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SCATTERING AND S-MATRIX 1/42 J. Rosner – Reaction Theory Summer Workshop Indiana University – June 2015 Lecture 1: Review of Lecture 2: S-matrix and Lecture 3: Some simple scattering theory related physics applications S, T , and K matrices 1-,2-channel examples Optical analogs Wave packet scattering Transmission resonances Eikonal approximation Scattering amplitude Bound states Diffractive scattering Partial wave expansion S wave properties Adding resonances Phase shifts Resonances Dalitz plot applications R-matrix Absorption Historical notes Unitarity circle Inelastic cross section Example from electronics For simplicity some of the formalism will be nonrelativistic “In” state i : Free particle in remote past | iin “Out” state j : Free particle in remote future outh | S-matrix takes “in” states to “out” (with wave packets: Goldberger and Watson, Scattering Theory, 1964.) UNITARITY; T AND K MATRICES 2/42 Sji out j i in is unitary: completeness and orthonormality of “in”≡ andh | i “out” states S†S = SS† = ½ Just an expression of probability conservation S has a piece corresponding to no scattering Can write S = ½ +2iT Notation of S. Spanier, BaBar Analysis Document #303, based on S. U. Chung et al. Ann. d. Phys. 4, 404 (1995). Unitarity of S-matrix T T † =2iT †T =2iT T †. ⇒ − 1 1 1 1 (T †)− T − =2i½ or (T − + i½)† = (T − + i½). − 1 1 1 Thus K [T − + i½]− is hermitian; T = K(½ iK)− ≡ − WAVE PACKETS 3/42 i~q ~r 3/2 Normalized plane wave states: χ~q = e · /(2π) 3 3 i(~q q~′) ~r 3 (χ , χ )=(2π)− d ~re − · = δ (~q q~ ) . q~′ ~q − ′ R Expansion of wave packet: ψ (~r, t =0)= d3q χ φ(~q ~p) ~p ~q − where φ is a weight function peaked aroundR 0 3 i~k ~r Fourier transform of φ: G(~r)= d k e · φ(~k) 3 R ψ~p(~r, t =0)= d q χ~q ~p+~p φ(~q ~p)= χ~p G(~r) . − − Norm: (ψ, ψ)=R d3q φ(~q ~p) 2 = 1 d3r G(~r) 2 =1 . | − | (2π)3 | | R R iHt 3 iEqt ψ (~r, t)= e− ψ (~r, 0) = d q φ(~q ~p)χ e− , ~p ~p − ~q 2 R 2 2 1/2 where Eq = q /2m (NR) or (q + m ) (Relativistic) . SPREADING WAVE 4/42 Expand about ~q = ~p; define ~k ~q ~p and v ∂E /∂p ≡ − i ≡ p i 2 1 ∂ Ep Eq = Ep + ~k ~v + kikj + .... · 2 ∂pi∂pj 2 iEpt 3 i~k (~r ~vt) it ∂ Ep ψ~p(~r, t)= e− χ~p d kφ(~k)e · − (1 kikj +...) − 2 ∂pi∂pj R 2 iEpt it ∂ Ep = e− χ~p(1 + i j + ...)G(~r ~vt) . 2 ∂pi∂pj ∇ ∇ − r2/2w2 Gaussian packet: G(r)= Ne− δ (r v t)(r v t) ij i− i j− j i jG(~r ~vt)= w2 + (w2)2 G(~r ~vt) . ∇ ∇ − h− i − 2 NR: ∂ Ep/(∂pi∂pj)= δij/m ; parameter describing 2 2 t ∂ Ep t L(∆k) spreading is ǫ = i jG(~r ~vt) 2 = . 