A Pathwise Ergodic Theorem for Quantum Trajectories
Abstract algebra, projective geometry and time encoding of quantum information
Michel Planat, Metod Saniga
411
The next important object is the concept of a residue class of a modulo I , which consists of all elements [a] = {a + c, ∀c ∈ I} and is useful to partition the ring into disjoint classes (or cosets). The set of classes has the property to be a ring, called the residue class ring and denoted R/I . For the integers Z modulo the ideal (3), one gets the three classes [0] = 0 + (3), [1] = 1 + (3) and [2] = 2 + (3), and the residue class ring is Z /(3) = F3 , where F3 is the unique field with 3 elements. It is known that for a prime number p, Z/(p) = Zp = Fp , where Zp is the set of integers modulo p and Fp the field with p elements. But, for example, Z/(4) is not a field since 2.2 = 4 = 0 and thus 2 divides 0. 2.2 Polynomial rings, Galois fields and their roduction
量子计算外文翻译中英文2019
量子计算中英文2019英文FROM BITS TO QUBITS, FROM COMPUTING TO QUANTUM COMPUTING: AN EVOLUTION ON THE VERGE OF A REVOLUTION IN THE COMPUTINGLANDSCAPEPi rjan Alexandru; Petroşanu Dana-Mihaela.ABSTRACTThe "Quantum Computing" concept has evolved to a new paradigm in the computing landscape, having the potential to strongly influence the field of computer science and all the fields that make use of information technology. In this paper, we focus first on analysing the special properties of the quantum realm, as a proper hardware implementation of a quantum computing system must take into account these properties. Afterwards, we have analyzed the main hardware components required by a quantum computer, its hardware structure, the most popular technologies for implementing quantum computers, like the trapped ion technology, the one based on superconducting circuits, as well as other emerging technologies. Our study offers important details that should be taken into account in order to complement successfully the classical computer world of bits with the enticing one of qubits.KEYWORDS: Quantum Computing, Qubits, Trapped Ion Technology, Superconducting Quantum Circuits, Superposition, Entanglement, Wave-Particle Duality, Quantum Tunnelling1. INTRODUCTIONThe "Quantum Computing" concept has its roots in the "Quantum Mechanics" physics subdomain that specifies the way how incredibly small particles, up to the subatomic level, behave. Starting from this concept, the Quantum Computing has evolved to a new paradigm in the computing landscape. Initially, the concept was put forward in the 1980s as a mean for enhancing the computing capability required tomodel the way in which quantum physical systems act. Afterwards, in the next decade, the concept has drawn an increased level of interest due to the Shor's algorithm, which, if it had been put into practice using a quantum computing machine, it would have risked decrypting classified data due to the exponential computational speedup potential offered by quantumcomputing [1].However, as the development of the quantum computing machines was infeasible at the time, the whole concept was only of theoretical value. Nowadays, what was once thought to be solely a theoretical concept, evolved to become a reality in which quantum information bits (entitled "qubits") can be stored and manipulated. Both governmental and private companies alike have an increased interest in leveraging the advantages offered by the huge computational speedup potential provided by the quantum computing techniques in contrast to traditional ones [2].One of the aspects that make the development of quantum computers attractive consists in the fact that the shrinkage of silicon transistors at the nanometer scale that has been taking place for more than 50 years according to Moore's law begins to draw to a halt, therefore arising the need for an alternate solution [3].Nevertheless, the most important factor that accounts for boosting the interest in quantum computing is represented by the huge computational power offered by these systems and the fact that their development from both hardware and software perspectives has become a reality. Quantum computing managed to surpass the computability thesis of ChurchTuring, which states that for any computing device, its power computation could increase only in a polynomial manner when compared to a "standard" computer, entitled the Turing machine [4].During the time, hardware companies have designed and launched "classical" computing machines whose processing performance has been improving over the time using two main approaches: firstly, the operations have been accelerated through an increased processing clock frequency and secondly, through an increase in the number of operations performed during each processing clock's cycle [5].Although the computing processing power has increased substantially after having applied the above-mentioned approaches, the overall gain has remained inaccordance with the thesis of Church-Turing. Afterwards, in 1993, Bernstein and Vazirani have published in [6] a theoretical analysis stating that the extended Church-Turing thesis can be surpassed by means of quantum computing. In the following year, Peter Shor has proved in his paper that by means of quantumcomputing the factorization of a large number can be achieved with an exponentially computing speedup when compared to a classical computing machine [7-9]. Astonishing as the theoretical framework was, a viable hardware implementation was still lacking at the time.The first steps for solving this issue have been made in 1995, when scientists have laid the foundations for a technology based on a trapped ion system [10] and afterwards, in 1999, for a technology employing superconducting circuits [11]. Based on the advancement of technology, over the last decades, researchers have obtained huge progress in this field, therefore becoming able to build and employ the