Force, Quantum Mechanics and Approximate Energy Eigenstates
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Key Concepts for Future QIS Learners Workshop Output Published Online May 13, 2020
Key Concepts for Future QIS Learners Workshop output published online May 13, 2020 Background and Overview On behalf of the Interagency Working Group on Workforce, Industry and Infrastructure, under the NSTC Subcommittee on Quantum Information Science (QIS), the National Science Foundation invited 25 researchers and educators to come together to deliberate on defining a core set of key concepts for future QIS learners that could provide a starting point for further curricular and educator development activities. The deliberative group included university and industry researchers, secondary school and college educators, and representatives from educational and professional organizations. The workshop participants focused on identifying concepts that could, with additional supporting resources, help prepare secondary school students to engage with QIS and provide possible pathways for broader public engagement. This workshop report identifies a set of nine Key Concepts. Each Concept is introduced with a concise overall statement, followed by a few important fundamentals. Connections to current and future technologies are included, providing relevance and context. The first Key Concept defines the field as a whole. Concepts 2-6 introduce ideas that are necessary for building an understanding of quantum information science and its applications. Concepts 7-9 provide short explanations of critical areas of study within QIS: quantum computing, quantum communication and quantum sensing. The Key Concepts are not intended to be an introductory guide to quantum information science, but rather provide a framework for future expansion and adaptation for students at different levels in computer science, mathematics, physics, and chemistry courses. As such, it is expected that educators and other community stakeholders may not yet have a working knowledge of content covered in the Key Concepts. -
Unit 1 Old Quantum Theory
UNIT 1 OLD QUANTUM THEORY Structure Introduction Objectives li;,:overy of Sub-atomic Particles Earlier Atom Models Light as clectromagnetic Wave Failures of Classical Physics Black Body Radiation '1 Heat Capacity Variation Photoelectric Effect Atomic Spectra Planck's Quantum Theory, Black Body ~diation. and Heat Capacity Variation Einstein's Theory of Photoelectric Effect Bohr Atom Model Calculation of Radius of Orbits Energy of an Electron in an Orbit Atomic Spectra and Bohr's Theory Critical Analysis of Bohr's Theory Refinements in the Atomic Spectra The61-y Summary Terminal Questions Answers 1.1 INTRODUCTION The ideas of classical mechanics developed by Galileo, Kepler and Newton, when applied to atomic and molecular systems were found to be inadequate. Need was felt for a theory to describe, correlate and predict the behaviour of the sub-atomic particles. The quantum theory, proposed by Max Planck and applied by Einstein and Bohr to explain different aspects of behaviour of matter, is an important milestone in the formulation of the modern concept of atom. In this unit, we will study how black body radiation, heat capacity variation, photoelectric effect and atomic spectra of hydrogen can be explained on the basis of theories proposed by Max Planck, Einstein and Bohr. They based their theories on the postulate that all interactions between matter and radiation occur in terms of definite packets of energy, known as quanta. Their ideas, when extended further, led to the evolution of wave mechanics, which shows the dual nature of matter -
Exercises in Classical Mechanics 1 Moments of Inertia 2 Half-Cylinder
Exercises in Classical Mechanics CUNY GC, Prof. D. Garanin No.1 Solution ||||||||||||||||||||||||||||||||||||||||| 1 Moments of inertia (5 points) Calculate tensors of inertia with respect to the principal axes of the following bodies: a) Hollow sphere of mass M and radius R: b) Cone of the height h and radius of the base R; both with respect to the apex and to the center of mass. c) Body of a box shape with sides a; b; and c: Solution: a) By symmetry for the sphere I®¯ = I±®¯: (1) One can ¯nd I easily noticing that for the hollow sphere is 1 1 X ³ ´ I = (I + I + I ) = m y2 + z2 + z2 + x2 + x2 + y2 3 xx yy zz 3 i i i i i i i i 2 X 2 X 2 = m r2 = R2 m = MR2: (2) 3 i i 3 i 3 i i b) and c): Standard solutions 2 Half-cylinder ϕ (10 points) Consider a half-cylinder of mass M and radius R on a horizontal plane. a) Find the position of its center of mass (CM) and the moment of inertia with respect to CM. b) Write down the Lagrange function in terms of the angle ' (see Fig.) c) Find the frequency of cylinder's oscillations in the linear regime, ' ¿ 1. Solution: (a) First we ¯nd the distance a between the CM and the geometrical center of the cylinder. With σ being the density for the cross-sectional surface, so that ¼R2 M = σ ; (3) 2 1 one obtains Z Z 2 2σ R p σ R q a = dx x R2 ¡ x2 = dy R2 ¡ y M 0 M 0 ¯ 2 ³ ´3=2¯R σ 2 2 ¯ σ 2 3 4 = ¡ R ¡ y ¯ = R = R: (4) M 3 0 M 3 3¼ The moment of inertia of the half-cylinder with respect to the geometrical center is the same as that of the cylinder, 1 I0 = MR2: (5) 2 The moment of inertia with respect to the CM I can be found from the relation I0 = I + Ma2 (6) that yields " µ ¶ # " # 1 4 2 1 32 I = I0 ¡ Ma2 = MR2 ¡ = MR2 1 ¡ ' 0:3199MR2: (7) 2 3¼ 2 (3¼)2 (b) The Lagrange function L is given by L('; '_ ) = T ('; '_ ) ¡ U('): (8) The potential energy U of the half-cylinder is due to the elevation of its CM resulting from the deviation of ' from zero. -
