Advanced Quantum Mechanics

Transcription

1 Advanced Quantum Mechanics Rajdeep Sensama Scatteing Theoy Ref : Sakuai, Moden Quantum Mechanics Tayo, Quantum Theoy of Non-Reativistic Coisions Landau and Lifshitz, Quantum Mechanics

2 Recap of Pevious Casses S Matix and decomposition of scatteed state in fee patice basis Fee patice with momentum k and enegy E Fee patice with momentum k and enegy E (, ) Detecto S = h (+) (t!)i Popagato in pesence of scattee d ẑ G R = G R + G R VG R + G R VG R VG R +... = G R + G R VG R G G = + = + G VG + G VG VG +... Pobabiity consevation ----> Unitaity of S

3 Recap of Pevious Casses T = V [ G V ] Fee Pop. with G = V + VG V + VG VG V +... = V + VG T T Fee Pop with G T captues a the effects of V inc. mutipe scatteings T V VG V VG VG V = = + D Scatteed State A e ikx C e ikx B e -ikx D e -ikx 3D Scatteed State (+) (~) = (2 ) 3/2 appee i~p ~ + eip f( ~p, ~p)

4 Recap of Pevious Casses Bon Appoximation T = V + VG V + VG VG V +... Vaidity: Bon. Appox/ st Bon Appox. Weak potentias High Enegy of Incident patices 2 nd Bon Appox. q = ~p ~p =2p sin 2 3 d Bon Appox. V () =V <<a A) Squae We f( ) = 2mV.5 = >a q 2 appe sin(qa) q a cos(qa) f(θ) 2 /a 2 qa=.,.5,.,.5. 2mV a 2 = Diffeentia Scatteing Coss Section goes down monotonicay.5 p 2 B) Yukawa Potentia f(θ) 2 λ 2..5 p 2 θ p p/λ=.,.5,.,.5 2mV /λ 2 = p f( ) = 2mV q V () = V e As λ --->, with V /λ fixed, the Yukawa potentia goes ove to the Couomb potentia. In this imit, ecove the Ruthefod cosssection. d d = (2m)2 (ZZ e 2 ) 2 6p 4 sin 4 ( /2)

5 Recap of Pevious Casses Expansion in Patia waves f(~p, ~p )= 4 2 p X m T (E) Y m ( ˆp )Y m (ˆp) f(~p, ~p )= X (2 + )f (p)p (cos ) = θ Scatteed Wavefunction: (+) (~) = X (2 + ) P (cos ) [ + 2ipf (2 ) 3/2 (p)] eip 2ip outgoing spheica wave e i(p ) incoming spheica wave Consevation of Angua Momentum > Unitaity of S S Matix : S (p) =+2ipf (p) S (p) =) S (p) =e 2i (p) phase shift in channe T Matix: T (p) = ei (p) sin (p) = cot (p) i

6 Recap of Pevious Casses Expansion in Patia waves f(~p, ~p )= 4 2 p X m T (E) Y m ( ˆp )Y m (ˆp) f(~p, ~p )= X (2 + )f (p)p (cos ) = θ Scatteing Ampitude: f( ) = X (2 + )P (cos ) ei (p) sin (p) p Diffeentia Coss Section: d d = f( ) 2 = X (2 + )(2 + ) p 2 P (cos )P (cos )sin (p)sin (p)e i[ (p) (p)] intefeence of diffeent channes Tota Coss Section: = 4 X p 2 (2 + ) sin 2 (p) intefeence washed out

7 Low Enegy Scatteing and few Patia Waves Use the expansion of a pane wave into spheica waves in 3D Use othonomaity of Y m Use the fact that k is aong z and Y m (, ) = P (cos ) m Use fo x << to show fo sma k, k How sma shoud k be? Range of intega is R, the ange of potentia. So, kr << fo this to be vaid