first quantum computing systems.While in the case of a classical computing machine the data is stored and processed as bits (having the values 0 or 1), in the case of a quantum computingmachine, the basic unit of quantum information under which the data is stored and processed is represented by the quantum bits, or qubits that can have besides the values of 0 and 1, a combination of both these values in the same time, representing a "superposition" of them [12].At a certain moment in time, the binary values of the n bits corresponding to a classical computer define a certain state for it, while in the case of a quantumcomputer, at a certain moment in time, a number of n qubits have the possibility to define all the classical computer's states, therefore covering an exponential increased computational volume. Nevertheless, in order to achieve this, the qubits must be quantum entangled, a non-local property that makes it possible for several qubits to be correlated at a higher level than it was previously possible in classical computing. In this purpose, in order to be able to entangle two or several qubits, a specific controlled environment and special conditions must be met [13].During the last three decades, a lot of studies have been aiming to advance thestate of knowledge in order to attain the special conditions required to build functional quantum computing systems. Nowadays, besides the most popular technologies employed in the development of quantum computing systems, namely the ones based on trapped ion systems and superconducting circuits, a wide range of other alternative approaches are being extensively tested in complex research projects in order to successfully implement qubits and achieve quantum computing [14].One must take into account the fact that along with the new hardware architectures and implementations of quantum computing systems, new challenges arise from the fact that this new computing landscape necessitates new operations, computing algorithms, specialized software, all of these being different than the ones used in the case of classical computers.A proper hardware implementation of a quantum computing system must take into account the special properties of the quantum realm. Therefore, this paper focuses first on analyzing these characteristics and afterwards on presenting the main hardware components required by a quantum computer, its hardware structure, the most popular technologies for implementing quantum computers, like the trapped ion technology, the one based on superconducting circuits, as well as other emerging technologies. Our developed research offers important details that should be taken into account in order to complement successfully the classical computer world of bits with the enticing one of qubits.2.SPECIAL PROPERTIES OF THE QUANTUM REALMThe huge processing power of quantum computers results from the capacity of quantum bits to take all the binary values simultaneously but harnessing this vast amount of computational potential is a challenging task due to the special properties of the quantum realm. While some of these special properties bring considerable benefits towards quantum computing, there are others that can hinder the whole process.One of the most accurate and extensively tested theory that comprehensibly describes our physical world is quantum mechanics. While this theory offers intuitive explanations for large-scale objects, while still very accurate also at the subatomiclevel, the explanations might seem counterintuitive at the first sight. At the quantum level, an object does not have a certain predefined state, the object can behave like a particle when a measurement is performed upon it and like a wave if left unmeasured, this representing a special quantum property entitled wave-particle duality [15].The global state of a quantum system is determined by the interference of the multitude of states that the objects can simultaneously have at a quantum level, the state being mathematically described through a wave function. Actually, the system's state is often described by the sum of the different possible states of its components, multiplied by a coefficient consisting in a complex number, representing, for each state, its relative weight [16, 17]. For such a complex coefficient, by taking into consideration its trigonometric (polar) form, one can write it under the form Aew = A(cos6 + i sind), where A > 0 represents the module of this complex number and is denoted as the "amplitude", while в represents the argument of the complex number, being denoted as "the phase shift". Therefore, the complex coefficient is known if the two real numbers A and в are known.All the constitutive components of a quantum system have wave-like properties, therefore being considered "coherent". In the case of coherence, the different states of the quantum components interact between them, either in a constructive manner or in a destructive one [1]. If a quantum system is measured at a certain moment, the system exposes only a single component, the probability of this event being equal to the squared absolute value of the corresponding coefficient, multiplied by a constant. If the quantum system is measured, from that moment on it will behave like a classical system, therefore leading to a disruption of its quantum state. This phenomenon causes a loss of information, as the wave function is collapsed, and only a single state remains. As a consequence of the measurement, the wave function associated to the quantum obj ect corresponds only to the measured state [1, 17].Considering a qubit, one can easily demonstrate that its quantum state could be represented by a linear superposition of two vectors, in a space endowed with a scalar product having the dimension 2. The orthonormal basis in this space consists of thevectors denoted as |0 >= [Jj and |1 >= [°j. If one considers