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Berry phase in electronic structure theoryIn quantum mechanics,Berry phase is a very important concept that describes the geometric properties of a system in parameter space.In electronic structure theory,Berry phase also plays an important role.It is not only of great significance for understanding the wave function and energy level of electrons,but also plays a crucial role in many physical phenomena.In the theory of electronic structure,Berry phase usually refers to the phase that the electron wave function evolves in the parameter space in a periodic lattice.This phase is dependent on system parameters and can affect the energy of electrons and the shape of wave functions.By calculating the Berry phase,one can gain a deeper understanding of the quantum behavior of electrons and the geometric properties of the system.In many physical phenomena,Berry phase plays an important role.For example,in spintronics, Berry phase can affect the spin state and magnetization direction of electrons.In topological insulators,Berry phase and topological properties are closely related and can affect the band structure and surface state of electrons.In addition,Berry phase can also affect optical and magnetic properties,making it widely applicable in materials science and physics.In recent years,with the continuous development of computer technology,calculating Berry phase has become a hot research field.Many numerical methods and computational software have been developed for calculating Berry phases and related physical quantities.These methods and software can not only be used for theoretical research,but also for the analysis and simulation of experimental data.In summary,Berry phase is a very important concept in electronic structure theory.It is not only of great significance for understanding the wave function and energy level of electrons,but also plays a crucial role in many physical phenomena.With the continuous development of computer technology,calculating Berry phase has become a hot research field,providing important tools and means for theoretical and experimental research.。
有关量子力学的英语作文Quantum mechanics, a fundamental theory in physics, has been a subject of fascination and debate since its inception in the early 20th century. It describes the behavior of matter and energy at the smallest scales, where the classical laws of physics no longer apply. This essay aims to explore the key principles of quantum mechanics, its implications for our understanding of the universe, and the ongoing challenges it presents to scientists and philosophers alike.Firstly, the concept of wave-particle duality is central to quantum mechanics. This principle posits that all particles, such as electrons, can exhibit both wave-like and particle-like properties. This duality is demonstrated in the famous double-slit experiment, where particles create aninterference pattern when not observed, but act as discrete entities when measured. The act of observation, therefore, plays a critical role in determining the state of a quantum system.Secondly, the superposition principle is another cornerstone of quantum mechanics. It states that a quantum system can exist in multiple states simultaneously until it is measured. This is exemplified by the thought experiment known asSchrödinger's cat, where a cat in a sealed box is considered to be both alive and dead until the box is opened and thecat's state is observed.Entanglement, a phenomenon where particles become interconnected and the state of one instantaneously influences the state of another, regardless of the distance between them, is another intriguing aspect of quantum mechanics. This has led to the development of quantum computing, which promises to revolutionize information processing by performing calculations at speeds unattainable by classical computers.However, quantum mechanics also presents significant challenges. The interpretation of quantum theory is a subject of ongoing debate. The Copenhagen interpretation suggeststhat the act of measurement collapses the wave function, determining the outcome, while the many-worlds interpretation proposes that all possible outcomes of a quantum event exist in separate, non-interacting parallel universes.Moreover, the reconciliation of quantum mechanics with general relativity, the theory of gravity, remains an unsolved problem in physics. The two theories operate under fundamentally different principles, and finding a unified theory that encompasses both has been a holy grail for physicists.In conclusion, quantum mechanics has reshaped our understanding of the microscopic world and has profound implications for technology, philosophy, and the very fabric of reality. As research continues, it is likely that the mysteries of quantum mechanics will continue to inspire awe and provoke thought about the nature of existence itself.。
相位跃迁原理及作用Phase transition in quantum mechanics, also known as quantum phase transitions, refers to the abrupt changes in the ground state of a system driven by external parameters such as temperature or magnetic field strength. 相位跃迁是指由外部参数如温度或磁场强度驱动而导致系统基态发生突变的量子力学现象。
This phenomenon is of great interest to researchers as it can lead to the emergence of new phases of matter and exotic quantum states. 这一现象引起研究人员的极大兴趣，因为它可能导致新物质相和异域量子态的出现。
One of the key principles underlying phase transitions is known as the Landau theory, which describes the behavior of order parameters near a critical point in a phase transition. 一个关键原则是朗道理论，它描述了在相位跃迁的临界点附近，有序参数的行为。
The concept of phase transitions has wide-ranging applications in various fields, including condensed matter physics, quantum information theory, and even in understanding the behavior ofcomplex systems in biology and social sciences. 相位跃迁的概念在各个领域有着广泛的应用，包括凝聚态物理学、量子信息理论，甚至在理解生物学和社会科学中复杂系统的行为方面。
Title: Theoretical Study of Quantum Mechanical Effects in Condensed Matter SystemsAuthor: [Your Name]Keywords: Quantum Mechanics, Condensed Matter Physics, Theoretical PhysicsAbstract: This article presents a theoretical study of quantum mechanical effects in condensed matter systems. By using quantum field theory and density functional theory, we investigate the quantum properties of electrons in solids and their interactions with phonons, photons, and other particles. The results provide insights into the fundamental mechanisms of quantum transport, superconductivity, and other quantum phenomena in condensed matter systems.Keywords: Quantum Mechanics, Condensed Matter Physics, Theoretical Physics1.Introduction2.Quantum mechanics is a fundamental theory in physics that describesthe behavior of matter and light at the atomic and subatomic scales.In recent years, there has been increasing interest in applying quantum mechanical principles to the study of condensed matter systems. This is because many novel quantum phenomena, such as superconductivity and topological phases, are observed in these systems. In this article, we present a theoretical study of quantum mechanical effects in condensed matter systems.3.Quantum Mechanical Effects in Condensed Matter Systems4.In condensed matter systems, quantum mechanical effects are oftenobserved on a macroscopic scale. One of the most well-known examples is superconductivity. At temperatures close to absolute zero, some materials lose all electrical resistance and behave as if theywere superconductors. This phenomenon can be explained by the BCS theory, which describes the interaction between electrons and lattice vibrations (phonons) using quantum mechanical principles.Another example is the quantum Hall effect, which occurs in a strong magnetic field at low temperatures. In this case, the Hall resistance of a two-dimensional electron gas exhibits plateaus at certain values, indicating that the electrons behave as if they were confined to a single Landau level. This effect is understood in terms of quantum mechanical wave functions and their topological properties.5.Theoretical Methods for Studying Quantum Mechanical Effects6.To theoretically study quantum mechanical effects in condensedmatter systems, several methods are commonly employed. One of the most widely used methods is quantum field theory (QFT), which provides a framework for describing the interactions between particles and fields. Density functional theory (DFT) is also a widely used tool for studying the electronic structure and properties of materials. By combining QFT and DFT, it is possible to describe the quantum mechanical behavior of electrons in solids and their interactions with other particles.7.Conclusion8.In this article, we have presented a theoretical study of quantummechanical effects in condensed matter systems. We have discussed some of the fundamental mechanisms of quantum transport, superconductivity, and other quantum phenomena observed in these systems. By using quantum field theory and density functional theory, we have been able to gain insights into the quantum properties of electrons in solids and their interactions with phonons, photons, and other particles. These insights are crucial for understanding thebehavior of materials at the atomic and subatomic scales and may lead to new applications in technology and materials science.。
华中师范大学物理学院物理学专业英语仅供内部学习参考！2014一、课程的任务和教学目的通过学习《物理学专业英语》，学生将掌握物理学领域使用频率较高的专业词汇和表达方法，进而具备基本的阅读理解物理学专业文献的能力。
通过分析《物理学专业英语》课程教材中的范文，学生还将从英语角度理解物理学中个学科的研究内容和主要思想，提高学生的专业英语能力和了解物理学研究前沿的能力。
培养专业英语阅读能力，了解科技英语的特点，提高专业外语的阅读质量和阅读速度；掌握一定量的本专业英文词汇，基本达到能够独立完成一般性本专业外文资料的阅读；达到一定的笔译水平。
要求译文通顺、准确和专业化。
要求译文通顺、准确和专业化。
二、课程内容课程内容包括以下章节：物理学、经典力学、热力学、电磁学、光学、原子物理、统计力学、量子力学和狭义相对论三、基本要求1.充分利用课内时间保证充足的阅读量（约1200~1500词/学时），要求正确理解原文。
2.泛读适量课外相关英文读物，要求基本理解原文主要内容。
3.掌握基本专业词汇（不少于200词）。
4.应具有流利阅读、翻译及赏析专业英语文献，并能简单地进行写作的能力。
四、参考书目录1 Physics 物理学 (1)Introduction to physics (1)Classical and modern physics (2)Research fields (4)V ocabulary (7)2 Classical mechanics 经典力学 (10)Introduction (10)Description of classical mechanics (10)Momentum and collisions (14)Angular momentum (15)V ocabulary (16)3 Thermodynamics 热力学 (18)Introduction (18)Laws of thermodynamics (21)System models (22)Thermodynamic processes (27)Scope of thermodynamics (29)V ocabulary (30)4 Electromagnetism 电磁学 (33)Introduction (33)Electrostatics (33)Magnetostatics (35)Electromagnetic induction (40)V ocabulary (43)5 Optics 光学 (45)Introduction (45)Geometrical optics (45)Physical optics (47)Polarization (50)V ocabulary (51)6 Atomic physics 原子物理 (52)Introduction (52)Electronic configuration (52)Excitation and ionization (56)V ocabulary (59)7 Statistical mechanics 统计力学 (60)Overview (60)Fundamentals (60)Statistical ensembles (63)V ocabulary (65)8 Quantum mechanics 量子力学 (67)Introduction (67)Mathematical formulations (68)Quantization (71)Wave-particle duality (72)Quantum entanglement (75)V ocabulary (77)9 Special relativity 狭义相对论 (79)Introduction (79)Relativity of simultaneity (80)Lorentz transformations (80)Time dilation and length contraction (81)Mass-energy equivalence (82)Relativistic energy-momentum relation (86)V ocabulary (89)正文标记说明：蓝色Arial字体（例如energy）：已知的专业词汇蓝色Arial字体加下划线（例如electromagnetism）：新学的专业词汇黑色Times New Roman字体加下划线（例如postulate）：新学的普通词汇1 Physics 物理学1 Physics 物理学Introduction to physicsPhysics is a part of natural philosophy and a natural science that involves the study of matter and its motion through space and time, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy. Over the last two millennia, physics was a part of natural philosophy along with chemistry, certain branches of mathematics, and biology, but during the Scientific Revolution in the 17th century, the natural sciences emerged as unique research programs in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry,and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms of other sciences, while opening new avenues of research in areas such as mathematics and philosophy.Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs. For example, advances in the understanding of electromagnetism or nuclear physics led directly to the development of new products which have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.Core theoriesThough physics deals with a wide variety of systems, certain theories are used by all physicists. Each of these theories were experimentally tested numerous times and found correct as an approximation of nature (within a certain domain of validity).For instance, the theory of classical mechanics accurately describes the motion of objects, provided they are much larger than atoms and moving at much less than the speed of light. These theories continue to be areas of active research, and a remarkable aspect of classical mechanics known as chaos was discovered in the 20th century, three centuries after the original formulation of classical mechanics by Isaac Newton (1642–1727) 【艾萨克·牛顿】.University PhysicsThese central theories are important tools for research into more specialized topics, and any physicist, regardless of his or her specialization, is expected to be literate in them. These include classical mechanics, quantum mechanics, thermodynamics and statistical mechanics, electromagnetism, and special relativity.Classical and modern physicsClassical mechanicsClassical physics includes the traditional branches and topics that were recognized and well-developed before the beginning of the 20th century—classical mechanics, acoustics, optics, thermodynamics, and electromagnetism.Classical mechanics is concerned with bodies acted on by forces and bodies in motion and may be divided into statics (study of the forces on a body or bodies at rest), kinematics (study of motion without regard to its causes), and dynamics (study of motion and the forces that affect it); mechanics may also be divided into solid mechanics and fluid mechanics (known together as continuum mechanics), the latter including such branches as hydrostatics, hydrodynamics, aerodynamics, and pneumatics.Acoustics is the study of how sound is produced, controlled, transmitted and received. Important modern branches of acoustics include ultrasonics, the study of sound waves of very high frequency beyond the range of human hearing; bioacoustics the physics of animal calls and hearing, and electroacoustics, the manipulation of audible sound waves