Elastic Collisions in One Dimension

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Elastic Collisions in One Dimension Elastic Collisions in One Dimension Bởi: OpenStaxCollege Let us consider various types of two-object collisions These collisions are the easiest to analyze, and they illustrate many of the physical principles involved in collisions The conservation of momentum principle is very useful here, and it can be used whenever the net external force on a system is zero We start with the elastic collision of two objects moving along the same line—a onedimensional problem An elastic collision is one that also conserves internal kinetic energy Internal kinetic energy is the sum of the kinetic energies of the objects in the system [link] illustrates an elastic collision in which internal kinetic energy and momentum are conserved Truly elastic collisions can only be achieved with subatomic particles, such as electrons striking nuclei Macroscopic collisions can be very nearly, but not quite, elastic—some kinetic energy is always converted into other forms of energy such as heat transfer due to friction and sound One macroscopic collision that is nearly elastic is that of two steel blocks on ice Another nearly elastic collision is that between two carts with spring bumpers on an air track Icy surfaces and air tracks are nearly frictionless, more readily allowing nearly elastic collisions on them Elastic Collision An elastic collision is one that conserves internal kinetic energy Internal Kinetic Energy Internal kinetic energy is the sum of the kinetic energies of the objects in the system 1/5 Elastic Collisions in One Dimension An elastic one-dimensional two-object collision Momentum and internal kinetic energy are conserved Now, to solve problems involving one-dimensional elastic collisions between two objects we can use the equations for conservation of momentum and conservation of internal kinetic energy First, the equation for conservation of momentum for two objects in a one-dimensional collision is (Fnet = 0) p1 + p2 = p ′ 1+p ′ or (Fnet = 0), m1v1 + m2v2 = m1v ′ + m2v ′ where the primes (') indicate values after the collision By definition, an elastic collision conserves internal kinetic energy, and so the sum of kinetic energies before the collision equals the sum after the collision Thus, 2 m 1v 1 1 + m2v22 = m1v ′ 12 + m2v ′ 22 (two-object elastic collision) expresses the equation for conservation of internal kinetic energy in a one-dimensional collision Calculating Velocities Following an Elastic Collision Calculate the velocities of two objects following an elastic collision, given that 2/5 Elastic Collisions in One Dimension m1 = 0.500 kg, m2 = 3.50 kg, v1 = 4.00 m/s, and v2 = Strategy and Concept First, visualize what the initial conditions mean—a small object strikes a larger object that is initially at rest This situation is slightly simpler than the situation shown in [link] where both objects are initially moving We are asked to find two unknowns (the final velocities v ′ and v ′ 2) To find two unknowns, we must use two independent equations Because this collision is elastic, we can use the above two equations Both can be simplified by the fact that object is initially at rest, and thus v2 = Once we simplify these equations, we combine them algebraically to solve for the unknowns Solution For this problem, note that v2 = and use conservation of momentum Thus, p1 = p ′ + p ′ or m1v1 = m1v ′ + m2v ′ Using conservation of internal kinetic energy and that v2 = 0, 2 m 1v 1 = m1v ′ 12 + m2v ′ 22 Solving the first equation (momentum equation) for v ′ 2, we obtain v′2= m1 m2 (v1 − v ′ 1) Substituting this expression into the second equation (internal kinetic energy equation) eliminates the variable v ′ 2, leaving only v ′ as an unknown (the algebra is left as an exercise for the reader) There are two solutions to any quadratic equation; in this example, they are v ′ = 4.00 m/s and v ′ = − 3.00 m/s 3/5 Elastic Collisions in One Dimension As noted when quadratic equations were encountered in earlier chapters, both solutions may or may not be meaningful In this case, the first solution is the same as the initial condition The first solution thus represents the situation before the collision and is discarded The second solution (v ′ = − 3.00 m/s) is negative, meaning that the first object bounces backward When this negative value of v ′ is used to find the velocity of the second object after the collision, we get v′2= m1 m2 kg (v1 − v ′ 1) = 0.500 3.50 kg [4.00 − ( − 3.00)] m/s or v ′ = 1.00 m/s Discussion The result of this example is intuitively reasonable A small object strikes a larger one at rest and bounces backward The larger one is knocked forward, but with a low speed (This is like a compact car bouncing backward off a full-size SUV that is initially at rest.) As a check, try calculating the internal kinetic energy before and after the collision You will see that the internal kinetic energy is unchanged at 4.00 J Also check the total momentum before and after the ...Novel Aspects of Localization, Delocalization, and Anomalous Transport in One-dimensional Systems QIFANG ZHAO BSc (Hons), NUS A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN SCIENCE DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2014 Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Qifang Zhao 21 August 2014 i ii Dedicated to my parents and wife iii iv Acknowledgements First of all, I would like to thank my supervisor, A/Prof Gong Jiang Bin. Throughout my four years of PhD study, he provided me with guidance, inspirations and support. He also shared with me