two qubits, they could be represented as a linear combination of the 22 elements of the base, namely the ones denoted as .... Generally, in the case of n qubits, they could be represented by a superposition state vector in a space having the dimension 2n [2].Another special property of the quantum realm consists in the entanglement, a property that has the ability to exert a significant influence on quantumcomputing and open up a plethora of novel applications. The physical phenomenon of quantum entanglement takes place when two (or more) quantumobjects are intercorrelated and therefore the state of a quantum object influences instantaneously the state(s) of the other(s) entangled quantum object(s), no matter the distance(s) between these objects [16].Another important quantum mechanical phenomenon that plays a very important role in quantum computing is quantum tunneling that allows a subatomic particle to go through a potential barrier, which otherwise would have been impossible to achieve, if it were to obey only the physical laws of classical mechanics. An explanation of this different behavior consists in the fact that in quantum mechanics the matter is treated both as waves and particles, as we have described above, when we have presented the wave-particle duality concept [15].The Schrödinger equation describes the variation of the wave function, taking into account the energy environment that acts upon a quantum system, therefore highlighting the way in which this quantum system evolves. In order to obtain the mathematical description of the environment, of the energies corresponding to all the forces acting upon the system, one uses the Hamiltonian of the quantum system. Therefore, the control of a quantum system can be achieved by controlling its energy environment, which can be obtained by isolating the system from the external forces, and by subjecting the system to certain energy fields as to induce a specific behavior. One should note that a perfect isolation of the quantum system from the external world cannot be achieved, therefore in practice the interactions are minimized as much as possible. Over time, the quantum system is continuously influenced to a small extent by the external environment, through a process called "decoherence",process that modifies the wave function, therefore collapsing it to a certain degree [1].Figure 1 depicts the main special properties of the quantum realm, which, when precisely controlled, have the ability to influence to a large extent the performance of a quantum computer implementation, and open up new possibilities for innovation concerning the storing, manipulation and processing of data.In the following, we analyze a series of hardware components and existing technologies used for developing and implementing quantum computers.3.AN OVERVIEW OF THE NECESSARY HARDWARE AND OF THE EXISTING TECHNOLOGIES USED IN THE IMPLEMENTATIONS OF QUANTUM COMPUTERSA proper hardware architecture is vital in order to be able to program, manipulate, retrieve qubits and overall to achieve an appropriate and correct quantumcomputer implementation. When implementing a quantum computer at the hardware level, one must take into account the main hardware functions, a proper modularization of the equipment along with both similarities and differences between quantum and classic computer implementations. Conventional computers are an essential part in the successful implementation of a quantum computer, considering the fact that after having performed its computation, a quantumcomputer will have to interact with different categories of users, to store or transmit its results using classic computer networks. In order to be efficient, quantum computers need to precisely control the qubits, this being an aspect that can be properly achieved by making use of classic computing systems.The scientific literature [1, 18, 19] identifies four abstract layers in the conceptual modelling process of quantum computers. The first layer is entitled the "quantum data plane" and it is used for storing the qubits. The second layer, called "control and measurement plane", performs the necessary operations and measurement actions upon the qubits. The third layer entitled "control processor plane" sets up the particular order of operations that need to be performed along with the necessary measurement actions for the algorithms, while the fourth abstract layer, the "host processor", consists in a classical computer that manages the interface withthe different categories of personnel, the storage of data and its transmission over the networks.In the following, we present the two most popular technologies employed in the development of quantum computing systems, namely the ones based on trapped ion systems and superconducting circuits and, afterwards, other alternative approaches that are being extensively tested in complex research projects in order to successfully implement qubits and achieve quantum computing.By means of trapping atomic ions, based on the theoretical concepts presented by Cirac et al within [20], in 1995, Monroe et al [21] revealed the first quantumlogic gate. This was the starting point in implementing the first small scale quantum processing units, making it possible to design and implement a rich variety of basic quantum computing algorithms. However, the challenges to scale up the implementations of quantum computers based on the trapped ion technology are enormous because this process implies a synergy of complex technologies like coherent electronic controllers, laser, radio frequency, vacuum, microwave [1, 22].In the case of a quantum computer based on the trapped atomic ions technology, the qubits are represented by atomic ions contained within the quantum data plane by a mechanism that keeps them in a certain fixed location. The desired operations and measurement actions are performed