Semiclassical and quantum polarons in crystaline acetanilide

The Properties and Uses of QuantumDotsQuantum dots are small semiconductor crystals that have unique optical and electronic properties. They are nanoscale in size and can range from 2 to 20 nanometersin diameter. Quantum dots absorb and emit light at specific wavelengths, depending on their size and composition. Because of their size, they have quantum mechanical properties that differ from those of bulk materials. Quantum dots have diverse applications in areas such as biomedical imaging, solar energy, display technology, and solid-state lighting.Quantum dots have unique optical properties that make them useful in biomedical imaging. They can absorb light and emit it at a higher frequency, which makes them ideal for use in fluorescent imaging. They are also small enough to penetrate tissues and cells, allowing for high-resolution imaging of cellular structures. Moreover, quantum dots are photostable, meaning they can withstand repeated exposure to light without deteriorating, which makes them ideal for long-term imaging applications.Quantum dots have also been used in solar energy applications. Their photovoltaic properties make them ideal for use in solar cells. When exposed to sunlight, they can absorb photons and convert them into electrical energy. Because of their unique size-dependent properties, they can be tuned to absorb different wavelengths of light, making them more efficient at converting sunlight into electricity. Moreover, they are lightweight, flexible, and can be deposited onto a variety of materials, making them a promising material for use in flexible solar cells.Quantum dots also have applications in display technology. Because of their unique optical properties, they can produce highly efficient and bright colors, making them ideal for use in displays. They are also small enough to be used in ultra-thin displays, making them a promising material for use in flexible and foldable displays. Moreover, they havea longer lifespan than traditional display materials, making them ideal for use in electronic devices.Finally, quantum dots are also used in solid-state lighting applications. They can produce light at specific wavelengths, making them ideal for use in lighting applications that require high color quality and efficiency. They are also small enough to be used in LED technology, making them a promising material for use in energy-efficient lighting. Moreover, they have a longer lifespan than traditional lighting materials, making them a cost-effective and environmentally friendly option for lighting technology.In conclusion, quantum dots have unique optical and electronic properties that make them useful in a variety of applications. They can be tuned to absorb and emit light at specific wavelengths, making them ideal for use in biomedical imaging, solar energy, display technology, and solid-state lighting. Moreover, they are lightweight, flexible, and have a long lifespan, making them a promising material for the development of new technologies. With continued research and development, quantum dots have the potential to revolutionize these and other areas of technology.。

The mysteries of the atom QuantummechanicsQuantum mechanics, the branch of physics that deals with the behavior of particles at the atomic and subatomic levels, is a field filled with mysteries and complexities that continue to baffle and intrigue scientists and laypeople alike. At its core, quantum mechanics seeks to understand the fundamental building blocks of the universe and how they interact with each other. From the wave-particle duality of particles to the uncertainty principle, the concepts and principles of quantum mechanics challenge our conventional understanding of reality and push the boundaries of what we thought was possible. One of the most perplexing aspects of quantum mechanics is the concept of superposition, where particles can exist in multiple states at the same time until they are observed or measured. This idea, famously illustrated by Schr?dinger's thought experiment with the cat in a box, challenges our everyday experience of the world, where objects are clearly defined and exist in specific states. The notion that particles can exist in a state of uncertainty until they are observed raises profound questions about the nature of reality and the role of consciousness in shaping it. Another enigma of quantum mechanics is entanglement, where particles become linked in such a way that the state of one particle instantaneously influences the state of another, regardless of the distance between them. This phenomenon, which Albert Einstein famously referred to as "spooky action at a distance," defies our classical understanding of cause and effect and suggests a deeper interconnectedness between particlesthat transcends traditional notions of space and time. The implications of entanglement are far-reaching, with potential applications in quantum computing, cryptography, and communication. Furthermore, the uncertainty principle, formulated by Werner Heisenberg, asserts that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. This fundamental limit to our knowledge of the behavior of particles at the quantum level challenges our intuition and raises profound questions about the nature of reality and the limits of human understanding. The uncertainty principle has profound implications for our ability to predict and control the behavior ofparticles at the quantum level, with potential applications in fields such as nanotechnology and quantum engineering. The mysteries of quantum mechanics also extend to the behavior of particles themselves, which often exhibit wave-like properties in addition to their particle-like behavior. This wave-particle duality, first proposed by Louis de Broglie, suggests that particles such as electrons and photons can exhibit both wave-like and particle-like behavior, depending on how they are observed or measured. This duality challenges our classical intuitions about the nature of matter and energy and has profound implications for our understanding of the fundamental building blocks of the universe. In conclusion, the mysteries of quantum mechanics continue to fascinate and confound scientists and laypeople alike, challenging our understanding of the fundamental nature of reality and pushing the boundaries of what we thought was possible. From the concept of superposition and entanglement to the uncertainty principle and wave-particle duality, the principles and phenomena of quantum mechanics defy our classical intuitions and invite us to reexamine our understanding of the universe. As we continue to unravel the mysteries of the atom and delve deeper into the quantum realm, we are confronted with profound questions about the nature ofreality and our place within it, inspiring awe and wonder at the intricate and enigmatic nature of the quantum world.。

PERSPECTIVES National Science Review2:9–15,2015PHYSICSMacroscopic mechanical systems are entering the quantum world Yong-Chun Liu and Yun-Feng Xiao∗In the classic Chinese novel Journey to the West,the Monkey King has a special power that enables him to be in two or more places simultaneously.Normally, an amazing thing such as this could never happen in real life.However,in the quantum world,where the motions of objects obey the rules of quantum me-chanics,this could happen.These weird quantum rules usually only apply in the microscopic world to atoms and molecules,because quantum properties are generally very fragile in the macro-scopic world.However,scientists have recently been making efforts to push macroscopic and mesoscopic mechanical systems into the quantum world,i.e.they are putting conventional mechanical systems into the quantum mechanical regime.Spring oscillators,swings,and pendu-lum clocks are typical mechanical sys-tems that we often see in daily lives, and they can be described by a basic physical model:the harmonic oscilla-tor.In the quantum regime,harmonic oscillators have some quantum proper-ties,including discrete energy spectra and zero-point motion.To observe these quantum properties,the environmental thermal noise must be suppressed so that the energy of the quantum motion domi-nates over the energy of the thermal mo-tion.To this end,a generic model of a cavity optomechanical system is used, as shown in Fig.1.The system consists of an optical cavity with a fixed mirror and a movable mirror,where the lat-ter is attached to a spring and under-goes harmonic oscillation.A laser beam is launched into the cavity and is reflected multiple times between the two mirrors,and the cavity field thus builds up,re-sulting in a greatly enhanced optical field,which then exerts a considerable force onthe movable mirror.The interaction be-tween the optical field and the mechan-ical motion can reduce the thermal mo-tion noise to enable the fragile quantumproperties to be observed.The study of such systems has pro-duced an emerging field called cavityoptomechanics[1–3].The pioneeringwork was conducted by Braginsky andco-workers in the1960s[4].In recentyears,various experimental systemshave been proposed and investigated,including Fabry–P´e rot cavities[bygroups including MIT/Caltech LIGOLaboratory,the Mavalvala group at MIT,the Aspelmeyer group at the Universityof Vienna,the Heidmann group at theUniversity