8 Low Enegy Scatteing and few Patia Waves Let us use the sef-consistent eqn. fo T matix Note that we want k, k to be sma, but q sum is unesticted Howeve, we have and So, Expand the seies to show that each tem has the fom V kq V pk Now use the fact that fo eastic scatteing k=k and T (E)/k ~ T kk (E) Fo ow E << /(2m R 2 ) Low E scatteing is dominated by a few patia waves OK to conside ony = channe fo E >, s-wave scatteing

9 Cacuating Phase Shifts in simpe potentias The staight fowad appoach (woks ony in few seective cases) is to sove the Schodinge equation with the potentia, and obtain the phase shift fom the asymptotic fom of the wave-function fa fom the oigin by compaing it with (+) () = X (2 + ) P (cos ) (2 ) 3/2 2ip e 2i (p) eip e i(p ) Spheicay Symmetic Squae Potentia We/Baie: V() V Potentia Baie V() Potentia We V() Had Sphee a a V a V () =V <<a = >a Scatteing states behave ike fee patice fa fom the potentia egion. Since they have KE, we ae ooking fo states with E> The potentia we can sustain bound states, but we ae not inteested in them (fo now).

10 Spheicay Symmetic Potentia We/Baie: Use spheica co-od angua pat given by Y (+) (, ) = X i (2 + )R ()P (cos ) = X i (2 + ) u ()P (cos ) d 2 appe u () d 2 + ( + ) 2 u () = Radia Equation: 2mV () p 2 + V () =V <<a = >a >a Fee patice soutions fa fom oigin R () =c j (p)+c 2 n (p) =c () h () (p)+c (2) h (2) (p) Spheica Besse Functions Spheica Hanke Functions h ((2)) (p) =j (p) ± in (p) p h ((2)) /2) e±i(p (p) ± ip Compaing with (+) X () = (2 + ) P (cos ) (2 ) 3/2 2ip e 2i (p) eip e i(p ) c () = 2 e2i (p) c (2) = 2 R () =e i [cos j (p) sin n (p)]

11 Spheicay Symmetic Potentia We/Baie: R () =c j (p)+c 2 n (p) =c () h () (p)+c (2) h (2) (p) Spheica Besse Functions Spheica Hanke Functions Compaing with (+) () = X (2 + ) P (cos ) (2 ) 3/2 2ip e 2i (p) eip e i(p ) c () = 2 e2i (p) c (2) = 2 R () =e i [cos j (p) sin n (p)] c () and c (2) ae obtained fom the continuity of the ogaithmic deivative at =a. = appe R dr d =a tan (p) = paj (pa) j (pa) pan (pa) n (pa) Note that ti now we have not used the specific squae-wave fom of the potentia. This esut is vaid fo any potentia that vanishes at a finite ange

12 Spheicay Symmetic Potentia We/Baie: To find the paamete β, we need the soution inside the potentia egion A) Had Sphee Potentia Bounday Condition : wfn. vanishes at =a V() = appe dr R d =a! a tan (p) = paj (pa) pan (pa) j (pa) n (pa) tan (p) = j (pa) n (pa) s-wave phase shift d d (p) = = sin2 k 2 ' a 2 fo ka pa Negative phase shift fo epusive potentia Geneicay tue fo finite potentia baie as we. Indicates that the wave-fn is pushed out

13 Spheicay Symmetic Potentia We/Baie: To find the paamete β, we need the soution inside the potentia egion B) Potentia We/ Baie V() V Potentia Baie V() Potentia We a a V Inside Son.: egua at = j (q) q 2 2m = E V = appe dr R d =a = qaj (qa) j (qa) tan (p) = pj (pa) pn (pa) qj (qa)j (pa)/j (qa) qj (qa)n (pa)/j (qa) (p) p 2+

14 Squae Potentia We: s-wave Scatteing s-wave scatteing Conside the = channe in the ow enegy imit tan (p) pa ength s Fo squae we = qaj (qa) j (qa) Using j (x) = sin x = qa cot(qa) x Scatteing Length a s = a a s diveges when qa = (2n+) π/2 Scatteing Coss-Section + =4 a 2 s Scatteing Ampitude f (p) = p cot (p) ip = a s + ip = a s +ipa s f () = a s f does not divege when qa = (2n+) π/2, f is -/ ip at this point --- Unitay imit β 5 a s ma 2 V 2ma 2 V

