Scattering Theorytheory

Total Page:16

File Type:pdf, Size:1020Kb

Scattering Theorytheory LectureLecture 44 ScatteringScattering theorytheory SS2011: ‚Introduction to Nuclear and Particle Physics, Part 2‘ SS2011: ‚Introduction to Nuclear and Particle Physics, Part 2‘ 1 I.I. ScatteringScattering experimentsexperiments Scattering experiment: A beam of incident scatterers with a given flux or intensity (number of particles per unit area dA per unit time dt ) impinges on the target (described by a scattering potential); the flux can be written as The number of particles per unit time which are detected in a small region of the solid angle, dΩ, located at a given angular deflection specified by (θ, φ), can be counted as 2 ScatteringScattering crosscross sectionsection The differential cross-section for scattering is defined as the number of particles scattered into an element of solid angle dΩ in the direction (θ,φ) per unit time : dσ ( ϑ ,φ ) 1 dN (1.1) = sc dΩ J inc dΩ [dimensions of an area] Jinc - incident flux The total cross-section corresponds to scatterings through any scattering angle: (1.2) Most scattering experiments are carried out in the laboratory (Lab) frame in which the target is initially at rest while the projectiles are moving. Calculations of the cross sections are generally easier to perform within the center-of-mass (CM) frame in which the center of mass of the projectiles–target system is at rest (before and after collision) Î one has to know how to transform the cross sections from one frame into the other. Note: the total cross section σ is the same in both frames, since the total number of collisions that take place does not depend on the frame in which the observation is carried out. However, the differential cross sections dσ/dΩ are not the same in both frames, since the scattering angles (θ,φ) are frame dependent. 3 ConnectingConnecting thethe aanglesngles inin thethe LabLab andand CMCM framesframes Elastic scattering of two structureless particles in the Lab and CM frames: Note: consider for simplicity NON-relativistic kinematics! r r V CM || V 1 L To find the connection between the Lab and CM cross sections, we need first to find how the scattering angles in one frame are related to their counterparts in the other. If denote the position of m1 in the Lab and CM frames, respectively, and if denotes the position of the center of mass with respect to the Lab frame, we have A time derivative of this relation leads to (1.3) where are the velocities of m1 in the Lab and CM frames before collision and is the velocity of the CM with respect to the Lab frame. Similarly, the velocity of m after collision is 1 (1.4) 4 ConnectingConnecting thethe aanglesngles inin thethe LabLab andand CMCM framesframes Since r r (1.5) V CM || V 1 L (1.6) Dividing (1.6) by (1.5), we end up with (1.7) Use that CM momenta r r r (1.8) pCM = p1 L + p 2 L r r r ( m 1 + m 2 )V CM = m 1V 1 L + m 2V 2 L Since from (1.8) Î (1.9) or from (1.9) Î (1.10) (1.11) 5 ConnectingConnecting thethe aanglesngles inin thethe LabLab andand CMCM framesframes Since the center of mass is at rest in the CM frame, the total momenta before and after collisions are separately zero: r r v r r r pC = p1C + p 2 C = pC′ = p1′ C + p′2 C = 0 (1.12) (1.13) Since the kinetic energy is conserved: (1.14) (1.15) In the case of elastic collisions, the speeds of the particles in the CM frame are the same before and after the collision; From (1.11) Î (1.16) Dividing (1.9) by (1.16) Î (1.17) 6 ConnectingConnecting thethe aanglesngles inin thethe LabLab andand CMCM framesframes Finally, a substitution of (1.17) into (1.7) yields (1.17) and using we obtain (1.18) Note: In a similar way we can establish a connection between θ2 and θ. From (1.4) we have + using Î the x and y components of this relation are (1.19) (1.20) 7 ConnectingConnecting thethe LabLab andand CMCM CrossCross SectionsSections The connection between the differential cross sections in the Lab and CM frames can be obtained from the fact that the number of scattered particles passing through an infinitesimal cross section is the same in both frames: (1.21) What differs is the solid angle dΩ : in the Lab frame: in the CM frame: (1.22) Since there is a cylindrical symmetry around the direction of the incident beam (1.23) From (1.18) Î (1.24) 8 ConnectingConnecting thethe LabLab andand CMCM CrossCross SectionsSections Thus : (1.25) From (1.23) and (1.20) Î (1.26) Limiting cases: 1) the Lab and CM results are the same, since (1.17) leads to θ1=θ Î (1.27) 2) then from (1.20) Î from (1.25) Æ (1.28) 9 FromFrom classicalclassical toto quantumquantum scatteringscattering theorytheory 1. Classical theory: - scattering of hard ‚spheres‘ - individual well-defined trajectories 2. Quantum theory: - scattering of wave pakeges Å wave–particle duality - probabilistic origin of scattering process 10 II.II. ClassicalClassical ttrajectoriesrajectories andand ccrossross--ssectionsections Scattering trajectories, corresponding to different impact parameters, b give different scattering angles θ. All of the particles in the beam in the hatched region of area dσ = 2π bdb are scattered into the angular region (θ, θ + dθ) The equations of motion for incident particle: (2.1) + initial conditions: Î defines the trajectory (2.2) (2.3) since initial kinetic energy For a particle obeying classical mechanics: the trajectory for the unbound motion, corresponding to a scattering event, is deterministically predictable, given by the interaction potential and the initial conditions; the path of any scatterer in the incident beam can be followed, and its angular deflection is determined as precisely as required 11 ClassicalClassical ttrajectoriesrajectories andand ccrossross--ssectionsections The number of particles scattered per unit time into the angular region (θ, θ+dθ) with any value of φ can be written as (2.4) (2.5) The knowledge of b(θ), obtained directly from Newton’s laws or other methods, is then sufficient to calculate the scattering cross-section. For the scattering from nontrivial central forces, the trajectory can be obtained from the equations of motion by using energy- and angular momentum- conservation methods E.g., one can rewrite L - angular momentum in the form (2.6) 12 ClassicalClassical ttrajectoriesrajectories andand ccrossross--ssectionsections The angular momentum can be written via the angular velocity Îfrom (2.6) (2.7) Î the angle through which the particle moves between two radial distances r1 and r2 : (2.8) Scattering trajectories in a central potential : rmin - the distance of closest approach, Θ – deflection angle (2.9) Use that the initial angular momentum and (2.10) Î from (2.8), (2.9) 13 ClassicalClassical CoulombCoulomb scatteringscattering The potential corresponding to the scattering of charged particles via Coulomb’s law can be written in the general form (2.11) where the constant corresponds to the electrostatic interaction of point-like charges Z1e and Z2e Î from (2.10) (2.12) Note that for large energies (E→∞) or vanishing charges (A → 0), one has rmin → b Use that (2.13) Î from (2.12) (2.14) 14 Classical Coulomb scattering Use that (2.15) Î Rutherford formula - characteristic for Coulomb scattering from a point-like charge: (2.16) Note: If one attempts to evaluate the total cross-section using Eq. (2.16), one obtains an infinite result. This divergence is due to the infinite range of the 1/r potential, and the “infinity” in the integral comes from scatterings as θ → 0 which corresponds to arbitrarily large impact parameters where the Coulomb potential causes a scatter through an arbitrarily small angle, which still contributes to the total cross-section. 15 III.III. QuantumQuantum ScatteringScattering The scattering amplitude of spinless, nonrelativistic particles: Consider the elastic scattering between two spinless, nonrelativistic particles of masses m1 and m2. During the scattering process, the particles interact with each other. If the interaction is time independent, we can describe the two-particle system with stationary states (3.1) where ET is the total energy and is a solution of the time-independent Schrödinger equation: (3.2) is the potential representing the interaction between the two particles. In the case where the interaction between m1 and m2 depends only on their relative distance one can reduce the eigenvalue problem (3.2) to two decoupled eigenvalue problems: one for the center of mass (CM), which moves like a free particle of mass M= m1+m2 (which is of no concern to us here) and another for a fictitious particle with a reduced mass which moves in the potential (3.3) 16 ScatteringScattering ofof spinlessspinless particlesparticles In quantum mechanics the incident particle is described by means of a wave packet that interacts with the target. After scattering, the wave function consists of an unscattered part propagating in the forward direction and a scattered part that propagates along some direction (θ,ϕ) 17 ScatteringScattering ofof spinlessspinless particlesparticles One can consider (3.3) as representing the scattering of a particle of mass μ from a fixed scattering center described by V(r), where r is the distance from the particle μ to the center of V(r): (3.3) ⎧ ≠ 0 ,if r ≤ a assume that V(r) has a finite range a: Vˆ ( r ) = ⎨ ⎩ = 0 ,if r > a Outside the range, r > a, the potential vanishes, V(r)= 0; the eigenvalue problem (3.3) then becomes (3.4) In this case μ behaves as a free particle before the collision and can be described by a plane wave (3.5) where is the wave vector associated with the incident particle;A is a normalization factor.