of Pierre and Marie Curie,and the Bouwmeester group at theUniversity of California,Santa Barbara(UCSB)],whispering-gallery cavities(by groups including the Vahala group atCaltech,the Kippenberg group at´EcolePolytechnique F´e d´e rale de Lausanne,the Wang group at the University ofOregon,the Lipson group at CornellUniversity,the Carmon group at theUniversity of Michigan,the BowenL a s e r in p u tFixed mirror Movable mirrorFigure1.Illustration of a generic cavity op-tomechanical system.The left mirror is fixedand the right mirror is attached to a spring.group at the University of Queensland,and the Xiao group at Peking Univer-sity),photonic crystal cavities(by groupsincluding the Painter group at Caltech,the Wong group at Columbia Univer-sity,and the Tang group at Yale Uni-versity),separated mechanical oscillatorsinside or near a cavity(by groups in-cluding the Harris group at Yale Univer-sity,the Regal group at Joint Institutefor Laboratory Astrophysics(JILA),theFavero group at Paris Diderot Univer-sity,and the Weig group at the Univer-sity of Munich),superconducting circuits(by groups including the Schwab groupat Caltech,the Lehnert group at JILA,theCleland group at UCSB,and the Teufelgroup at National Institute of Science andTechnology,Boulder),and cold atoms(by groups including the Kimble group atCaltech,the Stamper-Kurn group at theUniversity of California,Berkeley,andthe Treutlein group at the University ofBasel).Cavity optomechanics has beenfeatured as the most recent milestone inphoton history in Nature.The researchin cavity optomechanics is importantto both fundamental physics studiesand the applied sciences.First,cav-ity optomechanics provides a uniqueplatform for the study of fundamentalquantum physics,including macroscopicquantum phenomena,decoherence,and quantum–classical transitions.Thefield offers the best test bed to studyeffects like gravity-induced decoherence,which is important to the understandingof macroscopic quantum phenomena.Second,cavity optomechanics is promis-ing for high-precision measurements ofsmall forces,masses,displacements,andC The Author(s)2014.Published by Oxford University Press on behalf of China Science Publishing&Media Ltd.All rights reserved.For Permissions,please email:journals. permissions@10Natl Sci Rev ,2015,Vol.2,No.1PERSPECTIVESYear (a)(b)N u m b e r o f p u b l i c a t i o n sN u m b e r o f c i t a t i o n s200200200200200201201201201Year200200200200200201201201201Figure 2.Numbers of publications (a)and citations (b)per year in the period from 2005to 2013with the keyword ‘cavity optomechanics’,retrieved from the Web of Science.accelerations,and is considered to be capable of surpassing the standard quan-tum limit.Third,cavity optomechanics provides resources for both classical and quantum information processing.For instance,optomechanical devices can serve as information storage devices and act as interfaces between light beams with different wavelengths or even microwaves.In the last few years,researchers have made considerable efforts to put mechanical systems into their quan-tum ground states [5].Recent efforts have demonstrated optomechanically induced transparency [6],normal-mode splitting [7],quantum-coherent coupling [8],wavelength conversion [9],and measurements performed below the standard quantum limit [10].Future de-velopments will aim to integrate different quantum systems to form hybrid quan-tum devices,e.g.hybrid optomechanical and electromechanical systems.In this way,we can enable phonons,photons,and electrons to work together in the quantum world.Recent years have seen rapidly grow-ing interest in the field of cavity optome-chanics.As shown in Fig.2,the publica-tions and citations in this field have grown exponentially.The now booming devel-opment in this field will turn the dream of manipulating macroscopic mechani-cal systems in a quantum manner into reality.Yong-Chun Liu and Yun-Feng Xiao ∗School of Physics,Peking University,China;Collaborative Innovation Center of Quantum Matter,China∗Corresponding author.E-mail:ycliu@REFERENCES1.Kippenberg,TJ and Vahala,KJ.Science 2008;321:1172–6.2.Aspelmeyer,M,Kippenberg,TJ and Marquardt,F.2013;arXiv:1303.0733.3.Liu,YC,Hu,YW and Wong,CW et al.Chin Phys B 2013;22:114213.4.Braginsky,VB and Manukin,AB.Sov Phys-JETP 1967;25:653–5.5.Chan,J,Alegre,TPM and Safavi-Naeini,AH et al.Nature 2011;478:89–92.6.Weis,S,Rivi`e re,R and Del´e glise,S et al.Science 2010;330:1520–3.7.Gr¨o blacher,S,Hammerer,K and Vanner,MR et al.Nature 2009;460:724–7.8.Verhagen,E,Del´e glise,S and Weis,S et al.Nature 2012;482:63–7.9.Dong,C,Fiore,V and Kuzyk,MC et al.Science 2012;338:1609–13.10.Gavartin,E,Verlot,P and Kippenberg,TJ.Nat Nanotechnol 2012;7:509–14.doi:10.1093/nsr/nwu050Advance access publication 21August 2014IMMUNOLOGYTargeting the immune system:a new horizon of cancer therapiesChen DongINTRODUCTIONThe immune system is our trustworthy army of defense against invasion of mi-croorganisms.As the first line in this army,the innate immune system,com-posed of myeloid cells (dendritic cells and macrophages),some lymphocytes (NK cells and innate lymphocytes as well as some types of T lymphocytes such as NKT cells and γδT cells)and other cells in our body,can quickly mountinflammatory responses after the spe-cific receptors in these cells recognize the pathogen-associated pattern molecules inside or outside host cells.The adap-tive immune system,consisting of B and T lymphocytes,is slower in their acti-vation,but it is more specific in their antigenic recognition and long-lasting,owing to the generation of memory lym-phocytes.Two major types of T lym-phocytes that carry αβT cell receptorshave different functions—those express the CD4molecule secrete cytokines to regulate immune function and are called helper T cells,while those express CD8co-receptor are called cytotoxic T lym-phocytes (CTL)and can directly kill cells infected by viruses.The immune system has an intimate relationship with cancer.Immune cells,such as myeloid cells and lymphocytes,are frequently found in the tumor。

Chapter6Many-Particle Systemsc 2010by Harvey Gould and Jan Tobochnik8March2010We apply the general formalism of statistical mechanics to systems of many particles and discuss the semiclassical limit of the partition function,the equipartition theorem for classical systems,and the general applicability of the Maxwell velocity distribution.We then consider noninteracting quantum systems and discuss the single particle density of states,the Fermi-Dirac and Bose-Einstein distribution functions,the thermodynamics of ideal Fermi and Bose gases,blackbody radiation,and the specific heat of crystalline solids among other applications.6.1The Ideal Gas in the Semiclassical LimitWefirst apply the canonical ensemble to an ideal gas in the semiclassical limit.Because the thermodynamic properties of a system are independent of the choice of ensemble,we willfind the same thermal and pressure equations of state as we found in Section4.5.Although we will not obtain any new results,this application will give us more experience in working with the canonical ensemble and again show the subtle nature of the semiclassical limit.In Section6.6we will derive the classical equations of state using the grand canonical ensemble without any ad hoc assumptions.In Sections4.4and4.5we derived the thermodynamic properties of the ideal classical gas1 using the microcanonical ensemble.If the gas is in thermal equilibrium with a heat bath at temperature T,it is more natural and convenient to treat the ideal gas in the canonical ensemble. Because the particles are not localized,they cannot be distinguished from each other as were the harmonic oscillators considered in Example4.3and the spins in Chapter5.Hence,we cannot simply focus our attention on one particular particle.The approach we will take here is to treat the particles as distinguishable,and then correct for the error approximately.As before,we will consider a system of noninteracting particles starting from their fundamental description according to quantum mechanics.If the temperature is sufficiently high,we expectCHAPTER6.MANY-PARTICLE SYSTEMS293 that we can treat the particles classically.To do so we cannot simply take the limit →0 wherever it appears because the counting of microstates is different in quantum mechanics and classical mechanics.That is,particles of the same type are indistinguishable according to quantum mechanics.So in the following we will consider the semiclassical limit,and the particles will remain indistinguishable even in the limit of high temperatures.To take the semiclassical limit the mean de Broglie wavelengthλ,thefirst condition will always be satisfied.As shown in Problem6.1,the mean distance between particles in three dimensions isρ−1/3.Hence,the semiclassical limit requires thatλ3≪1(semiclassical limit).(6.1) Problem6.1.Mean distance between particles(a)Consider a system of N particles confined to a line of length L.What is the definition ofthe particle densityρ?The mean distance between particles is L/N.How does this distance depend onρ?(b)Consider a system of N particles confined to a square of linear dimension L.How does themean distance between particles depend onρ?(c)Use similar considerations to determine the density dependence of the mean distance betweenparticles in three dimensions.To estimate the magnitude ofp2/2m= 3kT/2.