Quantizing the damped harmonic oscillator

一个含时谐振子的精确波函数和相干态(英文)A coherent state is a quantum state in which the oscillation of a harmonic oscillator is amazingly close tothe expected value of a classical harmonic oscillator. Coherent states are important in quantum mechanics, including a wide range of applications, from quantum information theory to quantum foundations and from quantum cryptography to quantum optics.A coherent state can be described as a wavefunction that obeys the Schrodinger equation for a harmonic oscillator. The wavefunction is a superposition of all the possible energy eigenstates and is highly localized in both space and time.It is often referred to as a 'clock state' because it accurately reflects the behavior of a classical harmonic oscillator.In quantum mechanics, coherent states can be used to describe the oscillations of a system with a specific frequency and amplitude. For example, the coherent states ofa harmonic oscillator are used to describe the motion of particles in a crystal lattice and the motion of particles in a cavity. In electro-optics, coherent states are used to describe the interference of light waves, where the lightfield is superimposed on the electric potential of the medium.In recent years, coherent states have also found application in quantum information theory. They allow for the manipulation of quantum states with great precision and accuracy, and can be used to generate and detect entangled states. Furthermore, coherent states are useful for therealization of quantum gates, which are gates of individual qubits, as well as for efficient error correction protocols.In summary, a coherent state is a quantum state in which the oscillations of a harmonic oscillator are amazingly close to the expected value of a classical harmonic oscillator. It is an important concept in quantum mechanics and finds application in a variety of areas, from quantum information theory to quantum optics. It is the basis of many important advances in quantum technologies.。