his personal experiences in doing independent research, teaching me efficient ways to achieve fruitful outcomes through meaningful processes. During the early stage, his insightful suggestions laid the paths to successful projects. In the later stages of the course, his emphasis on independent research has successfully forged a young research mind. It is a great honour and blessing to be his student. His knowledge, wisdom and kindness have always been and will always be what I look up to. Next, I would like to thank my co-supervisor, A/Prof Cord A. Müller. We collaborated for the past three years, during which I picked up useful skills such as presentation, poster-designing and many more from him. I would like to express my deepest gratitude to him for his guidance in my first project which really means a lot to me in the early stage of PhD study. I am also very grateful to my fellow group members Derek, Hailong, Long Wen, Adam, Gao Yang, Yon Shin and Da Yang for all the beneficial and delightful discussions. I would like to offer my special thanks to Derek who helped me clarify some doubts through rigorous derivations, Hailong for introducing me to the research of periodically driven systems, and Long Wen who elaborated some physics concepts to me, sharing his profound knowledge. I also thank fellow PhD students in block S16, Taolin, Qinglin, Feng Ling, Liu Sha, Lina, Qin Chu, Qiao Zhi and so on, for their companionship. v Last but not least, I would like to thank my family members, my wife Chen Zhixiu, my father Wang Zaiqin, mother Zhao Zonghua and little brother Wang Guangwei for their continuous support and encouragement throughout my PhD study. vi Contents Introduction 1.1 Physics of Disorder in Quantum Systems . . . . . . . . . . . . . . . . . 1.2 Physics of Disorder in Current Research Frontiers . . . . . . . . . . . . 1.3 The Structure of Research Topics and the Outline of Thesis . . . . . . Physics Background and Mathematical Preliminaries 2.1 The Transfer Matrix Formalism in 1D . . . . . . . . . . . . . . . . . . 9 2.1.1 Properties of the Reflection and Transmission Coefficients . . . 10 2.1.2 The Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 The Application of the Transfer Matrix Formalism in Kronig-Penney Model 13 2.3 The Formalisms for Periodically Driven Systems in Classical and Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 The Kicked Rotor in Classical Mechanics . . . . . . . . . . . . . 16 2.3.2 The Kicked Rotor in Quantum Mechanics . . . . . . . . . . . . 17 Localization Behavior of Dirac Particles in Disordered Graphene Superlattices 20 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.1 Graphene Superlattices . . . . . . . . . . . . . . . . . . . . . . 21 3.1.2 Disordered Graphene Superlattices . . . . . . . . . . . . . . . . 22 Localization Length in Disordered Graphene Superlattices . . . . . . . . 24 3.2 vii Contents 3.2.1 Modeling Disordered Graphene Superlattices . . . . . . . . . . . 24 3.2.2 Transfer Matrix QUANTUM SIMULATIONS WITH PHOTONS IN ONE-DIMENSIONAL NONLINEAR WAVEGUIDES MING-XIA HUO A thesis submitted for the Degree of Doctor of Philosophy CENTRE FOR QUANTUM TECHNOLOGIES NATIONAL UNIVERSITY OF SINGAPORE 2013 DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. MING-XIA HUO 24 July 2013 Acknowledgments First and foremost, I am deeply grateful to Kwek Leong Chuan. To work with him has been a real pleasure to me. He has been oriented and supported me with promptness and care. He has always been patient and encouraging in times of new ideas and difficulties. He has also provided insightful discussions and suggestions. I appreciate all his contributions of time, ideas, and funding to make my PhD experience excellent. Above all, he made me feel a friend, which I appreciate from my heart. Furthermore, I am immensely grateful to Dimitris Angelakis. His high level of comprehension and sharpness on physical subjects have taught me to be rigorous and to tackle aspects of physics with a level of confidence that I never had before. I am also very grateful for his scientific advice and knowledge and many insightful discussions and suggestions. In addition, I have been very privileged to get to know and to collaborate with David Hutchinson. I learned a lot from him about research, language, how to tackle new problems and how to develop techniques to solve them. He has been a pleasure to work with. Thanks for helping me a lot. I would also like to give thanks to Wenhui Li for her unwavering support professionally and personally at every important moment of my PhD experience. I will truly miss those discussions and conversations. Thanks for all the good times. I also had the great pleasure of meeting Christian Miniatura. From the very beginning of my PhD career, he supported me. Thanks for the fun and encouraging discussions over the last several years. I would like to thank my collaborator Darrick Chang, whose physical understanding in the research area is tremendous and whose scientific work inspired me a lot. Our collaborated work has also benefited from suggesi tions and kind encouragement from Vladimir Korepin. His technical depth and attention to details are amazing. I also want to thank David Wilkowski for providing me the opportunity to have fruitful collaborations with his experimental group in a near future. I would like to thank the collaborators I had the pleasure to work with. Special thanks to the postdocs Changsuk Noh, Blas M. Rodriguez-Lara, Elica Kyoseva, and the student Nie Wei. They have helped me a lot. I am grateful to my committee members: Berge Englert, Wenhui Li, and Chorng Haur Sow. I have also