2∂pi∂pj ∇ ∇ − ∼ 2mw 2p WAVE PACKET SCATTERING 5/42 E. Merzbacher, Quantum Mechanics (3rd Ed. Ch. 13) has a good discussion which will be abbreviated here 2 Free Hamiltonian: H0 = p /(2m); full: H = H0 + V 1 3 i~k (~r ~r0) Packet: ψ (~r, 0) = d k φ(~k ~k )e · − , φ(~k) ~k0 (2π)3/2 − 0 centered about 0, width ∆Rk, center of packet ~r0 At t = 0 packet is headed to target, momentum ~k0, distance ~r0 from it; want its shape for large t Expand ψ~k0(~r, 0) in eigenfunctions of H: iEnt ψ~k0(~r, 0) = n cnψn(~r) , ψ~k0(~r, t)= n cnψn(~r)e− . P P Need to find the eigenfunctions ψn(~r) GREEN’S FUNCTION 6/42 2 ~p 2 2m + V ψ = Eψ , or with k 2mE, U 2mV , ≡ ≡ Schr¨odinger equation is ( 2 + k2)ψ = Uψ ∇ ~ Define a Green’s function G(~r, r′) satisfying ( 2 + k2)G(~r, r~ )= 4πδ(~r r~ ) ∇ ′ − − ′ 1 3 ψ(~r)= d r′ G(~r, r~ )U(r~ )ψ(r~ ): particular sol’n. −4π ′ ′ ′ R i~k ~r 3/2 Add solution e · /(2π) of homogeneous equation: ~ eik·~r 1 3 ψ (~r)= d r′ G(~r, r~ )U(r~ )ψ (r~ ) ~k (2π)3/2 − 4π ′ ′ ~k ′ R Differential eqn. + boundary condx. integral equation ⇔ SPHERICAL WAVES 7/42 ~′ ~ e±ik|~r−r | Two Green’s functions: G (~r, r′)= ± ~r r~′ | − | with (+, ) (outgoing,incoming) spherical waves − ⇔ ~ Can take r′ =0; G are solutions for ~r =0 ± 6 For ~r = 0, integrate test function f(~r) times ( 2 + k2)G over small region surrounding the origin; use Gauss’∇ Law Green’s functions G define two sets of solutions: ± ~ ~′ ( ) eik·~r 1 3 e±ik|~r−r | ( ) ψ ± (~r)= d r′ U(r~ )ψ ± (r~ ) ~k (2π)3/2 − 4π ~r r~′ ′ ~k ′ R | − | where r′ is limited if U is of short range ~ 2 ~ 2 ~r r~′ Expand k ~r r′ = k r 2~r r′ + r′ kr 1 r·2 + ... | − | p − · ≃ − 1/ ~r r~ 1/r | − ′|≃ SCATTERING AMPLITUDE 8/42 ~ ( ) eik·~r 1 ikr 3 ( ) ik~′ r~′ ψ ± (~r) e± d r′ U(r~ )ψ ± (r~ )e∓ · ~k ∼ (2π)3/2 − 4πr ′ ~k ′ ~ R k′ krˆ ≡ 3 2 ( ) (2π) / 3 ( ) ik~′ r~′ Define f ± (ˆr) d r′ U(r~ )ψ ± (r~ )e∓ · ~k ≡− 4π ′ ~k ′ h i R ( ) ~ e±ikr ( ) ± 3/2 ik ~r ± Then ψ~ (~r) (2π)− e · + r f~ (ˆr) (r ) k ∼ h k i →∞ Initial flux per unit area I k 0 ∼ Final fluence I in cone of solid angle dΩ: ( ) I k(r2dΩ) f ± (ˆr)/r 2 ∼ | ~k | ( ) dσ/dΩ= I/I = f ± (ˆr) 2 differential cross section 0 | ~k | ≡ 9/42 PHASE SHIFTS δℓ ikr Large-r Schr. eq. solution: ψ eikr cos θ + f (θ)e (1) ∼ k r uℓ,k(r) m Connect with central-force solutions r Yℓ (θ,φ) where d2 ℓ(ℓ+1) 2 2 dr2 + r2 +2mV (r) k uℓ,k(r)=0 (k 2mE) h− − i ≡ Free u (r)/r R (r) solutions j (kr), n (kr) ℓ,k ≡ ℓ,k ℓ ℓ Outside range of V : Rℓ,k(r)= Aℓ jℓ(kr)+ Bℓ nℓ(kr) sin(kr ℓπ/2) cos(kr ℓπ/2) As kr R (r) A − B − →∞ ℓ,k → ℓ kr − ℓ kr