a r X i v :q u a n t -p h /0606222v 1 27 J u n 2006QUANTUM DECOHERENCE OF THE DAMPED HARMONIC OSCILLATORA.IsarInstitute of Physics and Nuclear Engineering,Bucharest-Magurele,Romaniae-mail:isar@theory.nipne.roAbstract In the framework of the Lindblad theory for open quantum systems,we de-termine the degree of quantum decoherence of a harmonic oscillator interacting with a thermal bath.It is found that the system manifests a quantum de-coherence which is more and more significant in time.We also calculate the decoherence time and show that it has the same scale as the time after which thermal fluctuations become comparable with quantum fluctuations.PACS numbers:03.65.Yz,05.30.-d 1Introduction The quantum to classical transition and classicality of quantum systems continue to be among the most interesting problems in many fields of physics,for both conceptual and experimental reasons [1,2].Two conditions are essential for the classicality of a quantum system [3]:a)quantum decoherence (QD),that means the irreversible,uncontrollable and persistent formation of a quantum correlation (entanglement)of the system with its environment [4],expressed by the damping of the coherences present in the quantum state of the system,when the off-diagonal elements of the density matrix decay below a certain level,so that this density matrix becomes approximatelydiagonal and b)classical correlations,expressed by the fact that the Wigner function of the quantum system has a peak which follows the classical equations of motion in phase space with a good degree of approximation,that is the quantum state becomes peaked along a classical trajectory.Classicality is an emergent property of open quantum systems,since both main features of this process –QD and classical correlations –strongly depend on the interaction between the system and its external environment[1,2].The role of QD became relevant in many interesting physical problems.In manycases one is interested in understanding QD because one wants to prevent decoherence from damaging quantum states and to protect the information stored in quantum states from the degrading effect of the interaction with the environment.Decoherence is also responsible for washing out the quantum interference effects which are desirableto be seen as signals in experiments.QD has a negative influence on many areas relying upon quantum coherence effects,in particular it is a major problem in quantum optics and physics of quantum information and computation[5].In this work we study QD of a harmonic oscillator interacting with an environ-ment,in particular with a thermal bath,in the framework of the Lindblad theory for open quantum systems.We determine the degree of QD and then we apply the criterion of QD.We consider different regimes of the temperature of environment and it is found that the system manifests a QD which in general increases with time and temperature.The organizing of the paper is as follows.In Sec.2we review the Lindblad master equation for the damped harmonic oscillator and solve the master equation in coordinate representation.Then in Sec.3we investigate QD and in Sec.4we calculate the decoherence time of the system.We show that this time has the same scale as the time after which thermalfluctuations become comparable with quantum fluctuations.A summary and concluding remarks are given in Sec.5.2Master equation and density matrixIn the Lindblad axiomatic formalism based on quantum dynamical semigroups,the irreversible time evolution of an open system is described by the following general quantum Markovian master equation for the density operatorρ(t)[6]:dρ(t)¯h [H,ρ(t)]+12(qp+pq),H0=12q2(2)and the operators V j,V†j,which model the environment,are taken as linear polynomials in coordinate q and momentum p.Then the master equation(1)takes the following form[7]:dρ¯h [H0,ρ]−i2¯h(λ−µ)[p,ρq+qρ]−D pp¯h2[p,[p,ρ]]+D pqcase when the asymptotic state is a Gibbs stateρG(∞)=e−H0kT,these coeffi-cients becomeD pp=λ+µ2kT,D qq=λ−µmωcoth¯hω2kT≥λ2(5)and the asymptotic valuesσqq(∞),σpp(∞),σpq(∞)of the dispersion(variance),re-spectively correlation(covariance),of the coordinate and momentum,reduce to[7]σqq(∞)=¯h2kT,σpp(∞)=¯h mω2kT,σpq(∞)=0.(6)We consider a harmonic oscillator with an initial Gaussian wave function(σq(0) andσp(0)are the initial averaged position and momentum of the wave packet)Ψ(q)=(14exp[−1¯hσpq(0))(q−σq(0))2+i 2mω,σpp(0)=¯h mω2√∂t=i¯h∂q2−∂22¯h(q2−q′2)ρ−1∂q−∂2(λ−µ)[(q+q′)(∂∂q′)+2]ρ−D pp∂q+∂∂q+∂a damping effect(exchange of energy with environment).The last three are noise (diffusive)terms and producefluctuation effects in the evolution of the system.D pp promotes diffusion in momentum and generates decoherence in coordinate q–it re-duces the off-diagonal terms,responsible for correlations between spatially separated pieces of the wave packet.Similarly D qq promotes diffusion in coordinate and gener-ates decoherence in momentum p.The D pq term is the so-called”anomalous diffusion”term and it does not generate decoherence.The density matrix solution of Eq.(9)has the general Gaussian form<q|ρ(t)|q′>=(12exp[−12−σq(t))2−σ(t)¯hσqq (t)(q+q′¯hσp(t)(q−q′)],(10)whereσ(t)≡σqq(t)σpp(t)−σ2pq(t)is the Schr¨o dinger generalized uncertainty function. In the case of a thermal bath we obtain the following steady state solution for t→∞(ǫ≡¯hω/2kT):<q|ρ(∞)|q′>=(mω2exp{−mωcothǫ+(q−q′)2cothǫ]}.