DOCSLIB.ORG

Scattering and S-Matrix 1/42 J

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Scattering and S-Matrix 1/42 J 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)).
Load more

Recommended publications
  • A Study of Fractional Schrödinger Equation-Composed Via Jumarie Fractional Derivative
    A Study of Fractional Schrödinger Equation-composed via Jumarie fractional derivative Joydip Banerjee1, Uttam Ghosh2a , Susmita Sarkar2b and Shantanu Das3 Uttar Buincha Kajal Hari Primary school, Fulia, Nadia, West Bengal, India email- [email protected] 2Department of Applied Mathematics, University of Calcutta, Kolkata, India; 2aemail : [email protected] 2b email : [email protected] 3 Reactor Control Division BARC Mumbai India email : [email protected] Abstract One of the motivations for using fractional calculus in physical systems is due to fact that many times, in the space and time variables we are dealing which exhibit coarse-grained phenomena, meaning that infinitesimal quantities cannot be placed arbitrarily to zero-rather they are non-zero with a minimum length. Especially when we are dealing in microscopic to mesoscopic level of systems. Meaning if we denote x the point in space andt as point in time; then the differentials dx (and dt ) cannot be taken to limit zero, rather it has spread. A way to take this into account is to use infinitesimal quantities as ()Δx α (and ()Δt α ) with 01<α <, which for very-very small Δx (and Δt ); that is trending towards zero, these ‘fractional’ differentials are greater that Δx (and Δt ). That is()Δx α >Δx . This way defining the differentials-or rather fractional differentials makes us to use fractional derivatives in the study of dynamic systems. In fractional calculus the fractional order trigonometric functions play important role. The Mittag-Leffler function which plays important role in the field of fractional calculus; and the fractional order trigonometric functions are defined using this Mittag-Leffler function.
    [Show full text]
  • Statistics of a Free Single Quantum Particle at a Finite
    STATISTICS OF A FREE SINGLE QUANTUM PARTICLE AT A FINITE TEMPERATURE JIAN-PING PENG Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China Abstract We present a model to study the statistics of a single structureless quantum particle freely moving in a space at a finite temperature. It is shown that the quantum particle feels the temperature and can exchange energy with its environment in the form of heat transfer. The underlying mechanism is diffraction at the edge of the wave front of its matter wave. Expressions of energy and entropy of the particle are obtained for the irreversible process. Keywords: Quantum particle at a finite temperature, Thermodynamics of a single quantum particle PACS: 05.30.-d, 02.70.Rr 1 Quantum mechanics is the theoretical framework that describes phenomena on the microscopic level and is exact at zero temperature. The fundamental statistical character in quantum mechanics, due to the Heisenberg uncertainty relation, is unrelated to temperature. On the other hand, temperature is generally believed to have no microscopic meaning and can only be conceived at the macroscopic level. For instance, one can define the energy of a single quantum particle, but one can not ascribe a temperature to it. However, it is physically meaningful to place a single quantum particle in a box or let it move in a space where temperature is well-defined. This raises the well-known question: How a single quantum particle feels the temperature and what is the consequence? The question is particular important and interesting, since experimental techniques in recent years have improved to such an extent that direct measurement of electron dynamics is possible.1,2,3 It should also closely related to the question on the applicability of the thermodynamics to small systems on the nanometer scale.4 We present here a model to study the behavior of a structureless quantum particle moving freely in a space at a nonzero temperature.
    [Show full text]
  • The Heisenberg Uncertainty Principle*
    OpenStax-CNX module: m58578 1 The Heisenberg Uncertainty Principle* OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 Abstract By the end of this section, you will be able to: • Describe the physical meaning of the position-momentum uncertainty relation • Explain the origins of the uncertainty principle in quantum theory • Describe the physical meaning of the energy-time uncertainty relation Heisenberg's uncertainty principle is a key principle in quantum mechanics. Very roughly, it states that if we know everything about where a particle is located (the uncertainty of position is small), we know nothing about its momentum (the uncertainty of momentum is large), and vice versa. Versions of the uncertainty principle also exist for other quantities as well, such as energy and time. We discuss the momentum-position and energy-time uncertainty principles separately. 1 Momentum and Position To illustrate the momentum-position uncertainty principle, consider a free particle that moves along the x- direction. The particle moves with a constant velocity u and momentum p = mu. According to de Broglie's relations, p = }k and E = }!. As discussed in the previous section, the wave function for this particle is given by −i(! t−k x) −i ! t i k x k (x; t) = A [cos (! t − k x) − i sin (! t − k x)] = Ae = Ae e (1) 2 2 and the probability density j k (x; t) j = A is uniform and independent of time. The particle is equally likely to be found anywhere along the x-axis but has denite values of wavelength and wave number, and therefore momentum.