upon the qubits using accurate lasers or a source of microwave electromagnetic radiation in order to alter the states of the quantum objects, namely the atomic ions. In order to reduce the velocity of the quantum objects and perform measurements upon them, one uses a laser beam, while for assessing the state of the ions one uses photon detectors [14, 23, 24]. Figure 2 depicts an implementation of the quantum trapping atomic ions technology.Another popular technology used in the development and implementation of quantum computers is based on superconducting quantum circuits. These quantum circuits have the property of emitting quantized energy when exposed to temperatures of 10-3K order, being referred in the literature as "superconducting artificial atoms" [25]. In contrast to classic integrated circuits, the superconducting quantum circuits incorporate a distinctive characteristic, namely a"Josephson junction" that uses wires made of superconducting materials in order to achieve a weak connection. The common way of implementing the junction consists in using an insulator that exposes a very thin layer and is created through the Niemeyer-Dolan technique which is a specialized lithographic method that uses thin layers of film in order to achieve overlapping structures having a nanometer size [26].Superconducting quantum circuits technology poses a series of important advantages, offering red3uced decoherence and an improved scale up potential, being compatible with microwaves control circuits, operating with time scales of the nanosecond order [1]. All of these characteristics make the superconducting quantum circuits an attractive and performant technique in developing quantum computers. A superconducting quantum circuit developed by D-Wave Systems Inc. is depicted in Figure 3.In order to overcome the numerous challenges regarding the scaling of quantum computers developed based on trapped ion systems and superconducting circuits, many scientists focus their research activity on developing emerging technologies that leverage different approaches for developing quantumcomputers.One of the alternatives that scientists investigate consists in making use of the photons' properties, especially of the fact that photons have a weak interaction between each other and also with the environment. The photons have been tested in a series of quantum experiments and the obtained results made the researchers remark that the main challenge in developing quantum computers through this approach is to obtain gates that operate on spaces of two qubits, as at the actual moment the photons offer very good results in terms of single qubit gates. In order to obtain the two-qubit gates, two alternative approaches are extensively being investigated as these have provided the most promising results.The first approach is based on operations and measurements of a single photon, therefore creating a strong interaction, useful in implementing a probabilistic gate that operates on a space of two qubits [1]. The second alternative approach employs semiconductor crystals structures of small dimensions in order to interact with the photons. These small structures can be found in nature, case in which they are called"optically active defects", but can also be artificially created, case in which they are called "quantum dots". An important challenge that must be overcome when analyzing quantum computers based on photons is their size. Until now, the development of this type of computers has been possible only for small dimensions, as a series of factors limit the possibility to increase the dimensions of photon quantum computers: the very small wavelengths of the photons (micron-size), their very high speed (the one of the light), the direction of their movement being along a certain dimension of the optical chip. Therefore, trying to significantly increase the number of qubits (represented by the photons) proves to be a difficult task in the case of a photonic device, much more difficult than in the case of other systems, in which the qubits are located in space. Nevertheless, the evolution of this emerging technology promises efficient implementations in the near future [27].Another technology that resembles the one of "trapping atomic ions" for obtaining qubits consists in the use and manipulation of neutral atoms by means of microwave radiation, lasers and optics. Just like in the case of the trapping atomic ions technology, the "cooling" process is achieved using laser sources. According to [1, 28], in 2018 there were implemented successfully quantum systems having 50 qubits that had a reduced space between them. By means of altering the space between the qubits, these quantum systems proved to be a successful analog implementation of quantum computers. In what concerns the error rates, according to [29], in 2018 there have been registered values as low as 3% within two-qubit quantum systems that managed to isolate properly the operations performed by nearby qubits. Since there are many similarities between the two technologies, the scaling up process faces a lot of the problems of the "trapping atomic ions" technology. However, the use of the neutral atoms technology offers the possibility of creating multidimensional arrays.A classification of semiconductor qubits is made according to the method used to manipulate the qubits that can be achieved either by photon manipulation or by using electrical signals. Quantum dots are used in the case of semiconductor qubits that are gated by optical means in order to assure a strong coupling of the photons while in the case of semiconductor qubits manipulated via electrical signals, voltages are usedupon lithographically metal gates for manipulating the qubits [1]. This quantum technology, although being less popular than other alternatives, resembles the existing classical electronic circuits, therefore