Advanced Quantum Mechanics

Transcription

1 Advanced Quantum Mechanics Rajdeep Sensarma Scattering Theory Ref : Sakurai, Modern Quantum Mechanics Taylor, Quantum Theory of Non-Relativistic Collisions Landau and Lifshitz, Quantum Mechanics

2 Recap of Last Class Basic Set up of Scattering > cross-section Time Evolution in presence of potential G R = G R 0 + G R 0 VG R 0 + G R 0 VG R 0 VG R = G R 0 + G R 0 VG R (G R ) 1 =(G R 0 ) 1 V G G 0 = + S Matrix G 0 VG 0 + G 0 VG 0 VG S = h (+) (t!1)i Elastic Scattering and S(E) Incident wave Involves G rather than G 0 Scattered to different out states

3 Scattering (S) and Transition (T) Matrices Incident wave Involves G rather than G 0 Scattered to different out states Let us define the Transition (T) matrix such that : (VG R )αβ=(tg 0R )αβ T kk (E) incorporates all the effects of scattering in the interction region Relation between S and T matrix Free Prop. with k Free Prop with k

4 The scattered state (1D) Incident particles with mom k A e ikx C e ikx Outgoing particles with mom k B e -ikx Outgoing particles with mom -k D e -ikx Incident particles with mom -k Nice Properties: Unitarity, Symmetry S Matrix T Matrix t 2 is transmission coeff for left incidence r 2 is reflection coeff for left incidence t 2 is transmission coeff for right incidence Easily related to reflection and transmission co-effs r 2 is reflection coeff for right incidence Note: Scattering states have particular boundary cond. The eigenstate with hard walls at x= ± L is different from scattering soln. since it follows different boundary cond.

5 The scattered state (1D) Unitarity of S r 2 + t 2 = r 2 + t 2 =1 tr * + rt * = 0 Time Reversal Invariance A e ikx C e ikx B* e ikx D* e ikx B e -ikx D e -ikx A* e -ikx C* e -ikx If this is a solution So is this S*S =1 r = r Reflection (Parity) Invariance r=r, t = t

6 The scattered state (1D) Transfer Matrix Easy to calculate S Matrix Calculation for a repulsive square well V Parity + TR invariance > S 11 = S 22 Follow any Std. QM Book (Schiff, Merzbacher etc.) A e ikx V0 C e ikx B e -ikx D e -ikx -a a x k = [2m(E-V 0 )] 1/2 k = [2mE] 1/2 ε = i (k 2 +k 2 )/2kk η = -i (k 2 - k 2 )/2kk

7 The scattered state (3D) Assume: incident particles have momentum k along z direction i.e. energy of particles is E=k 2 /2m By definition of the propagator φ is the incident state. Now, the S matrix connects the incoming and the outgoing states Using Now in 3D magnitude of momentum k = p 2mE

8 The scattered state (3D) ~r For geometry of scattering For elastic scattering Now, incident state So, far from the interaction region So and

9 T Matrix and cross section (3D) Incident wave Outgoing spherical wavefront Scattering amplitude Differential Cross Section S Matrix > T Matrix > Cross Section (Measureable Quantity)

10 Unitarity of S and Optical Theorem Statement of Optical Theorem Im[f( = 0)] = p tot 4 Imaginary part of fwd scattering amplitude, which measures how many particles are lost in this dirn., is equal to total number of scattered particles. This is just a restatement of probability conservation. θ Unitarity of S Take expectation in p states Use Note: Prob Conservation > σ tot should include both elastic and inelastic cross section

11 Singular Potentials and T matrix V(r) The defining Equation: VG= TG 0 Hard Sphere Model: The FT of V does not exist, impossible to work with V kk 0 = hk 0 V ki = hk 0 V ki V (r) = 1 = 0 0 <r<a r>a a r T kk 0 = hk 0 V (+) k i Actual Solution in presence of potential The wfn in presence of the potential vanishes at r=a and is finite for r>a. So T kk is well defined The system responds to the presence of the infinite potential by avoiding the region where the potential is infinite. The T matrix incorporates this information and is non-singular.