using electronics.Optics, the study of light, is concerned not only with visible light but also with infrared and ultraviolet radiation, which exhibit all of the phenomena of visible light except visibility, e.g., reflection, refraction, interference, diffraction, dispersion, and polarization of light.Heat is a form of energy, the internal energy possessed by the particles of which a substance is composed; thermodynamics deals with the relationships between heat and other forms of energy.Electricity and magnetism have been studied as a single branch of physics since the intimate connection between them was discovered in the early 19th century; an electric current gives rise to a magnetic field and a changing magnetic field induces an electric current. Electrostatics deals with electric charges at rest, electrodynamics with moving charges, and magnetostatics with magnetic poles at rest.Modern PhysicsClassical physics is generally concerned with matter and energy on the normal scale of1 Physics 物理学observation, while much of modern physics is concerned with the behavior of matter and energy under extreme conditions or on the very large or very small scale.For example, atomic and nuclear physics studies matter on the smallest scale at which chemical elements can be identified.The physics of elementary particles is on an even smaller scale, as it is concerned with the most basic units of matter; this branch of physics is also known as high-energy physics because of the extremely high energies necessary to produce many types of particles in large particle accelerators. On this scale, ordinary, commonsense notions of space, time, matter, and energy are no longer valid.The two chief theories of modern physics present a different picture of the concepts of space, time, and matter from that presented by classical physics.Quantum theory is concerned with the discrete, rather than continuous, nature of many phenomena at the atomic and subatomic level, and with the complementary aspects of particles and waves in the description of such phenomena.The theory of relativity is concerned with the description of phenomena that take place in a frame of reference that is in motion with respect to an observer; the special theory of relativity is concerned with relative uniform motion in a straight line and the general theory of relativity with accelerated motion and its connection with gravitation.Both quantum theory and the theory of relativity find applications in all areas of modern physics.Difference between classical and modern physicsWhile physics aims to discover universal laws, its theories lie in explicit domains of applicability. Loosely speaking, the laws of classical physics accurately describe systems whose important length scales are greater than the atomic scale and whose motions are much slower than the speed of light. Outside of this domain, observations do not match their predictions.Albert Einstein【阿尔伯特·爱因斯坦】contributed the framework of special relativity, which replaced notions of absolute time and space with space-time and allowed an accurate description of systems whose components have speeds approaching the speed of light.Max Planck【普朗克】, Erwin Schrödinger【薛定谔】, and others introduced quantum mechanics, a probabilistic notion of particles and interactions that allowed an accurate description of atomic and subatomic scales.Later, quantum field theory unified quantum mechanics and special relativity.General relativity allowed for a dynamical, curved space-time, with which highly massiveUniversity Physicssystems and the large-scale structure of the universe can be well-described. General relativity has not yet been unified with the other fundamental descriptions; several candidate theories of quantum gravity are being developed.Research fieldsContemporary research in physics can be broadly divided into condensed matter physics; atomic, molecular, and optical physics; particle physics; astrophysics; geophysics and biophysics. Some physics departments also support research in Physics education.Since the 20th century, the individual fields of physics have become increasingly specialized, and today most physicists work in a single field for their entire careers. "Universalists" such as Albert Einstein (1879–1955) and Lev Landau (1908–1968)【列夫·朗道】, who worked in multiple fields of physics, are now very rare.Condensed matter physicsCondensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. In particular, it is concerned with the "condensed" phases that appear whenever the number of particles in a system is extremely large and the interactions between them are strong.The most familiar examples of condensed phases are solids and liquids, which arise from the bonding by way of the electromagnetic force between atoms. More exotic condensed phases include the super-fluid and the Bose–Einstein condensate found in certain atomic systems at very low temperature, the superconducting phase exhibited by conduction electrons in certain materials,and the ferromagnetic and antiferromagnetic phases of spins on atomic lattices.Condensed matter physics is by far the largest field of contemporary physics.Historically, condensed matter physics grew out of solid-state physics, which is now considered one of its main subfields. The term condensed matter physics was apparently coined by Philip Anderson when he renamed his research group—previously solid-state theory—in 1967. In 1978, the Division of Solid State Physics of the American Physical Society was renamed as the Division of Condensed Matter Physics.Condensed matter physics has a large overlap with chemistry, materials science, nanotechnology and engineering.Atomic, molecular and optical physicsAtomic, molecular, and optical physics (AMO) is the study of matter–matter and light–matter interactions on the scale of single atoms and molecules.1 Physics 物理学The three areas are grouped together because of their interrelationships, the similarity of methods used, and the commonality of the energy scales that are relevant. All three areas include both classical, semi-classical and quantum treatments; they can treat their subject from a microscopic view (in contrast to a macroscopic view).Atomic physics studies the electron shells of atoms. Current research focuses on activities in quantum control, cooling and trapping of atoms and ions, low-temperature collision dynamics and the effects of electron correlation on structure and dynamics. Atomic physics is influenced by the nucleus (see, e.g., hyperfine splitting), but intra-nuclear phenomena such as fission and fusion are considered part of high-energy physics.Molecular physics focuses on multi-atomic structures and their internal and external interactions with matter and light.Optical physics is distinct from optics in that it tends to focus not on the control of classical light fields by macroscopic objects, but on the fundamental properties of optical fields and their interactions with matter in the microscopic realm.High-energy physics (particle physics) and nuclear physicsParticle physics is the study of the elementary constituents of matter and energy, and the interactions between them.In addition, particle physicists design and develop the high energy accelerators,detectors, and computer programs necessary for this research. The field is also called "high-energy physics" because many elementary particles do not occur naturally, but are created only during high-energy collisions of other particles.Currently, the interactions of elementary particles and fields are described by the Standard Model.●The model accounts for the 12 known particles of matter (quarks and leptons) thatinteract via the strong, weak, and electromagnetic fundamental forces.●Dynamics are described in terms of matter particles exchanging gauge bosons (gluons,W and Z bosons, and photons, respectively).●The Standard Model also predicts a particle known as the Higgs boson. In July 2012CERN, the European laboratory for particle physics, announced the detection of a particle consistent with the Higgs boson.Nuclear Physics is the field of physics that studies the constituents and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those in nuclear medicine and magnetic resonance imaging, ion implantation in materials engineering, and radiocarbon dating in geology and archaeology.University PhysicsAstrophysics and Physical CosmologyAstrophysics and astronomy are the application of the theories and methods of physics to the study of stellar structure, stellar evolution, the origin of the solar system, and related problems of cosmology. Because astrophysics is a broad subject, astrophysicists typically apply many disciplines of physics, including mechanics, electromagnetism, statistical mechanics, thermodynamics, quantum mechanics, relativity, nuclear and particle physics, and atomic and molecular physics.The discovery by Karl Jansky in 1931 that radio signals were emitted by celestial bodies initiated the science of radio astronomy. Most recently, the frontiers of astronomy have been expanded by space exploration. Perturbations and interference from the earth's atmosphere make space-based observations necessary for infrared, ultraviolet, gamma-ray, and X-ray astronomy.Physical cosmology is the study of the formation and evolution of the universe on its largest scales. Albert Einstein's theory of relativity plays a central role in all modern cosmological theories. In the early 20th century, Hubble's discovery that the universe was expanding, as shown by the Hubble diagram, prompted rival explanations known as the steady state universe and the Big Bang.The Big Bang was confirmed by the success of Big Bang nucleo-synthesis and the discovery of the cosmic microwave background in 1964. The Big Bang model rests on two theoretical pillars: Albert Einstein's general relativity and the cosmological principle (On a sufficiently large scale, the properties of the Universe are the same for all observers). Cosmologists have recently established the ΛCDM model (the standard model of Big Bang cosmology) of the evolution of the universe, which includes cosmic inflation, dark energy and dark matter.Current research frontiersIn condensed matter physics, an important unsolved theoretical problem is that of high-temperature superconductivity. Many condensed matter experiments are aiming to fabricate workable spintronics and quantum computers.In particle physics, the first pieces of experimental evidence for physics beyond the Standard Model have begun to appear. Foremost among these are indications that neutrinos have non-zero mass. These experimental results appear to have solved the long-standing solar neutrino problem, and the physics of massive neutrinos remains an area of active theoretical and experimental research. Particle accelerators have begun probing energy scales in the TeV range, in which experimentalists are hoping to find evidence for the super-symmetric particles, after discovery of the Higgs boson.Theoretical attempts to unify quantum mechanics and general relativity into a single theory1 Physics 物理学of quantum gravity, a program ongoing for over half a century, have not yet been decisively resolved. The current leading candidates are M-theory, superstring theory and loop quantum gravity.Many astronomical and cosmological phenomena have yet to be satisfactorily explained, including the existence of ultra-high energy cosmic rays, the baryon asymmetry, the acceleration of the universe and the anomalous rotation rates of galaxies.Although much progress has been made in high-energy, quantum, and astronomical physics, many everyday phenomena involving complexity, chaos, or turbulence are still poorly understood. Complex problems that seem like they could be solved by a clever application of dynamics and mechanics remain unsolved; examples include the formation of sand-piles, nodes in trickling water, the shape of water droplets, mechanisms of surface tension catastrophes, and self-sorting in shaken heterogeneous collections.These complex phenomena have received growing attention since the 1970s for several reasons, including the availability of modern mathematical methods and computers, which enabled complex systems to be modeled in new ways. Complex physics has become part of increasingly interdisciplinary research, as exemplified by the study of turbulence in aerodynamics and the observation of pattern formation in biological systems.Vocabulary★natural science 自然科学academic disciplines 学科astronomy 天文学in their own right 凭他们本身的实力intersects相交，交叉interdisciplinary交叉学科的，跨学科的★quantum 量子的theoretical breakthroughs 理论突破★electromagnetism 电磁学dramatically显著地★thermodynamics热力学★calculus微积分validity★classical mechanics 经典力学chaos 混沌literate 学者★quantum mechanics量子力学★thermodynamics and statistical mechanics热力学与统计物理★special relativity狭义相对论is concerned with 关注，讨论，考虑acoustics 声学★optics 光学statics静力学at rest 静息kinematics运动学★dynamics动力学ultrasonics超声学manipulation 操作，处理，使用University Physicsinfrared红外ultraviolet紫外radiation辐射reflection 反射refraction 折射★interference 干涉★diffraction 衍射dispersion散射★polarization 极化，偏振internal energy 内能Electricity电性Magnetism 磁性intimate 亲密的induces 诱导，感应scale尺度★elementary particles基本粒子★high-energy physics 高能物理particle accelerators 粒子加速器valid 有效的，正当的★discrete离散的continuous 连续的complementary 互补的★frame of reference 参照系★the special theory of relativity 狭义相对论★general theory of relativity 广义相对论gravitation 重力，万有引力explicit 详细的，清楚的★quantum field theory 量子场论★condensed matter physics凝聚态物理astrophysics天体物理geophysics地球物理Universalist博学多才者★Macroscopic宏观Exotic奇异的★Superconducting 超导Ferromagnetic铁磁质Antiferromagnetic 反铁磁质★Spin自旋Lattice 晶格，点阵，网格★Society社会，学会★microscopic微观的hyperfine splitting超精细分裂fission分裂，裂变fusion熔合，聚变constituents成分，组分accelerators加速器detectors 检测器★quarks夸克lepton 轻子gauge bosons规范玻色子gluons胶子★Higgs boson希格斯玻色子CERN欧洲核子研究中心★Magnetic Resonance Imaging磁共振成像，核磁共振ion implantation 离子注入radiocarbon dating放射性碳年代测定法geology地质学archaeology考古学stellar 恒星cosmology宇宙论celestial bodies 天体Hubble diagram 哈勃图Rival竞争的★Big Bang大爆炸nucleo-synthesis核聚合，核合成pillar支柱cosmological principle宇宙学原理ΛCDM modelΛ-冷暗物质模型cosmic inflation宇宙膨胀1 Physics 物理学fabricate制造，建造spintronics自旋电子元件，自旋电子学★neutrinos 中微子superstring 超弦baryon重子turbulence湍流，扰动，骚动catastrophes突变，灾变，灾难heterogeneous collections异质性集合pattern formation模式形成University Physics2 Classical mechanics 经典力学IntroductionIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology.Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. Besides this, many specializations within the subject deal with gases, liquids, solids, and other specific sub-topics.Classical mechanics provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When the objects being dealt with become sufficiently small, it becomes necessary to introduce the other major sub-field of mechanics, quantum mechanics, which reconciles the macroscopic laws of physics with the atomic nature of matter and handles the wave–particle duality of atoms and molecules. In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity. General relativity unifies special relativity with Newton's law of universal gravitation, allowing physicists to handle gravitation at a deeper level.The initial stage in the development of classical mechanics is often referred to as Newtonian mechanics, and is associated with the physical concepts employed by and the mathematical methods invented by Newton himself, in parallel with Leibniz【莱布尼兹】, and others.Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newton's work, particularly through their use of analytical mechanics. Ultimately, the mathematics developed for these were central to the creation of quantum mechanics.Description of classical mechanicsThe following introduces the basic concepts of classical mechanics. For simplicity, it often2 Classical mechanics 经典力学models real-world objects as point particles, objects with negligible size. The motion of a point particle is characterized by a small number of parameters: its position, mass, and the forces applied to it.In reality, the kind of objects that classical mechanics can describe always have a non-zero size. (The physics of very small particles, such as the electron, is more accurately described by quantum mechanics). Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom—for example, a baseball can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite objects, made up of a large number of interacting point particles. The center of mass of a composite object behaves like a point particle.Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space and its speed. It also assumes that objects may be directly influenced only by their immediate surroundings, known as the principle of locality.In quantum mechanics objects may have unknowable position or velocity, or instantaneously interact with other objects at a distance.Position and its derivativesThe position of a point particle is defined with respect to an arbitrary fixed reference point, O, in space, usually accompanied by a coordinate system, with the reference point located at the origin of the coordinate system. It is defined as the vector r from O to the particle.In general, the point particle need not be stationary relative to O, so r is a function of t, the time elapsed since an arbitrary initial time.In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute, i.e., the time interval between any given pair of events is the same for all observers. In addition to relying on absolute time, classical mechanics assumes Euclidean geometry for the structure of space.Velocity and speedThe velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time. In classical mechanics, velocities are directly additive and subtractive as vector quantities; they must be dealt with using vector analysis.When both objects are moving in the same direction, the difference can be given in terms of speed only by ignoring direction.University PhysicsAccelerationThe acceleration , or rate of change of velocity, is the derivative of the velocity with respect to time (the second derivative of the position with respect to time).Acceleration can arise from a change with time of the magnitude of the velocity or of the direction of the velocity or both . If only the magnitude v of the velocity decreases, this is sometimes referred to as deceleration , but generally any change in the velocity with time, including deceleration, is simply referred to as acceleration.Inertial frames of referenceWhile the position and velocity and acceleration of a particle can be referred to any observer in any state of motion, classical mechanics assumes the existence of a special family of reference frames in terms of which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames .An inertial frame is such that when an object without any force interactions (an idealized situation) is viewed from it, it appears either to be at rest or in a state of uniform motion in a straight line. This is the fundamental definition of an inertial frame. They are characterized by the requirement that all forces entering the observer's physical laws originate in identifiable sources (charges, gravitational bodies, and so forth).A non-inertial reference frame is one accelerating with respect to an inertial one, and in such a non-inertial frame a particle is subject to acceleration by fictitious forces that enter the equations of motion solely as a result of its accelerated motion, and do not originate in identifiable sources. These fictitious forces are in addition to the real forces recognized in an inertial frame.A key concept of inertial frames is the method for identifying them. For practical purposes, reference frames that are un-accelerated with respect to the distant stars are regarded as good approximations to inertial frames.Forces; Newton's second lawNewton was the first to mathematically express the relationship between force and momentum . Some physicists interpret Newton's second law of motion as a definition of force and mass, while others consider it a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law":a m t v m t p F ===d )(d d dThe quantity m v is called the (canonical ) momentum . The net force on a particle is thus equal to rate of change of momentum of the particle with time.So long as the force acting on a particle is known, Newton's second law is sufficient to。
有关我的科学发明英语作文英文回答：Science has always been a subject that has fascinated me. The idea of being able to understand and explain the world around us is something that I find truly captivating. As a result, I have always been interested in pursuing a career in the field of science.In high school, I took as many science courses as I could. I enjoyed learning about all aspects of science, but I was particularly drawn to the physical sciences. I was fascinated by the laws of physics and how they could be used to explain the behavior of the world around us.After graduating from high school, I decided to pursue a degree in physics at the University of California, Berkeley. I chose Berkeley because it is one of the top universities in the world for physics and because it has a strong research program.During my time at Berkeley, I conducted research in the field of condensed matter physics. My research focused on the development of new materials with novel properties. I was able to publish my research in several peer-reviewed journals and I presented my work at several international conferences.After graduating from Berkeley, I went on to earn a PhD in physics from the Massachusetts Institute of Technology. My research at MIT focused on the development of new quantum materials. I was able to develop several new materials with potential applications in electronics, energy storage, and medicine.I am now a professor of physics at the University of California, Los Angeles. My research interests include the development of new quantum materials, the exploration of new phases of matter, and the development of new techniques for manipulating and controlling quantum systems.I am proud of the work that I have done in the field ofscience. I believe that my research has the potential to make a significant contribution to our understanding of the world around us and to improve the lives of people aroundthe world.中文回答：科学一直是我感兴趣的一门学科。
凝聚态物理英语全文共3篇示例，供读者参考篇1Condensed matter physics, also known as solid-state physics, is a branch of physics that deals with the physical properties of solids and liquids, such as conductors, semiconductors, and insulators. It is a highly multidisciplinary field that encompasses elements of quantum mechanics, statistical mechanics, and electromagnetism.One of the key concepts in condensed matter physics is the idea of collective behavior. In a solid or liquid, the individual atoms or molecules interact with each other in complex ways, leading to emergent phenomena that cannot be understood by studying the individual particles alone. For example, the behavior of electrons in a crystal lattice is governed by the collective interactions between them, leading to phenomena such as superconductivity and magnetism.Another important concept in condensed matter physics is symmetry breaking. In a system with particular symmetry, certain properties of the system are preserved under transformationssuch as rotations or translations. However, in a phase transition, the symmetry of the system is broken, leading to the emergence of new properties. For example, when a material undergoes a phase transition from a normal state to a superconducting state, the symmetry of the system is broken, leading to the emergence of perfect conductivity.Condensed matter physics is a highly active area of research, with ongoing investigation into phenomena such as topological insulators, quantum spin liquids, and high-temperature superconductors. These materials exhibit novel physical properties that could have profound implications for technology, such as faster computers and more efficient energy storage devices.In conclusion, condensed matter physics is a fascinating field that studies the physical properties of solids and liquids at the atomic and subatomic level. By understanding the complex interactions between particles in condensed matter systems, physicists can develop new materials with unique properties and advance our understanding of the fundamental laws of nature.篇2Condensed matter physics is a branch of physics that deals with the study of the physical properties of matter in its condensed phases, which include solids and liquids. This field of physics plays a crucial role in understanding the behavior of materials and their interactions at the atomic and subatomic level.One of the key concepts in condensed matter physics is the idea of symmetry breaking. Symmetry breaking refers to the process by which a system transitions from a symmetric state to a less symmetric state, leading to the emergence of new properties and phenomena. This phenomenon is responsible for the formation of structures such as crystals and magnets, which exhibit specific patterns and behaviors due to broken symmetries.Another important aspect of condensed matter physics is the study of phase transitions. Phase transitions occur when a material undergoes a change in its physical state, such as melting, freezing, or vaporization. Understanding the mechanisms and properties of phase transitions is crucial for developing new materials with unique properties and applications.One of the most exciting areas of research in condensed matter physics is the study of quantum materials. Quantummaterials display unique quantum mechanical properties, such as superconductivity and topological insulators, which have the potential to revolutionize technologies in fields such as electronics and energy storage.Overall, condensed matter physics is a diverse and dynamic field that continues to push the boundaries of our understanding of matter and its properties. By studying the behavior of materials at the atomic and subatomic level, physicists are able to develop new technologies and materials that have the potential to impact a wide range of industries. So yeah, Condensed matter physics is truly fascinating and holds immense potential for future discoveries.篇3Condensed matter physics is a branch of physics that deals with the physical properties of condensed phases of matter, such as solids and liquids. It is a highly interdisciplinary field that combines elements of physics, chemistry, materials science, and engineering to study the behavior of matter at the macroscopic level.One of the key goals of condensed matter physics is to understand the underlying principles that govern the behavior ofmaterials and to develop new materials with specific properties. This is important for a wide range of applications, from developing new electronic devices and sensors to designing new materials for energy storage and conversion.One of the key concepts in condensed matter physics is symmetry breaking. Symmetry breaking is a phenomenon in which the symmetry of a system is spontaneously broken, leading to the emergence of new properties and behaviors. For example, in a crystalline solid, the atoms are arranged in a regular, repeating pattern that exhibits translational symmetry. However, at low temperatures, the crystal may undergo a phase transition in which the symmetry is broken, leading to the emergence of new properties such as magnetic ordering or superconductivity.Another important concept in condensed matter physics is collective behavior. Condensed matter systems often exhibit complex collective behavior that emerges from the interactions between individual particles. For example, in a superconductor, the electrons pair up to form Cooper pairs, which then move through the material without resistance. This collective behavior is responsible for the unique properties of superconductors, such as zero electrical resistance and the expulsion of magnetic fields.Condensed matter physics also plays a critical role in advancing our understanding of quantum mechanics. Quantum mechanics is the branch of physics that describes the behavior of particles at the smallest scales, such as atoms and subatomic particles. In condensed matter systems, the behavior of electrons, atoms, and other particles is governed by the principles of quantum mechanics. Understanding how these principles manifest at the macroscopic level in condensed matter systems is essential for developing new materials with novel properties and applications.One of the key challenges in condensed matter physics is the complexity of the systems being studied. Condensed matter systems contain a large number of interacting particles, which can exhibit a wide range of behaviors and properties. This complexity makes it difficult to predict the behavior of condensed matter systems from first principles alone, and often requires the use of advanced theoretical and computational methods to model and simulate these systems.Despite these challenges, condensed matter physics has made significant contributions to our understanding of the physical world and has paved the way for numerous technological advances. From the discovery of new materialswith exotic properties to the development of new electronic devices and sensors, condensed matter physics continues to play a crucial role in shaping the future of science and technology.In conclusion, condensed matter physics is a vibrant and interdisciplinary field that studies the physical properties of condensed phases of matter. By exploring the principles of symmetry breaking, collective behavior, and quantum mechanics in condensed matter systems, researchers are able to develop new materials with novel properties and applications. As we continue to unlock the mysteries of condensed matter physics, we open up new possibilities for advancing science and technology in the 21st century.。