(11)3Quantum decoherenceAn isolated system has an unitary evolution and the coherence of the state is not lost–pure states evolve in time only to pure states.The QD phenomenon,that is the loss of coherence or the destruction of off-diagonal elements representing coherences between quantum states in the density matrix,can be achieved by introducing an interaction between the system and environment:an initial pure state with a density matrix which contains nonzero off-diagonal terms can non-unitarily evolve into afinal mixed state with a diagonal density matrix.Using new variablesΣ=(q+q′)/2and∆=q−q′,the density matrix(10) becomesρ(Σ,∆,t)= πexp[−αΣ2−γ∆2+iβΣ∆+2ασq(t)Σ+i(σp(t)2σqq(t),γ=σ(t)¯hσqq(t).(13)The representation-independent measure of the degree of QD[3]is given by the ratio of the dispersion1/√2/αof the diagonal elementρ(Σ,0,t):δQD(t)=1α24{e−4λt[1−(δ+1δ(1−r2)−2cothǫ)ω2−µ2cos(2Ωt)δ(1−r2))µsin(2Ωt)Ω2√2kT,(16)which for high T becomesδQD(∞)=¯hω/2kT.We see thatδQD decreases,and therefore QD increases,with time and temperature,i.e.the density matrix becomes more and more diagonal at higher T and the contributions of the off-diagonal elements get smaller and smaller.At the same time the degree of purity decreases and the degree of mixedness increases with T.For T=0the asymptotic(final)state is pure andδQD reaches its initial maximum value1.δQD=0when the quantum coherence is completely lost,and ifδQD=1there is no QD.Only ifδQD<1we can say that the considered system interacting with the thermal bath manifests QD,when the magnitude of the elements of the density matrix in the position basis are peaked preferentially along the diagonal q=q′.Dissipation promotes quantum coherences, whereasfluctuation(diffusion)reduces coherences and promotes QD.The balance of dissipation andfluctuation determines thefinal equilibrium value ofδQD.The initial pure state evolves approximately following the classical trajectory in phase space and becomes a quantum mixed state during the irreversible process of QD.4Decoherence timeIn order to obtain the expression of the decoherence time,we consider the coefficientγ(13),which measures the contribution of non-diagonal elements in the density matrix(12).For short times(λt≪1,Ωt≪1),we have:γ(t)=−mωδ(1−r2))cothǫ+µ(δ−r2δ√2[λ(δ+r2δ(1−r2))cothǫ−λ−µ−ωr1−r2].(18)The decoherence time depends on the temperature T and the couplingλ(dissipation coefficient)between the system and environment,the squeezing parameterδand the initial correlation coefficient r.We notice that the decoherence time is decreasing with increasing dissipation,temperature and squeezing.For r=0we obtain:t deco=12λ(δ−1).(20) We see that when the initial state is the usual coherent state(δ=1),then the deco-herence time tends to infinity.This corresponds to the fact that for T=0andδ=1 the coefficientγis constant in time,so that the decoherence process does not occur in this case.At high temperature,expression(18)becomes(τ≡2kT/¯hω)t deco=1δ(1−r2))+µ(δ−r24(λ+µ)δkT.(22) The generalized uncertainty functionσ(t)(15)has the following behaviour for short times:σ(t)=¯h2δ(1−r2))cothǫ+µ(δ−1This expression shows explicitly the contribution for small time of uncertainty that is intrinsic to quantum mechanics,expressed through the Heisenberg uncertainty prin-ciple and uncertainty due to the coupling to the thermal environment.From Eq.(23)we can determine the time t d when thermalfluctuations become comparable with quantumfluctuations.At high temperature we obtain1t d=)+µ(δ−1δ(1−r2)squeezing and correlation,it depends on temperature only.QD is expressed by theloss of quantum coherences in the case of a thermal bath atfinite temperature.(2)We determined the general expression of the decoherence time,which showsthat it is decreasing with increasing dissipation,temperature and squeezing.We havealso shown that the decoherence time has the same scale as the time after which thermalfluctuations become comparable with quantumfluctuations and the valuesof these scales become closer with increasing temperature and squeezing.After the decoherence time,the decohered system is not necessarily in a classical regime.There exists a quantum statistical regime in between.Only at a sufficiently high temperaturethe system can be considered in a classical regime.The Lindblad theory provides a self-consistent treatment of damping as a general extension of quantum mechanics to open systems and gives the possibility to extendthe model of quantum Brownian motion.The results obtained in the framework ofthis theory are a useful basis for the description of the connection between uncertainty, decoherence and correlations(entanglement)of open quantum systems with their en-vironment,in particular in the study of Gaussian states of continuous variable systemsused in quantum information processing to quantify the similarity or distinguishabilityof quantum states using distance measures,like trace distance and quantumfidelity. AcknowledgmentsThe author acknowledges thefinancial support received within the Project PN06350101/2006. References[1]E.Joos,H.D.Zeh,C.Kiefer,D.Giulini,J.Kupsch,I.O.Stamatescu,Decoherenceand the Appearance of a Classical World in Quantum Theory(Springer,Berlin,2003).[2]W.H.Zurek,Rev.Mod.Phys.75,715,2003.[3]M.Morikawa,Phys.Rev.D42,2929(1990).[4]R.Alicki,Open Sys.and Information Dyn.11,53(2004).[5]M.A.Nielsen and I.L.Chuang,Quantum Computation and Quantum Information(Cambridge Univ.Press,Cambridge,2000).[6]G.Lindblad,Commun.Math.Phys.48,119(1976).[7]A.Isar,A.Sandulescu,H.Scutaru,E.Stefanescu,W.Scheid,Int.J.Mod.Phys.E3,635(1994).[8]A.Isar,W.Scheid,Phys.Rev.A66,042117(2002).[9]B.L.Hu,Y.Zhang,Int.J.Mod.Phys.A10,4537(1995).[10]A.Isar,W.Scheid,Physica A,in press(2006).。

