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18 Pages

### Scattering_II

Course: PHYSICS 751, Fall 2007
School: UVA
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Word Count: 3722

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Scattering: More the Partial Wave Expansion Michael Fowler 1/17/08 Plane Waves and Partial Waves We are considering the solution to Schrdingers equation for scattering of an incoming plane wave in the z-direction by a potential localized in a region near the origin, so that the total wave function beyond the range of the potential has the form ( r , , ) = eikr cos + f ( , ) eikr . r The overall...

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UVA - PHYSICS - 751
Relating Scattering Amplitudes to Bound StatesMichael Fowler, UVa. 1/17/08Low Energy Approximations for the S MatrixIn this section, we examine the properties of the partial-wave scattering matrixSl ( k ) = 1 + 2ikfl ( k )for complex values of the mo
UVA - PHYSICS - 751
Scattering TheoryMichael Fowler 1/16/08References:Baym, Lectures on Quantum Mechanics, Chapter 9.Sakurai, Modern Quantum Mechanics, Chapter 7.Shankar, Principles of Quantum Mechanics, Chapter 19.IntroductionAlmost everything we know about nuclei an
UVA - PHYSICS - 751
MainSchrdingers Equation for the Simple Harmonic OscillatorMichael FowlerThis spreadsheet integrates f&quot;(x) = (V(x)-E)f(x) with V(x) a simple harmonic oscillator.The table of values and calculation are on Sheet 2.The &quot;Eigenvalue&quot; button activates the
UVA - PHYSICS - 751
TheSimpleHarmonicOscillatorMichaelFowler 11/13/06EinsteinsSolutionoftheSpecificHeatPuzzleThesimpleharmonicoscillator,anonrelativisticparticleinapotential 1 kx2, isan2excellentmodelforawiderangeofsystemsinnature.Infact,notlongafterPlancksdiscoverytha
UVA - PHYSICS - 751
SpinMichael Fowler 11/26/06IntroductionThe Stern Gerlach experiment for the simplest possible atom, hydrogen in its ground state,demonstrated unambiguously that the component of the magnetic moment of the atom along thez-axis could only have two valu
UVA - PHYSICS - 751
MainSchrdingers Equation for the Finite Square WellThis spreadsheet integrates f&quot;(x) = (V(x)-E)f(x) with V(x) a finite-depth square well.We take V(x) = 0 inside the well, V(x) = D outside the well.The &quot;look for bound state&quot; button activates a macro,w
UVA - PHYSICS - 751
Time-Dependent Perturbation TheoryMichael Fowler 7/6/07Introduction: General FormalismWe look at a Hamiltonian H = H 0 + V ( t ) , with V ( t ) some time-dependent perturbation, so nowthe wave function will have perturbation-induced time dependence.O
UVA - PHYSICS - 751
Time-Independent Perturbation TheoryMichael Fowler 2/16/06IntroductionIf an atom (not necessarily in its ground state) is placed in an external electric field, the energylevels shift, and the wave functions are distorted. This is called the Stark effe
UVA - PHYSICS - 751
UVA - PHYSICS - 1425
Physics 1425: General Physics ISpring 2010Michael FowlerRoom 307, PhysicsHome PageTodays Topics Course arrangements, syllabus outline. Nature of science: observation andmeasurement. Accuracy, significant figures. Units, mass of water, estimation
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One-Dimensional Motion:Displacement, Velocity, AccelerationPhysics 1425 Lecture 2Michael Fowler, UVa.Todays Topics The previous lecture covered measurement,units, accuracy, significant figures, estimation. Today well focus on motion along a straigh
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Freely Falling ObjectsPhysics 1425 Lecture 3Michael Fowler, UVa.Todays Topics In the previous lecture, we analyzed onedimensional motion, defining displacement,velocity, and acceleration and finding formulasfor motion at constant acceleration. Toda
UVA - PHYSICS - 1425