科学家的故事英语作文800字The Unwavering Quest for Knowledge: An Odyssey of Scientific Pioneers.Throughout history, the pursuit of scientific knowledge has been propelled by the relentless efforts of extraordinary individuals who dared to challenge the boundaries of human understanding. These pioneers, driven by an unquenchable thirst for enlightenment, have dedicated their lives to unraveling the mysteries of nature, expanding the frontiers of science, and shaping the course of human civilization.Archimedes, the Inventor of Buoyancy:In the ancient world, Archimedes of Syracuse emerged as a brilliant polymath whose groundbreaking discoveries left an enduring legacy. Renowned for his ingenuity, he revolutionized mathematics, physics, and engineering. His most famous accomplishment, the discovery of buoyancy,revolutionized naval engineering and laid the foundationfor modern shipbuilding. By meticulously observing the behavior of sinking objects, Archimedes formulated the principle of buoyancy, which states that an object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object.Galileo Galilei, the Father of Modern Physics:Centuries later, the Italian astronomer, physicist, and engineer Galileo Galilei emerged as a cornerstone of the Scientific Revolution. His unwavering commitment to empirical observation and experimentation challenged prevailing Aristotelian doctrines. Armed with his newly invented telescope, Galileo made groundbreaking observations of the solar system, including the fourlargest moons of Jupiter and the phases of Venus. His celestial discoveries shattered the geocentric worldview and paved the way for the heliocentric theory proposed by Nicolaus Copernicus.Isaac Newton, the Architect of Classical Mechanics:In the 17th century, Sir Isaac Newton, one of the most influential scientists of all time, emerged as thearchitect of classical mechanics. His seminal work, Principia Mathematica, laid the foundation for understanding the laws of motion, gravity, and calculus. Newton's contributions revolutionized physics, ushering in an era of scientific inquiry characterized by rigorous mathematics and predictive power. His theories of motion and gravitation became the bedrock of modern physics and engineering, shaping our understanding of the universe from the smallest atoms to the grandest celestial bodies.Marie Curie, the Pioneer of Radioactivity:In the early 20th century, Marie Curie's groundbreaking research in radioactivity changed the course of modern physics and medicine. Together with her husband, Pierre Curie, she dedicated her life to studying the phenomena of radioactivity and its applications in medicine. Their tireless experiments led to the discovery of radium and polonium, two radioactive elements that have had profoundmedical implications, particularly in the treatment of cancer. Curie's pioneering work earned her the distinction of being the first woman to receive a Nobel Prize and the only person to win Nobel Prizes in two different scientific fields, physics, and chemistry.Albert Einstein, the Genius of Relativity:No discussion of scientific pioneers would be complete without mentioning Albert Einstein, the Swiss-bornphysicist whose revolutionary theories of relativity forever altered our perception of space, time, and the universe. Einstein's landmark publications on special and general relativity challenged the foundations of classical physics, introducing new concepts such as spacetime curvature, mass-energy equivalence, and the finite speed of light. His groundbreaking theories have had profound implications in astrophysics, cosmology, and the development of modern technologies such as GPS and laser-based devices.Stephen Hawking, the Unifier of Cosmology and QuantumPhysics:In the contemporary era, theoretical physicist Stephen Hawking emerged as a brilliant mind who dedicated his life to unlocking the mysteries of the universe. Despite being diagnosed with amyotrophic lateral sclerosis (ALS) at a young age, Hawking's unwavering determination fueled his groundbreaking work in theoretical physics. His contributions to cosmology, black hole theory, and quantum gravity have revolutionized our understanding of the early universe and the fundamental nature of space and time. His best-selling book, "A Brief History of Time," became a global sensation, inspiring countless minds to embark on scientific pursuits.Conclusion:The stories of these scientific pioneers serve as a testament to the indomitable human spirit and the transformative power of scientific inquiry. Through their unwavering dedication, intellectual prowess, and relentless curiosity, they have expanded the boundaries of humanknowledge, pushing back the frontiers of understanding and shaping the world we live in. Their legacy continues to inspire future generations of scientists to strive for excellence, challenge the status quo, and unlock the mysteries that lie ahead.。
英文原版量子论科普If you're interested in reading about quantum theory in English, here are some recommendations:1. "Quantum Mechanics: The Theoretical Minimum" by Leonard Susskind and Art Friedman. This textbook provides a thorough and accessible introduction to quantum mechanics, starting from the basics and building up to more advanced concepts. It's written in a lively and engaging style, making it suitable for self-study or as a classroom textbook.2. "Quantum Physics: A First Encounter" by John Taylor. This book is aimed at undergraduates and covers the key ideas of quantum theory, including wavefunctions, operators, measurement, and entanglement. It provides plenty of examples and exercises to help readers understand and apply the theory.3. "Quantum Computation and Quantum Information" by Michael Nielsen and Isaac Chuang. This textbook provides a comprehensive introduction to the field of quantum information science, covering quantum computing, quantum algorithms, quantum error correction, and quantum cryptography. It's suitable for graduate students and researchers in the field.4. "The Quantum World" by Christopher French and Carlo Michelli. This book provides a broad overview of modern quantum theory and its applications, including quantum computing, quantum cryptography, and quantum metrology. It's written in a clear and accessible style, making it suitable for non-experts who want to understand the basics of quantum theory.Remember that reading about quantum theory can be challenging because it involves concepts that are counterintuitive and打破常识。
高一科学探索英语阅读理解25题1<背景文章>The Big Bang Theory is one of the most important scientific theories in modern cosmology. It attempts to explain the origin and evolution of the universe. According to the Big Bang theory, the universe began as an extremely hot and dense singularity. Then, a tremendous explosion occurred, releasing an enormous amount of energy and matter. This event marked the beginning of time and space.In the early moments after the Big Bang, the universe was filled with a hot, dense plasma of subatomic particles. As the universe expanded and cooled, these particles began to combine and form atoms. The first atoms to form were hydrogen and helium. Over time, gravity caused these atoms to clump together to form stars and galaxies.The discovery of the cosmic microwave background radiation in 1964 provided strong evidence for the Big Bang theory. This radiation is thought to be the residual heat from the Big Bang and is uniformly distributed throughout the universe.The Big Bang theory has had a profound impact on modern science. It has helped us understand the origin and evolution of the universe, as well as the formation of stars and galaxies. It has also led to the development ofnew technologies, such as telescopes and satellites, that have allowed us to study the universe in greater detail.1. According to the Big Bang theory, the universe began as ___.A. a cold and empty spaceB. an extremely hot and dense singularityC. a collection of stars and galaxiesD. a large cloud of gas and dust答案：B。
九章三号量子计算机英语Chapter 9: Quantum ComputingQuantum computing is a revolutionary approach to computation that utilizes the principles of quantum mechanics to perform complex computations. In traditional computers, information is processed using bits, which can exist in a state of either 0 or 1. However, in quantum computers, the basic unit of information is a quantum bit, or qubit, which can exist in a superposition of both 0 and 1 states simultaneously. This unique property allows quantum computers to perform computations much faster than classical computers.The third generation of quantum computers, known as the Nine Chapter Three (九章三号) quantum computer, represents a significant advancement in the field of quantum computing. It is built upon the principles of quantum mechanics and employs advanced quantum algorithms to solve complex problems efficiently.The Nine Chapter Three quantum computer is equipped with a large number of qubits, allowing for more complex calculations and simulations. These qubits are extremely delicate and need to be protected from environmental interference. Therefore, the computer is designed with a highly controlled environment, including low temperatures and minimal electromagnetic interference.The Nine Chapter Three quantum computer has the potential to solve problems that are currently computationally infeasible forclassical computers. Its computational power can be harnessed for a wide range of applications, including cryptography, optimization, materials science, drug discovery, and machine learning.However, the field of quantum computing is still in its early stages, and there are many significant challenges that need to be overcome before practical quantum computers become a reality. These challenges include qubit stability, error correction, scaling up the number of qubits, and reducing noise in quantum systems.In conclusion, the Nine Chapter Three quantum computer represents a major step forward in the development of quantum computing technology. With its advanced capabilities and potential applications, it has the potential to revolutionize fields ranging from science and technology to finance and healthcare.。
《材料科学与工程基础》英文习题及思考题及答案第二章习题和思考题Questions and Problems2.6 Allowed values for the quantum numbers ofelectrons are as follows:The relationships between n and the shell designationsare noted in Table 2.1.Relative tothe subshells,l ＝0 corresponds to an s subshelll ＝1 corresponds to a p subshelll ＝2 corresponds to a d subshelll ＝3 corresponds to an f subshellFor the K shell, the four quantum numbersfor each of the two electrons in the 1s state, inthe order of nlmlms , are 100(1/2 ) and 100(-1/2 ).Write the four quantum numbers for allof the electrons inthe L and M shells, and notewhich correspond to the s, p, and d subshells.2.7 Give the electron configurations for the followingions: Fe2+, Fe3+, Cu+, Ba2+,Br-, andS2-.2.17 (a) Briefly cite the main differences betweenionic, covalent, and metallicbonding.(b) State the Pauli exclusion principle.2.18 Offer an explanation as to why covalently bonded materials are generally lessdense than ionically or metallically bonded ones.2.19 Compute the percents ionic character of the interatomic bonds for the followingcompounds: TiO2 , ZnTe, CsCl, InSb, and MgCl2 .2.21 Using Table 2.2, determine the number of covalent bonds that are possible foratoms of the following elements: germanium, phosphorus, selenium, and chlorine.2.24 On the basis of the hydrogen bond, explain the anomalous behavior of waterwhen it freezes. That is, why is there volume expansion upon solidification?3.1 What is the difference between atomic structure and crystal structure?3.2 What is the difference between a crystal structure and a crystal system?3.4Show for the body-centered cubic crystal structure that the unit cell edge lengtha and the atomic radius R are related through a =4R/√3.3.6 Show that the atomic packing factor for BCC is 0.68. .3.27* Show that the minimum cation-to-anion radius ratio for a coordinationnumber of 6 is 0.414. Hint: Use the NaCl crystal structure (Figure 3.5), and assume that anions and cations are just touching along cube edges and across face diagonals.3.48 Draw an orthorhombic unit cell, and within that cell a [121] direction and a(210) plane.3.50 Here are unit cells for two hypothetical metals:(a)What are the indices for the directions indicated by the two vectors in sketch(a)?(b) What are the indices for the two planes drawn in sketch (b)?3.51* Within a cubic unit cell, sketch the following directions:.3.53 Determine the indices for the directions shown in the following cubic unit cell:3.57 Determine the Miller indices for the planesshown in the following unit cell:3.58Determine the Miller indices for the planes shown in the following unit cell: 3.61* Sketch within a cubic unit cell the following planes:3.62 Sketch the atomic packing of (a) the (100)plane for the FCC crystal structure, and (b) the (111) plane for the BCC crystal structure (similar to Figures 3.24b and 3.25b).3.77 Explain why the properties of polycrystalline materials are most oftenisotropic.5.1 Calculate the fraction of atom sites that are vacant for lead at its meltingtemperature of 327_C. Assume an energy for vacancy formation of 0.55eV/atom.5.7 If cupric oxide (CuO) is exposed to reducing atmospheres at elevatedtemperatures, some of the Cu2_ ions will become Cu_.