小学上册英语第四单元测验卷(有答案)考试时间：80分钟（总分:100）B卷一、综合题(共计100题共100分)1. 填空题：I enjoy riding my ______ around the neighborhood.2. 填空题：When I help others, I feel ______ (满足). It’s important to be kind and ______ (乐于助人).3. 选择题：What is the largest organ in the human body?A. BrainB. HeartC. SkinD. Liver答案:C4. 听力题：Friction can slow down a ______.5. 填空题：The ancient Romans established ________ to provide public services.6. 选择题：What is the name of the fairy tale character who had long hair?A. CinderellaB. RapunzelC. Sleeping BeautyD. Snow White答案: B7. 填空题：I like to play ______ (视频游戏).8. 填空题：The frog croaks loudly during the ______ (春天).9. 听力题：The color of cabbage juice changes with pH; it can be red or ______.10. 选择题：What is 5 + 5?A. 8B. 9C. 10D. 1111. 填空题：The _____ (果树) produces sweet fruit.12. 填空题：My friend is very __________ (乐观的) about life.13. 选择题：What is 15 7?A. 6B. 7C. 8D. 9答案:C14. 听力题：The flower pot is ______ (colorful) and bright.15. 填空题：I love to watch ______ movies.16. 听力题：The _______ can be used for decoration.17. 选择题：What do you call a baby jackal?A. KitB. PupC. CalfD. Cub18. 听力题：The _______ of a wave can be described as its height.19. ssance was marked by advancements in ________ (科学). 填空题：The Rock20. 选择题：What is the name of the first spacecraft to fly by Jupiter?A. Pioneer 10B. Voyager 1C. Voyager 2D. Galileo21. 填空题：My mom loves to _______ (动词) to relax. 她觉得这个很 _______ (形容词).22. 选择题：What is a synonym for "fast"?A. SlowB. QuickC. LazyD. Tired23. 填空题：My mom is a wonderful __________ (家长) who teaches me well.24. 选择题：What do you call the sound a cow makes?A. MeowB. BarkC. MooD. Quack25. of Hammurabi is one of the oldest known ______ (法律). 填空题：The Cold26. 听力题：I see a ___ in the sky. (bird)27. 听力题：A ____ is often seen leaping gracefully through the air.28. 听力题：The chameleon changes ______ to blend in.29. 听力题：The dog is ______ at the squirrel. (barking)The flowers are ___. (colorful)31. 选择题：What do we call the person who teaches students?A. EngineerB. TeacherC. DoctorD. Chef答案:B32. 选择题：Which food is made from milk?A. BreadB. CheeseC. RiceD. Pasta答案: B33. 选择题：What do we call the act of making something happen?A. CreationB. InnovationC. ProductionD. Action答案:D34. 填空题：My brother loves __________ (学习) different instruments.35. 听力题：A mineral's ______ refers to the color of its powder when scraped on a surface.36. 听力题：They go to _____ (school/market) every day.37. 选择题：What is the capital of Brunei?A. Bandar Seri BegawanB. Kuala BelaitC. TutongD. Seria答案:A38. 听力题：The pizza is ______ and cheesy. (hot)The ____ is a favorite among children and loves to play in the grass.40. 听力题：The capital city of Sweden is __________.41. 听力题：The process of forming a precipitate occurs in a _______ reaction.42. 填空题：_____ (farming) can be both rewarding and challenging.43. 选择题：What is the capital city of Germany?A. MunichB. BerlinC. FrankfurtD. Hamburg44. 听力题：The children are _______ (drawing) pictures.45. 听力题：The _______ of a pendulum can be affected by air resistance.46. 听力题：When water freezes, it becomes ______.47. 选择题：What is the capital of Portugal?A. LisbonB. MadridC. ParisD. Rome答案:A48. 选择题：What do we call the process of changing from a liquid to a solid?A. MeltingB. FreezingC. BoilingD. Evaporating答案:B49. 选择题：What do you call a group of fish?A. SchoolB. FlockC. PackD. Pride答案: A50. 选择题：What is the chemical symbol for gold?A. AuB. AgC. PbD. Fe51. 听力题：She has ___ (ten) fingers.52. 填空题：The ______ (狐狸) is very clever and sly.53. 听力题：I like to _____ on weekends. (relax)54. 填空题：A chicken lays ______.55. 听力题：She is _______ (studying) for her exam.56. 填空题：My cat loves to chase after ______ (线).57. 填空题：I enjoy drawing _____ (树木) in art class.58. 选择题：What is the primary color of a pumpkin?A. GreenB. OrangeC. YellowD. Brown答案: B. Orange59. 听力题：The squirrel is very ___ (quick).The __________ (热带雨林) is rich in biodiversity.61. 听力题：The apple tree is _______ (full) of fruit.62. 选择题：What is the capital of Sweden?A. StockholmB. OsloC. HelsinkiD. Copenhagen答案: A. Stockholm63. 选择题：What do we call the hard outer layer of the Earth?A. CrustB. MantleC. CoreD. Lithosphere64. 填空题：The ________ (生态研究) reveals insights.65. 填空题：The crow is known for its black ______ (羽毛).66. 填空题：The _____ (birch) tree has beautiful bark.67. 听力题：The chemical symbol for francium is ______.68. 听力题：A saturated solution can no longer dissolve ______.69. 听力题：The __________ point is the temperature at which a solid becomes a liquid.70. 选择题：What do you call the person who studies stars and planets?A. BiologistB. GeologistC. AstronomerD. Physicist答案:CI have a toy _______ that makes me giggle.72. 选择题：What do we call a person who studies history?A. BiologistB. HistorianC. ScientistD. Researcher答案:B73. 听力题：My ______ likes to cook delicious food.74. 听力题：My mom makes _____ for breakfast. (pancakes)75. 填空题：The ________ grow in the garden.76. 选择题：Which sport uses a net and a ball?A. SoccerB. TennisC. BaseballD. Golf答案: B77. 填空题：在古代，________ (leaders) 的决策对国家未来有着重大影响。