Recommended publications
  • Lecture 6: Spectroscopy and Photochemistry II
    Lecture 6: Spectroscopy and Photochemistry II Required Reading: FP Chapter 3 Suggested Reading: SP Chapter 3 Atmospheric Chemistry CHEM-5151 / ATOC-5151 Spring 2005 Prof. Jose-Luis Jimenez Outline of Lecture • The Sun as a radiation source • Attenuation from the atmosphere – Scattering by gases & aerosols – Absorption by gases • Beer-Lamber law • Atmospheric photochemistry – Calculation of photolysis rates – Radiation fluxes – Radiation models 1 Reminder of EM Spectrum Blackbody Radiation Linear Scale Log Scale From R.P. Turco, Earth Under Siege: From Air Pollution to Global Change, Oxford UP, 2002. 2 Solar & Earth Radiation Spectra • Sun is a radiation source with an effective blackbody temperature of about 5800 K • Earth receives circa 1368 W/m2 of energy from solar radiation From Turco From S. Nidkorodov • Question: are relative vertical scales ok in right plot? Solar Radiation Spectrum II From F-P&P •Solar spectrum is strongly modulated by atmospheric scattering and absorption From Turco 3 Solar Radiation Spectrum III UV Photon Energy ↑ C B A From Turco Solar Radiation Spectrum IV • Solar spectrum is strongly O3 modulated by atmospheric absorptions O 2 • Remember that UV photons have most energy –O2 absorbs extreme UV in mesosphere; O3 absorbs most UV in stratosphere – Chemistry of those regions partially driven by those absorptions – Only light with λ>290 nm penetrates into the lower troposphere – Biomolecules have same bonds (e.g. C-H), bonds can break with UV absorption => damage to life • Importance of protection From F-P&P provided by O3 layer 4 Solar Radiation Spectrum vs. altitude From F-P&P • Very high energy photons are depleted high up in the atmosphere • Some photochemistry is possible in stratosphere but not in troposphere • Only λ > 290 nm in trop.