(We will rederive this result more generally in Section6.2.1.)Henceλ∼h/ p2∼h/√2πmkT 1/2= 2π 22πwill allow us to express the partition function in a convenientform[see(6.11)].The calculation of the partition function of an ideal gas in the semiclassical limit proceeds as follows.First,we assume thatλ≪ρ−1/3so that we can pick out one particle if we make the additional assumption that the particles are distinguishable.(Ifλ∼ρ−1/3,the wavefunctions of the particles overlap.)Because identical particles are intrinsically indistinguishable,we will have to correct for the latter assumption later.With these considerations in mind we now calculate Z1,the partition function for one particle, in the semiclassical limit.As we found in(4.40),the energy eigenvalues of a particle in a cube of side L are given byh2ǫn=CHAPTER 6.MANY-PARTICLE SYSTEMS 294where the subscript n represents the set of quantum numbers n x ,n y ,and n z ,each of which can be any nonzero,positive integer.The corresponding partition function is given byZ 1= n e −βǫn =∞ n x =1∞ n y =1∞ n z =1e −βh 2(n x 2+n y 2+n z 2)/8mL 2.(6.4)Because each sum is independent of the others,we can rewrite(6.4)asZ 1=∞n x =1e −α2n x 2 ∞ n y =1e −αn y 2 ∞ n z =1e −αn z 2 =S 3,(6.5)whereS =∞ n x =1e −α2n x 2.(6.6)andα2=βh 24λ2α ∞0e −u 2du −1=L2πmβh 23/2.(6.10)The result (6.10)is the partition function associated with the translational motion of one particle in a box.Note that Z 1can be conveniently expressed asZ 1=V 2ln β+3h 2.(6.12)CHAPTER6.MANY-PARTICLE SYSTEMS295 The mean pressure due to one particle is given byβ∂ln Z1βV=kTe=−∂ln Z12β=3P=NkTE=3∂T V=3he−βp2/2m.(6.18)The integral over p in(6.18)extends from−∞to+∞.The entropy of an ideal classical gas of N particles.Although it is straightforward to calculate the mean energy and pressure of an ideal classical gas by considering the partition function for one particle,the calculation of the entropy is more subtle.To understand the difficulty,consider the calculation of the partition function of an ideal gas of two particles.Because there are noCHAPTER6.MANY-PARTICLE SYSTEMS296microstate s blue1ǫaǫb2ǫb3ǫcǫaǫa+ǫb5ǫaǫaǫa+ǫc7ǫaǫbǫb+ǫc9ǫbCHAPTER6.MANY-PARTICLE SYSTEMS297 than there are particles(see Problem4.14,page190).(In our simple example,each particle can be in one of only three microstates,and the number of microstates is comparable to the number of particles.)If we assume that the particles are indistinguishable and that microstates with multiple occupancy can be ignored,then Z2is given byZ2=e−β(ǫa+ǫb)+e−β(ǫa+ǫc)+e−β(ǫb+ǫc)(indistinguishable,no multiple occupancy).(6.23)We see that if we ignore multiple occupancy there are three microstates for indistinguishable particles and six microstates for distinguishable particles.Hence,in the semiclassical limit we can write Z2=Z21/2!where the factor of2!corrects for overcounting.For three particles(each of which can be in one of three possible microstates)and no multiple occupancy,there would be one microstate of the system for indistinguishable particles and no multiple occupancy,namely, the microstate a,b,c.However,there would be six such microstates for distinguishable particles. Thus if we count microstates assuming that the particles are distinguishable,we would overcount the number of microstates by N!,the number of permutations of N particles.We conclude that if we begin with the fundamental quantum mechanical description of matter, then identical particles are indistinguishable at all temperatures.If we make the assumption that single particle microstates with multiple occupancy can be ignored,we can express the partition function of N noninteracting identical particles asZ N=Z1NN! 2πmkTN +3h2 +1 .(6.26)In Section6.6we will use the grand canonical ensemble to obtain the entropy of an ideal classical gas without any ad hoc assumptions such as assuming that the particles are distinguish-able and then correcting for overcounting by including the factor of N!.That is,in the grand canonical ensemble we will be able to automatically satisfy the condition that the particles are indistinguishable.Problem6.4.Equations of state of an ideal classical gasUse the result(6.26)tofind the pressure equation of state and the mean energy of an ideal gas.Do the equations of state depend on whether the particles are indistinguishable or distinguishable? Problem6.5.Entropy of an ideal classical gasCHAPTER6.MANY-PARTICLE SYSTEMS298 (a)The entropy can be found from the relations F=E−T S or S=−∂F/∂T.Show thatS(T,V,N)=Nk ln V2ln 2πmkT2 .(6.27)The form of S in(6.27)is known as the Sackur-Tetrode equation(see Problem4.20,page197).Is this form of S applicable for low temperatures?(b)Express kT in terms of E and show that S(E,V,N)can be expressed asS(E,V,N)=Nk ln V2ln 4πmE2 ,(6.28) in agreement with the result(4.63)found using the microcanonical ensemble.Problem6.6.The chemical potential of an ideal classical gas(a)Use the relationµ=∂F/∂N and the result(6.26)to show that the chemical potential of anideal classical gas is given byµ=−kT ln V h2 3/2 .(6.29)(b)We will see in Chapter7that if two systems are placed into contact with different initialchemical potentials,particles will go from the system with higher chemical potential to the system with lower chemical potential.(This behavior is analogous to energy going from high to low temperatures.)Does“high”chemical potential for an ideal classical gas imply“high”or“low”density?(c)Calculate the entropy and chemical potential of one mole of helium gas at standard temperatureand pressure.Take V=2.24×10−2m3,N=6.02×1023,m=6.65×10−27kg,and T= 273K.Problem6.7.Entropy as an extensive quantity(a)Because the entropy is an extensive quantity,we know that if we double the volume and doublethe number of particles(thus keeping the density constant),the entropy must double.This condition can be written formally asS(T,λV,λN)=λS(T,V,N).(6.30) Although this behavior of the entropy is completely general,there is no guarantee that an approximate calculation of S will satisfy this condition.Show that the Sackur-Tetrode form of the entropy of an ideal gas of identical particles,(6.27),satisfies this general condition. (b)Show that if the N!term were absent from(6.25)for Z N,S would be given byS=Nk ln V+3h2 +3CHAPTER6.MANY-PARTICLE SYSTEMS299(a)(b)Figure6.1:(a)A composite system is prepared such that there are N argon atoms in container A and N argon atoms in container B.The two containers are at the same temperature T and have the same volume V.What is the change of the entropy of the composite system if the partition separating the two containers is removed and the two gases are allowed to mix?(b)A composite system is prepared such that there are N argon atoms in container A and N helium atoms in container B.The other conditions are the same as before.The change in the entropy when the partition is removed is equal to2Nk ln2.(c)The fact that(6.31)yields an entropy that is not extensive does not indicate that identicalparticles must be indistinguishable.Instead the problem arises from our identification of S with ln Z as mentioned in Section4.6,page199.Recall that we considered a system withfixed N and made the identification that[see(4.106)]dS/k=d(ln Z+βE).(6.32) It is straightforward to integrate(6.32)and obtainS=k(ln Z+βE)+g(N),(6.33) where g(N)is an arbitrary function only of N.Although we usually set g(N)=0,it is important to remember that g(N)is arbitrary.What must be the form of g(N)in order that the entropy of an ideal classical gas be extensive?Entropy of mixing.Consider two containers A and B each of volume V with two identical gases of N argon atoms each at the same temperature T.What is the change of the entropy of the combined system if we remove the partition separating the two containers and allow the two gases to mix[see Figure6.1)(a)]?Because the argon atoms are identical,nothing has really changed and no information has been lost.Hence,∆S=0.In contrast,suppose that one container is composed of N argon atoms and the other is composed of N helium atoms[see Figure6.1)(b)].What is the change of the entropy of theCHAPTER6.MANY-PARTICLE SYSTEMS300 combined system if we remove the partition separating them and allow the two gases to mix? Because argon atoms are distinguishable from helium atoms,we lose information about the system, and therefore we know that the entropy must increase.Alternatively,we know that the entropy must increase because removing the partition between the two containers is an irreversible process. (Reinserting the partition would not separate the two gases.)We conclude that the entropy of mixing is nonzero:∆S>0(entropy of mixing).(6.34) In the following,we will derive these results for the special case of an ideal classical gas.Consider two ideal gases at the same temperature T with N A and N B particles in containers of volume V A and V B,respectively.The gases are initially separated by a partition.We use(6.27) for the entropy andfindS A=N A k ln V AN B+f(T,m B) ,(6.35b)where the function f(T,m)=3/2ln(2πmkT/h2)+5/2,and m A and m B are the particle masses in system A and system B,respectively.We then allow the particles to mix so that theyfill the entire volume V=V A+V B.If the particles are identical and have mass m,the total entropy after the removal of the partition is given byS=k(N A+N B) ln V A+V BN A+N B−N A ln V AN B (identical gases).