sin(kr ℓπ/2+δ ) For tan δ B /A , R (r) − ℓ as kr ℓ ≡− ℓ ℓ ℓ,k ∼ kr →∞ Then ψ C (k)P (cos θ)[sin(kr ℓπ/2+ δ )]/r (2) → ℓ ℓ − ℓ ikr cosPθ ℓ Bauer: e = ℓ∞=0 i (2ℓ + 1)jℓ(kr)Pℓ(cos θ) (3) P PARTIAL WAVE EXPANSION 10/42 Compare incoming spherical wave coefficients in (2,3): ℓ iδ ψ ∞ (2ℓ + 1)i e ℓ[sin(kr ℓπ/2+ δ )]P (cos θ)/kr ≃ ℓ=0 − ℓ ℓ CompareP coeff. of outgoing spherical wave in this and (1): 2iδ (k) f (θ)= ∞ (2ℓ + 1)[(e ℓ 1)/(2ik)]P (cos θ) k ℓ=0 − ℓ P 1 iδℓ(k) = k− ℓ∞=0(2ℓ + 1)e sin δℓ(k)Pℓ(cos θ) Total cross section:P dσ 2 4π 2 σ = dΩ = dΩ f (θ) = 2 ∞ (2ℓ + 1)sin δ (k) dΩ | k | k ℓ=0 ℓ R R using dΩ Pℓ(cos θ)Pℓ′(cos θP)=4πδℓℓ′/(2ℓ + 1) R Optical theorem: σ = (4π/k)Im fk(0) Convenient to define f (k) [e2iδℓ(k) 1]/(2ik) ℓ ≡ − Then optical theorem takes the form Im f (k)= k f (k) 2 ℓ | ℓ | THE R-MATRIX 11/42 Want a real function reducing to fℓ(k) for small f Stereographic projection: R (k) (1/k)tan δ (k) ℓ ≡ ℓ 1 iδ (k) fℓ(k)= k− e ℓ sin δℓ(k) =1/k (maximum) for δℓ(k)= π [1+ikRℓ(k)]Rℓ(k) fℓ(k)= 2 2 1+k Rℓ (k) 2iδℓ(k) 1+ikRℓ(k) Sℓ(k) e = 1 ikR (k) ≡ − ℓ Many channels: real R is useful because it has eigenvalues S-matrix is useful because it is unitary: S†S =1 S = ½ +2iT = (½ + iK)/(½ iK) − UNITARITY CIRCLE 12/42 For single channel, phase shift defined by S = e2iδ. Then T = (S ½)/(2i)= − eiδ sin δ; K = tan δ = kR. The T amplitude must lie on boundary of the circle For inelastic processes 2iδ T = (ηe ½)/(2i) , η< 1. − Consider scattering in one dimension with no reflections Class of potentials giving rise to full transmission of a plane wave ψ(x) eikx incident from the left, so that as x , ψ(x) S∼(k)eikx with S(k) =1. Take 2m =1. →∞ → | | 1,2–CHANNEL EXAMPLES 13/42 2 These potls. have bound states at energies Ej = αj (1 N − ≤ j N and one can write S(k)=Πj=1[(ik αj)/(ik + ≤ 2iδ N − αj)] = e , where δ = j=1 δj and tan δj = αj/k. P Simplest one-level potential: V (x)= 2α2/cosh2α(x x ) − − 0 One-level K-matrix is just K = α/k. If we define Kj = α /k then K N K as k , but not in general. j → j=1 j →∞ Now let the potentialP permit reflections, so that ikx ikx ψ(x) Ae + Be− (x ) → ikx ikx → −∞ ψ(x) F e + Ge− (x ) → →∞ With channel 1 eikx, channel 2 e ikx, can write ∼ ∼ − F = S11A + S12G ; B = S21A + S22G TRANSMISSION RESONANCES14/42 The incoming and outgoing fluxes must be equal: F 2 + 2 2 2 | | B = A + G . This implies S†S = SS† =1. | | | | | | Square well, V (x) = V0 for x a, V (x)=0 for x > a shows transmission− resonances:| | ≤ S = S = 1 | | | 11| | 22| and S12 = S21 =0 when 2k′a = nπ (k′ = 2m(E + V0)).