    [Show full text]
  • Compton Scattering from the Deuteron at Low Energies
    SE0200264 LUNFD6-NFFR-1018 Compton Scattering from the Deuteron at Low Energies Magnus Lundin Department of Physics Lund University 2002 W' •sii" Compton spridning från deuteronen vid låga energier (populärvetenskaplig sammanfattning på svenska) Vid Compton spridning sprids fotonen elastiskt, dvs. utan att förlora energi, mot en annan partikel eller kärna. Kärnorna som användes i detta försök består av en proton och en neutron (sk. deuterium, eller tungt väte). Kärnorna bestrålades med fotoner av kända energier och de spridda fo- tonerna detekterades mha. stora Nal-detektorer som var placerade i olika vinklar runt strålmålet. Försöket utfördes under 8 veckor och genom att räkna antalet fotoner som kärnorna bestålades med och antalet spridda fo- toner i de olika detektorerna, kan sannolikheten för att en foton skall spridas bestämmas. Denna sannolikhet jämfördes med en teoretisk modell som beskriver sannolikheten för att en foton skall spridas elastiskt mot en deuterium- kärna. Eftersom protonen och neutronen består av kvarkar, vilka har en elektrisk laddning, kommer dessa att sträckas ut då de utsätts för ett elek- triskt fält (fotonen), dvs. de polariseras. Värdet (sannolikheten) som den teoretiska modellen ger, beror på polariserbarheten hos protonen och neu- tronen i deuterium. Genom att beräkna sannolikheten för fotonspridning för olika värden av polariserbarheterna, kan man se vilket värde som ger bäst överensstämmelse mellan modellen och experimentella data. Det är speciellt neutronens polariserbarhet som är av intresse, och denna kunde bestämmas i detta arbete. Organization Document name LUND UNIVERSITY DOCTORAL DISSERTATION Department of Physics Date of issue 2002.04.29 Division of Nuclear Physics Box 118 Sponsoring organization SE-22100 Lund Sweden Author (s) Magnus Lundin Title and subtitle Compton Scattering from the Deuteron at Low Energies Abstract A series of three Compton scattering experiments on deuterium have been performed at the high-resolution tagged-photon facility MAX-lab located in Lund, Sweden.
    [Show full text]
  • Solving the Quantum Scattering Problem for Systems of Two and Three Charged Particles
    Solving the quantum scattering problem for systems of two and three charged particles Solving the quantum scattering problem for systems of two and three charged particles Mikhail Volkov c Mikhail Volkov, Stockholm 2011 ISBN 978-91-7447-213-4 Printed in Sweden by Universitetsservice US-AB, Stockholm 2011 Distributor: Department of Physics, Stockholm University In memory of Professor Valentin Ostrovsky Abstract A rigorous formalism for solving the Coulomb scattering problem is presented in this thesis. The approach is based on splitting the interaction potential into a finite-range part and a long-range tail part. In this representation the scattering problem can be reformulated to one which is suitable for applying exterior complex scaling. The scaled problem has zero boundary conditions at infinity and can be implemented numerically for finding scattering amplitudes. The systems under consideration may consist of two or three charged particles. The technique presented in this thesis is first developed for the case of a two body single channel Coulomb scattering problem. The method is mathe- matically validated for the partial wave formulation of the scattering problem. Integral and local representations for the partial wave scattering amplitudes have been derived. The partial wave results are summed up to obtain the scat- tering amplitude for the three dimensional scattering problem. The approach is generalized to allow the two body multichannel scattering problem to be solved. The theoretical results are illustrated with numerical calculations for a number of models. Finally, the potential splitting technique is further developed and validated for the three body Coulomb scattering problem. It is shown that only a part of the total interaction potential should be split to obtain the inhomogeneous equation required such that the method of exterior complex scaling can be applied.
    [Show full text]
  • 7. Gamma and X-Ray Interactions in Matter