小学下册英语第1单元测验试卷考试时间：90分钟（总分:120）B卷一、综合题(共计100题共100分)1. 听力题：My dad works in an ________.2. 选择题：What do we call the time when the sun sets?A. DawnB. DuskC. NoonD. Midnight3. 填空题：I have a collection of ________ (邮票). Each one tells a ________ (故事) about different places.4. 听力题：The color of a solution can indicate its ______.5. 选择题：What do we call the process of a caterpillar becoming a butterfly?a. Metamorphosisb. Evolutionc. Growthd. Development答案:a6. 听力题：The chemical formula for calcium hypochlorite is ______.7. n River flows through __________. (巴西) 填空题：The Amaz8. 选择题：What do you call a person who plays a sport professionally?A. AmateurB. ProfessionalC. CoachD. Referee答案: B9. 填空题：I want to learn how to ________ (画画) better.10. 听力题：A _______ is a type of machine that can make work easier.11. 听力题：The bat flies at _____.12. 听力题：Sugar dissolves in water to form a ______.13. 填空题：The ancient civilization of ________ is revered for its artistic achievements.14. 选择题：What is the name of the imaginary line that divides the Earth into the Eastern and Western Hemispheres?A. EquatorB. Prime MeridianC. International Date LineD. Tropic of Cancer答案:B15. 填空题：The __________ (历史的回顾) informs our present.16. 填空题：A _______ (小嗓子) sings beautifully during spring.17. 填空题：A ____(national landmark) holds historical significance.18. 选择题：What do we call the time it takes for the Earth to complete one orbit around the sun?A. DayB. MonthC. YearD. Season答案: C19. 选择题：What do bees produce?a. Milkb. Honeyc. Eggsd. Silk答案:b20. 听力题：A ____ is often kept as a pet and enjoys being around people.21. 听力题：The _______ of light can create shadows.22. 填空题：I like to bake ______ (蛋糕) for my f riends’ birthdays.23. 填空题：The sunflower always turns towards the _____ (太阳).24. 填空题：I have a toy _______ that can bounce high.25. 填空题：The ______ (松鼠) collects acorns in the fall.26. 选择题：What is the name of the boundary surrounding a black hole?A. Event HorizonB. SingularityC. Accretion DiskD. Photon Sphere27. 填空题：A _____ (植物成长) can lead to community beautification.28. 填空题：A ______ (风景如画) garden can be a great place to relax.29. 听力题：The study of Earth's geological history is important for understanding ______.30. 选择题：What do we call the process of making a new plant from a seed?A. GerminationB. PropagationC. CultivationD. Pollination31. 选择题：What do we call the study of the relationship between organisms and their environment?A. EcologyB. BiologyC. ZoologyD. Botany32. 选择题：What do we call the process of removing trees from a forest?A. DeforestationB. ReforestationC. AfforestationD. Conservation答案: A. Deforestation33. 听力题：The chemical formula for carbon monoxide is __________.34. 填空题：I feel happy when my mom calls me . (当我妈妈叫我____时，我感到很快乐。