(a) Under these conditions, name one crystalline defect that you would expect toform in order to maintain charge neutrality.(b) How many Cu_ ions are required for the creation of each defect?5.8 Below, atomic radius, crystal structure, electronegativity, and the most commonvalence are tabulated, for several elements; for those that are nonmetals, only atomic radii are indicated.Which of these elements would you expect to form the following with copper:(a) A substitutional solid solution having complete solubility?(b) A substitutional solid solution of incomplete solubility?(c) An interstitial solid solution?5.9 For both FCC and BCC crystal structures, there are two different types ofinterstitial sites. In each case, one site is larger than the other, which site isnormally occupied by impurity atoms. For FCC, this larger one is located at the center of each edge of the unit cell; it is termed an octahedral interstitial site. On the other hand, with BCC the larger site type is found at 0, __, __ positions—that is, lying on _100_ faces, and situated midway between two unit cell edges on this face and one-quarter of the distance between the other two unit cell edges; it is termed a tetrahedral interstitial site. For both FCC and BCC crystalstructures, compute the radius r of an impurity atom that will just fit into one of these sites in terms of the atomic radius R of the host atom.5.10 (a) Suppose that Li2O is added as an impurity to CaO. If the Li_ substitutes forCa2_, what kind of vacancies would you expect to form? How many of thesevacancies are created for every Li_ added?(b) Suppose that CaCl2 is added as an impurity to CaO. If the Cl_ substitutes forO2_, what kind of vacancies would you expect to form? How many of thevacancies are created for every Cl_ added?5.28 Copper and platinum both have the FCC crystal structure and Cu forms asubstitutional solid solution for concentrations up to approximately 6 wt% Cu at room temperature. Compute the unit cell edge length for a 95 wt% Pt-5 wt% Cu alloy.5.29 Cite the relative Burgers vector–dislocation line orientations for edge, screw, andmixed dislocations.6.1 Briefly explain the difference between selfdiffusion and interdiffusion.6.3 (a) Compare interstitial and vacancy atomic mechanisms for diffusion.(b) Cite two reasons why interstitial diffusion is normally more rapid thanvacancy diffusion.6.4 Briefly explain the concept of steady state as it applies to diffusion.6.5 (a) Briefly explain the concept of a driving force.(b) What is the driving force for steadystate diffusion?6.6Compute the number of kilograms of hydrogen that pass per hour through a5-mm thick sheet of palladium having an area of 0.20 m2at 500℃. Assume a diffusion coefficient of 1.0×10- 8 m2/s, that the concentrations at the high- and low-pressure sides of the plate are 2.4 and 0.6 kg of hydrogen per cubic meter of palladium, and that steady-state conditions have been attained.6.7 A sheet of steel 1.5 mm thick has nitrogen atmospheres on both sides at 1200℃and is permitted to achieve a steady-state diffusion condition. The diffusion coefficient for nitrogen in steel at this temperature is 6×10-11m2/s, and the diffusion flux is found to be 1.2×10- 7kg/m2-s. Also, it is known that the concentration of nitrogen in the steel at the high-pressure surface is 4 kg/m3. How far into the sheet from this high-pressure side will the concentration be 2.0 kg/m3?Assume a linear concentration profile.6.24. Carbon is allowed to diffuse through a steel plate 15 mm thick. Theconcentrations of carbon at the two faces are 0.65 and 0.30 kg C/m3 Fe, whichare maintained constant. If the preexponential and activation energy are 6.2 _10_7 m2/s and 80,000 J/mol, respectively, compute the temperature at which the diffusion flux is 1.43 _ 10_9 kg/m2-s.6.25 The steady-state diffusion flux through a metal plate is 5.4_10_10 kg/m2-s at atemperature of 727_C (1000 K) and when the concentration gradient is _350kg/m4. Calculate the diffusion flux at 1027_C (1300 K) for the sameconcentration gradient and assuming an activation energy for diffusion of125,000 J/mol.10.2 What thermodynamic condition must be met for a state of equilibrium to exist? 10.4 What is the difference between the states of phase equilibrium and metastability?10.5 Cite the phases that are present and the phase compositions for the followingalloys:(a) 90 wt% Zn–10 wt% Cu at 400℃(b) 75 wt% Sn–25wt%Pb at 175℃(c) 55 wt% Ag–45 wt% Cu at 900℃(d) 30 wt% Pb–70 wt% Mg at 425℃(e) 2.12 kg Zn and 1.88 kg Cu at 500℃(f ) 37 lbm Pb and 6.5 lbm Mg at 400℃(g) 8.2 mol Ni and 4.3 mol Cu at 1250℃.(h) 4.5 mol Sn and 0.45 mol Pb at 200℃10.6 For an alloy of composition 74 wt% Zn–26 wt% Cu, cite the phases presentand their compositions at the following temperatures: 850℃, 750℃, 680℃, 600℃, and 500℃.10.7 Determine the relative amounts (in terms of mass fractions) of the phases forthe alloys and temperatures given inProblem 10.5.10.9 Determine the relative amounts (interms of volume fractions) of the phases forthe alloys and temperatures given inProblem 10.5a, b, and c. Below are given theapproximate densities of the various metalsat the alloy temperatures:10.18 Is it possible to have a copper–silveralloy that, at equilibrium, consists of a _ phase of composition 92 wt% Ag–8wt% Cu, and also a liquid phase of composition 76 wt% Ag–24 wt% Cu? If so, what will be the approximate temperature of the alloy? If this is not possible,explain why.10.20 A copper–nickel alloy of composition 70 wt% Ni–30 wt% Cu is slowly heatedfrom a temperature of 1300_C .(a) At what temperature does the first liquid phase form?(b) What is the composition of this liquid phase?(c) At what temperature does complete melting of the alloy occur?(d) What is the composition of the last solid remaining prior to completemelting?10.28 .Is it possible to have a copper–silver alloy of composition 50 wt% Ag–50 wt%Cu, which, at equilibrium, consists of _ and _ phases having mass fractions W_ _0.60 and W_ _ 0.40? If so, what will be the approximate temperature of the alloy?If such an alloy is not possible, explain why.10.30 At 700_C , what is the maximum solubility (a) of Cu in Ag? (b) Of Ag in Cu?第三章习题和思考题3.3If the atomic radius of aluminum is 0.143nm, calculate the volume of its unitcell in cubic meters.3.8 Iron has a BCC crystal structure, an atomic radius of 0.124 nm, and an atomicweight of 55.85 g/mol. Compute and compare its density with the experimental value found inside the front cover.3.9 Calculate the radius of an iridium atom given that Ir has an FCC crystal structure,a density of 22.4 g/cm3, and an atomic weight of 192.2 g/mol.3.13 Using atomic weight, crystal structure, and atomic radius data tabulated insidethe front cover, compute the theoretical densities of lead, chromium, copper, and cobalt, and then compare these values with the measured densities listed in this same table. The c/a ratio for cobalt is 1.623.3.15 Below are listed the atomic weight, density, and atomic radius for threehypothetical alloys. For each determine whether its crystal structure is FCC,BCC, or simple cubic and then justify your determination. A simple cubic unitcell is shown in Figure 3.40.3.21 This is a unit cell for a hypotheticalmetal:(a) To which crystal system doesthis unit cell belong?(b) What would this crystal structure be called?(c) Calculate the density of the material, given that its atomic weight is 141g/mol.3.25 For a ceramic compound, what are the two characteristics of the component ionsthat determine the crystal structure?3.29 On the basis of ionic charge and ionic radii, predict the crystal structures for thefollowing materials: (a) CsI, (b) NiO, (c) KI, and (d) NiS. Justify your selections.3.35 Magnesium oxide has the rock salt crystal structure and a density of 3.58 g/cm3.(a) Determine the unit cell edge length. (b) How does this result compare withthe edge length as determined from the radii in Table 3.4, assuming that theMg2_ and O2_ ions just touch each other along the edges?3.36 Compute the theoretical density of diamond given that the CUC distance andbond angle are 0.154 nm and 109.5°, respectively. How does this value compare with the measured density?3.38 Cadmium sulfide (CdS) has a cubic unit cell, and from x-ray diffraction data it isknown that the cell edge length is 0.582 nm. If the measured density is 4.82 g/cm3 , how many Cd 2+ and S 2—ions are there per unit cell?3.41 A hypothetical AX type of ceramic material is known to have a density of 2.65g/cm 3 and a unit cell of cubic symmetry with a cell edge length of 0.43 nm. The atomic weights of the A and X elements are 86.6 and 40.3 g/mol, respectively.On the basis of this information, which of the following crystal structures is (are) possible for this material: rock salt, cesium chloride, or zinc blende? Justify your choice(s).3.42 The unit cell for Mg Fe2O3 (MgO-Fe2O3) has cubic symmetry with a unit celledge length of 0.836 nm. If the density of this material is 4.52 g/cm 3 , compute its atomic packing factor. For this computation, you will need to use ionic radii listed in Table 3.4.3.44 Compute the atomic packing factor for the diamond cubic crystal structure(Figure 3.16). Assume that bonding atoms touch one another, that the angle between adjacent bonds is 109.5°, and that each atom internal to the unit cell is positioned a/4 of the distance away from the two nearest cell faces (a is the unit cell edge length).3.45 Compute the atomic packing factor for cesium chloride using the ionic radii inTable 3.4 and assuming that the ions touch along the cube diagonals.3.46 In terms of bonding, explain why silicate materials have relatively low densities.3.47 Determine the angle between covalent bonds in an SiO44—tetrahedron.3.63 For each of the following crystal structures, represent the indicated plane in themanner of Figures 3.24 and 3.25, showing both anions and cations: (a) (100)plane for the rock salt crystal structure, (b) (110) plane for the cesium chloride crystal structure, (c) (111) plane for the zinc blende crystal structure, and (d) (110) plane for the perovskite crystal structure.3.66 The zinc blende crystal structure is one that may be generated from close-packedplanes of anions.(a) Will the stacking sequence for this structure be FCC or HCP? Why?(b) Will cations fill tetrahedral or octahedral positions? Why?(c) What fraction of the positions will be occupied?3.81* The metal iridium has an FCC crystal structure. If the angle of diffraction forthe (220) set of planes occurs at 69.22°(first-order reflection) when monochromatic x-radiation having a wavelength of 0.1542 nm is used, compute(a) the interplanar spacing for this set of planes, and (b) the atomic radius for aniridium atom.4.10 What is the difference between configuration and conformation in relation topolymer chains? vinyl chloride).4.22 (a) Determine the ratio of butadiene to styrene mers in a copolymer having aweight-average molecular weight of 350,000 g/mol and weight-average degree of polymerization of 4425.(b) Which type(s) of copolymer(s) will this copolymer be, considering thefollowing possibilities: random, alternating, graft, and block? Why?4.23 Crosslinked copolymers consisting of 60 wt% ethylene and 40 wt% propylenemay have elastic properties similar to those for natural rubber. For a copolymer of this composition, determine the fraction of both mer types.4.25 (a) Compare the crystalline state in metals and polymers.(b) Compare thenoncrystalline state as it applies to polymers and ceramic glasses.4.26 Explain briefly why the tendency of a polymer to crystallize decreases withincreasing molecular weight.4.27* For each of the following pairs of polymers, do the following: (1) state whetheror not it is possible to determine if one polymer is more likely to crystallize than the other; (2) if it is possible, note which is the more likely and then cite reason(s) for your choice; and (3) if it is not possible to decide, then state why.(a) Linear and syndiotactic polyvinyl chloride; linear and isotactic polystyrene.(b) Network phenol-formaldehyde; linear and heavily crosslinked ci s-isoprene.(c) Linear polyethylene; lightly branched isotactic polypropylene.(d) Alternating poly(styrene-ethylene) copolymer; randompoly(vinylchloride-tetrafluoroethylene) copolymer.4.28 Compute the density of totally crystalline polyethylene. The orthorhombic unitcell for polyethylene is shown in Figure 4.10; also, the equivalent of two ethylene mer units is contained within each unit cell.5.11 What point defects are possible for MgO as an impurity in Al2O3? How manyMg 2+ ions must be added to form each of these defects?5.13 What is the composition, in weight percent, of an alloy that consists of 6 at% Pband 94 at% Sn?5.14 Calculate the composition, in weight per-cent, of an alloy that contains 218.0 kgtitanium, 14.6 kg of aluminum, and 9.7 kg of vanadium.5.23 Gold forms a substitutional solid solution with silver. Compute the number ofgold atoms per cubic centimeter for a silver-gold alloy that contains 10 wt% Au and 90 wt% Ag. The densities of pure gold and silver are 19.32 and 10.49 g/cm3 , respectively.8.53 In terms of molecular structure, explain why phenol-formaldehyde (Bakelite)will not be an elastomer.10.50 Compute the mass fractions of αferrite and cementite in pearlite. assumingthat pressure is held constant.10.52 (a) What is the distinction between hypoeutectoid and hypereutectoid steels?(b) In a hypoeutectoid steel, both eutectoid and proeutectoid ferrite exist. Explainthe difference between them. What will be the carbon concentration in each?10.56 Consider 1.0 kg of austenite containing 1.15 wt% C, cooled to below 727_C(a) What is the proeutectoid phase?(b) How many kilograms each of total ferrite and cementite form?(c) How many kilograms each of pearlite and the proeutectoid phase form?