    Photon interactions in matter Gamma- and X-Ray • Compton effect • Photoelectric effect Interactions in Matter • Pair production • Rayleigh (coherent) scattering Chapter 7 • Photonuclear interactions F.A. Attix, Introduction to Radiological Kinematics Physics and Radiation Dosimetry Interaction cross sections Energy-transfer cross sections Mass attenuation coefficients 1 2 Compton interaction A.H. Compton • Inelastic photon scattering by an electron • Arthur Holly Compton (September 10, 1892 – March 15, 1962) • Main assumption: the electron struck by the • Received Nobel prize in physics 1927 for incoming photon is unbound and stationary his discovery of the Compton effect – The largest contribution from binding is under • Was a key figure in the Manhattan Project, condition of high Z, low energy and creation of first nuclear reactor, which went critical in December 1942 – Under these conditions photoelectric effect is dominant Born and buried in • Consider two aspects: kinematics and cross Wooster, OH http://en.wikipedia.org/wiki/Arthur_Compton sections http://www.findagrave.com/cgi-bin/fg.cgi?page=gr&GRid=22551 3 4 Compton interaction: Kinematics Compton interaction: Kinematics • An earlier theory of -ray scattering by Thomson, based on observations only at low energies, predicted that the scattered photon should always have the same energy as the incident one, regardless of h or • The failure of the Thomson theory to describe high-energy photon scattering necessitated the • Inelastic collision • After the collision the electron departs
    [Show full text]
  • Zero-Point Energy of Ultracold Atoms
    Zero-point energy of ultracold atoms Luca Salasnich1,2 and Flavio Toigo1 1Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM, Universit`adi Padova, via Marzolo 8, 35131 Padova, Italy 2CNR-INO, via Nello Carrara, 1 - 50019 Sesto Fiorentino, Italy Abstract We analyze the divergent zero-point energy of a dilute and ultracold gas of atoms in D spatial dimensions. For bosonic atoms we explicitly show how to regularize this divergent contribution, which appears in the Gaussian fluctuations of the functional integration, by using three different regular- ization approaches: dimensional regularization, momentum-cutoff regular- ization and convergence-factor regularization. In the case of the ideal Bose gas the divergent zero-point fluctuations are completely removed, while in the case of the interacting Bose gas these zero-point fluctuations give rise to a finite correction to the equation of state. The final convergent equa- tion of state is independent of the regularization procedure but depends on the dimensionality of the system and the two-dimensional case is highly nontrivial. We also discuss very recent theoretical results on the divergent zero-point energy of the D-dimensional superfluid Fermi gas in the BCS- BEC crossover. In this case the zero-point energy is due to both fermionic single-particle excitations and bosonic collective excitations, and its regu- larization gives remarkable analytical results in the BEC regime of compos- ite bosons. We compare the beyond-mean-field equations of state of both bosons and fermions with relevant experimental data on dilute and ultra- cold atoms quantitatively confirming the contribution of zero-point-energy quantum fluctuations to the thermodynamics of ultracold atoms at very low temperatures.
    [Show full text]
  • 12 Scattering in Three Dimensions
    12 Scattering in three dimensions 12.1 Cross sections and geometry Most experiments in physics consist of sending one particle to collide with another, and looking at what comes out. The quantity we can usually measure is the scattering cross section: by analogy with classical scattering of hard spheres, we assuming that scattering occurs if the particles ‘hit’ each other. The cross section is the apparent ‘target area’. The total scattering cross section can be determined by the reduction in intensity of a beam of particles passing through a region on ‘targets’, while the differential scattering cross section requires detecting the scattered particles at different angles. We will use spherical polar coordinates, with the scattering potential located at the origin and the plane wave incident flux parallel to the z direction. In this coordinate system, scattering processes dσ are symmetric about φ, so dΩ will be independent of φ. We will also use a purely classical concept, the impact parameter b which is defined as the distance of the incident particle from the z-axis prior to scattering. S(k) δΩ I(k) θ z φ Figure 11: Standard spherical coordinate geometry for scattering 12.2 The Born Approximation We can use time-dependent perturbation theory to do an approximate calculation of the cross- section. Provided that the interaction between particle and scattering centre is localised to the region around r = 0, we can regard the incident and scattered particles as free when they are far from the scattering centre. We just need the result that we obtained for a constant perturbation, Fermi’s Golden Rule, to compute the rate of transitions between the initial state (free particle of momentum p) to the final state (free particle of momentum p0).