one might argue that it has a better chance in attracting considerable investments that eventually will help speed up the scaling up process of quantum computers implementation.In order to scale up qubits that are optically gated, one needs a high degree of consistency and has to process every qubit separately at the optical level. In [30], Pla et al. state that even if the qubits that are gated electrically can be very dense, the material related problems posed not long-ago serious quality problems up to single qubits gates level. Although the high density provided by this type of quantum technology creates opportunities for integrating a lot of qubits on a single processor, complex problems arise when one has to manipulate this kind of qubits because the wiring will have to assure an isolation of the control signals as to avoid interference and crosstalk.Another ongoing approach in developing quantum computers consists in using topological qubits within which the operations to be performed upon are safeguarded due to a microscopically incorporated topological symmetry that allows the qubit to correct the errors that may arise during the computing process [1]. If in the future this approach materializes, the computational cost associated with correcting the quantum errors will diminish considerably or even be eliminated altogether. Although this type of technology is still in its early stages, if someday one is able to implement it and prove its technical feasibility, the topological quantum computers will become an important part of the quantum computing landscape.4. CONCLUSIONSQuantum computing represents a field in a continuous evolution and development, a huge challenge in front of researchers and developers, having the potential to influence and revolutionize the development of a wide range of domains like the computing theory, information technology, communications and, in a general framework, regarding from the time perspective, even the evolution and progress of society itself. Therefore, each step of the quantum computers' evolution has thepotential to become of paramount importance for the humanity: from bits to qubits, from computing to quantum computing, an evolution on the verge of a revolution in the computing landscape.中文从比特到量子比特，从计算到量子计算：计算机革命的演变抽象“量子计算”的概念已发展成为计算领域的一个新范例，具有极大地影响计算机科学领域和所有利用信息技术的领域的潜力。
英语作文介绍科学家
When introducing a scientist in an English essay,it is essential to provide a comprehensive overview of their life,achievements,and contributions to their field.Here is a detailed guide on how to write such an essay:1.Introduction:Begin with a captivating introduction that includes the scientists full name and a brief mention of their most significant contributions.For example,Albert Einstein,a theoretical physicist,is best known for his theory of relativity,which revolutionized our understanding of space,time,and gravity.2.Early Life and Education:Describe the scientists early life,including their birthplace, family background,and any early indications of their interest in science.Mention their education,including any notable schools or universities they attended.For instance,Born in Ulm,in the Kingdom of Württemberg in the German Empire,Einstein showed an early aptitude for mathematics and physics.3.Career and Major Discoveries:Detail the scientists professional journey,focusing on their major discoveries or inventions.Explain these contributions in a way that is accessible to readers who may not have a background in the subject.For example, Einsteins1905paper on the photoelectric effect laid the foundation for the development of quantum theory.4.Impact on the Field:Discuss the impact of the scientists work on their field and beyond. Explain how their discoveries have influenced other areas of science or have had practical applications.For example,Einsteins massenergy equivalence formula,Emc², has had profound implications for nuclear energy and weapons development.5.Awards and Recognition:Mention any awards or recognitions the scientist has received for their work.This could include Nobel Prizes,honorary degrees,or other prestigious accolades.For example,In recognition of his contribution to theoretical physics,Einstein was awarded the1921Nobel Prize in Physics.6.Personal Life and Legacy:Provide insights into the scientists personal life,including their beliefs,values,and any significant relationships.Discuss their legacy and how they are remembered today.For example,A pacifist,Einstein was deeply affected by the use of his scientific theories in the development of nuclear weapons,and he became an advocate for disarmament and world peace.7.Conclusion:Conclude the essay by summarizing the scientists most important contributions and reflecting on their enduring influence.For example,Einsteins work continues to inspire scientists and thinkers around the world,and his theories remain atthe forefront of modern physics.8.Citations and References:If you have used specific sources or quotes,be sure to cite them properly to avoid plagiarism and to give credit to the original authors. Remember to maintain a formal and respectful tone throughout the essay,as you are discussing the life and work of a respected figure in the scientific e clear, concise language and avoid overly technical jargon that might confuse readers who are not specialists in the field.。
微积分介值定理的英文