Quantum Mechanics
Quantum Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 5 1.1 Intendedaudience................................ 5 1.2 MajorSources .................................. 5 1.3 AimofCourse .................................. 6 1.4 OutlineofCourse ................................ 6 2 Probability Theory 7 2.1 Introduction ................................... 7 2.2 WhatisProbability?.............................. 7 2.3 CombiningProbabilities. ... 7 2.4 Mean,Variance,andStandardDeviation . ..... 9 2.5 ContinuousProbabilityDistributions. ........ 11 3 Wave-Particle Duality 13 3.1 Introduction ................................... 13 3.2 Wavefunctions.................................. 13 3.3 PlaneWaves ................................... 14 3.4 RepresentationofWavesviaComplexFunctions . ....... 15 3.5 ClassicalLightWaves ............................. 18 3.6 PhotoelectricEffect ............................. 19 3.7 QuantumTheoryofLight. .. .. .. .. .. .. .. .. .. .. .. .. .. 21 3.8 ClassicalInterferenceofLightWaves . ...... 21 3.9 QuantumInterferenceofLight . 22 3.10 ClassicalParticles . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 3.11 QuantumParticles............................... 25 3.12 WavePackets .................................. 26 2 QUANTUM MECHANICS 3.13 EvolutionofWavePackets . 29 3.14 Heisenberg’sUncertaintyPrinciple . ........ 32 3.15 Schr¨odinger’sEquation . 35 3.16 CollapseoftheWaveFunction . 36 4 Fundamentals of Quantum Mechanics 39 4.1 Introduction .................................. -
Quantum Mechanics in One Dimension
Solved Problems on Quantum Mechanics in One Dimension Charles Asman, Adam Monahan and Malcolm McMillan Department of Physics and Astronomy University of British Columbia, Vancouver, British Columbia, Canada Fall 1999; revised 2011 by Malcolm McMillan Given here are solutions to 15 problems on Quantum Mechanics in one dimension. The solutions were used as a learning-tool for students in the introductory undergraduate course Physics 200 Relativity and Quanta given by Malcolm McMillan at UBC during the 1998 and 1999 Winter Sessions. The solutions were prepared in collaboration with Charles Asman and Adam Monaham who were graduate students in the Department of Physics at the time. The problems are from Chapter 5 Quantum Mechanics in One Dimension of the course text Modern Physics by Raymond A. Serway, Clement J. Moses and Curt A. Moyer, Saunders College Publishing, 2nd ed., (1997). Planck's Constant and the Speed of Light. When solving numerical problems in Quantum Mechanics it is useful to note that the product of Planck's constant h = 6:6261 × 10−34 J s (1) and the speed of light c = 2:9979 × 108 m s−1 (2) is hc = 1239:8 eV nm = 1239:8 keV pm = 1239:8 MeV fm (3) where eV = 1:6022 × 10−19 J (4) Also, ~c = 197:32 eV nm = 197:32 keV pm = 197:32 MeV fm (5) where ~ = h=2π. Wave Function for a Free Particle Problem 5.3, page 224 A free electron has wave function Ψ(x; t) = sin(kx − !t) (6) •Determine the electron's de Broglie wavelength, momentum, kinetic energy and speed when k = 50 nm−1. -
Newtonian Mechanics Is Most Straightforward in Its Formulation and Is Based on Newton’S Second Law
CLASSICAL MECHANICS D. A. Garanin September 30, 2015 1 Introduction Mechanics is part of physics studying motion of material bodies or conditions of their equilibrium. The latter is the subject of statics that is important in engineering. General properties of motion of bodies regardless of the source of motion (in particular, the role of constraints) belong to kinematics. Finally, motion caused by forces or interactions is the subject of dynamics, the biggest and most important part of mechanics. Concerning systems studied, mechanics can be divided into mechanics of material points, mechanics of rigid bodies, mechanics of elastic bodies, and mechanics of fluids: hydro- and aerodynamics. At the core of each of these areas of mechanics is the equation of motion, Newton's second law. Mechanics of material points is described by ordinary differential equations (ODE). One can distinguish between mechanics of one or few bodies and mechanics of many-body systems. Mechanics of rigid bodies is also described by ordinary differential equations, including positions and velocities of their centers and the angles defining their orientation. Mechanics of elastic bodies and fluids (that is, mechanics of continuum) is more compli- cated and described by partial differential equation. In many cases mechanics of continuum is coupled to thermodynamics, especially in aerodynamics. The subject of this course are systems described by ODE, including particles and rigid bodies. There are two limitations on classical mechanics. First, speeds of the objects should be much smaller than the speed of light, v c, otherwise it becomes relativistic mechanics. Second, the bodies should have a sufficiently large mass and/or kinetic energy. -