(6.37)Problem6.8.Entropy of mixing of identical particles(a)Use(6.37)to show that∆S=0if the two gases have equal densities before separation.WriteN A=ρV A and N B=ρV B.(b)Why is the entropy of mixing nonzero if the two gases initially have different densities eventhough the particles are identical?If the two gases are not identical,the total entropy after mixing isS=k N A ln V A+V B N B+N A f(T,m A)+N B f(T,m B) .(6.38) Then the entropy of mixing becomes∆S=k N A ln V A+V B N B−N A ln V A N B .(6.39) For the special case of N A=N B=N and V A=V B=V,wefind∆S=2Nk ln2.(6.40)CHAPTER6.MANY-PARTICLE SYSTEMS301 Problem6.9.More on the entropy of mixing(a)Explain the result(6.40)for nonidentical particles in simple terms.(b)Consider the special case N A=N B=N and V A=V B=V and show that if we use the result(6.31)instead of(6.27),the entropy of mixing for identical particles is nonzero.This incorrectresult is known as Gibbs paradox.Does it imply that classical physics,which assumes that particles of the same type are distinguishable,is incorrect?6.2Classical Statistical MechanicsFrom our discussions of the ideal gas in the semiclassical limit we found that the approach to the classical limit must be made with care.Planck’s constant appears in the expression for the entropy even for the simple case of an ideal gas,and the indistinguishability of the particles is not a classical concept.If we work entirely within the framework of classical mechanics,we would replace the sum over microstates in the partition function by an integral over phase space,that is,Z N,classical=C N e−βE(r1,...,r N,p1,...,p N)d r1...d r N d p1...d p N.(6.41)The constant C N cannot be determined from classical mechanics.From our counting of microstates for a single particle and the harmonic oscillator in Section4.3and the arguments for including the factor of1/N!on page295we see that we can obtain results consistent with starting from quantum mechanics if we choose the constant C N to be1C N=N! e−βE(r1,...,r N,p1,...,p N)d r1...d r N d p1...d p NCHAPTER6.MANY-PARTICLE SYSTEMS302 For a classical system in equilibrium with a heat bath at temperature T,the meanvalue of each contribution to the total energy that is quadratic in a coordinate equals1f= f(r1,...,r N,p1,...,p N)e−βE(r1,...,r N,p1,...,p N)d r1...d r N d p1...d p NCHAPTER6.MANY-PARTICLE SYSTEMS303whereǫ1=ap21with a equal to a constant.We have separated out the quadratic dependence of the energy of particle one on its momentum.We use(6.45)and express the mean value ofǫ1as∞−∞e−βE(x1,x2,p1,p2)dx1dx2dp1dp2(6.47a)= ∞−∞ǫ1e−β[ǫ1+˜E(x1,x2,p2)]dx1dx2dp1dp2∞−∞e−βǫ1dp1 e−β˜E dx1dx2dp2.(6.47c) The integrals over all the coordinates except p1cancel,and we have∞−∞e−βǫ1dp1.(6.48) As we have done in other contexts[see(4.84),page202]we can writeǫ1=−∂ǫ1=−∂2kT.(6.51)Equation(6.51)is an example of the equipartition theorem of classical statistical mechanics.The equipartition theorem is applicable only when the system can be described classically, and is applicable only to each term in the energy that is proportional to a coordinate squared. This coordinate must take on a continuum of values from−∞to+∞.Applications of the equipartition theorem.A system of particles in three dimensions has 3N quadratic contributions to the kinetic energy,three for each particle.From the equipartition theorem,we know that the mean kinetic energy is3NkT/2,independent of the nature of the interactions,if any,between the particles.Hence,the heat capacity at constant volume of an ideal classical monatomic gas is given by C V=3Nk/2as we have found previously.Another application of the equipartition function is to the one-dimensional harmonic oscillator in the classical limit.In this case there are two quadratic contributions to the total energy andCHAPTER6.MANY-PARTICLE SYSTEMS304 hence the mean energy of a one-dimensional classical harmonic oscillator in equilibrium with a heat bath at temperature T is kT.In the harmonic model of a crystal each atom feels a harmonic or spring-like force due to its neighboring atoms(see Section6.9.1).The N atoms independently perform simple harmonic oscillations about their equilibrium positions.Each atom contributes three quadratic terms to the kinetic energy and three quadratic terms to the potential energy. Hence,in the high temperature limit the energy of a crystal of N atoms is E=6NkT/2,and the heat capacity at constant volume isC V=3Nk(law of Dulong and Petit).(6.52)The result(6.52)is known as the law of Dulong and Petit.This result wasfirst discovered empiri-cally and is valid only at sufficiently high temperatures.At low temperatures a quantum treatment is necessary and the independence of C V on T breaks down.The heat capacity of an insulating solid at low temperatures is discussed in Section6.9.2.We next consider an ideal gas consisting of diatomic molecules(see Figure6.5on page345). Its pressure equation of state is still given by P V=NkT,because the pressure depends only on the translational motion of the center of mass of each molecule.However,its heat capacity differs from that of a ideal monatomic gas because a diatomic molecule has additional energy associated with its vibrational and rotational motion.Hence,we expect that C V for an ideal diatomic gas to be greater than C V for an ideal monatomic gas.The temperature dependence of the heat capacity of an ideal diatomic gas is explored in Problem6.47.6.2.2The Maxwell velocity distributionSo far we have used the tools of statistical mechanics to calculate macroscopic quantities of in-terest in thermodynamics such as the pressure,the temperature,and the heat capacity.We now apply statistical mechanics arguments to gain more detailed information about classical systems of particles by calculating the velocity distribution of the particles.Consider a classical system of particles in equilibrium with a heat bath at temperature T.We know that the total energy can be written as the sum of two parts:the kinetic energy K(p1,...,p N) and the potential energy U(r1,...,r N).The kinetic energy is a quadratic function of the momenta p1,...,p N(or velocities),and the potential energy is a function of the positions r1,...,r N of the particles.The total energy is E=K+U.The probability density of a microstate of N particles defined by r1,...,r N,p1,...,p N is given in the canonical ensemble byp(r1,...,r N;p1,...,p N)=A e−[K(p1,p2,...,p N)+U(r1,r2,...,r N)]/kT(6.53a)=A e−K(p1,p2,...,p N)/kT e−U(r1,r2,...,r N)/kT,(6.53b)where A is a normalization constant.The probability density p is a product of two factors,one that depends only on the particle positions and the other that depends only on the particle momenta. This factorization implies that the probabilities of the momenta and positions are independent. The probability of the positions of the particles can be written asf(r1,...,r N)d r1...d r N=B e−U(r1,...,r N)/kT d r1...d r N,(6.54)and the probability of the momenta is given byf(p1,...,p N)d p1...d p N=C e−K(p1,...,p N)/kT d p1...d p N.(6.55)CHAPTER6.MANY-PARTICLE SYSTEMS305 For notational simplicity,we have denoted the two probability densities by f,even though their meaning is different in(6.54)and(6.55).The constants B and C in(6.54)and(6.55)can be found by requiring that each probability be normalized.We stress that the probability distribution for the momenta does not depend on the nature of the interaction between the particles and is the same for all classical systems at the same temper-ature.This statement might seem surprising because it might seem that the velocity distribution should depend on the density of the system.An external potential also does not affect the velocity distribution.These statements do not hold for quantum systems,because in this case the position and momentum operators do not commute.That is,e−β(ˆK+ˆU)=e−βˆK e−βˆU for quantum systems, where we have used carets to denote operators in quantum mechanics.Because the total kinetic energy is a sum of the kinetic energy of each of the particles,the probability density f(p1,...,p N)is a product of terms that each depend on the momenta of only one particle.This factorization implies that the momentum probabilities of the various particles are independent.These considerations imply that we can write the probability that a particle has momentum p in the range d p asf(p x,p y,p z)dp x dp y dp z=c e−(p2x+p2y+p2z)/2mkT dp x dp y dp z.(6.56) The constant c is given by the normalization conditionc ∞−∞ ∞−∞ ∞−∞e−(p2x+p2y+p2z)/2mkT dp x dp y dp z=c ∞−∞e−p2/2mkT dp 3=1.(6.57)If we use the fact that ∞−∞e−αx2dx=(π/α)1/2(see the Appendix),wefind that c=(2πmkT)−3/2. Hence the momentum probability distribution can be expressed as1f(p x,p y,p z)dp x dp y dp z=2πkT 3/2e−m(v2x+v2y+v2z)/2kT dv x dv y dv z.(6.59) Equation(6.59)is known as the Maxwell velocity distribution.Note that its form is a Gaussian. The probability distribution for the speed is discussed in Section6.2.3.Because f(v x,v y,v z)is a product of three independent factors,the probability of the velocity of a particle in a particular direction is independent of the velocity in any other direction.For example,the probability that a particle has a velocity in the x-direction in the range v x to v x+dv x isf(v x)dv x= mCHAPTER6.MANY-PARTICLE SYSTEMS306Problem6.10.Is there an upper limit to the velocity?The upper limit to the velocity of a particle is the velocity of light.Yet the Maxwell velocity distribution imposes no upper limit to the velocity.Does this contradiction lead to difficulties? Problem6.11.Simulations of the Maxwell velocity distribution(a)Program LJ2DFluidMD simulates a system of particles interacting via the Lennard-Jones poten-tial(1.1)in two dimensions by solving Newton’s equations of motion numerically.The program computes the distribution of velocities in the x-direction among other pare the form of the velocity distribution to the form of the Maxwell velocity distribution in(6.60).How does its width depend on the temperature?