(d) Schematically sketch and label the resulting microstructure.10.60 The mass fractions of total ferrite and total cementite in an iron–carbon alloyare 0.88 and 0.12, respectively. Is this a hypoeutectoid or hypereutectoid alloy?Why?10.64 Is it possible to have an iron–carbon alloy for which the mass fractions of totalferrite and proeutectoid cementite are 0.846 and 0.049, respectively? Why orwhy not?第四章习题和思考题7.3 A specimen of aluminum having a rectangular cross section 10 mm _ 12.7 mmis pulled in tension with 35,500 N force, producing only elastic deformation. 7.5 A steel bar 100 mm long and having a square cross section 20 mm on an edge ispulled in tension with a load of 89,000 N , and experiences an elongation of 0.10 mm . Assuming that the deformation is entirely elastic, calculate the elasticmodulus of the steel.7.7 For a bronze alloy, the stress at which plastic deformation begins is 275 MPa ,and the modulus of elasticity is 115 Gpa .(a) What is the maximum load that may be applied to a specimen with across-sectional area of 325mm, without plastic deformation?(b) If the original specimen length is 115 mm , what is the maximum length towhich it may be stretched without causing plastic deformation?7.8 A cylindrical rod of copper (E _ 110 GPa, Stress (MPa) ) having a yield strengthof 240Mpa is to be subjected to a load of 6660 N. If the length of the rod is 380 mm, what must be the diameter to allow an elongation of 0.50 mm?7.9 Consider a cylindrical specimen of a steel alloy (Figure 7.33) 10mm in diameterand 75 mm long that is pulled in tension. Determine its elongation when a load of 23,500 N is applied.7.16 A cylindrical specimen of some alloy 8 mm in diameter is stressed elasticallyin tension. A force of 15,700 N produces a reduction in specimen diameter of 5 _ 10_3 mm. Compute Poisson’s ratio for this material if its modulus of elasticity is 140 GPa .7.17 A cylindrical specimen of a hypothetical metal alloy is stressed in compression.If its original and final diameters are 20.000 and 20.025 mm, respectively, and its final length is 74.96 mm, compute its original length if the deformation is totally elastic. The elastic and shear moduli for this alloy are 105 Gpa and 39.7 GPa,respectively.7.19 A brass alloy is known to have a yield strength of 275 MPa, a tensile strength of380 MPa, and an elastic modulus of 103 GPa . A cylindrical specimen of thisalloy 12.7 mm in diameter and 250 mm long is stressed in tension and found to elongate 7.6 mm . On the basis of the information given, is it possible tocompute the magnitude of the load that is necessary to produce this change inlength? If so, calculate the load. If not, explain why.7.20A cylindrical metal specimen 15.0mmin diameter and 150mm long is to besubjected to a tensile stress of 50 Mpa; at this stress level the resulting deformation will be totally elastic.(a)If the elongation must be less than 0.072mm,which of the metals in Tabla7.1are suitable candidates? Why ?(b)If, in addition, the maximum permissible diameter decrease is 2.3×10-3mm,which of the metals in Table 7.1may be used ? Why?7.22 Cite the primary differences between elastic, anelastic, and plastic deformationbehaviors.7.23 diameter of 10.0 mm is to be deformed using a tensile load of 27,500 N. It mustnot experience either plastic deformation or a diameter reduction of more than7.5×10-3 mm. Of the materials listed as follows, which are possible candidates?Justify your choice(s).7.24 A cylindrical rod 380 mm long, having a diameter of 10.0 mm, is to besubjected to a tensile load. If the rod is to experience neither plastic deformationnor an elongation of more than 0.9 mm when the applied load is 24,500 N,which of the four metals or alloys listed below are possible candidates?7.25 Figure 7.33 shows the tensile engineering stress–strain behavior for a steel alloy.(a) What is the modulus of elasticity?(b) What is the proportional limit?(c) What is the yield strength at a strain offset of 0.002?(d) What is the tensile strength?7.27 A load of 44,500 N is applied to a cylindrical specimen of steel (displaying thestress–strain behavior shown in Figure 7.33) that has a cross-sectional diameter of 10 mm .(a) Will the specimen experience elastic or plastic deformation? Why?(b) If the original specimen length is 500 mm), how much will it increase inlength when t his load is applied?7.29 A cylindrical specimen of aluminumhaving a diameter of 12.8 mm and a gaugelength of 50.800 mm is pulled in tension. Usethe load–elongation characteristics tabulatedbelow to complete problems a through f.(a)Plot the data as engineering stressversusengineering strain.(b) Compute the modulus of elasticity.(c) Determine the yield strength at astrainoffset of 0.002.(d) Determine the tensile strength of thisalloy.(e) What is the approximate ductility, in percent elongation?(f ) Compute the modulus of resilience.7.35 (a) Make a schematic plot showing the tensile true stress–strain behavior for atypical metal alloy.(b) Superimpose on this plot a schematic curve for the compressive truestress–strain behavior for the same alloy. Explain any difference between thiscurve and the one in part a.(c) Now superimpose a schematic curve for the compressive engineeringstress–strain behavior for this same alloy, and explain any difference between this curve and the one in part b.7.39 A tensile test is performed on a metal specimen, and it is found that a true plasticstrain of 0.20 is produced when a true stress of 575 MPa is applied; for the same metal, the value of K in Equation 7.19 is 860 MPa. Calculate the true strain that results from the application of a true stress of 600 Mpa.7.40 For some metal alloy, a true stress of 415 MPa produces a plastic true strain of0.475. How much will a specimen of this material elongate when a true stress of325 MPa is applied if the original length is 300 mm ? Assume a value of 0.25 for the strain-hardening exponent n.7.43 Find the toughness (or energy to cause fracture) for a metal that experiences bothelastic and plastic deformation. Assume Equation 7.5 for elastic deformation,that the modulus of elasticity is 172 GPa , and that elastic deformation terminates at a strain of 0.01. For plastic deformation, assume that the relationship between stress and strain is described by Equation 7.19, in which the values for K and n are 6900 Mpa and 0.30, respectively. Furthermore, plastic deformation occurs between strain values of 0.01 and 0.75, at which point fracture occurs.7.47 A steel specimen having a rectangular cross section of dimensions 19 mm×3.2mm (0.75in×0.125in.) has the stress–strain behavior shown in Figure 7.33. If this specimen is subjected to a tensile force of 33,400 N (7,500lbf ), then(a) Determine the elastic and plastic strain values.(b) If its original length is 460 mm (18 in.), what will be its final length after theload in part a is applied and then released?7.50 A three-point bending test was performed on an aluminum oxide specimenhaving a circular cross section of radius 3.5 mm; the specimen fractured at a load of 950 N when the distance between the support points was 50 mm . Another test is to be performed on a specimen of this same material, but one that has a square cross section of 12 mm length on each edge. At what load would you expect this specimen to fracture if the support point separation is 40 mm ?7.51 (a) A three-point transverse bending test is conducted on a cylindrical specimenof aluminum oxide having a reported flexural strength of 390 MPa . If the speci- men radius is 2.5 mm and the support point separation distance is 30 mm ,predict whether or not you would expect the specimen to fracture when a load of 620 N is applied. Justify your prediction.(b) Would you be 100% certain of the prediction in part a? Why or why not?7.57 When citing the ductility as percent elongation for semicrystalline polymers, it isnot necessary to specify the specimen gauge length, as is the case with metals.Why is this so?7.66 Using the data represented in Figure 7.31, specify equations relating tensilestrength and Brinell hardness for brass and nodular cast iron, similar toEquations 7.25a and 7.25b for steels.8.4 For each of edge, screw, and mixed dislocations, cite the relationship between thedirection of the applied shear stress and the direction of dislocation line motion.8.5 (a) Define a slip system.(b) Do all metals have the same slip system? Why or why not?8.7. One slip system for theBCCcrystal structure is _110__111_. In a manner similarto Figure 8.6b sketch a _110_-type plane for the BCC structure, representingatom positions with circles. Now, using arrows, indicate two different _111_ slip directions within this plane.8.15* List four major differences between deformation by twinning and deformationby slip relative to mechanism, conditions of occurrence, and final result.8.18 Describe in your own words the three strengthening mechanisms discussed inthis chapter (i.e., grain size reduction, solid solution strengthening, and strainhardening). Be sure to explain how dislocations are involved in each of thestrengthening techniques.8.19 (a) From the plot of yield strength versus (grain diameter)_1/2 for a 70 Cu–30 Zncartridge brass, Figure 8.15, determine values for the constants _0 and ky inEquation 8.5.(b) Now predict the yield strength of this alloy when the average grain diameteris 1.0 _ 10_3 mm.8.20. The lower yield point for an iron that has an average grain diameter of 5 _ 10_2mm is 135 MPa . At a grain diameter of 8 _ 10_3 mm, the yield point increases to 260MPa. At what grain diameter will the lower yield point be 205 Mpa ?8.24 (a) Show, for a tensile test, thatif there is no change in specimen volume during the deformation process (i.e., A0 l0 _Ad ld).(b) Using the result of part a, compute the percent cold work experienced bynaval brass (the stress–strain behavior of which is shown in Figure 7.12) when a stress of 400 MPa is applied.8.25 Two previously undeformed cylindrical specimens of an alloy are to be strainhardened by reducing their cross-sectional areas (while maintaining their circular cross sections). For one specimen, the initial and deformed radii are 16 mm and11 mm, respectively. The second specimen, with an initial radius of 12 mm, musthave the same deformed hardness as the first specimen; compute the secondspecimen’s radius after deformation.8.26 Two previously undeformed specimens of the same metal are to be plasticallydeformed by reducing their cross-sectional areas. One has a circular cross section, and the other is rectangular is to remain as such. Their original and deformeddimensions are as follows:Which of these specimens will be the hardest after plastic deformation, and why?8.27 A cylindrical specimen of cold-worked copper has a ductility (%EL) of 25%. Ifits coldworked radius is 10 mm (0.40 in.), what was its radius beforedeformation?8.28 (a) What is the approximate ductility (%EL) of a brass that has a yield strengthof 275 MPa ?(b) What is the approximate Brinell hardness of a 1040 steel having a yieldstrength of 690 MPa?8.41 In your own words, describe the mechanisms by which semicrystalline polymers(a) elasticallydeform and (b) plastically deform, and (c) by which elastomerselastically deform.8.42 Briefly explain how each of the following influences the tensile modulus of asemicrystallinepolymer and why:(a) molecular weight;(b) degree of crystallinity;(c) deformation by drawing;(d) annealing of an undeformed material;(e) annealing of a drawn material.8.43* Briefly explain how each of the following influences the tensile or yieldstrength of a semicrystalline polymer and why:(a) molecular weight;。
高三物理研究英语阅读理解30题1<背景文章>Newton's three laws of motion are fundamental principles in physics. The first law, also known as the law of inertia, states that an object at rest will remain at rest and an object in motion will continue in motion with a constant velocity unless acted upon by an external force. This law was a major breakthrough in understanding the nature of motion.The second law describes the relationship between force, mass, and acceleration. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically, it can be expressed as F = ma, where F is the force, m is the mass, and a is the acceleration.The third law, known as the law of action and reaction, states that for every action, there is an equal and opposite reaction. This means that when one object exerts a force on another object, the second object exerts an equal and opposite force on the first object.Newton's laws were developed in the 17th century and had a profound impact on the development of modern science and technology. They have been used to explain the motion of planets, the behavior of machines, and the flight of rockets. In modern technology, Newton's laws are applied invarious fields such as engineering, aerospace, and robotics.1. What is Newton's first law also known as?A. The law of gravityB. The law of inertiaC. The law of accelerationD. The law of action and reaction答案：B。
量子物理的秘密英语作文Title: Unveiling the Secrets of Quantum Physics。
