tanαa β B and C D whence Asnαa Ce β a αacosαa Cβe β a tanαa α β e.. α cotαa β PCD STTACS Unt Quantum Theoy of Cosons 8

219 a α + V V( x a me ( E mv β + V > E (bound states α tanαa β A and C D both ξ & η ae postve α tanαa β A and C D x put tan a α cotαa β B and C D tanαa α α β α β α [ tan a ] α : magnay β < : contadcton ethe o : & ξ αa η β a ξ tanξ η ξ mv + η a mva PCD STTACS Unt Quantum Theoy of Cosons 9

220 Schff/QM/page 4/Fg.8 ξ tanξ η ξ < π ξ tanξ η π 3π < ξ < adus of the cce s detemned by the "stength" V a tanθ Va m Va m Va 4 m π ξ ξ π π π 3π m ξ + η Va cces Va m ξ PCD STTACS Unt Quantum Theoy of Cosons 3 π

221 Schff/QM/page 4/Fg.8 ξ tanξ η Va ξ < π m Intesectons of ξ tanξ η π 3π < ξ < ξ tanξ m and cces of adus V a gve the bound state eneges Va 4 m π ξ ξ π ξ & η ae postve tanθ π π 3π m ξ + η Va cces Va m ξ PCD STTACS Unt Quantum Theoy of Cosons 3 π

222 Schff/QM/page 4/Fg.9 α cotαa β B and C D ξ cotξ η Va m NO souton PCD STTACS Unt Quantum Theoy of Cosons

223 U ( a β V( x V x a a PCD STTACS Unt Quantum Theoy of Cosons 3

224 DISCRETE BOUND STATES of a SPHERICAL squae we attactve potenta U ( a β m U ( V ( ξ tanξ η E γ dscete bound state contnuty at tan ( a β γ a β γ ξ a β γ & η aγ Bound state dscete enegy eves ae gven by the ntesecton of the cuves descbed by these two equatons. ξ + η a β γ + a γ a β U a γ PCD STTACS Unt Quantum Theoy of Cosons 4

225 ξ tanξ η η ξ tanξ Each cce epesents a patcua potenta of a gven stength. The numbe of tmes t cosses the cuve gves η ξ tanξ the numbe of bound states the potenta hods π.57 6 π 3π < aβ <.57 < aβ < 4.7 n ξ + η a β γ + a γ a Ua β aβ a U Cacuatons and gaphs by Sayantan Auddy and Panav Manangath 3π 5π < aβ < 4.7 < aβ < 7.85 n ξ PCD STTACS Unt Quantum Theoy of Cosons 5

226 U ( β a ξ + η a β γ + a γ a β Ua tanθ dscete bound state tan a β γ tanθ 3 β γ γ Dscete bound states < a 3π π π yε ( Asn β γ > y ( B ε e γ PCD STTACS Unt Quantum Theoy of Cosons 3π π ~.57 π π 3 π π ~3.4 ~4.7 π ~6.8 5π 3π dscete bound states π 3π tanξ < aβ < 7π 4 θ aβ dscete bound states ξ η ξ a β γ & η aγ Etc. 6

227 We consdeed the bound states of a SPHERICAL we attactve potenta U ( a β m U ( V ( U β fo < a fo > a yε( R ε( Now, we consde scatteng ; contnuum states E > m d ( + + V( E y ( + ε md m d d PCD STTACS Unt Quantum Theoy of Cosons me m + { U (} + E yε ( d d + { U (} + yε ( 7

228 U ( β a m U ( V ( U β fo < a fo > a d d me > yε( R ε( + { U (} + yε ( d d d d + β + y ( < a y ( Csn + s a PCD STTACS Unt Quantum Theoy of Cosons ( β ε ε + n yε > a yε D + Contnuty at β + ( + δ Csn a Dsn a ( ( δ C β + a β + D a+ δ cos cos 8

229 β + ( + δ Csn a Dsn a ( C β + a β + D a+ δ cos cos ( a β + ( a+ δ tan tan β + + ( δ ( δ tan( a tan tan a tan ( ( a β ( a β ( a ( δ tan + tan + tan tan β + β + ( δ tan( a + tan δ a β a a ( a β tan + tan + tan tan tan + β + β + PCD STTACS Unt Quantum Theoy of Cosons 9

230 ( β tan( a tan a + β + tan ( δ( + tan + tan β + ( δ tan ( a β ( a β + β + tan ( a β tan ( a tan a tan( a β δ tan a β + tan( a β + + tan a β + tan ( a tan tanδ α m scatteng ength ( β + α ( β atan a βa tan α a β βa ( aβ PCD STTACS Unt Quantum Theoy of Cosons 3