Motion in Two and Three Dimensions:VectorsPhysics 1425 Lecture 4Michael Fowler, UVa.Todays Topics In the previous lecture, we analyzed themotion of a particle moving vertically undergravity. In this lecture and the next, well generalize tothe cas
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ProjectilesPhysics 1425 Lecture 5Michael Fowler, UVa.Reminder: Galileos Laws of Motion Neglecting air resistance and friction, Galileoclaimed:1. Horizontal motion: (ball rolling on table) anobject would continue to move at constantvelocity unless
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Newtons LawsPhysics 1425 lecture 6Michael Fowler, UVa.Newton Extended Galileos Picture ofMotion to Include ForcesGalileo said: Natural horizontal motion is at constant velocityunless a force acts: a push from behind will causeacceleration, frictio
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FrictionPhysics 1425 Lecture 8Michael Fowler, UVa.Warm Up Question A brass cube and a flat brass disk of the sameweight are on a flat board. The board isgradually tilted until sliding begins. Whichslides first?A. The brass cubeB. The flat brass d
UVA - PHYSICS - 1425
Circular MotionPhysics 1425 Lecture 9Michael Fowler, UVa.A Cannon on a Mountain Back to Galileo one more time imagine apowerful cannon shooting horizontally from ahigh mountaintop: The path falls 5 m below a horizontal line inone second.Newtons I
UVA - PHYSICS - 1425
More Circular MotionPhysics 1425 Lecture 10Michael Fowler, UVa.The Conical Pendulum A mass moving in ahorizontal circle, suspendedby a string or rod from afixed point above. If the tension in the stringor rod is T, and the string is degrees from
UVA - PHYSICS - 1425
GravitationPhysics 1425 Lecture 11Michael Fowler, UVaThe Inverse Square Law Newtons idea: the centripetal force keeping theMoon circling the Earth is the same gravitationalforce that pulls us to the ground. BUT: the Moons centripetal acceleration i
UVA - PHYSICS - 1425
Work and EnergyPhysics 1425 Lecture 12Michael Fowler, UVaWhat is Work and What Isnt? In physics, work has a very restricted meaning! Doing homework isnt work. Carrying somebody a mile on a level road isntwork Lifting a stick of butter three feet i
UVA - PHYSICS - 1425
Kinetic Energy and Energy ConservationPhysics 1425 Lecture 13Michael Fowler, UVaMoving Things Have Energy Energy is the ability to do work: todeliver a force that acts through adistance. Placing a weight gently on a nail doesnothing. Dropping the
UVA - PHYSICS - 1425
More Energy TopicsPhysics 1425 Lecture 14Michael Fowler, UVaTopics for TodayOverall Energy ConservationGravitation and Escape VelocityPowerEquilibriumOverall Energy Conservation In the real world, theres lots of friction, airresistance, etc., so
UVA - PHYSICS - 1425
MomentumPhysics 1425 Lecture 15Michael Fowler, UVaPhysics Definition of Momentum Momentum is another word (like work,energy, etc.) from everyday life that has aprecise meaning when used in physics. To begin with, we discuss point particles (orsmal
UVA - PHYSICS - 1425
More about MomentumPhysics 1425 Lecture 15Michael Fowler, UVaElastic One-Dimensional Collisions 1 An elastic collision is one in which mechanical energyis conserved. First example: two equal masses with oppositevelocities:v-v Conservation of mom
UVA - PHYSICS - 1425
Center of MassPhysics 1425 Lecture 17Michael Fowler, UVaCenter of Mass and Center of Gravity Everyone knows that if onekid has twice the weight, theother kid must sit twice as farfrom the axle to balance. Each kid then has the sametorque about th
UVA - PHYSICS - 1425
Circular MotionPhysics 1425 Lecture 18Michael Fowler, UVaHow Far is it Around a Circle? A regular hexagon (6 sides) can abe made by putting together 6equilateral triangles (all sidesequal). The radius of the circle = 1. The distance all the way r
UVA - PHYSICS - 1425
Rotational DynamicsPhysics 1425 Lecture 19Michael Fowler, UVaRotational Dynamics Newtons First Law: a rotatingbody will continue to rotate atconstant angular velocity aslong as there is no torque actingon it. Picture a grindstone on asmooth axle