Quantum physics, often described as the most revolutionary theory in science, delves into the mysterious realm of the very small, challenging our understanding of reality itself. In this essay, we embark on a journey to explore the enigmatic world of quantum physics and uncover its secrets.At the heart of quantum physics lies the concept of quantum mechanics, which governs the behavior of particles at the atomic and subatomic levels. Unlike classical physics, where objects obey deterministic laws, quantum mechanics introduces probabilistic behavior, whereparticles exist in multiple states simultaneously until observed.One of the fundamental principles of quantum mechanics is superposition. This principle suggests that particlescan exist in a combination of multiple states until a measurement is made, collapsing the superposition into a single outcome. This phenomenon has profound implications, leading to the development of technologies like quantum computing, which harnesses the power of superposition to perform complex calculations exponentially faster than classical computers.Another intriguing concept is entanglement, where particles become interconnected in such a way that thestate of one particle instantaneously influences the stateof another, regardless of the distance between them. This phenomenon, famously referred to as "spooky action at a distance" by Albert Einstein, has been experimentally verified and is being explored for applications in quantum communication and cryptography.Furthermore, Heisenberg's uncertainty principle asserts that certain pairs of physical properties, such as position and momentum, cannot be precisely determined simultaneously. This inherent uncertainty at the quantum level challenges our classical intuition and underscores the probabilisticnature of reality.The Copenhagen interpretation, proposed by Niels Bohr and Werner Heisenberg, is one of the most widely accepted interpretations of quantum mechanics. It asserts that particles exist in a state of probability until observed, at which point their wave function collapses into adefinite state. This interpretation emphasizes the role of the observer in shaping reality, highlighting the profound philosophical implications of quantum physics.However, the mysteries of quantum physics extend beyond our current understanding. The quest to reconcile quantum mechanics with general relativity, the theory of gravity, remains one of the greatest challenges in modern physics. Theoretical frameworks like string theory and loop quantum gravity offer potential avenues for unifying these two pillars of physics, but definitive answers remain elusive.In conclusion, quantum physics unveils a reality far stranger and more mysterious than we could have imagined. From the mind-bending concepts of superposition andentanglement to the profound implications for technology and philosophy, the secrets of quantum physics continue to captivate and inspire scientists and thinkers alike. As we delve deeper into this enigmatic realm, we may uncover even more astonishing truths about the nature of the universe.。
a r X i v :h e p -t h /0503121v 1 15 M a r 2005Phases of Quantum Gravity inAdS 3and Linear Dilaton Backgrounds A.Giveon 1,D.Kutasov 2,E.Rabinovici 1and A.Sever 11Racah Institute of Physics,The Hebrew University,Jerusalem,91904,Israel 2EFI and Department of Physics,University of Chicago,5640S.Ellis Ave.,Chicago,IL 60637,USA We show that string theory in AdS 3has two distinct phases depending on the radius of curvature R AdS =√C)invariant vacuum of the spacetime conformal field theory is normalizable,the high energy density of states is given by the Cardy formula with c eff=c ,and generic high energy states look like large BTZ black holes.For k <1,the SL (2,22).The states responsiblefor this growth are two dimensional black holes for k >1,and highly excited perturbative strings living in the linear dilaton throat for k <1.The change of behavior at k =1in the two cases is an example of a string/black hole transition.The entropies of black holes and strings coincide at k =1.3/051.IntroductionIt is widely believed that in weakly coupled string theory inflat spacetime,a generic high energy state with particular values of mass,angular momentum and other charges looks from afar like a black hole with those charges.As the energy of the state increases, the curvature at the horizon decreases,and the black hole description becomes more and more reliable.Conversely,as the mass of a black hole decreases,string(α′)corrections in the vicinity of the horizon grow and the black hole picture becomes less and less appropriate.When α′effects at the horizon are large,e.g.if the string frame curvature at the horizon is much larger than the string scale,a more useful description of the state is in terms of weakly coupled strings and D-branes.The transition between the black hole picture valid for small horizon curvature,and the perturbative string picture valid for large curvature, which occurs when the curvature at the horizon is of order the string scale,is smooth.In particular,the black hole and perturbative string entropies match,up to numerical coeffi-cients independent of the string coupling and charges.1This is known as the string/black hole correspondence principle[1,2].The above discussion relies crucially on the fact that the curvature at the horizon of a black hole depends on its mass,and goes to zero for large mass.In this paper we will study situations where this is not the case.Two prototypical examples which we will discuss are the BTZ black hole in AdS3,and the SL(2,I R)/U(1)black hole in I R1,1with asymptotically linear(spacelike)dilaton.In the former case,the curvature is the same everywhere,and is proportional to the cosmological constant of the underlying anti-de-Sitter spacetime.In the latter,the mass of the black hole determines the value of the dilaton at the horizon,while the curvature there as well as the gradient of the dilaton depend solely on the cosmological constant.Thus,for these black holes varying the mass does not change the size of theα′corrections.It is natural to ask whether there is an analog of the string/black hole transition in these cases.We will see below that the answer is affirmative.To study the transition, we will take a slightly different approach than that offlat spacetime.Instead of varying the energy of the string or black hole,we will study the dependence of the high energy spectrum on the cosmological constant,which controls theα′corrections.For AdS3,this means varying the radius of curvature of the anti-de-Sitter space,which is related to the cosmological constantΛvia1Λ=−,(1.2)2l pwhere l p is the three dimensional Planck length.The expression for the central charge,√(1.2),is valid when R AdS is much larger than l p,l s(l s=c effL0¯c eff¯L0;J=L0−¯L0.(1.4)12Modular invariance of the spacetime CFT,which is generally expected to be a property of quantum gravity in AdS3,implies that the effective central charge c effis given by[5,6]c eff=c−24∆min,(1.5) where∆min is the lowest scaling dimension in the spacetime theory.Unitarity of the theory implies that∆min≥0,so that c eff≤c;c eff=c iffthe SL(2,R AdSk=We will see that the structure of the theory depends crucially on the magnitude of k.For k>1,the SL(2,C)invariant vacuum,as well as BTZ black holes are non-normalizable;hence,c eff<c.The high energy spectrum is dominated by highly excited states of long strings,which become more and more weakly interacting as they approach the boundary.These states are well described by perturbative string theory in AdS3.The transition between the black hole and string phases occurs at the point where the radius of curvature,R AdS,is equal to l s(1.6),in agreement with general expectations from the correspondence principle.At the transition point,the black hole and perturbative string entropies match exactly.A similar structure is found for the asymptotically linear dilaton case.The high energy behavior of the entropy is in general Hagedorn,but the nature of the high energy states depends on the linear dilaton slope,Q.Below a certain critical value of Q,the generic high energy states correspond to SL(2,I R)/U(1)black holes,or their charged cousins.Above this value,the black holes become non-normalizable,and the high energy states correspond to highly excited perturbative strings living in the linear dilaton region.Again,at the transition between the non-perturbative(black hole)and perturbative(string)phases,the entropies match,for both charged and uncharged states.The fact that the AdS3and linear dilaton backgrounds exhibit very similar string/black hole transitions is not accidental.To see that,consider a background of the form I R1,1×I Rφ×N,where I Rφis the real line along which the dilaton varies linearly andN is a compact unitary CFT.String theory in this background has in general a Hagedorn density of states and exhibits the transition described above as a function of Q,the gradient of the dilaton.One can add to the background Q1fundamental strings stretched in I R1,1, and study the low energy dynamics of the combined system.This theory is related by the AdS/CFT correspondence to string theory in AdS3×N,with string coupling g2s∼1/Q1, and curvature(1.6)k=2/Q2.The string/black hole transition in AdS3×N occurs atk=1for all Q1.Thus,it is natural to expect that the same transition should also occur √at k=1(or Q=The AdS3and linear dilaton theories are also related via matrix theory[9,10].The low energy theory on the fundamental strings living in the linear dilaton throat can be thought of as the discrete lightcone description of the throat,with the discretization parameter equal to Q1.This relation implies that the transitions seen in the two cases should occur at the same value of k.2.AdS3In this section we will study string theory on AdS3times a compact space.A large class of such backgrounds can be constructed as follows2.Consider type II string theory onI R1,1×I Rφ×S1×M.(2.1) Here,I Rφis the real line labeled byφ,with a dilaton linear inφQΦ=−2We will restrict here to a certain class of supersymmetric backgrounds,but most of what we say can be generalized to any stable linear dilaton or AdS3background.The radius of curvature of AdS3,R AdS,is related to the level of the SL(2,I R)current algebra of AdS3,k,via the relation(1.6).The worldsheet central charge of the CFT on AdS3is given by36c AdS=3+2.(2.9)Q1Thus,weakly coupled string theory in AdS3requires a large number of fundamental strings, Q1≫1.A familiar example of(2.5)is the system of k NS5-branes wrapped around a four-manifold M4(M4=T4or K3),and fundamental strings stretched in the uncompactified direction of thefivebranes.In this case one has M=SU(2)k,such that the background(2.5)becomesU(1)AdS3×S3×M4.(2.10) the levels of SL(2,I R),(2.6),and SU(2)are both equal to the number offivebranes,k, which is an integer≥2.Thus,in the class of backgrounds(2.10),the question what happens for k<1does not arise.Nevertheless,there are cases where k<1in(2.5).For example,we can take M to be the N=2minimal model with c M=3−6n+1<1.(2.11) Increasing the central charge of M beyond c M=3takes one to k>1[12].The spacetime dynamics in an AdS3background of the form(2.5)corresponds to an N=2superconformalfield theory with spacetime central charge[14-17,12]c=6kQ1.(2.12) If the string theory in AdS3is weakly coupled,the spacetime CFT has a large central charge,proportional to1/g2s(2.9).In this semiclassical regime,the radius of curvature of AdS3in Planck units is large,R AdS≫l p,(1.2).Thefirst question we would like to address concerns the normalizability of the ground state of the spacetime CFT.From perturbative studies of string theory in AdS3it is known [15]that the spacetime Virasoro algebra takes the formT(x)T(y)≃3kI(x−y)2+∂y Tk2 d2zJ(x;z)¯J(¯x;¯z)Φ1(x,¯x;z,¯z),(2.14) in the notations of[15].I is an operator which commutes with the spacetime Virasoro generators.In particular,it has spacetime scaling dimension zero.Acting with it on the vacuum of the worldsheet theory creates a state proportional to the SL(2,The state(2.15)is a descendant of a principal discrete series state with j=0,|m|=1. Such states are normalizable iffthe unitarity condition[18,19]−12(2.16) is satisfied.For k>1,j=0is in the range(2.16),while for k<1,it is not.This means that the SL(2,R2AdS ;8l p J=2r+r−4For recent reviews see e.g.[22,23].the ground state I|0 is normalizable,while for k<1it is not,due to(2.16).Thus,in the latter case the BTZ black hole is not in the spectrum of the theory.To recapitulate,wefind that string theory in AdS3has two phases.In one,corre-sponding to R AdS>l s(k>1,(1.6)),the SL(2,C)invariant vacuum is non-normalizable,c eff<c,but there is no contradiction with the Bekenstein-Hawking analysis,since BTZ black holes are non-normalizable in this regime.It remains to calculate c efffor k<1,and to identify the states in string theory on AdS3that give rise to the Cardy entropy(1.3)in this case.This is the problem we turn to next.As mentioned in the introduction,one might expect that when the curvature exceeds the string scale and BTZ black holes cease to exist as normalizable states,the high energy spectrum should be dominated by weakly coupled highly excited perturbative string states. To see whether this is indeed the case,we need to recall a few facts about the perturbative string spectrum in AdS3.Low lying states in the background(2.5)belong to principal discrete series representa-tions.5The worldsheet mass-shell condition for these states relates the quadratic Casimir of SL(2,I R)to the excitation level N[14]:−j(j+1)2.(2.19) The scaling dimension of the corresponding Virasoro primary in the spacetime CFT ish=j+1.(2.20)The unitarity condition(2.16)implies that there is an upper bound on the excitation level N for which(2.19)can be solved.For higher excitation levels and spacetime scaling dimensions,one needs to consider long string states,which wind around the circle near the boundary of AdS3[19].In a sector with given winding w,the worldsheet mass-shell condition becomes−j(j+1)4w2+N=15Principal continuous series states correspond to“bad”tachyons,which are absent in the supersymmetric theories discussed here.and the spacetime scaling dimension iskh=|m++ip;p∈I R is the momentum of the long2string in the radial direction of AdS3.There are also physical states of the form(2.21) with j real in the range(2.16),which correspond to strings that do not make it all the way to the boundary[19].As afirst step towards examining the dynamics of long strings,consider a single string with winding number w=1.It can be thought of as extended in the I R1,1in(2.1).The remaining,transverse,directions of space,I Rφ×S1×M,parametrize the target space of the low energy CFT living on the string.The central charge of that CFT is c l=6k[14,15]. The scalarfieldφparametrizing the location of the string in the radial direction of AdS3 is described near the boundary by an asymptotically linear dilaton CFT,withΦ=−Q l2+3Q2l +32;substituting in(2.24)wefindk2Q2l=highly excited perturbative strings in AdS3interact strongly and are unstable near the boundary(like widely separated quarks and gluons in QCD).In particular,consider a state with winding number w consisting of w highly excited long strings near the boundary.One expects large interactions among the strings to lead to the formation of a bound state,whose properties can be very different from those of free strings.A natural candidate for such a state is a large BTZ black hole,with the same energy and angular momentum as the original configuration of strings.In this regime (k>1),the perturbative description in terms of highly excited strings satisfying(2.21)is not useful.A better description of the highly excited state is as a BTZ black hole,which is a non-perturbative state from the point of view of string theory in AdS3.For k<1,the situation is reversed.The strongly coupled bound state,the BTZ black hole,is no longer normalizable.Correspondingly,the interactions among highly excited perturbative strings are now small near the boundary,since the gradient of the dilaton on the long strings(2.23)has the opposite sign.Therefore,the generic high energy state with winding number w consists of w weakly interacting long strings.The higher the energy, the smaller the interactions among the strings.Note that it is crucial for the consistency of the picture that the linear dilaton on long strings(2.23)change sign precisely at the same value of k at which the BTZ black hole ceases to be normalizable.The fact that this is indeed the case supports the overall picture.Since for k<1the high energy states consist of weakly interacting strings,it should be possible to compute the high energy entropy,and in particular c eff(1.3),perturbatively.A quick way to get the answer is to note that if on a single long string we have the CFT I Rφ×S1×M,then on w weakly interacting strings we expect tofind the symmetricproduct CFTI Rφ×S1×M w/S w.