    [Show full text]
  • The S-Matrix Formulation of Quantum Statistical Mechanics, with Application to Cold Quantum Gas
    THE S-MATRIX FORMULATION OF QUANTUM STATISTICAL MECHANICS, WITH APPLICATION TO COLD QUANTUM GAS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Pye Ton How August 2011 c 2011 Pye Ton How ALL RIGHTS RESERVED THE S-MATRIX FORMULATION OF QUANTUM STATISTICAL MECHANICS, WITH APPLICATION TO COLD QUANTUM GAS Pye Ton How, Ph.D. Cornell University 2011 A novel formalism of quantum statistical mechanics, based on the zero-temperature S-matrix of the quantum system, is presented in this thesis. In our new formalism, the lowest order approximation (“two-body approximation”) corresponds to the ex- act resummation of all binary collision terms, and can be expressed as an integral equation reminiscent of the thermodynamic Bethe Ansatz (TBA). Two applica- tions of this formalism are explored: the critical point of a weakly-interacting Bose gas in two dimensions, and the scaling behavior of quantum gases at the unitary limit in two and three spatial dimensions. We found that a weakly-interacting 2D Bose gas undergoes a superfluid transition at T 2πn/[m log(2π/mg)], where n c ≈ is the number density, m the mass of a particle, and g the coupling. In the unitary limit where the coupling g diverges, the two-body kernel of our integral equation has simple forms in both two and three spatial dimensions, and we were able to solve the integral equation numerically. Various scaling functions in the unitary limit are defined (as functions of µ/T ) and computed from the numerical solutions.
    [Show full text]
  • L the Probability Current Or Flux
    Atomic and Molecular Quantum Theory Course Number: C561 L The Probability Current or flux 1. Consider the time-dependent Schr¨odinger Equation and its complex conjugate: 2 ∂ h¯ ∂2 ıh¯ ψ(x, t)= Hψ(x, t)= − 2 + V ψ(x, t) (L.26) ∂t " 2m ∂x # 2 2 ∂ ∗ ∗ h¯ ∂ ∗ −ıh¯ ψ (x, t)= Hψ (x, t)= − 2 + V ψ (x, t) (L.27) ∂t " 2m ∂x # 2. Multiply Eq. (L.26) by ψ∗(x, t), andEq. (L.27) by ψ(x, t): ∂ ψ∗(x, t)ıh¯ ψ(x, t) = ψ∗(x, t)Hψ(x, t) ∂t 2 2 ∗ h¯ ∂ = ψ (x, t) − 2 + V ψ(x, t) (L.28) " 2m ∂x # ∂ −ψ(x, t)ıh¯ ψ∗(x, t) = ψ(x, t)Hψ∗(x, t) ∂t 2 2 h¯ ∂ ∗ = ψ(x, t) − 2 + V ψ (x, t) (L.29) " 2m ∂x # 3. Subtract the two equations to obtain (see that the terms involving V cancels out) 2 2 ∂ ∗ h¯ ∗ ∂ ıh¯ [ψ(x, t)ψ (x, t)] = − ψ (x, t) 2 ψ(x, t) ∂t 2m " ∂x 2 ∂ ∗ − ψ(x, t) 2 ψ (x, t) (L.30) ∂x # or ∂ h¯ ∂ ∂ ∂ ρ(x, t)= − ψ∗(x, t) ψ(x, t) − ψ(x, t) ψ∗(x, t) (L.31) ∂t 2mı ∂x " ∂x ∂x # 4. If we make the variable substitution: h¯ ∂ ∂ J = ψ∗(x, t) ψ(x, t) − ψ(x, t) ψ∗(x, t) 2mı " ∂x ∂x # h¯ ∂ = I ψ∗(x, t) ψ(x, t) (L.32) m " ∂x # (where I [···] represents the imaginary part of the quantity in brackets) we obtain ∂ ∂ ρ(x, t)= − J (L.33) ∂t ∂x or in three-dimensions ∂ ρ(x, t)+ ∇·J =0 (L.34) ∂t Chemistry, Indiana University L-37 c 2003, Srinivasan S.
    [Show full text]
  • Published Version
    PUBLISHED VERSION G. Aad ... P. Jackson ... L. Lee ... A. Petridis ... N. Soni ... M.J. White ... et al. (ATLAS Collaboration) Measurement of the total cross section from elastic scattering in pp collisions at √s = 7TeV with the ATLAS detector Nuclear Physics B, 2014; 889:486-548 © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license Originally published at: http://doi.org/10.1016/j.nuclphysb.2014.10.019 PERMISSIONS http://creativecommons.org/licenses/by/3.0/ 24 April 2017 http://hdl.handle.net/2440/102041 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.