Class 2: Operators, Stationary States
Class 2: Operators, stationary states Operators In quantum mechanics much use is made of the concept of operators. The operator associated with position is the position vector r. As shown earlier the momentum operator is −iℏ ∇ . The operators operate on the wave function. The kinetic energy operator, T, is related to the momentum operator in the same way that kinetic energy and momentum are related in classical physics: p⋅ p (−iℏ ∇⋅−) ( i ℏ ∇ ) −ℏ2 ∇ 2 T = = = . (2.1) 2m 2 m 2 m The potential energy operator is V(r). In many classical systems, the Hamiltonian of the system is equal to the mechanical energy of the system. In the quantum analog of such systems, the mechanical energy operator is called the Hamiltonian operator, H, and for a single particle ℏ2 HTV= + =− ∇+2 V . (2.2) 2m The Schrödinger equation is then compactly written as ∂Ψ iℏ = H Ψ , (2.3) ∂t and has the formal solution iHt Ψ()r,t = exp − Ψ () r ,0. (2.4) ℏ Note that the order of performing operations is important. For example, consider the operators p⋅ r and r⋅ p . Applying the first to a wave function, we have (pr⋅) Ψ=−∇⋅iℏ( r Ψ=−) i ℏ( ∇⋅ rr) Ψ− i ℏ( ⋅∇Ψ=−) 3 i ℏ Ψ+( rp ⋅) Ψ (2.5) We see that the operators are not the same. Because the result depends on the order of performing the operations, the operator r and p are said to not commute. In general, the expectation value of an operator Q is Q=∫ Ψ∗ (r, tQ) Ψ ( r , td) 3 r . -
Rotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn that where the force is applied and how the force is applied is just as important as how much force is applied when we want to make something rotate. This tutorial discusses the dynamics of an object rotating about a fixed axis and introduces the concepts of torque and moment of inertia. These concepts allows us to get a better understanding of why pushing a door towards its hinges is not very a very effective way to make it open, why using a longer wrench makes it easier to loosen a tight bolt, etc. This module begins by looking at the kinetic energy of rotation and by defining a quantity known as the moment of inertia which is the rotational analog of mass. Then it proceeds to discuss the quantity called torque which is the rotational analog of force and is the physical quantity that is required to changed an object's state of rotational motion. Moment of Inertia Kinetic Energy of Rotation Consider a rigid object rotating about a fixed axis at a certain angular velocity. Since every particle in the object is moving, every particle has kinetic energy. To find the total kinetic energy related to the rotation of the body, the sum of the kinetic energy of every particle due to the rotational motion is taken. The total kinetic energy can be expressed as .. -
On a Classical Limit for Electronic Degrees of Freedom That Satisfies the Pauli Exclusion Principle
On a classical limit for electronic degrees of freedom that satisfies the Pauli exclusion principle R. D. Levine* The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91904, Israel; and Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095 Contributed by R. D. Levine, December 10, 1999 Fermions need to satisfy the Pauli exclusion principle: no two can taken into account, this starting point is not necessarily a be in the same state. This restriction is most compactly expressed limitation. There are at least as many orbitals (or, strictly in a second quantization formalism by the requirement that the speaking, spin orbitals) as there are electrons, but there can creation and annihilation operators of the electrons satisfy anti- easily be more orbitals than electrons. There are usually more commutation relations. The usual classical limit of quantum me- states than orbitals, and in the model of quantum dots that chanics corresponds to creation and annihilation operators that motivated this work, the number of spin orbitals is twice the satisfy commutation relations, as for a harmonic oscillator. We number of sites. For the example of 19 sites, there are 38 discuss a simple classical limit for Fermions. This limit is shown to spin-orbitals vs. 2,821,056,160 doublet states. correspond to an anharmonic oscillator, with just one bound The point of the formalism is that it ensures that the Pauli excited state. The vibrational quantum number of this anharmonic exclusion principle is satisfied, namely that no more than one oscillator, which is therefore limited to the range 0 to 1, is the electron occupies any given spin-orbital. -
Statics Syllabus