微积分介值定理的英文The Intermediate Value Theorem in CalculusCalculus, a branch of mathematics that has revolutionized the way we understand the world around us, is a vast and intricate subject that encompasses numerous theorems and principles. One such fundamental theorem is the Intermediate Value Theorem, which plays a crucial role in understanding the behavior of continuous functions.The Intermediate Value Theorem, also known as the Bolzano Theorem, states that if a continuous function takes on two different values, then it must also take on all values in between those two values. In other words, if a function is continuous on a closed interval and takes on two different values at the endpoints of that interval, then it must also take on every value in between those two endpoint values.To understand this theorem more clearly, let's consider a simple example. Imagine a function f(x) that represents the height of a mountain as a function of the distance x from the base. If the function f(x) is continuous and the mountain has a peak, then theIntermediate Value Theorem tells us that the function must take on every height value between the base and the peak.Mathematically, the Intermediate Value Theorem can be stated as follows: Let f(x) be a continuous function on a closed interval [a, b]. If f(a) and f(b) have opposite signs, then there exists a point c in the interval (a, b) such that f(c) = 0.The proof of the Intermediate Value Theorem is based on the properties of continuous functions and the completeness of the real number system. The key idea is that if a function changes sign on a closed interval, then it must pass through the value zero somewhere in that interval.One important application of the Intermediate Value Theorem is in the context of finding roots of equations. If a continuous function f(x) changes sign on a closed interval [a, b], then the Intermediate Value Theorem guarantees that there is at least one root (a value of x where f(x) = 0) within that interval. This is a powerful tool in numerical analysis and the study of nonlinear equations.Another application of the Intermediate Value Theorem is in the study of optimization problems. When maximizing or minimizing a continuous function on a closed interval, the Intermediate Value Theorem can be used to establish the existence of a maximum orminimum value within that interval.The Intermediate Value Theorem is also closely related to the concept of connectedness in topology. If a function is continuous on a closed interval, then the image of that interval under the function is a connected set. This means that the function "connects" the values at the endpoints of the interval, without any "gaps" in between.In addition to its theoretical importance, the Intermediate Value Theorem has practical applications in various fields, such as economics, biology, and physics. For example, in economics, the theorem can be used to show the existence of equilibrium prices in a market, where supply and demand curves intersect.In conclusion, the Intermediate Value Theorem is a fundamental result in calculus that has far-reaching implications in both theory and practice. Its ability to guarantee the existence of values between two extremes has made it an indispensable tool in the study of continuous functions and the analysis of complex systems. Understanding and applying this theorem is a crucial step in mastering the powerful concepts of calculus.。
Was Einstein a Space Alien原文及翻译
Was Einstein a Space Alien?1 Albert Einstein was exhausted. For the third night in a row, his baby son Hans, crying, kept the household awake until dawn. When Albert finally dozed off ... it was time to get up and go to work. He couldn't skip a day. He needed the job to support his young family.1. 阿尔伯特.爱因斯坦精疲力竭。
他幼小的儿子汉斯连续三个晚上哭闹不停，弄得全家人直到天亮都无法入睡。
阿尔伯特总算可以打个瞌睡时，已是他起床上班的时候了。
他不能一天不上班，他需要这份工作来养活组建不久的家庭。
2 Walking briskly to the Patent Office, where he was a "Technical Expert, Third Class," Albert worried about his mother. She was getting older and frail, and she didn't approve of his marriage to Mileva. Relations were strained. Albert glanced at a passing shop window. His hair was a mess; he had forgotten to comb it again.2. 阿尔伯特是专利局三等技术专家。
在快步去专利局上班的路上，他为母亲忧心忡忡。
母亲年纪越来越大，身体虚弱。
她不同意儿子与迈尔娃的婚事，婆媳关系紧张。
Empirical processes of dependent random variables
2
Preliminaries
n i=1
from R to R. The centered G -indexed empirical process is given by (P n − P )g = 1 n
n
the marginal and empirical distribution functions. Let G be a class of measurabrocesses that have been discussed include linear processes and Gaussian processes; see Dehling and Taqqu (1989) and Cs¨ org˝ o and Mielniczuk (1996) for long and short-range dependent subordinated Gaussian processes and Ho and Hsing (1996) and Wu (2003a) for long-range dependent linear processes. A collection of recent results is presented in Dehling, Mikosch and Sorensen (2002). In that collection Dedecker and Louhichi (2002) made an important generalization of Ossiander’s (1987) result. Here we investigate the empirical central limit problem for dependent random variables from another angle that avoids strong mixing conditions. In particular, we apply a martingale method and establish a weak convergence theory for stationary, causal processes. Our results are comparable with the theory for independent random variables in that the imposed moment conditions are optimal or almost optimal. We show that, if the process is short-range dependent in a certain sense, then the limiting behavior is similar to that of iid random variables in that the limiting distribution is a Gaussian process and the norming √ sequence is n. For long-range dependent linear processes, one needs to apply asymptotic √ expansions to obtain n-norming limit theorems (Section 6.2.2). The paper is structured as follows. In Section 2 we introduce some mathematical preliminaries necessary for the weak convergence theory and illustrate the essence of our approach. Two types of empirical central limit theorems are established. Empirical processes indexed by indicators of left half lines, absolutely continuous functions, and piecewise differentiable functions are discussed in Sections 3, 4 and 5 respectively. Applications to linear processes and iterated random functions are made in Section 6. Section 7 presents some integral and maximal inequalities that may be of independent interest. Some proofs are given in Sections 8 and 9.