(b)Program IdealThermometerIdealGas implements the demon algorithm for an ideal classicalgas in one dimension(see Section4.9).All the particles have the same initial velocity.The program computes the distribution of velocities among other quantities.What is the form of the velocity distribution?Give an argument based on the central limit theorem(see Section3.7) to explain why the distribution has the observed form.Is this form consistent with(6.60)? 6.2.3The Maxwell speed distributionWe have found that the distribution of velocities in a classical system of particles is a Gaussian and is given by(6.59).To determine the distribution of speeds for a three-dimensional system we need to know the number of microstates between v and v+∆v.This number is proportional to the volume of a spherical shell of width∆v or4π(v+∆v)3/3−4πv3/3→4πv2∆v in the limit ∆v→0.Hence,the probability that a particle has a speed between v and v+dv is given byf(v)dv=4πAv2e−mv2/2kT dv,(6.61) where A is a normalization constant,which we calculate in Problem6.12.Problem6.12.Maxwell speed distribution(a)Compare the form of the Maxwell speed distribution(6.61)with the form of the Maxwellvelocity distribution(6.59).(b)Use the normalization condition ∞0f(v)dv=1to calculate A and show thatf(v)dv=4πv2 m v,the most probable speed˜v,and the root-mean-square speed v rmsand discuss their relative magnitudes.(d)Make the change of variables u=v/ π)u2e−u2du,(6.63)where we have again used the same notation for two different,but physically related probability densities.The(dimensionless)speed probability density f(u)is shown in Figure6.2.CHAPTER 6.MANY-PARTICLE SYSTEMS 3070.00.20.40.60.81.00.00.5 1.0 1.5 2.0 2.5 3.0u max uu rmsu f(u)Figure 6.2:The probability density f (u )=4/√u ≈1.13,and theroot-mean-square speed u rms ≈1.22.The dimensionless speed u is defined by u ≡v/(2kT/m )1/2.Problem 6.13.Maxwell speed distribution in one or two dimensionsFind the Maxwell speed distribution for particles restricted to one and two dimensions.6.3Occupation Numbers and Bose and Fermi StatisticsWe now develop the formalism for calculating the thermodynamic properties of ideal gases for which quantum effects are important.We have already noted that the absence of interactions between the particles of an ideal gas enables us to reduce the problem of determining the energy levels of the gas as a whole to determining ǫk ,the energy levels of a single particle.Because the particles are indistinguishable,we cannot specify the microstate of each particle.Instead a microstate of an ideal gas is specified by the occupation number n k ,the number of particles in the single particle state k with energy ǫk .2If we know the value of the occupation number for each single particle microstate,we can write the total energy of the system in microstate s asE s = kn k ǫk .(6.64)The set of n k completely specifies a microstate of the system.The partition function for an ideal gas can be expressed in terms of the occupation numbers asZ (V,T,N )= {n k }e −βP k n k ǫk ,(6.65)CHAPTER6.MANY-PARTICLE SYSTEMS308 where the occupation numbers n k satisfy the conditionN= k n k.(6.66)The condition(6.66)is difficult to satisfy in practice,and we will later use the grand canonical ensemble for which the condition of afixed number of particles is relaxed.As discussed in Section4.3.6,one of the fundamental results of relativistic quantum mechanics is that all particles can be classified into two groups.Particles with zero or integral spin such as4He are bosons and have wavefunctions that are symmetric under the exchange of any pair of particles. Particles with half-integral spin such as electrons,protons,and neutrons are fermions and have wavefunctions that are antisymmetric under particle exchange.The Bose or Fermi character of composite objects can be found by noting that composite objects that have an even number of fermions are bosons and those containing an odd number of fermions are themselves fermions.For example,an atom of3He is composed of an odd number of particles:two electrons,two protons, and one neutron each of spin13In spite of its fundamental importance,it is only a slight exaggeration to say that“everyone knows the spin-statistics theorem,but no one understands it.”See Duck and Sudarshan(1998).CHAPTER6.MANY-PARTICLE SYSTEMS309n1n301011010Table6.2:The possible states of a three-particle fermion system with four single particle energy microstates(see Example6.1).The quantity n1represents the number of particles in the single particle microstate labeled1,etc.Note that we have not specified which particle is in a particular microstate.From Table6.2we see that the partition function is given byZ3=e−β(ǫ2+ǫ3+ǫ4)+e−β(ǫ1+ǫ3+ǫ4)+e−β(ǫ1+ǫ2+ǫ4)+e−β(ǫ1+ǫ2+ǫ3).(6.69)♦Problem6.14.Calculate。

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。

山西省长治二中等五校2024学年全国卷Ⅲ英语试题高考模拟题注意事项1．考生要认真填写考场号和座位序号。

2．试题所有答案必须填涂或书写在答题卡上，在试卷上作答无效。

第一部分必须用2B 铅笔作答；第二部分必须用黑色字迹的签字笔作答。

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第一部分（共20小题，每小题1.5分，满分30分）1．Teachers in primary schools ______ influence the kids fall under should be role models.A．whose B．whoC．where D．which2．My friend warned me ______ going to the East Coast because it was crowded with tourists.A．by B．against C．on D．for3．I ________ my cellphone last night. Now the battery is running out.A．could have charged B．might chargeC．should have charged D．would charge4．In contrast with the liberal social climate of the present, traditions in the past were relatively ______. A．competitive B．comprehensiveC．creative D．conservative5．I had been betrayed by those who I trusted several times, ______ in a suspicious attitude towards everything and everyone.A．resulted B．having resulted C．resulting D．to result6．The educational reform is now under way throughout the country, ________ the students more opportunities to develop to their greatest potential.A．to grant B．having grantedC．granting D．granted7．When the admission letter from Harvard University arrived, Ben's parents were and threw a big party.A．in the red B．tickled pinkC．as white as a sheet D．in a blue mood8．Only when _________hard __________ make your dream come true.A．do you work; you can B．you work; you canC．do you work; can you D．you work; can you9．Join us and you will discover an environment ______ you can make the most of your skills and talents.A．that B．whereC．how D．what10．Instead of making choices for their children, liberal parents usually say, “Go where you ________ .”A．will B．shouldC．can D．must11．he newly-discovered star was named _____ a Chinese astronomer ________his contributions to astronomy. A．for; in favor of B．after; in honor ofC．by; in memory of D．as; in praise of12．Much to my ______, my vocabulary has expanded a great deal.A．delighting B．delighted C．delight13．— I want to learn tennis. Would you like to help me?—. But learning tennis is no walk in the park.A．No kidding B．No wonder C．No problem D．No way14．I really don’t know _________ she gets by on such a modest salary.A．what B．whyC．how D．that15．My mom once worked in a very small village school, which is__________only on foot.A．acceptable B．adequate C．accessible D．appropriate16．Don’t leave the water ______while you brush your teeth.A．racing B．rushingC．running D．rolling17．Eggs, meat, vegetables and other foods can easily be poisoned by microorganisms such as ______. A．phenomena B．dilemma C．diploma D．bacteria18．Mankind must first of all eat, drink, have shelter and clothing ________ it can pursue politics, science, art and religion.A．until B．unlessC．before D．since19．Mark has lived in China for many years, yet he still can't ________ himself to the Chinese customs. A．observe B．adaptC．lead D．devote20．—What did you say you were reluctant to risk just now?—_________ to high levels of radiation.A．Being exposed B．Having been exposedC．To be exposed D．Exposed第二部分阅读理解（满分40分）阅读下列短文，从每题所给的A、B、C、D四个选项中，选出最佳选项。