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  • Measurement of the Total Cross Section from Elastic Scattering in Pp Collisions at √S = 7 Tev with the ATLAS Detector

    Measurement of the Total Cross Section from Elastic Scattering in Pp Collisions at √S = 7 Tev with the ATLAS Detector

    Measurement of the total cross section from elastic scattering in pp collisions at √s = 7 TeV with the ATLAS detector The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation “Measurement of the Total Cross Section from Elastic Scattering in Pp Collisions at √s = 7 TeV with the ATLAS Detector.” Nuclear Physics B 889 (December 2014): 486–548. As Published http://dx.doi.org/10.1016/j.nuclphysb.2014.10.019 Publisher Elsevier Version Final published version Citable link http://hdl.handle.net/1721.1/94503 Terms of Use Creative Commons Attribution Detailed Terms http://creativecommons.org/licenses/by/4.0/ Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 889 (2014) 486–548 www.elsevier.com/locate/nuclphysb Measurement of the total cross section√ from elastic scattering in pp collisions at s = 7TeV with the ATLAS detector .ATLAS Collaboration CERN, 1211 Geneva 23, Switzerland Received 25 August 2014; received in revised form 17 October 2014; accepted 21 October 2014 Available online 28 October 2014 Editor: Valerie Gibson Abstract √ A measurement of the total pp cross section at the LHC at s = 7TeVis presented. In a special run with − high-β beam optics, an integrated luminosity of 80 µb 1 was accumulated in order to measure the differ- ential elastic cross section as a function of the Mandelstam momentum transfer variable t. The measurement is performed with the ALFA sub-detector of ATLAS. Using a fit to the differential elastic cross section in 2 2 the |t| range from 0.01 GeV to 0.1GeV to extrapolate to |t| → 0, the total cross section, σtot(pp → X), is measured via the optical theorem to be: σtot(pp → X) = 95.35 ± 0.38 (stat.) ± 1.25 (exp.) ± 0.37 (extr.) mb, where the first error is statistical, the second accounts for all experimental systematic uncertainties and the last is related to uncertainties in the extrapolation to |t| → 0.
  • Photon Wave Mechanics: a De Broglie - Bohm Approach

    Photon Wave Mechanics: a De Broglie - Bohm Approach

    PHOTON WAVE MECHANICS: A DE BROGLIE - BOHM APPROACH S. ESPOSITO Dipartimento di Scienze Fisiche, Universit`adi Napoli “Federico II” and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli Mostra d’Oltremare Pad. 20, I-80125 Napoli Italy e-mail: [email protected] Abstract. We compare the de Broglie - Bohm theory for non-relativistic, scalar matter particles with the Majorana-R¨omer theory of electrodynamics, pointing out the impressive common pecu- liarities and the role of the spin in both theories. A novel insight into photon wave mechanics is envisaged. 1. Introduction Modern Quantum Mechanics was born with the observation of Heisenberg [1] that in atomic (and subatomic) systems there are directly observable quantities, such as emission frequencies, intensities and so on, as well as non directly observable quantities such as, for example, the position coordinates of an electron in an atom at a given time instant. The later fruitful developments of the quantum formalism was then devoted to connect observable quantities between them without the use of a model, differently to what happened in the framework of old quantum mechanics where specific geometrical and mechanical models were investigated to deduce the values of the observable quantities from a substantially non observable underlying structure. We now know that quantum phenomena are completely described by a complex- valued state function ψ satisfying the Schr¨odinger equation. The probabilistic in- terpretation of it was first suggested by Born [2] and, in the light of Heisenberg uncertainty principle, is a pillar of quantum mechanics itself. All the known experiments show that the probabilistic interpretation of the wave function is indeed the correct one (see any textbook on quantum mechanics, for 2 S.
  • Chapter 3 Wave Properties of Particles

    Chapter 3 Wave Properties of Particles

    Chapter 3 Wave Properties of Particles Overview of Chapter 3 Einstein introduced us to the particle properties of waves in 1905 (photoelectric effect). Compton scattering of x-rays by electrons (which we skipped in Chapter 2) confirmed Einstein's theories. You ought to ask "Is there a converse?" Do particles have wave properties? De Broglie postulated wave properties of particles in his thesis in 1924, based partly on the idea that if waves can behave like particles, then particles should be able to behave like waves. Werner Heisenberg and a little later Erwin Schrödinger developed theories based on the wave properties of particles. In 1927, Davisson and Germer confirmed the wave properties of particles by diffracting electrons from a nickel single crystal. 3.1 de Broglie Waves Recall that a photon has energy E=hf, momentum p=hf/c=h/, and a wavelength =h/p. De Broglie postulated that these equations also apply to particles. In particular, a particle of mass m moving with velocity v has a de Broglie wavelength of h λ = . mv where m is the relativistic mass m m = 0 . 1-v22/ c In other words, it may be necessary to use the relativistic momentum in =h/mv=h/p. In order for us to observe a particle's wave properties, the de Broglie wavelength must be comparable to something the particle interacts with; e.g. the spacing of a slit or a double slit, or the spacing between periodic arrays of atoms in crystals. The example on page 92 shows how it is "appropriate" to describe an electron in an atom by its wavelength, but not a golf ball in flight.