Harmonic Oscillators1IntroductionThis is a simple overview of the basic properties of damped harmonic oscillators.It is meant to provide a simple guide for the various types of harmonic oscillators,including undamped, underdamped,overdamped,and critically damped.2Undamped OscillatorsConsider a mass on a spring that slides on a frictionless surface.At rest,define its position at equilibrium to be y=0.We will represent the position of the mass at time t by y(t).If y(t)<0 then the spring is compressed and if y(t)>0the spring is being stretched,as illustrated below.Recall from Newton’s Second Law thatF=ma(1) where F is Force,m is the mass of an object,and a is the acceleration of the mass in motion.We know thatv(t)=ddty(t)is the velocity of the mass at time t and also,a(t)=d2y dt2is the acceleration of the mass at time t.Substituting this value of a into Equation1gives usF=m d2ydt2(2)But we must keep in mind that our system also falls under Hooke’s Law,which saysF=−kywhere k is the spring constant.Summing the forces on the mass yieldsF=m d2ydt2=−ky(3)We can rearrange this equation to obtaind2y dt2+kmy=0(4)This second-order differential equation models the motion of an undamped mass-spring system. It can be broken down into a system offirst-order differential equations as follows:dydt=v(5)dv dt =d2ydt2=−kmySo how do we solve this system of differential equations?We begin by taking y(t)=eλt as our ansatz.We calculate y (t)=λeλt and y (t)=λ2eλt.We can now try substituting our ansatz into the original equation to solve forλ:d2y dt2+kmy=λ2eλt+λeλt=0eλt(λ2+1)=0λ2=−1λ=−i,iThis yields the general solutiony(t)=G1e it+G2e−itUsing Euler’s Formula we can give the general solution asy(t)=C1cos t+C2i sin t(6) The values of the constants C1and C2will depend on the initial conditions of the system(we can solve for them if we know what the initial conditions are).Since the solution is a function of cosine and sine the mass will oscillate forever with a constant amplitude.3Damped Harmonic Oscillators3.1Deriving the EquationNow that we know how to model an undamped oscillator(that is,one that does not take resistive forces into account)let us now look at damped harmonic oscillators.To take resistive (i.e.damping)forces into account,we will say that they are proportional to the velocity of the mass.We will represent them by a”coefficient of damping”which we will call b.We will assume b is positive.We can modify Equation3to model the forces on the system:m d2ydt2=−ky−bdydt(7)This equation can be expressed asm d2ydt2+bdydt+ky=0(8)and we shall refer to this equation as the equation for a damped harmonic oscillator.It can be broken into a system offirst-order equations,similar to the undamped system:dydt=v(9)dv dt =−kmy−bmvSolving this equation is similar to solving the undamped case.We again guess our ansatz to be y(t)=eλt.Substituting this into the equations results in:mλ2eλt+bλeλt+keλt=0eλt(mλ2+bλ+k)=0and by the Zero Property,eλt can never be zero somλ2+bλ+k=0λ=−b±√b2−4mk2mThere are three types of damped harmonic oscillators,and they will be determined by the value of discriminant ofλ,since this determines what type of number the root will be(i.e.Real or Complex).3.2Underdamped OscillatorsWhen b2−4mk<0,we are dealing with an underdamped oscillator.In this case,we havecomplex roots whose real part is−b2m .The motion of the mass is oscillation about the restposition,but unlike the undamped system,the amplitude decreases as time goes on.To help visualize the motion,consider the following example.We have an underdamped system with the following initial conditions:m=1,b=1,k=1,y(0)=1,and v(0)=1Using the program Maple we canfind the general solution,the values ofλ,plot the motion of the mass as a function of time,and also view the directionfield(or phase plane)of the system with the given initial conditions.This is all shown on the following two pages.Thefigure on the previous page was the phase plane of the system.The phase plane plots both y(t)and v(t)over time.Note how the vectors in the phase plane move in a gradual spiral towards the origin.This indicates that the mass is gradually coming to rest.If we wanted to get an idea of the motion of the mass from this phase plane,we would start at an initial point and follow the direction of the vectors.This would give us a good feel for how fast the mass returns to it’s equilibrium point(by seeing how fast the vectors tend towards the origin).Notice from the y-t plot how the mass initially moves upwards.This is because of our choice of initial conditions.We chose to give the system an initially positive velocity.Also not how the mass only goes through one and a half full oscillations.This is because of the relatively high damping constant(relative to4mk).If we decrease the value of b we willfind that the mass experiences less resistance and so it will oscillate more times.This can be seen in the following Maple outputs,where everything from thefirst example remains the same except we change b=0.3.Notice how the spiral in the phase plane becomes more gradual.That is,the spirals becomes more”circle-like”in shape.This shows that if we were to start on the same initial point in both examples(as indeed we did),then following the path of the second set of arrows would take us longer to reach the origin than thefirst set(hence more oscillations about the initial state).So it should now be clear to see how the phase plane helps us to determine the mass’behavior over time.3.3Overdamped OscillatorsWe will now move on to the case of overdamped oscillators,where b2−4mk>0.In this case there are two distinct real roots.There is no real oscillation for an overdamped system.the mass just returns to equilibrium.It may be helpful to visualize the mass as being suspended in a big jar of honey.The density of the honey obviously provides a great resisting force on the mass.If we were to remove the mass from its equilibrium point,it would simply return to its original position without oscillating past equilibrium.This idea can be seen in the following example,again illustrated with the help of Maple.We have an overdamped system with the following initial conditions:m=1,b=3,k=1,y(0)=1,and v(0)=1We canfind the general solution,the values ofλ,plot the motion of the mass as a function of time,and also view the directionfield of the system with the given initial conditions.This is all shown on the following two pages.Notice how the mass made a gradual return to its equilibrium point in the position graph.This motion can be further illustrated in the phase plane graph.Unlike the underdamped system that eventually spiraled to the origin,this overdamped system approaches a line in the plane that the vectors approach and then follow directly to the origin.In order to see how the system behaves for higher damping constants,let us examine what happens when b=9.All the other initial conditions remain the same.Note that the motion of the mass does not change dramatically,it only returns to origin much slower than when b=3. Also,the slope of the line the vectors approach in the phase plane is different.3.4Critically Damped OscillatorsThe last type of damped oscillator is one that is said to be critically damped.So far,we have considered the systems for the cases of b2−4mk<0and b2−4mk>0.Only one case remains, and that is when b2−4mk=0.The system is said to be”critically”damped because a slight change of b’s value in either direction will cause the system to become either overdamped(if b is increased)or underdamped(if b is decreased).Systems of this type have only one root andit is−b2m .The motion of a critically damped oscillator is very similar to that of the overdampedoscillator.It does not oscillate about the origin,it simply returns to its original position.As we have done with the other systems,we will look at an example with the help of Maple. Consider the following system:m=1,b=2,k=1,y(0)=1,and v(0)=1As before,we willfind the general solution,the values ofλ,plot the motion of the mass as a function of time,and also view the directionfield of the system with the given initial conditions.We can see that this system is very similar to the overdamped examples.But perhaps the biggest difference to notice is that the line the vectors approach in the critical system has a slope of-1, whereas the lines from the overdamped systems did not.4ConclusionWe have now covered all the types of harmonic oscillators,how to derive their equations,and what their basic properties are.I hope this has provided a useful guide in understanding the differences in the various systems and what makes them each unique.。