AME-201 Statics Spring 2011 Instructor Charles Radovich [email protected] Lecture T Th 2:00 pm – 3:20 PM, VKC 152 Office Hours Tue. 3:30 – 5 pm, Wed. 4:30 – 5:30 pm, also by appointment – RRB 202 Textbook Beer, F. and E. Johnston. Vector Mechanics for Engineers: Statics, 9th Edition. McGraw-Hill, Boston, 2009. ISBN-10: 007727556X; ISBN-13: 978-0077275563. Course Description AME-201 STATICS Units: 3 Prerequisite: MATH-125 Recommended preparation: AME-101, PHYS-151 Analysis of forces acting on particles and rigid bodies in static equilibrium; equivalent systems of forces; friction; centroids and moments of inertia; introduction to energy methods. (from the USC Course Catalogue) The subject of Statics deals with forces acting on rigid bodies at rest covering coplanar and non- coplanar forces, concurrent and non-concurrent forces, friction forces, centroid and moments of inertia. Much time will be spent finding resultant forces for a variety of force systems, as well as analyzing forces acting on bodies to find the reacting forces supporting those bodies. Students will develop critical thinking skills necessary to formulate appropriate approaches to problem solutions. Course Objectives Throughout the semester students will develop an understanding of, and demonstrate their proficiency in the following concepts and principles pertaining to vector mechanics, statics. 1. Components of a force and the resultant force for a systems of forces 2. Moment caused by a force acting on a rigid body 3. Principle of transmissibility and the line of action 4. Moment due to several concurrent forces 5. Force and moment reactions at the supports and connections of a rigid body 6. -
Quantum Information Science: Emerging No More
Quantum Information Science: Emerging No More Carlton M. Caves Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico 87131-0001, USA Final: 2013 January 30 Quantum information science (QIS) is a new field of enquiry, nascent in the 1980s, founded firmly in the 1990s, exploding in the 2010s, now established as a discipline for the 21st Century. Born in obscurity, then known as the foundations of quantum mechanics, the field began in the 60s and 70s with studies of Bell inequalities. These showed that the predictions of quantum mechanics cannot be squared with the belief, called local realism, that physical systems have realistic properties whose pre-existing values are revealed by measurements. The predictions of quantum mechanics for separate systems, correlated in the quantum way that we now call entanglement, are at odds with any version of local realism. Experiments in the early 80s demonstrated convincingly that the world comes down on the side of quantum mechanics. With local realism tossed out the window, it was natural to dream that quantum correlations could be used for faster-than-light communication, but this speculation was quickly shot down, and the shooting established the principle that quantum states cannot be copied. A group consisting of quantum opticians, electrical engineers, and mathematical physicists spent the 60s and 70s studying quantum measurements, getting serious about what can be measured and how well, going well beyond the description of observables that was (and often still is) taught in quantum-mechanics courses. This was not an empty exercise: communications engineers needed a more general description of quantum measurements to describe communications channels and to assess their performance. -
Mechanics Lecture 3 Static Forces, Resultants, Equilibrium of a Particle Dr Philip Jackson
C.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 3 Static forces, resultants, equilibrium of a particle Dr Philip Jackson http://www.ee.surrey.ac.uk/Teaching/Courses/ee1.el3/ C.2 Mechanics • Mechanics is the study of the relationship between the motion of bodies and the forces applied to them It describes, measures and relates forces with motion • Statics: study of forces, resultant forces, bodies in equilibrium, no acceleration • Kinematics: study of motion, position and time • Dynamics: study of forces and motion C.3 Forces and equilibrium Forces magnitude and direction components of a force Equilibrium static or dynamic forces in balance Preparation What are the fundamental forces? List them What is a component of force? Give one example What is equilibrium? Draw a diagram with 3-4 forces C.4 Four fundamental forces All forces can be described by four fundamental forces: 1. Gravity: long range, v. weak 2. Electromagnetic: responsible for many electrical, magnetic and EM wave phenomena 3. Weak nuclear: short range, radioactive decay 4. Strong nuclear: short range C.5 Statics – what is a force? • A force is a directed action of one body on another • The action “tends to move” the body and depends on the magnitude, direction and point of application of the force • A force is represented by a vector: C.6 Properties of forces: opposition • Forces occur in equal and opposite pairs (Newton’s 3rd law), each having the same line of action and acting on different bodies • In mechanics, we consider two types of forces: – contact forces