immeasurably benefited from the course "Quantum Information and Computation", for which I thank the instructor Dagomir Kaszlikowski. I wish to thank Rosario Fazio and Davide Rossini for making DMRG available to me. I have been interested in DMRG for a long time, and they provided me with a big help. I also want to thank Kerson Huang for interesting and illuminating discussions when I first started my PhD career. I am grateful to our physics group members for providing me with the best working environment. In particular, I like to thank Dai Li, Setiawan, Thi Ha Kyaw for their willingness to share their research experience and many helpful information. I also want to thank Chunfeng Wu for her advice, support, and encouragement. A special acknowledgement goes to Hui Min Evon Tan and Ethan Lim, who were very nice and always ready to help. I will forever be thankful to my former Bachelor and Master Degree advisor Zhi Song. He has been helpful in providing advice many times during my stay there. He remains my best role model for a scientist and teacher. I am also very grateful to Chapter Pinning Quantum Phase Transition of Photons 3.1 Bose-Hubbard and Sine-Gordon Models The use of suitably controlled simple quantum systems to mimic the behavior of complex ones provides an exciting possibility to confirm existing physical theories and explore new physics. Two of the extremely successful models for describing a range of quantum many body effects and especially quantum phase transitions (QPT) [2, 44], are the Bose-Hubbard (BH) [6] and sine-Gordon (sG) models [2]. Cold atoms in optical lattices have so far been the most famous platform to implement these models, where it is possible to observe the MI to SF QPT for a weakly interacting gas in a deep lattice potential [6, 7]. More recently, it was made possible to tune up the interactions between the atoms in the gas leading to the realization of the sG model and the Pinning QPT [16, 45]. Alternative platforms in the field of quantum simulations of many body effects involve ions for quantum magnets [46], and photonic lattices for the understanding of in-and-out of equilibrium quantum many-body effects [47, 48, 49]. The photon-based ideas have initiated a stream of works in the many body properties of both closed and lossy cavity arrays [50, 51, 52, 53, 54, 55]. More recently, a new direction has appeared in 21 Chapter 3. Pinning Quantum Phase Transition of Photons the field of strongly correlated photons where hollowcore fibers filled with cold atomic gases or tapered fibers with cold atoms brought close to the surface of the fiber were considered [25, 26, 27, 28, 29, 30, 31, 32]. At low temperature, the speed of light can be substantially reduced, and the dephasing due to atomic collisions can be small enough to be ignored. The strong light confinement and the resulting large optical nonlinearities in the single photon level predicted for similar systems [22, 23, 24, 56] have motivated new proposals in the continuous polaritonic systems [57]. We will show here that it is possible to impose an effective lattice potential on the strongly interacting polaritonic gas in the fiber. This opens the possibilities for a large range of Hamiltonians to be simulated with photons. As examples we will study the simulation of the sG and BH models. We will show that the whole phase diagram of the MI to SF transitions for both models can be reproduced including a corresponding photonic “pinning transition”. We conclude with a discussion on the available tunability of the quantum optical parameters for the observation of the strongly correlated phases. The latter is possible by releasing the trapped polaritons and measuring the correlation on the photons emitted at the other end of the fiber. 3.2 Quantum Optical Simulator with OneSpecies Four-Level Atoms The considered atomic level structure is shown in Fig. 3.1. The 1D cold atomic ensemble is prepared outside the fiber using standard cold atom techniques and is transferred into or brought close to the surface of the fiber using techniques described in [25, 26, 27, 28, 29, 30, 31, 32]. The atoms are 22 3.2. Quantum Optical Simulator with One-Species Four-Level Atoms (a) (b) (c) (d) Figure 3.1: In (a) and (b) an ensemble of cold atoms with a 4-level structure is interacting with a pair of classical fields Ω± which create an effective stationary Bragg grating. The photons carried by the input pulse E+ coming in from the left are mapped to stationary excitations (polaritons) which are trapped in the grating. The strong photon nonlinearity which is induced by the 4th level leads to the creation of a strongly interacting gas (grey areas in (c) and (d). By modulating the density of the cold atomic ensemble, an effective lattice potential for polaritons/photons can be created-shown by red lines in (c) and (d). Tuning to the regime of weak interactions between polaritons and adjusting the lattice depth, the system undergoes a BH phase transition from SF (upper one) to MI (lower one) phase (c). In the opposite .. .Elastic Collisions in One Dimension An elastic one- dimensional two-object collision Momentum and internal kinetic energy are conserved Now, to solve problems involving one- dimensional elastic. .. conservation of internal kinetic energy in a one- dimensional collision Calculating Velocities Following an Elastic Collision Calculate the velocities of two objects following an elastic collision,... and initial conditions Is momentum conserved? Is kinetic energy conserved? Vary the elasticity and see what happens 4/5 Elastic Collisions in One Dimension Collision Lab Section Summary • An elastic