a r X i v :0708.3017v 1 [h e p -p h ] 22 A u g 2007Black Holes and Quantum Gravity at the LHCPatrick Meade and Lisa RandallJefferson Physical LaboratoryHarvard UniversityCambridge,MA 02138,USAmeade@,randall@AbstractWe argue that the highly studied black hole signatures based on thermal multiparticle final states are very unlikely and only occur in a very limited parameter regime if at all.However,we show that if the higher-dimensional quantum gravity scale is low,it should be possible to study quantum gravity in the context of higher dimensions through detailed compositeness-type searches.1IntroductionOne of the most exciting possibilities for the LHC is the discovery of small higher-dimensional [1]black holes that could be formed when two sufficiently energetic particles collide[2,3,4,5]. Ideally,such black holes would decay isotropically to many energetic particles,in keeping with the prediction of thermal Hawking radiation[6].However,over most of the viable parameter space,this expectation is not very realistic.Once inelasticity and black hole entropy are accounted for,it is clear that multiparticlefinal states are very suppressed,since only black holes produced well above threshold have sufficient entropy.The falling parton distribution functions(PDFs)more than compensate for the rise in black hole production with energy so most strong gravity events will occur at the lowest possible energy scale.Nonetheless,all is not lost.Even when the energy is too low to produce truly thermal black holes,which require sufficiently high entropy and energy,we would nevertheless expect signs of quantum gravity if higher dimensional gravity gets strong at a scale not too far above a TeV.Strong gravity is likely to result in more sphericalfinal states,even for thosefinal states with low multiplicity,which would therefore be measured as much more transverse than background.As we will show,over most regions of expected parameter space for higher dimensional models,we expect a significant change in the rate of highly transverse two particlefinal states to occur at the quantum gravity scale,both jet-like and leptonic, although the latter rate which is smaller spans a smaller region of parameter space.Strong gravity should be testable through standard compositeness tests.In fact,the threshold for a rise in the2→2scattering cross section is almost inevitably lower than the black hole production threshold.Though not necessarily a true thermal black hole,thesefinal states,if they occur,will nonetheless tell us about quantum gravity.In fact, in the thermal regime,black holes wouldn’t give us any insight into quantum gravity(except to confirm existing theoretical predictions).In the region at or below the true thermal black hole threshold,assuming strong gravity effects don’t turn on or offsuddenly at the black hole scale,we could in principle learn a lot by studying the two particlefinal states,in particular the angular distribution and the energy dependence of the angular distribution which would truly be quantum gravity results,not interpretable in terms of a classical calculation.Furthermore we will see that there is sufficient information to distinguish not only black hole type effects,but different forms of string amplitudes.This can in principle probe the effects of curvature or non-string objects in the theory as well.Moreover,we don’t expect only strong gravity effects if higher dimensional theories are right.We should in that casefind indications of KKfinal states at lower energy.In that case there would be indications whether composite-type effects might be associated with quantum gravity to help us disentangle it from other strongly interacting physics.In what follows,we will see other possible distinctive features of gravitational physics that might help distinguish among possibilities.Thus what we are saying is that even existing compositeness searches don’t just tell about strong gauge dynamics-they could in principle tell us about gravity as well.We show how we can hope to learn about black hole production and quantum gravity by studying the energy dependence of the high p T dijet or leptonic cross section.We consider the implications of a rise or fall in the cross section and what the energy dependence might teach us about quantum gravity.1We stress that although the two particlefinal state signal is unlikely to probe thermal black holes in the accessible energy range,it is of great interest as a way of probing quantum gravity.The rate as a function of energy as well as the angular distribution can differ significantly in various scenarios of quantum gravity.Furthermore in almost any scenario we expect the two particlefinal state to demonstrate effects of quantum gravity well before the proposed multiparticlefinal states characteristic of thermal black holes.Furthermore whereas we know the predictions for the semiclassical regime,independent of the particular theory of quantum gravity,the threshold regime can potentially distinguish among them.Others have considered the effects of specific gravitational effects on higher-dimensional operators and how they can be constrained by existing searches.Ref.[7]considered a dimension-8operator,Ref.[8]considered graviton loops generating a dimension-6operator, Ref.[9,10]considered string-generated dimension-8operators and string resonances,Ref.[11]considered dimension-6operators from string theory.Our point is to view compositeness searches more generally and to learn how to distinguish among the possibilities rather than to constrain the scale of any one particular model.Furthermore we emphasize that the gap between the quantum gravity scale and the true black hole threshold should be a good source of deviations in2→2scattering and probably yields a much better reach and more insight than multiparticle searches.2Black Hole Production and DecayThe large black hole cross section estimate stems from the classical cross section that is proportional to the geometrical area set by the Schwarzschild radius r S:σ(E)∼πr S(E)2.(2.1) This geometrical cross section impliesσ(E)∼1M α(2.2)where M is the effective scale of quantum gravity andα≤1for higher-dimensional black holes.Thus for instance at the LHC one might expect a parton-parton cross section of size at least∼1hole signatures,but used more optimistic assumptions for parameter space than are now experimentally allowed and neglected the inelasticity that we will soon discuss.Ref.[13]considered black holes that might arise in warped five-dimensional space in the context of cosmic ray searches.For further referencessee Appendix A.We will see in Appendix A that in the energy range between ˜M and (M/k )2˜M ,where M is the five dimensional Planckscale(˜M is M reduced by a warp factor)and k is related to the AdS curvature,we expect to a good approximation conventional five-dimensional black holes.Of course,in the RS case where approximately flat space black holes occur only over a limited energy range,we would need M/k large enough to permit high entropy black holes.2.1Criteria for Black HolesThe production cross section in (2.1)depends only on the mass scales involved and thus appears to be a very simple quantity to understand.Unfortunately however there are am-biguities associated with both of the two scales in the problem,M and M BH .Since one makes rough estimates assuming black holes start forming at a scale M ,and due to the falling PDFs (1)the rate changes dramatically depending on the scale at which black holes start to form,it is critical to keep track of the different conventions for the Planck scale and the relationships among them so that we can unambiguously compare rate predictions.See Figure 1to see the different relative contributions to 2→2scattering from the pdfs.These will be helpful in understanding results throughout the paper.0123451510501005001000qqbpqqpqqbqgggqqFigure 1:Arbitrarily normalized parton-parton luminosity plot as a function of √(1)The effective scaling of the PDFs can be summarized in terms of a parton luminosity.See for instance Figure 69of [14].The drop in the parton luminosity at the LHC depends on the mass range of interest,for instance for qq and√define G D with the Myers-Perry convention[15]1gR(2.3) and define L N as the normalization of the Einstein-Hilbert action for which(2.3)gives1/16πG D.In the case of n extra dimensions,the PDG convention[17]is L N=M n+2D /2(2π)nwhereas the early analysis of Dimopoulos and Landsberg[2]used M n+2P /16π.Although nei-ther analysis was done for case of one extra dimension due to the constraints on n=1ADD type set ups[16],there is a range of mass scales for which approximatefive-dimensional flat space black holes would be the most appropriate description for RS models(see Ap-pendix A).To illustrate the convention dependencies we give their formulae for n=1so as to compare to RS,in which case their formulae reduce to˜M3P/16πand˜M3D/4π,which should be compared to˜M3/2,which is the RS convention,where the tilde indicates the warped version of the various Planck scales.Although just conventions,it is important to bear these conventions in mind when interpreting results.The Schwarzschild radius of the black hole given in[15]for the(4+n)-dimensional case isr S= M BHΓ n+3L N(n+2)2πn+3n+1(2.4)where the scale is understood to be appropriately warped in the RS case(for details see the Appendix),which reduces toM BH˜M33π2 1/2.