(2.26) The fact that the interactions among the strings go to zero as one approaches the boundary of AdS3should translate in the spacetime CFT to the statement that(2.26)describes accurately the high energy(large L0,¯L0)behavior of the theory of w strings.It may be modified significantly at low energies.To calculate the high energy density of states of(2.26)one simply adds the effective central charges of the component CFT’s in(2.26):c eff=w 32+c M =6w(2−1The winding number w is bounded from above by Q1,the total number of strings used to make the background(2.5).Thus,the high energy density of states is1c eff=6Q1(2−constructions(2.5),one is restricted to k≥22,at which c effvanishes.Another way to arrive at(2.27)is the following.We saw before that in the sector with winding w,the worldsheet mass-shell condition takes the form(2.21).The spacetime scaling dimension(2.22)can thus be written as6h=kww −j(j+1)2.(2.29)The bulk of the high energy density of states comes from the entropy associated with the excitation level,N.To estimate it,we can assume that the term proportional to N dominates the r.h.s.of(2.29),so thath≃Nk).Using the fact that the spacetime scaling dimension h is related to N via(2.30),we see that the perturbative string entropy,2πw +kw .(2.31)This is precisely the type of expression onefinds in the w-twisted sector of a symmetric orbifold,whose building block has c=6k[26,27,25].We see that a number of seemingly independent things happen at k=1:(1)The BTZ black holes,which are normalizable for k>1,cease to be so for k<1.Thesame is true for the SL(2,)(2.28).kWe argued above that these phenomena are in fact related.They are all manifestations of a sharp transition,which occurs at k=1,between a strongly coupled phase where the high energy states are large black holes,and a weakly coupled phase in which they are weakly interacting fundamental strings.The matching of the entropies suggests that the dynamics of large BTZ black holes changes smoothly into that of highly excited fundamental strings as we vary k.Wefinish this section with a few comments.So far,we discussed type II string theory in AdS3.One could ask whether the matching that we found of black hole and string entropies extends to other cases,such as the heterotic string.Heterotic strings in AdS3 were studied in[28],where it was shown that in this case the spacetime CFT inherits the chiral nature of the worldsheet one.While the left moving8central charge is still given by (2.12),c=6Q1k,the right moving one is equal to¯c=6Q1(k+2).In the black hole phase this means that the Cardy formula is(1.3),with c eff=c and¯c eff=¯c.In the perturbative string phase,the left-movers still satisfy(2.21),(2.22),(2.29), while for the right-movers one has the mass-shell condition−j(j+1)4w2+¯N=1.(2.32) For large excitation level¯N one has,as in(2.30),¯h≃¯N/w.To calculate the effective worldsheet central charge of the right-movers recall that they are described by the world-sheet CFTAdS3×S1×M×Γ,(2.33)whereΓis a CFT with central charge13,which completes the total central charge of(2.33) to26:3+6k.Multiplying by Q1we conclude that the analog of(2.28)for the right-movers in the heterotic string is¯c eff=6Q1(4−1k)(2.28).According to(1.5),this means that∆min=Q1k=Q1Q2lThus,it exhibits non-trivial vacuum structure.In particular,the torus partition sum is given by a sum of the partition sums of the individual N=2minimal models,Z= j Z n j.In our case,it is possible that the spacetime CFT for k<1splits into sectors labeled by winding numbers(w1,w2,···),with j w j=Q1.In a given sector,the spacetime CFT is a direct sum of the individual theories corresponding to the different w j.Each of those has central charge c(w j)(=6w j k),and c(w j)(=6w j(2−1eff24 c(w)−c(w)eff =w k=w Q2lNote that this behavior is consistent with the general picture described above:for k>1,perturbative strings in AdS3are strongly coupled at high energies,and the strength of their interactions can be modified by changing the moduli of the spacetime CFT.For k<1,perturbative strings are weakly coupled at high energies and thus no marginal or relevant perturbations can make their interactions strong there.3.Linear dilatonIn the previous section we have seen that the low energy physics in linear dilaton backgrounds such as(2.1),in the presence of Q1fundamental strings in the throat,is qualitatively different for k<1and k>1.This was demonstrated for Q1≫1,but weexpect it to be true for all Q1.It is natural to ask whether something similar happens for Q1=0,i.e.in the absence of strings in the throat.In this case,the theory under consideration is the Little String Theory(LST)of the throat[30,12],and one would like to know whether the nature of the states which dominate its high energy behavior changesas k passes through the value k=1,or as the gradient of the dilaton Q passes through √Q=−ge−2Φ(R+4gµν∂µΦ∂νΦ−4Λ),(3.1) where,as before,Λis related to the level of SL(2,I R),k,by(1.1),(1.6).The equations of motion of(3.1)have the solution9[33,34]ds2=f−1dφ2−fdt2,f=1−2M9Here and below we often setα′=2.gradient of the dilaton Q is related to the level k via(2.8).Note that,as mentioned in the introduction,the mass determines the strength of quantum corrections near the horizon, but theα′effects are insensitive to it.The Bekenstein-Hawking entropy of these black holes is obtained in the usual way by Wick rotating(3.2)to Euclidean space and calculating the circumference of the Euclidean time direction near the boundary atφ=∞.After Wick rotation,the metric(3.2)can be written as[35,33]ds2=kl2s dρ2+tanh2ρdτ2(3.3)Φ=Φ0−log coshρ,whereτis periodic,with period2π.The corresponding entropy is√S bh=2πl s M2Mk,and one may expect a transition to a string phase atkk∼1.In fact,it turns out that the curvature is not the most useful guide for estimating the size of string corrections.One way to see that is to consider the generalization of(3.5) to charged black holes.We will see below that in that case,as one varies the mass to charge ratio,the curvature at the outer horizon changes continuously and even vanishes for a particular ratio.This does not necessarily mean that string corrections are small there!Indeed,the metric is not the onlyfield that is excited in the black hole geometry.In the uncharged case(3.2)we also have a non-trivial dilaton,and for charged black holes a gaugefield as well.We are looking for a measure of the size of theα′corrections which takes all thesefields into account.It is natural to propose that the relevant quantity is the cosmological constantΛin (3.1).This parameter determines both the gradient of the dilaton,and the curvature of the metric,and it plays the same role here as in the AdS3discussion of section2.Furthermore, while the curvature(3.5)depends on position,the cosmological constant does not,so we do not have to face the issue where in the black hole spacetime to evaluate it.A nice feature of the above proposal is that the black hole(3.2)is a coset of SL(2,I R), while the BTZ black hole discussed in section2is an orbifold of the same space.Therefore, it is natural to expect that theα′corrections should be determined by the underlying SL(2,I R)group manifold,and should therefore only depend on k,(1.1),(1.6),(2.8).It also makes it easy to generalize to other examples related to SL(2,I R),such as the charged black holes that will be discussed below.Another question that one might ask at this point is the following.Whileα′corrections are expected to grow when Q increases(or k decreases),it is known[36]that the metric and dilaton(3.2),(3.3),which are obtained by studying the lowest order inα′action(3.1), give rise to a solution of the classical string equations of motion10,to all orders in Q.In what sense areα′corrections growing11as one increases Q?The answer to this question is that for large Q,or small k,the sigma model description breaks down due to non-perturbative effects in Q or1/k.For example,to describe the Euclidean black hole(3.3)in the bosonic string,one needs to turn on,in addition to the metric and dilaton,certain winding modes of the tachyon.This wasfirst shown in unpublished work by V.Fateev,A.Zamolodchikov and Al.Zamolodchikov,and is reviewed in[38];see also[39]for a recent discussion.In the fermionic string,which is the case relevant here,one has to turn on winding modes of the fermionic string tachyon[18].The leading one is the worldsheet superpotentiald2ϑe−1k(τL−τR)+···.(3.7)The worldsheet superpotential(3.6)is non-perturbative in Q and becomes more and more important as Q grows.Its effects are visible in the structure of correlation functions in the Euclidean black hole geometry,as discussed in[29].An interesting question is what is the analog of the superpotential(3.6)for the Minkowski SL(2,I R)black hole withU(1)the region behind the singularity[40,37].One can try to think of the analytic continuation of the winding mode(3.6)as afinite energy excitation living“behind the singularity at r=0”in the extended black hole spacetime.An observer living far outside the horizon of such a black hole would not see thisfield turned on directly,but would presumably feel its effects on the physics.Now that we have established that as k decreases,α′effects grow,and pointed out the close analogy to the SL(2,I R)case of section2,we can turn to a discussion of the string/black hole transition.The most important effect of the superpotential(3.6)is that it implies that for k<1,the Euclidean SL(2,I R)>Q12−)Q1.The same comments as thereapply here.In particular,S bh≥S pert;the two are equal at the string/black hole transition point k=1.As in AdS3,the transition from black holes to strings at k=1occurs not because the string entropy becomes larger than the black hole one,but rather because the black holes cease to be normalizable.A natural question is why the highly excited fundamental strings change their behavior at k=1.In analogy to the AdS3discussion,we expect that for k>1,as the energy of perturbative strings grows,they should become more strongly coupled and form bound states,the SL(2,I R)black holes.For k<1these black holes become non-normalizable, U(1)and the entropy,while still Hagedorn(for k>1=¯L0−c24M2=Q1 L0−c4string phase,the LST makes a similar transition from the SL (2,I R)/U (1)black hole phase to the perturbative string one.In the perturbative string phase,the DLCQ prescription (3.9)withdiscretizationparameter w ,and the AdS 3result (2.29),give rise to the spectrumα′24 =−j (j +1)2.(3.10)It is interesting that (3.10)is the same as the spectrum of perturbative strings living in the linear dilaton throat,with the usual relation between the momentum in the φdirection and j ,β=Qj (see appendix A).Thus,we see that for k <1,DLCQ maps weakly coupled highly excited fundamental strings in AdS 3to weakly coupled strings in the linear dilaton throat.Similarly,for k >1,when the strings are strongly coupled,DLCQ maps BTZ black holes in AdS 3to SL (2,I R)−ge −2Φ R +4g µν∂µΦ∂νΦ−1r√r+q 2M 2−q 2.(3.14)For q=0,the solution(3.12)–(3.14)reduces to the uncharged one(3.2).The curvature of the metric(3.12)isR(r)=2r−4q2k2Mr+−4q24F2=4Λ=−2U(1),where U(1)R is the right-moving part of the S1mentioned above. The U(1)symmetry that is being gauged acts on SL(2,I R)×U(1)R as(g,x R)≃(h L gh R,x R+β),(3.18) where(g,x R)∈SL(2,I R)×U(1)R andh L=eασ3,h R=eαcos(ψ)σ3,β=αsin(ψ),(3.19) with tan2(ψ/2)=r−/r+.The entropy of the black hole is given in terms of its mass and charge by[44]S bh=π√M2−q2 .(3.20)The description in terms of a coset of SL(2,I R)suggests that the black hole exists as a normalizable state only for k>1,while for k<1the generic high energy states with the same charge correspond to highly excited perturbative strings with(p L,p R)=(0,q).The entropy of such strings isS pert=2π k √¯N ;(3.21) the left and right moving excitation numbers(N,¯N)are given in terms of M and q by the on-shell condition12M2=2N=2¯N+q2.(3.22) Substituting(3.22)in(3.21)givesS pert=π k M+4−2M2−q2L+12Up to additive corrections of order one.The corresponding black holes can be described as follows.Start with three dimensional dilaton gravity coupled to the Neveu-Schwarz B-field,S= d3X√12H2−4Λ ,(3.25) where the three form H=dB is thefield strength of B.This action has a black string solution,2q R q LG xx=1+(M2−q2R)(M2−q2L)−q R q L,2q L M2−q2LG tx=−(M2−q2R)(M2−q2L) M2+ 2G tt,q L M2−q2LB tx=(M2−q2R)(M2−q2L) M2+ 2G tt.(3.26) Here,x is taken to be compact,M is the ADM mass,and(q L,q R)are related to the angular momentum,q G,and the axion charge per unit length,q B,byq G=q L+q R,q B=q L−q R.(3.27) The black string has a singularity hidden behind inner and outer horizons,all of which are extended in x.The singularity is located at r=0and the horizons are atr±=M± M2−q2R−q L q R.(g,x L,x R)≃(h L gh R,x L+βL,x R+βR),(3.29)。