    [Show full text]
  • Discrete Variable Representations and Sudden Models in Quantum
    Volume 89. number 6 CHLMICAL PHYSICS LEITERS 9 July 1981 DISCRETEVARlABLE REPRESENTATIONS AND SUDDEN MODELS IN QUANTUM SCATTERING THEORY * J.V. LILL, G.A. PARKER * and J.C. LIGHT l7re JarrresFranc/t insrihrre and The Depwrmcnr ofC7temtsrry. Tire llm~crsrry of Chrcago. Otrcago. l7lrrrors60637. USA Received 26 September 1981;m fin11 form 29 May 1982 An c\act fOrmhSm In which rhe scarrcnng problem may be descnbcd by smsor coupled cqumons hbclcd CIIIW bb bans iuncltons or quadrature pomts ISpresented USCof each frame and the srnrplyculuatcd unitary wmsformatlon which connects them resulis III an cfliclcnt procedure ror pcrrormrnpqu~nrum scxrcrrn~ ca~cubr~ons TWO ~ppro~mac~~~ arc compxcd wrh ihe IOS. 1. Introduction “ergenvalue-like” expressions, rcspcctnely. In each case the potential is represented by the potcntud Quantum-mechamcal scattering calculations are function Itself evahrated at a set of pomts. most often performed in the close-coupled representa- Whjle these models have been shown to be cffcc- tion (CCR) in which the internal degrees of freedom tive III many problems,there are numerousambiguities are expanded in an appropnate set of basis functions in their apphcation,especially wtth regardto the resulting in a set of coupled diiierentral equations UI choice of constants. Further, some models possess the scattering distance R [ 1,2] _The method is exact formal difficulties such as loss of time reversal sym- IO w&in a truncation error and convergence is ob- metry, non-physical coupling, and non-conservation tained by increasing the size of the basis and hence the of energy and momentum [ 1S-191. In fact, it has number of coupled equations (NJ While considerable never been demonstrated that sudden models fit into progress has been madein the developmentof efficient any exactframework for solutionof the scattering algonthmsfor the solunonof theseequations [3-71, problem.
    [Show full text]
  • Photon Cross Sections, Attenuation Coefficients, and Energy Absorption Coefficients from 10 Kev to 100 Gev*
    1 of Stanaaros National Bureau Mmin. Bids- r'' Library. Ml gEP 2 5 1969 NSRDS-NBS 29 . A111D1 ^67174 tioton Cross Sections, i NBS Attenuation Coefficients, and & TECH RTC. 1 NATL INST OF STANDARDS _nergy Absorption Coefficients From 10 keV to 100 GeV U.S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS T X J ". j NATIONAL BUREAU OF STANDARDS 1 The National Bureau of Standards was established by an act of Congress March 3, 1901. Today, in addition to serving as the Nation’s central measurement laboratory, the Bureau is a principal focal point in the Federal Government for assuring maximum application of the physical and engineering sciences to the advancement of technology in industry and commerce. To this end the Bureau conducts research and provides central national services in four broad program areas. These are: (1) basic measurements and standards, (2) materials measurements and standards, (3) technological measurements and standards, and (4) transfer of technology. The Bureau comprises the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Center for Radiation Research, the Center for Computer Sciences and Technology, and the Office for Information Programs. THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consistent system of physical measurement; coordinates that system with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation’s scientific community, industry, and com- merce. The Institute consists of an Office of Measurement Services and the following technical divisions: Applied Mathematics—Electricity—Metrology—Mechanics—Heat—Atomic and Molec- ular Physics—Radio Physics -—Radio Engineering -—Time and Frequency -—Astro- physics -—Cryogenics.
    [Show full text]
  • Calculation of Photon Attenuation Coefficients of Elements And
    732 ISSN 0214-087X Calculation of photon attenuation coeffrcients of elements and compound Roteta, M.1 Baró, 2 Fernández-Varea, J.M.3 Salvat, F.3 1 CIEMAT. Avenida Complutense 22. 28040 Madrid, Spain. 2 Servéis Científico-Técnics, Universitat de Barcelona. Martí i Franqués s/n. 08028 Barcelona, Spain. 3 Facultat de Física (ECM), Universitat de Barcelona. Diagonal 647. 08028 Barcelona, Spain. CENTRO DE INVESTIGACIONES ENERGÉTICAS, MEDIOAMBIENTALES Y TECNOLÓGICAS MADRID, 1994 CLASIFICACIÓN DOE Y DESCRIPTORES: 990200 662300 COMPUTER LODES COMPUTER CALCULATIONS FORTRAN PROGRAMMING LANGUAGES CROSS SECTIONS PHOTONS Toda correspondencia en relación con este trabajo debe dirigirse al Servicio de Información y Documentación, Centro de Investigaciones Energéticas, Medioam- bientales y Tecnológicas, Ciudad Universitaria, 28040-MADRID, ESPAÑA. Las solicitudes de ejemplares deben dirigirse a este mismo Servicio. Los descriptores se han seleccionado del Thesauro del DOE para describir las materias que contiene este informe con vistas a su recuperación. La catalogación se ha hecho utilizando el documento DOE/TIC-4602 (Rev. 1) Descriptive Cataloguing On- Line, y la clasificación de acuerdo con el documento DOE/TIC.4584-R7 Subject Cate- gories and Scope publicados por el Office of Scientific and Technical Information del Departamento de Energía de los Estados Unidos. Se autoriza la reproducción de los resúmenes analíticos que aparecen en esta publicación. Este trabajo se ha recibido para su impresión en Abril de 1993 Depósito Legal n° M-14874-1994 ISBN 84-7834-235-4 ISSN 0214-087-X ÑIPO 238-94-013-4 IMPRIME CIEMAT Calculation of photon attenuation coefíicients of elements and compounds from approximate semi-analytical formulae M.