(2.6) For the case of one extra dimensions,the DL and PDG conventions would giver DLS= 8M BH M3D3π 1/2(2.7)where M P and M D are the higher-dimensional Planck scales in the two cases. Although just a convention,the numerical relationships mean that if we take r S∼1/M as the threshold for black hole production,comparing the two formulations of the Schwarzschild radius in the case of[2]we wouldfind that black holes would be produced at energies ∼M P,while in[18]black holes would be produced at a scale of∼41/3∼1.6M D while the convention would yield(8π)1/3˜M∼2.9˜M.These conventions are clearly significant in interpreting the meaning of the black hole energy reach for the LHC and comparing to experimental constraints.Of course the physical answers are not convention dependent.4When we compare the scales relative to threshold production to the current experimental bounds on KK masses,the convention dependence drops out.The real question is the black hole threshold where black holes start to form.Of course at center of mass energies much greater than the higher-dimensional Planck scale,M,we know black holes will be produced.However,the precise threshold is ambiguous.M is after all convention dependent.Though we will assume E>M is necessary,it is clearly not sufficient.Since we don’t know the precise threshold for a truly thermal black hole,it is useful to define a parameter x min that tells how far above the relevant Planck scale the semiclassical prediction applies[3].This could be defined relative to an arbitrary threshold mass or relative to the convention-dependent Planck scale.We will use the latter with the understanding that x min is unknown either way and is simply a parameter.In our analysis we will give results as a function of M and x min.We consider criteria for x min below.Note that we would want x min for RS to be less than(M/k)2where the curvature becomes relevant as outlined in Appendix A.Keep in mind that in addition to significantly reducing the black hole production cross section,the existence of a nontrivial x min obscures our ability to extract fundamental pa-rameters from the black hole cross section.The overall cross section depends very strongly on x min since as we have already noted,the rapid fall-offof the PDFs makes us very sensitive to the mass threshold where black hole production can begin.This means that any potential bounds from an LHC experiment on black hole production rates is only indirectly related to the fundamental scale of quantum gravity.For instance if onefinds an excess of events attributed to black hole it is unclear how to translate back to the scale M involved if one is only looking on the tail of a distribution.Without knowing more about the threshold behavior of black hole production,the de-pendence of the cross section on the fundamental Planck scale is insufficient to extract that parameter,which can be mimicked by an alternate choice of x min.In principle,the energy-dependence of the cross section can be useful in extracting the number of dimensions(if we know the PDFs sufficiently accurately),although in practice this will be very challenging.In any case,this slope won’t determine the higher-dimensional Planck scale.In principle,the differential cross section can be used to extract the Planck scale since,once it has turned on, the cross section depends on black hole mass(not x min).But without the energy-dependent inelasticity factor(see below)this will be impossible.Furthermore,uncertainties in PDFs and the experimental determination of energy scale will also make this unlikely.2.2ThermalityAlthough difficult to quantify precisely,we now consider several possible criteria for the formation of a truly thermal black hole.Though not sufficient,we expect these to be some minimum necessary criteria that will give some sense of what x min should be.Thefirst criterion one might apply is that the Compton wavelength of the colliding particle of energy E/2lies within the Schwarzschild radius for a black hole of given energy E.If we define the threshold as the point where a wave with wavelength4π/E lies within5the Schwarzschild radius for a black hole of mass E ,we find for ADD n =6black holes this yields x min >4.1(in the M D convention).Had we simply required r S >1/E ,we would have the weaker criterion x min >0.44.In the RS case,we find with the stronger criterion that x >16,whereas with the weaker criterion it should be greater than about 3.We see that this criterion in and of itself if fairly strong,and already will make black hole production very small or nonexistent given LHC parameters.Even so,the above criterion is not necessarily sufficient to guarantee a black hole since we don’t expect the semiclassical formula to apply at the threshold determined above.For a black hole to be truly thermal,we expect higher entropy is required and therefore the threshold will be above the energy we just considered.There are several additional criteria that we would want to be satisfied,all roughly amounting to the fact that the black hole should be sizable enough that the entropy is large.Although for sufficiently large black holes,any criteria of the sort below will be amply satisfied,as we have emphasized,the falling PDFs tell us production is dominated by near-threshold objects.For the criteria below,the following formula will prove useful.For n extra dimensions we have r S =1+n M DM BH 1+n ,(2.8)wherek (n )= 2n πn −32 1+n(2.9)S =1+n T BH(2.10)It is also useful to consider the average number of particles assuming the decay is mostly on the brane [19].The prediction for black hole decays in experiments have been approached in a couple of ways,including treating the decay as instantaneous[2],evolving with mass[18,20],and sometimes including the appropriate grey body factors for the extra-dimensional black holes as well.These distinctions have an order one impact on the average number of particles comparing for instance to an instantaneous decay calculation withN = 2√n +1M BH 2+n1/(n +1)(2.11)compared to one that evolved the black hole with mass and included greybody factorsN =ρS 0=ρ 4πk (n )M D (n +2)/(n +1).(2.12)Allowing for the difference in the definitions of the Planck scales,the instantaneous decay gives particle number a factor of 1.44times that calculated by decaying over time.The mass scaling is in accordance with the mass-dependence of the entropy.6For the specific cases we will be interested we list the average number of particles emittedfor ADD n=6N ∼4πρk(6)MD 8c i f iΦiζ(4)Γ(4)(2.14)which defines a ratio of multiplicities and greybody factors defined in[18].For RS n=1(2) wefindN ∼4ρ3 M BH2(2.15) Notice that N =ρS.In what follows below,we will use the grey-body corrected time-dependent decay esti-mate.Of course,near threshold,all these formulae are unreliable but give an idea of what one might expect.•Preskill et al[21]give the criterion|∂T/∂M|<<1,which is equivalent to the change in Hawking temperature per particle emission should be small.This condition is equiv-alent to the entropy(2.10)being large.More specifically,∂T/∂M∼1/((n+2)S).The improvement of this bound scales as x2+n(2)We approximate the greybody factor for n=1as the same as that for n=67the black hole energy if we are to interpret the decaying object as a higher-dimensional black hole.This is a stricter criterion than above.We find one bulk degree of freedom carries almost all the energy when M BH ∼3˜Min the case of RS,and slightly exceeds it in the case of ADD M BH ∼2M D (n =6).Clearly we would want M BH >M in both cases asthe bound improves as x (2+n )/(1+n )min ,again scaling as the entropy.Of course we should keep in mind this is the criterion for one degree of freedom in the bulk to carry all the mass.Clearly for a thermal black hole,we would want many particles carrying the energy,so the bound would be much stronger.For example,the maximum experimental reach on x min for ADD n =6is about 6,which would corre-spond to only 3bulk particles!For RS,the maximum x min is about 10,corresponding to at most about 5or 6particles sharing the energy,which also seems inadequate for a truly thermal state.•We want the blackhole lifetime to be bigger than 1/M ,so that the black hole appears as a resonance [3].This criterion scales roughly as the number of degrees of freedom modified by grey-body factors.This is borderline for n =6and reasonably well satisfied for n =1.For completeness we give the formula for the lifetime in ADD:τ=(4π)4k (n )2M−2(2+n )1+nBH 2π c i f i Φi ζ(4)Γ(4)(2.17)where the factors in αare defined in [18],and correspond to multiplicities and greybody factors.For the specific case of n =6we findτ=.7x 9/7min ˜M.(2.19)Using these criteria we find that in ADD the criteria is satisfied for x min ∼1.3and in RS for x min ∼1.6.•A sometimes stricter criterion in the case of black holes that can decay on the brane is that the lifetime should exceed the black hole radius,so that the black hole can reequilibrate as the black hole decays primarily along the brane.This requires in the ADD case that x 3while for RS the constraint is satisfied for any x min .802468100.5151050100LT R 3 n 3 NMBH n 3 T MBH LT 0246810110100LT R 3 n 3 N MBH n 3 T MBH LT Figure 2:Possible criteria for x min plotted as a function of x min .ADD with n=6is plotted on the left and RS is plotted on the right.