231 U( a Refeence: Quantum Theoy of Scatteng by Ta-You Wu and Taash Ohmua (Pentce Ha, 96 page 73 β n numbe of bound states tanδ ( α m m cot δ α α a β ( β tan a β ( ( aβ tan mα a β aβ a β n βa cotδ x α δ π βa< > x> α > * π π + π < βa< π > x> > α > π 3π π < βa< > x> > α > π + * 3π 3π + 3π < βa< π > x> > α > π 5π π < βa< > x> > α > π PCD STTACS Unt Quantum Theoy of Cosons 3

232 U( a Refeence: Quantum Theoy of Scatteng by Ta-You Wu and Taash Ohmua (Pentce Ha, 96 page 73 β n numbe of bound states tanδ ( α m m cot δ α α a β PCD STTACS Unt Quantum Theoy of Cosons ( β tan a β ( ( aβ tan mα a β aβ a β n βa Levnson s cotδ x Theoem α δ π βa< > δ ( x> α > π * π π ( + δ + π π < βa< π > x α δ ( > > > π π 3π π < βa< > x> > α > δ π π + * 3π 3π ( δ + + π 3π < βa< π > δ x( > > α > π π 5π π < βa< > δ ( x> > α > π π 3

233 LEVINSON s THEOREM zeo of δ ( : δ (... fo : δ ( n π o δ n + π f thee s a (esonant bound state souton zeo enegy esonance bows π σtota ( when λ a Ua up π δ( δ π Kg. Danse Vdensab. Sasab. Mat. Fys. Medd.5 9 (949 haf-bound state at zeo enegy. n... fo PCD STTACS Unt Quantum Theoy of Cosons QUESTIONS? Wte to: pcd@physcs.tm.ac.n 33

234 Seect/Speca Topcs n Theoy of Atomc Cosons and Spectoscopy P. C. Deshmuh Depatment of Physcs Indan Insttute of Technoogy Madas Chenna 636 Lectue Numbe Unt : Quantum Theoy of Cosons Levnson s theoem 949 Scatteng ength Effectve ange Low enegy scatteng Uta-Cod atoms PCD STTACS Unt Quantum Theoy of Cosons 34

235 LEVINSON s THEOREM zeo of δ ( : δ (... fo : δ ( n π o δ n + π f thee s a (esonant bound state souton zeo enegy esonance bows π σtota ( when λ a Ua up π δ( δ π Kg. Danse Vdensab. Sasab. Mat. Fys. Medd.5 9 (949 haf-bound state at zeo enegy. n... fo PCD STTACS Unt Quantum Theoy of Cosons 35

236 U( a Refeence: Quantum Theoy of Scatteng by Ta-You Wu and Taash Ohmua (Pentce Ha, 96 page 73 β n numbe of bound states tanδ ( α m m cot δ α α a β ( β tan a β ( ( aβ tan mα a β aβ a β n βa Levnson s cotδ x Theoem α δ * π βa< > δ ( x> α > π π π ( + δ + π π < βa< π > x α δ ( > > > π π 3π π < βa< > x> > α > δ π π + * 3π 3π δ ( + + π 3π < βa< π > δ x( > > α > π π 5π π < βa< > x> > α > π δ π PCD STTACS Unt Quantum Theoy of Cosons 36

237 U( a Refeence: Quantum Theoy of Scatteng by Ta-You Wu and Taash Ohmua (Pentce Ha, 96 page 73 β n numbe of bound states tanδ ( α m m cot δ α α a α a ( β tan a β tan ( aβ aβ ( n βa cotδ x α δ π βa< > x> α > * π π + π < βa< π > x> > α > π 3π π < βa< > x> > α > π + * 3π 3π + 3π < βa< π > x> > α > π 5π π < βa< > x> > α > π PCD STTACS Unt Quantum Theoy of Cosons 37

238 U( Negatve tan ( aα β ndcates a β an attactve potenta. α β Postve α n numbe of ndcates bound statesa tanδ ( epusve α m potenta. m cot δ α a ( A new bound n βa cotδ x α δ π state gets βa< > x> negatve α > fomed π * when π to + the sgn π < βa< π of > xthe > > postve α > π scatteng 3π π < βa< > ength x> s > negatve α > π about to 3π 3π + * change to + fom 3π < βa< π negatve > x> > postve α > π 5π to π < βa< > x> > negatve α > π postve. PCD STTACS Unt Quantum Theoy of Cosons 38

239 U( β a How the s-wave phase shft changes wth the stength of the potenta δ ( nπ π.57 n numbe of bound states π 3π 4.7 π 5π π β PCD STTACS Unt Quantum Theoy of Cosons 39

240 U ( β when ( aβ ( aβ tan a U β fo < a fo > a >, α s negatve when does α get to be MOST negatve? π ( β < st when a bound state 3π nd when ( aβ < bound state 5π d when ( aβ < 3 bound state α a α a tan α a atan ( aβ β tan ( aβ aβ ( aβ aβ PCD STTACS Unt Quantum Theoy of Cosons 4