12 Born Approximation Start with the T matrix: Born. Approx/ 1 st Born Approx. 2 nd Born Approx. T = V + VG 0 V + VG 0 VG 0 V rd Born Approx. Scattering Amplitude: p θ q = ~p 0 ~p =2psin 2 p f(~p, ~p 0 )=f( ) = 4 2 mt ~p0,~p = 4 2 mv ~p0,~p = 2m q Z 1 0 Scattering Ampl. depends only on q Scattering Amplitude is real dσ/dω indep. of sign of V drrv (r)sinqr Validity: VG 0 V << V Weak potentials High Energy of Incident particles ( Time spent in interaction region is small, single scattering dominates) Violation of Unitarity: f pp is real: what happens to optical theorem? LHS ~ V 2 Need 2 nd Born Approx on RHS to restore optical theorem

13 Partial Wave Analysis Rotationally Invariant Potentials: Want to expand in angular momentum states Simultaneous eigenkets of H 0, L 2 and L z E,l,mi with he 0,l 0,m 0 E,l,mi = (E E 0 ) ll 0 mm 0 Use Wigner Eckart Theorem: he,l 0,m 0 T E,l,mi = T l ll 0 mm 0 Decouples in different l channels, independent of m Scattering Amplitude: Z f(~p, ~p 0 )= 4 2 mh~p 0 T ~pi = 4 2 m de X lm h~p 0 E,l,miT l he,l,m ~pi Using he,l,m ~pi = 1 p mp Y m l (ˆp) (E p 2 /2m) f(~p, ~p 0 )= 4 2 p X lm T l (E) Y m l ( ˆp 0 )Yl m (ˆp) The initial dirn. can be taken along z axis (θ=0, φ=0) and the final dirn. along (θ, φ=0) f(~p, ~p 0 )= f(~p, ~p 0 )= 4 2 p p X lm T l (E) Y m l ( ˆp 0 )Yl m (ˆp) 1X (2l + 1)T l (E)P l (cos ) Y m l (, 0) = r 2l +1 4 P l(cos ) m0 θ l=0

14 Partial Wave Analysis Define Partial Scattering Amplitude f l (p) Scattered Wavefunction: (+) (~r) = 1 (2 ) 3/2 T l(e = p 2 /2m) p applee i~p ~r + eipr r f( ~p 0, ~p) f(~p, ~p 0 )= 1X (2l + 1)f l (p)p l (cos ) l=0 e i~p ~r = e ipr cos = X i l (2l + 1)j l (pr)p l (cos ) l pr 1 j l (pr)! ei(pr l /2) e 2ipr i(pr l /2) Spherical Bessel Functions, consist of both outgoing and incoming waves. Solution of Radial Schrodinger Eqn. for free particles in 3D (+) (~r) = = 1 (2 ) 3/2 " X e ipr e (2l + 1)P l (cos ) 2ipr l l i(pr l ) 1 X (2l + 1) P l(cos ) [1 + 2ipf (2 ) 3/2 l (p)] eipr 2ip r + f l (p) eipr r e i(pr r l ) # Scatterer changes co-efficient of outgoing wave. Incoming wave is unaffected

15 Partial Wave Analysis: S and T Matrices S matrix is the overlap of the incoming free-particle state and the outgoing scattered state (+) (~r) = 1 X (2l + 1) P l(cos ) (2 ) 3/2 2ip l [1 + 2ipf l (p)] eipr r e i(pr r l ) S l (p) =1+2ipf l (p) Probability conservation ---> Incoming flux = Outgoing flux Spherical Symmetry ---> L conservation ---> For every l channel, flux in = flux out Unitarity of S l : S l (p) =1) S l (p) =e 2i l(p) phase shift in l channel The phase shifts encode all the information about the scattering potential. Partial Scattering Amplitude: f l (p) = S l(p) 1 2ip = e2i l(p) 1 2ip = ei l(p) sin l (p) p = 1 p cot l (p) ip T Matrix: T l (p) = ei l(p) sin l (p) = 1 1 cot l (p) i Scattering Amplitude: f( ) = X l (2l + 1)P l (cos ) ei l(p) sin l (p) p Interference of different l channels

16 Partial Wave Analysis: Cross Section Total Cross Section: = Z 2 0 d Z 1 1 d(cos ) f( ) 2 = 2 p 2 X ll 0 (2l + 1)(2l 0 + 1) sin l (p)sin l 0(p)e i[ l(p) l0 (p)] Z 1 1 d(cos )P l (cos )P l 0(cos ) = 4 X p 2 l (2l + 1) sin 2 Interference washed out l(p) in angular integration Quick Check of Optical Theorem: Im[f( = 0)] = X l (2l + 1)Im[e i l(p) ]sin l (p) P l (cos = 1) p = p 4

17 Partial Wave Analysis: Cross Section d d = f( ) 2 = X ll 0 (2l + 1)(2l 0 + 1) p 2 P l (cos )P l 0(cos )sin l (p)sin l 0(p)e i[ l(p) l0 (p)] Note that different l channels contribute additively to scattering amplitude. The differential cross-section includes interference between different channels. (a) (b) FIG. 1 (color online). Illustration of the process of using Interference of s and d partial scattering amplitudes in a collision of 2 Rb 87 atom clouds. Scattered atoms are in the halos. [ From : N. Thomas et. al, PRL 93, (2004) ]