    [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]
  • (2021) Transmon in a Semi-Infinite High-Impedance Transmission Line
    PHYSICAL REVIEW RESEARCH 3, 023003 (2021) Transmon in a semi-infinite high-impedance transmission line: Appearance of cavity modes and Rabi oscillations E. Wiegand ,1,* B. Rousseaux ,2 and G. Johansson 1 1Applied Quantum Physics Laboratory, Department of Microtechnology and Nanoscience-MC2, Chalmers University of Technology, 412 96 Göteborg, Sweden 2Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, 75005 Paris, France (Received 8 December 2020; accepted 11 March 2021; published 1 April 2021) In this paper, we investigate the dynamics of a single superconducting artificial atom capacitively coupled to a transmission line with a characteristic impedance comparable to or larger than the quantum resistance. In this regime, microwaves are reflected from the atom also at frequencies far from the atom’s transition frequency. Adding a single mirror in the transmission line then creates cavity modes between the atom and the mirror. Investigating the spontaneous emission from the atom, we then find Rabi oscillations, where the energy oscillates between the atom and one of the cavity modes. DOI: 10.1103/PhysRevResearch.3.023003 I. INTRODUCTION [43]. Furthermore, high-impedance resonators make it pos- sible for light-matter interaction to reach strong-coupling In the past two decades, circuit quantum electrodynamics regimes due to strong coupling to vacuum fluctuations [44]. (circuit QED) has become a field of growing interest for quan- In this article, we investigate the spontaneous emission tum information processing and also to realize new regimes of a transmon [45] capacitively coupled to a 1D TL that is in quantum optics [1–11].
    [Show full text]
  • The Equation of Radiative Transfer How Does the Intensity of Radiation Change in the Presence of Emission and / Or Absorption?
    The equation of radiative transfer How does the intensity of radiation change in the presence of emission and / or absorption? Definition of solid angle and steradian Sphere radius r - area of a patch dS on the surface is: dS = rdq ¥ rsinqdf ≡ r2dW q dS dW is the solid angle subtended by the area dS at the center of the † sphere. Unit of solid angle is the steradian. 4p steradians cover whole sphere. ASTR 3730: Fall 2003 Definition of the specific intensity Construct an area dA normal to a light ray, and consider all the rays that pass through dA whose directions lie within a small solid angle dW. Solid angle dW dA The amount of energy passing through dA and into dW in time dt in frequency range dn is: dE = In dAdtdndW Specific intensity of the radiation. † ASTR 3730: Fall 2003 Compare with definition of the flux: specific intensity is very similar except it depends upon direction and frequency as well as location. Units of specific intensity are: erg s-1 cm-2 Hz-1 steradian-1 Same as Fn Another, more intuitive name for the specific intensity is brightness. ASTR 3730: Fall 2003 Simple relation between the flux and the specific intensity: Consider a small area dA, with light rays passing through it at all angles to the normal to the surface n: n o In If q = 90 , then light rays in that direction contribute zero net flux through area dA. q For rays at angle q, foreshortening reduces the effective area by a factor of cos(q).
    [Show full text]