•The black hole’s mass should be large compared to the 3-brane tension.We leave this criterion open since it is highly model-dependent.The strongest criteria are plotted in Figure 2asa functionof x min (with the exception of Schwarschild pton wavelength which would just be a vertical line)where the ratios are chosen such that every curve plotted should be greater than one if the criteria is satisfied.These criteria highlight the uncertainty in defining a precise threshold,and also indicate the blackhole threshold might be well above the putative Planck scale.We stress here that even though the various criteria might be satisfied for x min 3or 4(except for the wavelength criterion),all these criteria are should really be held to being ≫1not just ∼1in which case x min should be much larger in principle.They also show that the values of x min that were used in previous analyses [3,18]might be too low to trust to be in the thermal regime (and of course brings into question those analyses that neglected x min entirely).As we will see however,higher values of x min yield too low a production rate to appear at the LHC.2.3InelasticityIn addition to the thermality criteria above that raise the black hole energy threshold,another critical effect is energy loss of the colliding partons before their energy is trapped behind a black hole horizon.One of the most important effects is to understand exactly how much energy of the initial parton parton system ends up going into the mass of the intermediate black hole.We can define an inelasticity parameter as in [18]y ≡M BH /√(3)There are other effects that modify the cross section,i.e.the maximum impact parameter that can still create a black hole in comparison to the Schwarzschild radius and O (1)factors in front of the putative cross section σ≈πr 2S however for the LHC these effects are not nearly as crucial as the actual mass scale that defines the black hole production.9b r 0M 2µb r 00.10.20.30.40.50.60.7M 2µFigure 3:From Fig.10of [26].The ratio of the mass of the putative black hole compared to the initial energy of the collision is plotted as a function of the impact parameter divided by a unit r 0that approximates the Schwarzschild radius if all the energy of the initial collision were to end up as a black hole.The lowest curve represents the calculation of [26],and previous estimates from [24,25]are also included.impact parameter,the fraction of energy emitted in gravitational waves when colliding to Aichelburg-Sexl shock waves representing two highly boosted massless particles.This work was extended to extra dimensions and non-zero impact parameters by the seminal work of Eardley and Giddings [24]and then further refined by [25,26].In Figure 3we present the relevant results of [26],for the ratio of the mass trapped in the apparent horizon compared to initial energy as a function of the impact parameter for a 10dimensional black hole (hereafter referred to as ADD)and 5dimensional black hole (hereafter referred to as RS)which are relevant for our discussion.As one can see from Figure 3the largest energy fraction entering the black hole for both ADD and RS is O (.6)occurring for zero impact parameter.However they have different functional dependencies with respect to the impact parameter,and the ADD fraction goes down to y ≈0while RS goes to about y ≈.2at the largest possible impact parameters where an apparent horizon still forms.These estimates are interpreted as lower bounds on the inelasticity but we stress that they are also calculated classically and for energies that are approaching the Planck scale it is not obvious how this will be modified.To quantitatively include this inelasticity,we need to include the impact parameter de-pendent effect of inelasticity in calculating the black hole production cross section.Implicitly when calculating the cross section of a proton proton event we have summed over the possible impact parameter already when using the parton parton cross sectionσ(pp →X )= i,jdx 1dx 2f i (x 1)f j (x 2)σ(ij →X ).(2.20)To include the effects of inelasticity we adopt the impact parameter weighted average of the100200040006000800010000101000100000.1. 1071. 1090200040006000800010000101000100000.1. 1071. 109Figure 4:Total black hole cross section in femtobarns,including(solid curves)and not in-cluding(dashed)inelasticity as a function of M D for ADD with n =6and ˜Mfor RS1.The different curves from highest to lowest correspond to x min =1−6.inelasticity used in [18]σ(pp →BH )≡ i,j 102zdz 1(x min M D )2v f i (v,Q )f j (u/v,Q )σi,j →BH (M BH =us ),(2.21)with z =b/b max .The function y (z )is given in our case by the results of [26],as shown in Figure 3.This weighting of the impact parameter obviously shows a difference between the RS and ADD cases,because in 10dimensions the inelasticity parameter is smaller at order one impact parameters,meaning relatively higher energy will be needed to make a black hole.The total black hole cross section with and without inelasticity for both ADD and RS is shown in Figure 4.As demonstrated in Figure 4the inclusion of inelasticity can reduce the total cross section by several orders of magnitude,which is consistent with the results of [18]who used [25]to define their inelasticity.It is interesting to note that these effects are more important for ADD than RS in terms of reduction of total cross section,as it is interesting that the inelasticity is higher for lower dimensions.While the rates presented in Figure 4for the inclusion of inelasticity are taken as a lower bound for the black hole cross section,one should keep in mind that x min lower than the criteria presented in Section 2.1have been plotted and it is unclear what the “effective”inelasticity will be when quantum gravity effects are taken into account.3Black Hole DecaysIn the previous sections we have argued that it is unlikely that the LHC will produce thermal black holes,since the thermality criteria require a black hole threshold above the putative higher-dimensional Planck scale and furthermore energy is lost through initial radiation.In this section we go a step further and argue that even if black holes were produced,they are11rarely if ever in a regime where they will produce the“fireball”explosions consisting of a high multiplicity isotropic distribution of particles that are the most highly emphasized[2,3] black hole signature and possibly even revealing the negative specific heat that characterizes black holes.Since this signature relies on high multiplicity events,it is worth checking over what parameter range one expects tofind high multiplicities.Although not necessarily reliable for low multiplicities,we quantify this consideration by exploring the average number of particles assuming standard classical black holes with a thermal distribution offinal state particles obeying Poisson statistics[18,27]to determine thefluctuation about this mean value.The point is to show the relative merits of low and high multiplicity states.We use as a target “high multiplicity”six or more particles.Although far from afireball,we are trying to allow the most optimistic assumption for a multiparticle state.We compare this reach to two body final states in thefigures below.In Figure5we plot the cross sections with and without inelasticity for both6or more particles(multiparticle)and2particles.To summarize and better demonstrate the relative potential strengths of multiparticle vs.two particlefinal states we plot in Figure6the region in parameter space for the multiparticle and2particlefinal states with a.1fb cross section.We see that the“reach”(4)of two particlefinal states is in all cases at least as good as the multiparticlefinal state.Therefore a study of low multiplicityfinal states might explore black hole-like objects even when x min is not high enough to guarantee a thermalfinal state or a black hole.To be as optimistic as possible,we also checked for the maximum number of particles assuming a.1fb cross section according to a Poisson distribution for a given˜M[M D]where we looked for the maximum number of particles with that cross section.The maximum particle number in the RS case with a.1fb cross section was on the order of20for˜M=500 GeV and is about9for˜M=1TeV,and is only6for˜M of1.5TeV.The maximum particle number in the ADD case for M D=900GeV was about20,for1.4TeV was about14,and for 1.9TeV was about10.Although the later case might sound adequate,it should be kept in mind that this number depends on decays onto the brane.If we asked about the distribution of energy among thermal bulk particles,that is how many bulk particles would we expect for this sized black hole,the answer would be divided by3.And this was for the best possible cases.So the black hole signature is not likely to be an isotropic burst of a large number of particles.Instead we expect low multiplicityfinal states to dominate.Given the relative weakness of the muliparticlefinal states the likely black hole signature will not be an isotropic burst of a large number of particles.Instead we expect low multi-plicityfinal states to dominate.We consider the consequences of this conclusion in the next section.。

有关量子力学的英语作文Quantum mechanics is a branch of physics that deals with the behavior of very small particles, such as atoms and subatomic particles.It's a mind-boggling theory that challenges our understanding of the universe and how things work at the most fundamental level.Quantum mechanics has led to the development of many important technologies, such as lasers, transistors, and MRI machines.One of the most famous principles of quantum mechanics is the Heisenberg Uncertainty Principle, which states that it is impossible to simultaneously know the exact position and momentum of a particle.Quantum mechanics also introduces the concept of superposition, which means that particles can exist inmultiple states at the same time.Another fascinating aspect of quantum mechanics is entanglement, where particles become connected in such away that the state of one particle is instantly correlated with the state of another, no matter how far apart they are.Despite its incredible success in explaining the behavior of particles at the quantum level, quantum mechanics is still not fully understood, and many of its implications are still being explored by physicists.The strange and counterintuitive nature of quantum mechanics has led to many debates and discussions about its philosophical and metaphysical implications.。