    [Show full text]
  • Relativistic Quantum Mechanics 1
    Relativistic Quantum Mechanics 1 The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The emphasis 1.1 SpecialRelativity 1 is given to those elements of the formalism which can be carried on 1.2 One-particle states 7 to Relativistic Quantum Fields (RQF), which underpins the theoretical 1.3 The Klein–Gordon equation 9 framework of high energy particle physics. We begin with a brief summary of special relativity, concentrating on 1.4 The Diracequation 14 4-vectors and spinors. One-particle states and their Lorentz transforma- 1.5 Gaugesymmetry 30 tions follow, leading to the Klein–Gordon and the Dirac equations for Chaptersummary 36 probability amplitudes; i.e. Relativistic Quantum Mechanics (RQM). Readers who want to get to RQM quickly, without studying its foun- dation in special relativity can skip the first sections and start reading from the section 1.3. Intrinsic problems of RQM are discussed and a region of applicability of RQM is defined. Free particle wave functions are constructed and particle interactions are described using their probability currents. A gauge symmetry is introduced to derive a particle interaction with a classical gauge field. 1.1 Special Relativity Einstein’s special relativity is a necessary and fundamental part of any Albert Einstein 1879 - 1955 formalism of particle physics. We begin with its brief summary. For a full account, refer to specialized books, for example (1) or (2). The- ory oriented students with good mathematical background might want to consult books on groups and their representations, for example (3), followed by introductory books on RQM/RQF, for example (4).
    [Show full text]
  • Class 6: the Free Particle
    Class 6: The free particle A free particle is one on which no forces act, i.e. the potential energy can be taken to be zero everywhere. The time independent Schrödinger equation for the free particle is ℏ2d 2 ψ − = Eψ . (6.1) 2m dx 2 Since the potential is zero everywhere, physically meaningful solutions exist only for E ≥ 0. For each value of E > 0, there are two linearly independent solutions ψ ( x) = e ±ikx , (6.2) where 2mE k = . (6.3) ℏ Putting in the time dependence, the solution for energy E is Ψ( x, t) = Aei( kx−ω t) + Be i( − kx − ω t ) , (6.4) where A and B are complex constants, and Eℏ k 2 ω = = . (6.5) ℏ 2m The first term on the right hand side of equation (6.4) describes a wave moving in the direction of x increasing, and the second term describes a wave moving in the opposite direction. By applying the momentum operator to each of the travelling wave functions, we see that they have momenta p= ℏ k and p= − ℏ k , which is consistent with the de Broglie relation. So far we have taken k to be positive. Both waves can be encompassed by the same expression if we allow k to take negative values, Ψ( x, t) = Ae i( kx−ω t ) . (6.6) From the dispersion relation in equation (6.5), we see that the phase velocity of the waves is ω ℏk v = = , (6.7) p k2 m which differs from the classical particle velocity, 1 pℏ k v = = , (6.8) c m m by a factor of 2.
    [Show full text]
  • Problem: Evolution of a Free Gaussian Wavepacket Consider a Free Particle
    Problem: Evolution of a free Gaussian wavepacket Consider a free particle which is described at t = 0 by the normalized Gaussian wave function µ ¶1=4 2a 2 ª(x; 0) = e¡ax : ¼ The normalization factor is easy to obtain. We wish to ¯nd its time evolution. How does one do this ? Remember the recipe in quantum mechanics. We expand the given wave func- tion in terms of the energy eigenfunctions and we know how individual energy eigenfunctions evolve. We then reconstruct the wave function at a later time t by superposing the parts with appropriate phase factors. For a free particle H = p2=(2m) and therefore, momentum eigenfunctions are also energy eigenfunctions. So it is the mathematics of Fourier transforms! It is straightforward to ¯nd ©(k; 0) the Fourier transform of ª(x; 0): Z 1 dk ª(x; 0) = p ©(k; 0) eikx : ¡1 2¼ Recall that the Fourier transform of a Gaussian is a Gaussian. Z 1 dx Á(k) ´ ©(k; 0) = p ª(x; 0) e¡ikx (1.1) ¡1 2¼ µ ¶ µ ¶ 1=4 Z 1 1=4 2 1 2a ¡ikx ¡ax2 1 ¡ k = p dx e e = e 4a (1.2) 2¼ ¼ ¡1 2¼a What this says is that the Gaussian spatial wave function is a superposition of di®erent momenta with the probability of ¯nding the momentum between k1 and k1 + dk being pro- 2 portional to exp(¡k1=(2a)) dk. So the initial wave function is a superposition of di®erent plane waves with di®erent coef- ¯cients (usually called amplitudes).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]