UVA - PHYSICS - 1425
More Rotational DynamicsPhysics 1425 Lecture 20Michael Fowler, UVaClicker QuestionA uniform rod is free to rotate in a vertical plane abouta frictionless hinge at one end. It is released from restat an angle of 30. (I = (1/3)ML2, = Mg(L/2)cos30)The
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Angular MomentumPhysics 1425 Lecture 21Michael Fowler, UVaA New Look for = I Weve seen how = I works for a bodyrotating about a fixed axis. = I is not true in general if the axis ofrotation is itself accelerating BUT it IS true if the axis is thro
UVA - PHYSICS - 1425
More Angular Momentum, thenStaticsPhysics 1425 Lecture 23Michael Fowler, UVaVector Angular Momentum of a Particle A particle with momentum p is atposition r from the origin O.z Its angular momentum about theorigin isOL= r p This is in line wit
UVA - PHYSICS - 1425
More StaticsPhysics 1425 Lecture 24Michael Fowler, UVaStatics: Conditions for Equilibrium For any body, MdvCM / dt = Fi , the net forcecauses the CM to accelerate. Hence, if the body isremaining at rest,F = 0ii To eliminate angular acceleration,
UVA - PHYSICS - 1425
HydrostaticsPhysics 1425 Lecture 25Michael Fowler, UVaBasic Concepts Density Pressure: Pascals PrincipleThe Crown and the Bathtub Around 250 BC, the king ofSyracuse commissioned a newcrown,and gave the goldsmithabout 1 kg of gold (size of a Dba
UVA - PHYSICS - 1425
More HydrostaticsPhysics 1425 Lecture 26Michael Fowler, UVaBasic Concepts Atmospheric Pressure Buoyancy: Archimedes PrincipleClicker Question Galileo once observed that even a carefullyconstructed pump, situated at ground level,was not able to dr
UVA - PHYSICS - 1425
HydrodynamicsPhysics 1425 Lecture 27Michael Fowler, UVaBasic Concepts Fluid conservation Bernoullis EquationYou are sitting in a rowing boat in a small pond. There are somebricks in the boat. You take the bricks and throw them into thepond. They s
UVA - PHYSICS - 1425
Simple Harmonic MotionPhysics 1425 Lecture 28Michael Fowler, UVaForce of a Stretched Spring If a spring is pulled toextend beyond itsnatural length by adistance x, it will pullback with a forceF = kxwhere k is called thespring constant.The sam
UVA - PHYSICS - 1425
Damped and Driven Harmonic MotionPhysics 1425 Lecture 29Michael Fowler, UVaDamped Harmonic Motion In the real world, oscillators CSprings force Drag forceexperience damping forces:F = kx F = bvfriction, air resistance, etc.m These forces always
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Temperature, Expansion, Ideal Gas LawPhysics 1425 Lecture 30Michael Fowler, UVaEverythings Made of Atoms This idea was only fully accepted about 100years agoin part because of Einsteinsanalysis of Brownian motion. Brown, who studied the sex life of
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Kinetic Theory of GasesPhysics 1425 Lecture 31Michael Fowler, UVaBernoullis Picture Daniel Bernoulli, in 1738, wasthe first to understand airpressure in terms of moleculeshe visualized them shootingaround very rapidly in a closedcontainer, suppor
UVA - PHYSICS - 1425
More Kinetic Theory of GasesPhysics 1425 Lecture 32Michael Fowler, UVaVapor Pressure and Humidity The H2O molecules in liquid water stronglyattract each other, holding the liquid together.But these molecules are still jiggling around,with a Maxwell
UVA - PHYSICS - 1425
Heat and Energy ConservationPhysics 1425 Lecture 33Michael Fowler, UVaHeat Flow If something warm is in contact with somethingqcooler, the warm thing cools down as the cool thinggets a little warmer. This flow of energy, called heat,was until the
UVA - PHYSICS - 1425
Gas Processes and Heat TransportPhysics 1425 Lecture 34Michael Fowler, UVaThe First Law of Thermodynamics A closed system has a total internal energy Eint. This energy can be changed in two different ways:A. The system can do work W, or have work do
UVA - PHYSICS - 1425