量子力学英文读物
量子力学英文读物以下是一些关于量子力学的英文读物推荐：1. "Quantum: Einstein, Bohr, and the Great Debate about the Nature of Reality" by Manjit Kumar2. "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory" by Brian Greene3. "The Quantum World: Quantum Physics for Everyone" by Kenneth W. Ford4. "Quantum Mechanics: Concepts and Applications" by Nouredine Zettili5. "Quantum Physics: A Beginner's Guide" by Alastair Rae6. "Quantum Computing for Computer Scientists" by Noson S. Yanofsky and Mirco A. Mannucci7. "Quantum Physics for Babies" by Chris Ferrie (a simplified introduction for children and adults)8. "The Strange World of Quantum Mechanics" by Daniel F. Styer这些书籍从不同的角度介绍了量子力学的基本原理、应用、历史以及相关的思想争论。
根据你的兴趣和程度，选择适合你的读物开始探索量子世界吧！。
物理经典书籍教材推介
物理经典书籍教材推介个人以为国内好的物理教材不太多，很多教材内容严谨可是易读性不强。
因此那个书单特意精选了一些适宜自学且架构严谨的国外大学热点教材分享~~最重要的是，这套教材组成一个完整的物理教材体系，都是教得专门深切浅出的专著，专门适合自学提高。
学物理是一件宝贵的乐事，什么缘故不学得乐在其中呢？这些教材在保留了趣味的情形下不失学术水平，因此专门推荐。
以下是依照学习推荐进度排序的《费曼物理学讲义》卷一、卷二、卷3及《习题解答》诺奖大师费曼乐趣横生的经典之作，读起来津津有趣，没有大段令人畏惧的公式和推导，几乎全文都在说明物理思想和有趣的现象，关于明白得物理思想的本质有极大帮忙。
《Mathematical methods for Physics and engineering》by Riley涵盖了几乎所有物理研究需要的数学知识，没有过度苛求证明严格性且说明形象有趣读起来超级有成绩感，不像国内的数理方式写得过于太抽象，用户界面不太友好呵呵。
《All the Mathematics you missed but need to know》by Garrity轻松的读物，高屋建瓴地整合了之前学过的数学知识，使读者很容易看透其中的数学本质。
举重假设轻地谈了很多深刻的数学领域，例如拓扑和“形式(form)”。
能够给大三看，也能够给研一看，必然会有专门大收成。
《Classical Mechanics- Systems of Particles and Hamiltonian Dynamics》by Greiner清楚地讲述了理论力学的内容，尽管书厚，但说明清楚没有冗余，超级适合自学。
《Introduction to Electrodynamics》及《习题解答》by Griffiths深切浅出的一本书，把人轻松地从电磁学带进电动力学的世界。
《Classical Electrodynamics》及《习题解答》by JacksonJackson那个词几乎已经专指这本神作了，难度高但循序渐进不失自学性。
英文原版薛定谔科普
英文原版薛定谔科普Quantum mechanics, as proposed by Austrian physicist ErwinSchrödinger, is a fundamental theory in physics that describes the behavior of particles at the smallest scales. It provides a mathematical framework for understanding the wave-particle duality of matter and the probabilistic nature of physical interactions. Although quantum mechanics is widely considered to be one of the most successful theories in science, it is also one of the most perplexing and counterintuitive.量子力学是由奥地利物理学家薛定谔提出的一种基本理论，描述了粒子在最小尺度上的行为。
它为理解物质的波粒二象性和物理相互作用的概率性提供了数学框架。
虽然量子力学被广泛认为是科学中最成功的理论之一，但也是最令人困惑和反直觉的。
One of the fundamental principles of quantum mechanics is the superposition principle, which states that a system can exist in multiple states simultaneously until it is measured. This idea challenges our classical understanding of physical reality, where objects are expected to have definite properties at all times. Thefamous thought experiment known as Schrödinger's cat illustrates this concept, where a cat trapped in a box is considered both alive and dead until the box is opened and observed.量子力学的一个基本原理是叠加原理，它表明一个系统可以同时存在多个状态，直到被测量出来。
science公布的全球最前沿的125个科学问题