迈克尔逊莫雷实验英语版摘要：I.实验背景与目的- 迈克尔逊莫雷实验的起源- 实验的目的：验证以太是否存在II.实验装置与过程- 迈克尔逊干涉仪的构成- 实验的具体操作过程III.实验结果与分析- 实验观察到的干涉条纹- 基于干涉条纹的分析：以太风的假设- 洛伦兹等人的相对论解释IV.实验的意义与影响- 实验对以太理论的质疑- 实验对物理学发展的贡献：狭义相对论的提出正文：迈克尔逊莫雷实验是一个著名的物理学实验，始于19 世纪末，由迈克尔逊和莫雷两位科学家合作完成。

该实验旨在验证以太是否存在，以太是当时科学界普遍认为存在的一种物质，它是一种无所不在的、绝对静止的介质。

实验的装置由迈克尔逊干涉仪构成，它包括一个光源、一个分光镜、一个半反射镜和一个接收屏。

实验过程如下：首先，将光源发出的光经过分光镜分成两束，一束光照射到半反射镜上，另一束光照射到接收屏上。

然后，将接收屏上的光再次经过半反射镜反射，与照射到半反射镜上的光合并。

最后，在接收屏上观察到的干涉条纹进行分析。

实验结果表明，无论光源与接收屏如何移动，干涉条纹都没有发生变化。

这个结果让科学家们非常惊讶，因为按照当时的以太理论，当光源与接收屏移动时，干涉条纹应该发生变化。

为了解释这个现象，科学家们提出了以太风的假设，即以太在真空中流动，其流动速度与光源和接收屏的速度无关。

然而，这个假设并没有得到证实。

后来，洛伦兹等科学家提出了狭义相对论，认为光速是恒定的，与光源和接收屏的速度无关。

这个理论解释了迈克尔逊莫雷实验的结果，并成为了现代物理学的基础。

总之，迈克尔逊莫雷实验对以太理论提出了质疑，促使科学家们重新思考光的传播机制。

无锡2024年07版小学3年级上册英语第3单元真题(含答案)考试时间：80分钟（总分:110）B卷考试人：_________题号一二三四五总分得分一、综合题(共计100题)1、填空题：The sun sets in the ______ (西边). The sky turns ______ (橙色).2、What do we call a person who studies stars?A. BiologistB. AstronomerC. ChemistD. Geologist答案:B3、What do we call a scientist who studies ancient civilizations?A. ArchaeologistB. HistorianC. AnthropologistD. Sociologist答案: A4、听力题：Astronomers use spectroscopy to analyze the ______ of stars.5、听力题：We will go to the _____ (zoo/museum) this weekend.6、听力题：The chemical process that occurs in our bodies to release energy is called ______.7、选择题：What is the main ingredient in a salad?A. BreadB. VegetablesC. RiceD. Cheese8、听力题：My friend is a ______. He enjoys building things.9、填空题：The ______ (阳光) is necessary for photosynthesis.10、Which of these is a wild animal?A. CatB. DogC. LionD. Hamster11、填空题：We have ______ (很多) plants at home.12、填空题：A dolphin can perform _______ (特技).13、What is the name of the animal that can swim and has fins?A. DogB. CatC. FishD. Bird答案: C14、选择题：What do we call a building where people live?A. SchoolB. HospitalC. HouseD. Office15、What is the name of the famous American holiday celebrated on the last Thursday in November?A. Independence DayB. ThanksgivingC. HalloweenD. Christmas答案: B. Thanksgiving16、填空题：A ________ (庭院) is a great place to relax.I love to watch ______ on TV. (cartoons)18、What do we call the line that divides the Earth into the Northern and Southern Hemispheres?A. EquatorB. Prime MeridianC. Tropic of CancerD. Tropic of Capricorn19、听力题：My friend is a ______. He loves to play games.20、填空题：My favorite plant is a ________ because it smells nice.21、填空题：A hedgehog rolls into a ______ (球) for protection.22、What is the name of the famous American singer known for "Rolling in the Deep"?A. AdeleB. Billie EilishC. Sam SmithD. Sia答案：A23、听力题：I can ________ (jump) very high.24、选择题：What is the largest mammal in the ocean?A. SharkB. DolphinC. WhaleD. Octopus25、听力题：The chemical symbol for arsenic is ______.26、填空题：My aunt has a pet ______ (鹦鹉) that talks.27、听力题：The ______ helps us learn about social studies.He is my __________. (叔叔)29、What do you call a house made of snow?A. IglooB. CabinC. ChaletD. Lodge答案:A30、听力题：The children are ___ in the park. (playing)31、填空题：The scientist conducts important _____ (研究) on health.32、What is the color of a typical peacock?A. BlueB. GreenC. RedD. Yellow答案:B33、选择题：What do we call the process of finding out how much something weighs?A. MeasuringB. WeighingC. CalculatingD. Estimating34、填空题：A ____(estuary) is where freshwater meets saltwater.35、听力题：The chemical formula for sodium carbonate is ______.36、听力题：Oxygen is necessary for _______.37、What is the name of the popular video game where you build and explore worlds?A. MinecraftB. TerrariaC. RobloxD. Stardew Valley答案: AI enjoy _______ (参加) talent shows.39、听力题：A ____ burrows into the ground and enjoys digging.40、听力题：The soup is _____ (hot/cold) today.41、填空题：The ________ (环境变化) impacts plants.42、填空题：My dog is very __________ (活泼的).43、听力题：The ________ is a major river in Egypt.44、听力题：__________ are essential for maintaining healthy soil.45、填空题：I think being a __________ (志愿者) is very important.46、听力题：I found a __________ on the ground.47、填空题：The ________ (国际合作) addresses global issues.48、听力题：She is _____ (playing/reading) a book right now.49、选择题：What do we call the time of year when it rains the most?A. SummerB. SpringC. AutumnD. Winter50、听力题：The man has a funny ________.51、填空题：A frog can blend in with ______ (环境) for safety.A solution that can conduct electricity is called an ______ solution.53、听力题：The chemical formula for potassium chloride is ______.54、听力题：The _____ (电话) is ringing.55、填空题：The hamster stores food in its _______ (脸颊).56、Which of these is not a planet?A. MercuryB. MarsC. SunD. Jupiter答案:C57、填空题：My favorite _____ is a little robot.58、听力题：The ______ helps with the regulation of body temperature.59、填空题：I have a toy ________ that dances.60、填空题：The _____ (植物感知) can be explored through art and literature.61、听力题：Jupiter is the largest ______ in our solar system.62、填空题：Planting flowers can beautify our _____ (社区).63、听力题：My ______ is an artist who paints beautiful pictures.64、How many colors are in a rainbow?A. FiveB. SixC. SevenD. EightThe ____ has a thick, warm coat for cold climates.66、What do we call the process of a plant making its own food using sunlight?A. PhotosynthesisB. RespirationC. DigestionD. Fermentation答案: A67、听力题：I am going to the ________.68、What is the name of the sweet treat made from flour and sugar?A. CakeB. CandyC. CookieD. Brownie答案: C69、听力题：I see a _____ (蜜蜂) in the garden.70、填空题：A _______ (小仓鼠) is often kept as a pet in a cage.71、填空题:I like to ride my ________ in the neighborhood.72、选择题：What do bees make?A. MilkB. HoneyC. ButterD. Jam73、听力题：My teacher is very _____. (nice/quick/slow)74、填空题：Egypt is famous for its ancient ________ (埃及以其古老的________) and pyramids.75、What do you call a person who studies rocks?A. BiologistB. GeologistC. ChemistD. Astronomer答案:B76、填空题：I enjoy creating stories with my toy ________ (玩具名称).77、听力填空题：I love spending time outdoors. Activities like __________ help me appreciate nature and stay active. It’s refreshing to breathe in the fresh air.78、听力题：A __________ is an animal that can survive in very hot environments.79、听力题：The ______ is a common sight in gardens.80、What is the capital of England?A. ParisB. LondonC. BerlinD. Madrid答案: B81、听力题：A compound that contains both carbon and nitrogen is called a ______.82、填空题：The discovery of ________ led to significant scientific advancements.83、Which instrument has keys and is played by pressing them?A. GuitarB. DrumsC. PianoD. Flute答案:C84、What do you call the part of the plant that absorbs water and nutrients from the soil?A. StemB. LeafC. RootD. Flower答案:C85、填空题：I like to go ______ (滑冰) at the rink with my friends.86、听力题：My favorite color is ______. (blue)87、填空题：The ______ (小鸡) is yellow and fluffy.88、填空题：I love to watch _____ flutter by.89、填空题：The rabbit hops around the _______ (兔子在_______四处跳).90、填空题：My friend is very __________ (自信).91、填空题：A ____(quicksand) can trap living creatures.92、What is the name of the superhero with a shield?A. Iron ManB. Captain AmericaC. ThorD. Hulk答案:B93、填空题：I saw a ________ in the park.94、听力题：A _______ can help to visualize the motion of an object under various forces.95、 Mountains are located in ________ (落基山脉位于________). 填空题：The Roma96、填空题：My brother enjoys __________ (参加) local festivals.97、听力题：The __________ is famous for its geysers and hot springs.98、听力题：The Grand Canyon was formed by the erosion of the Colorado ______.99、填空题：The ________ grows in water and floats.100、What is the name of the fairy tale character who kissed a frog?A. Snow WhiteB. CinderellaC. The Princess and the FrogD. Sleeping Beauty答案: C。