The Second Law of Thermodynamics:Heat EnginesPhysics 1425 Lecture 35Michael Fowler, UVaThe First Law of Thermodynamics In any process, total energy is always conserved. Once it was fully realized that heat is just anotherform of energy, it was esta
UVA - PHYSICS - 1425
EntropyPhysics 1425 Lecture 36Michael Fowler, UVaFirst and Second Laws ofThermodynamics A quick review. First Law: total energy conserved in anyprocess: joules in = joules out Second Law: heat only flows one way, and wecant turn heat into just wo
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Physics 1425: General Physics ISpring 2010Michael FowlerRoom 307, PhysicsHome PageTodays Topics Course arrangements, syllabus outline. Nature of science: observation andmeasurement. Accuracy, significant figures. Units, mass of water, estimation
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A proposed Magnetized Dusty Plasma User FacilityE. Thomas, Jr. (Auburn Univ.), R. L. Merlino (Univ. Iowa), M. Rosenberg (UCSD)Previous experiments on magnetized dusty plasmasFeasibility of a magnetized dusty plasma facilityMPE superconducting magnet s
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SOLUTIONSPHYS 2200 Introduction to Quantum Mechanics and RelativityEXAM 1Fall Semester, 2011Instructions:a) This exam is closed book and closed notes.b) You MUST show all of your work in order to obtain full credit on the problems.Part 1: Short Pro
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Solutions to HW 1: Ch. 1 - 3, 4, 7, 8, 9, 20, 22, 25, 41 The problem uses the Lorentz transform approach. More explicitly, they have done the following: Earth frame: Plane frame: Dlyra = +2500 cy = x1
Auburn - PHYS - 2200
oo234x(m)(b) When 10 seconds have passed on the rocket's clock, only 6 seconds have passed onthe laboratory clock.Solutions to HW 1: Ch. 1 18, 23, 27, 43; Ch. 2 1, 4, 5, 81-18.(a)U'=0U~=0xyx'vx=UxU'x+v1 + vU.:O+V=-=VIc1+ 02
Auburn - PHYS - 2200
Solutions to HW 3: Ch. 2 12, 14, 17, 19, 24, 27, 40 -9Note: Error in text solution: 0.34 ng and 0.68 ng (n = nano = 10 ) NOT !
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Spanish 102 Exam 1 Study Guide - Captulo 5 - (35 points)* Please bring a purple scantron and # 2 pencils.*Receiving or giving aid on a test or an exam is cause for dismissal from the University ofMississippi. Any questions should be directed to your In
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STUDY GUIDE - ESPAOL 102 - EXAMEN 2 - Captulo 6 (33 points)* Please bring a purple scantron and # 2 pencils.*ACTIVIDAD A. LOS ALIMENTOS.Las descripciones de alimentos. John est en un restaurante mexicano pero no comprende elmen. Su amigo Miguel le des
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ESPAOL 102 - EXAMEN 3 - Captulo 7 - STUDY GUIDE* Please bring a purple scantron and # 2 pencils.*ACTIVIDAD A. Qu tiempo hace? Juana tiene que hacer un informe meteorolgico.Escucha lo que dice y selecciona la opcin correcta. (4 puntos)Modelo:You hear:
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ESPAOL 102 EXAMEN 4 Captulo 8 (27 puntos)* Please bring number a purple scantron sheet and number 2 pencils.*Receiving or giving aid on a test or an exam is cause for dismissal from the University ofMississippi. Any questions should be directed to your
Ole Miss - SPAN - 102
ESPAOL 102 STUDY GUIDE - EXAMEN FINAL (100 puntos)* Please do not forget your PURPLE scantron sheet and no. 2 pencils *Receiving or giving aid on a test or an exam is cause for dismissal from the University. Anyquestions should be directed to your inst
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SPANISH 101 STUDY GUIDE TEST 1 Captulo 1 (36 points)Please, bring a Purple Scantron No. 16485 and # 2 pencils.Receiving or giving aid on a test or an exam is cause for dismissal from the University ofMississippi. Any questions should be directed to you
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