241 U ( β when π < ( aβ < π + ε α > at aβ π + ε : α changes sgn + to ( aβ ( aβ tan a U β fo < a fo > a 3π when π + ε < ( aβ < α < >, α s negatve when does α get to be MOST negatve? π ( β < Scatteng ength st when a bound state when does α go to zeo? α a α a tan α a atan ( aβ β tan ( aβ aβ ( aβ aβ when PCD STTACS Unt Quantum Theoy of Cosons 4 ( aβ ( aβ tan < <, α s postve

242 π < ( aβ < π + ε α > at aβ π + ε : α changes sgn + to Scatteng ength α 3π when π + ε < ( aβ < α < Repusve potenta α a tan ( aβ aβ scatteng ength α fo an attactve potenta wth a fnte ange a tan ( aβ α a aβ α α -ve π +ve Fg.3.6/p8/Bue & Joachan/ Theoy of Eecton-Atom Cosons +ve 3π 5π -ve π + ε π + ε ' ε ' < ε -ve Attactve potenta Potenta stength change of sgn of scatteng ength α PCD STTACS Unt Quantum Theoy of Cosons 4

243 U ( β when when α a a U β fo < a ( aβ ( aβ tan fo > a >, α s negatve 3π π + ε< ( aβ <, tan ( aβ aβ α s negatve but not stong enough to bnd the next bound state α (effectve attactve potenta.. π 3π 5π π + ε π + ε ' ε ' < ε PCD STTACS Unt Quantum Theoy of Cosons 43

244 U ( β β > β < U β fo < a β < :epusve a fo > a : attactve α a α a tan α a atan ( aβ β tan ( aβ aβ ( aβ aβ when α s postve (effectve epusve potenta. PCD STTACS Unt Quantum Theoy of Cosons 44

245 cot ( δ ( α ( δ u (, A ( sn + ( Low enegy mt ( δ u m A + (, sn tan ( δ ( α u (, m A ( α m A ( α d + d U( u ε, ( d d U ( u ( u ( d d a u ( m+ C... a u ( R ( ε, ε, asymptote Lnea eaton. α ntecept PCD STTACS Unt Quantum Theoy of Cosons 45

246 u ( m + C... a m A ( α asymptotc behavo α α > : epusve potenta : at u mα Geometca meanng of the scatteng ength α Fg../page88/C.J.Joachan Quantum Theoy of Cosons at α u : PCD STTACS Unt Quantum Theoy of Cosons 46

247 α < but bound state not possbe u ( R ( ε, ε, asymptote Attactve potenta suppotng bound state α: ntesecton of the asymptote wth -axs st bound state: zeo enegy esonance α - (most negatve ( δ u A + (, m sn m A ( α m A ( α Postve α ndcates no moe bound state epusve potenta. Scatteng ength α fo vaous attactve potentas Fg../page89/C.J.Joachan Quantum Theoy of Cosons PCD STTACS Unt Quantum Theoy of Cosons 47

248 α < but bound state not possbe u ( R ( ε, ε, asymptote Attactve potenta suppotng bound state α: ntesecton of the asymptote wth -axs st bound state: zeo enegy esonance α - (most negatve ( δ u A + (, m sn m A ( α m A ( α Postve α ndcates no moe bound state epusve potenta. Scatteng ength α fo vaous attactve potentas Fg../page89/C.J.Joachan Quantum Theoy of Cosons PCD STTACS Unt Quantum Theoy of Cosons 48

249 α < but bound state not possbe u ( R ( ε, ε, asymptote Attactve potenta suppotng bound state α: ntesecton of the asymptote wth -axs st bound state: zeo enegy esonance α - (most negatve ( δ u A + (, m sn m A ( α m A ( α PCD STTACS Unt Quantum Theoy of Cosons Postve α ndcates no moe bound state epusve potenta. Scatteng ength α fo vaous attactve potentas Fg../page89/C.J.Joachan Quantum Theoy of Cosons 49

250 The scatteng ength has n t vta nfomaton about the physca popetes of the potenta, but t does not ncude detas about the stuctue of the potenta. PCD STTACS Unt Quantum Theoy of Cosons 5

251 SLOW cosons h λ : de Boge waveength age mv Detaed stuctue of the scatteng potenta : IMPORTANT? Essenta focus s then on symmety (s wave scatteng and a paamete scatteng ength α. Negatve Postve Chage dstbutons PCD STTACS Unt Quantum Theoy of Cosons 5

252 U ( β a m U ( V ( u ( R ( ε, ε, EFFECTIVE RANGE U β fo < a fo > a me > I Neuton-Poton scatteng Spn dependent et a. PCD STTACS Unt Quantum Theoy of Cosons 5