小学上册英语下册试卷(有答案)英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.The _______ can be a great source of inspiration for art.2.What is the term for a baby chicken?A. PigletB. CalfC. ChickD. Foal答案:C3.I think friendship is very important. A good friend is someone who __________. I am lucky to have a friend named __________, who always makes me __________. We like to __________ together and have fun.4.The _____ (野兔) is very fast and hard to catch.5.The squirrel's diet consists mainly of ______ (坚果) and seeds.6.The __________ is a major city known for its art and culture. (巴黎)7.The chemical formula for nitric acid is ________.8.The ancient civilization of ________ is celebrated for its cultural richness.9.The sun rises in the ______ (east).10.Metals are usually ______ and can be shaped easily.11.My doll has pretty _______ and a lovely dress.12.Which of these is a mammal?A. FrogB. LizardC. WhaleD. Shark答案:C13.Planets are divided into terrestrial and ______ planets.14.What is the capital of the Cayman Islands?A. George TownB. West BayC. Bodden TownD. North Side答案:a15.I have _____ (one/two) sister(s).16.It is ________ (cold) outside in winter.17.What do you call a sweet drink made from fruit?A. SodaB. JuiceC. MilkshakeD. Water答案: B18.The kitten is _____ (cute/ugly).19.We visit our __________ during holidays. (家人)20.The capital of Sri Lanka is __________.21.My friend’s father is my __________. (朋友的父亲)22.The boiling point of a liquid is the temperature at which it turns into a ______.23.My teacher is __________ (关心学生).24.The first people to circumnavigate the globe were led by _______ Magellan.25.看图选词。