15 s-wave Scatteing Length Fo age u () e i sin(p + ) e i sin p( a s ) e i p( a s ) So a s has the intepetation of the fist point in space whee the extapoation of the fa soution hits zeo. Note that it is not a zeo of the actua soution. d 2 appe u () d 2 + 2mV () p 2 + ( + ) 2 u () = U() u() Conside p=, = a Fo puey epusive potentia cuvatue is away fom axis. Scatteing Length is aways positive

16 s-wave Scatteing Length Fo age u () e i sin(p + ) e i sin p( a s ) e i p( a s ) So as has the intepetation of the fist point in space whee the extapoation of the fa soution hits zeo. Note that it is not a zeo of the actua soution. a u () u () u () a U() V() U() V() V() U() Fo attactive potentia wes, the scatteing ength is initiay negative As we incease the we depth,the scatteing ength becomes moe and moe negative ti it eaches - Beyond this point, the scatteing ength stats at + and keeps deceasing This is the point whee we have the fist bound state in the system

17 Effective Range Expansion H. A. Bethe, Phys. Rev. 76, 38 (949) What happens when we go to age enegies, aka what is the next tem in f f (p) = p cot (p) ip = a s + ip = a s +ipa s Schodinge Eqn fo 2 diffeent momenta d 2 appe u () d 2 + 2mV () p 2 + ( + ) 2 u () = d 2 u d 2 +[p2 V ()]u () = d 2 u 2 d 2 +[p2 2 V ()]u 2 () = u 2 d 2 u d 2 u d 2 u 2 d 2 =(p2 p 2 2)u u 2 u 2 du d u du 2 d R =(p 2 p 2 2) Z R du u 2 Conside the asymptotic fom of the soutions at age p() = sin[p + (p)] sin (p) The asymptotic son. aso foows simia eqns as u 2 d d d 2 d R =(p 2 p 2 2) Z R d 2

18 Effective Range Expansion H. A. Bethe, Phys. Rev. 76, 38 (949) Subtact the equations fo u and ψ, 2 d d u 2 du d d 2 d + u du 2 d R =(p 2 p 2 2) At =R, LHS vanishes by continuity eqn.s. At =, tems in LHS invoving u vanish as u()=. Z R d 2 u u 2 d 2 d 2 Z d =(p 2 p 2 d 2) d 2 u u 2 Int extended to since the integand vanishes outside the ange of potentia Using expicit fom ψ at =, p 2 cot (p 2 ) p cot (p )=(p 2 p 2 2) p!, p 2! p p cot (p) = a s p 2 Z d p ' Z p 2 d 2 u 2 a s u u p Effective ange of potentia Z d 2 u u 2 p cot (p) = a s p 2 f (p) = a s + ip + p 2 Effective ange expansion

19 Univesaity of ow enegy scatteing We have seen that the ow enegy scatteing fom a potentia can be chaacteized by a few paametes E.g. s-wave scatteing can be paametized by a s,, etc. Ceay this cannot depend on a the detais of the shape of the potentia Fo squae we = qa cot(qa) Scatteing Length a s = a + q 2 2m = E V So we can have many diffeent potentias at the micoscopic eve, whose ow enegy scatteing (say a s, ) ae same. E.g. can choose diffeent V and a fo a squae we so that qa is fixed. Low enegy scatteing is same fo both. We can even get away with a simpe potentia (say deta fn) povided we manage to get the coect scatteing ength This is you fist gimpse into the genea phenomenon of univesaity: Many systems which ook diffeent on a micoscopic scae (i.e. diffeent V) can show same phenomena at ow enegy. This is at the heat of theoetica endeavous to cacuate popeties of compicated systems.