science公布的全球最前沿的125个科学问题Science公布的全球最前沿的125个科学问题一、数学1. What makes prime numbers so special?什么使素数如此特别？2. Will the Navier–Stokes problem ever be solved?纳维尔-斯托克斯问题会得到解决吗？3. Is the Riemann hypothesis true?黎曼猜想是真的吗？二、化学1. Are there more color pigments to discover?还有更多色彩元素可发现吗？2. Will the periodic table ever be complete?元素周期表会完整吗？3. How can we measure interface phenomena on the microscopic level? 如何在微观层面测量界面现象？4. What is the future for energy storage？能量存储的未来是怎样的？5. Why does life require chirality?为什么生命需要手性？6. How can we better manage the world's plastic waste?我们如何更好地管理世界上的塑料废物？7. Will AI redefine the future of chemistry?AI会重新定义化学的未来吗？8. How can matter be programmed into living materials?物质如何被编码而成为生命材料？9. What drives reproduction in living systems?是什么驱动生命系统的复制？三、医学与健康1. Can we predict the next pandemic?我们可以预测下一次流行病吗？2. Will we ever find a cure for the common cold?我们会找到治疗感冒的方法吗？3. Can we design and manufacture medicines customized for individual people? 我们可以设计和制造出为个人定制的药物吗？4. Can a human tissue or organ be fully regenerated?人体组织或器官可以完全再生吗？5. How is immune homeostasis maintained and regulated?如何维持和调节免疫稳态？6.Is there a scientific basis to the Meridian System in traditional Chinese medicine?中医的经络系统有科学依据吗？7. How will the next generation of vaccines be made?下一代疫苗将如何生产？8. Can we ever overcome antibiotic resistance?我们能否克服抗生素耐药性？9. What is the etiology of autism?自闭症的病因是什么？10. What role does our microbiome play in health and disease?我们的微生物组在健康和疾病中扮演什么角色？11. Can xenotransplantation solve the shortage of donor organs?异种移植能否解决供体器官的短缺问题？四、生命科学1. What could help conservation of the oceans?什么可以帮助保护海洋？2. Can we stop ourselves from aging?我们可以阻止自己衰老吗？3. Why can only some cells become other cells?为什么只有一些细胞会变成其他细胞？4. Why are some genomes so big and others very small?为什么有些基因组非常大而另一些却很小？5. Will it be possible to cure all cancers?有可能治愈所有癌症吗？6. What genes make us uniquely human?哪些基因使我们人类与众不同？7. How do migratory animals know where they're going?迁徙动物如何知道它们要去哪里？8. How many species are there on Earth?地球上有多少物种？9. How do organisms evolve?有机体是如何进化的？10. Why did dinosaurs grow to be so big?为什么恐龙长得如此之大？11. Did ancient humans interbreed with other human-like ancestors? 远古人类是否曾与其他类人祖先杂交？12. Why do humans get so attached to dogs and cats?人类为什么会对猫狗如此着迷？13. Will the world's population keep growing indefinitely?世界人口会无限增长吗？14. Why do we stop growing?我们为什么会停止生长？15. Is de-extinction possible?能否复活灭绝生物？16. Can humans hibernate?人类可以冬眠吗？17. Where do human emotions originate?人类的情感源于何处？18. Will humans look physically different in the future?未来人类的外貌会有所不同吗？19. Why were there species explosions and mass extinction?为什么会发生物种大爆发和大灭绝？20. How might genome editing be used to cure disease?基因组编辑将如何用于治疗疾病？21. Can a cell be artificially synthesized?可以人工合成细胞吗？22. How are biomolecules organized in cells to function orderly and effectively? 细胞内的生物分子是如何组织从而有序有效发挥作用的？五、天文学1. How many dimensions are there in space?空间中有多少个维度？2. What is the shape of the universe?宇宙的形状是怎样的？3. Where did the big bang start?大爆炸从何处开始？4. Why don't the orbits of planets decay and cause them to crash into each other? 为什么行星的轨道不衰减并导致它们相互碰撞？5. When will the universe die? Will it continue to expand?宇宙何时消亡？它会继续膨胀吗？6. Is it possible to live permanently on another planet?我们有可能在另一个星球上长期居住吗？7. Why do black holes exist?为什么存在黑洞？8. What is the universe made of?宇宙是由什么构成的？9. Are we alone in the universe?我们是宇宙中唯一的生命体吗？10. What is the origin of cosmic rays?宇宙射线的起源是什么？11.What is the origin of mass?物质的起源是什么？12. What is the smallest scale of space-time?时空的最小尺度是是多少？13. Is water necessary for all life in the universe, or just on Earth?水是宇宙中所有生命所必需的么，还是仅对地球生命？14. What is preventing humans from carrying out deep-space exploration? 是什么阻止了人类进行深空探测？15. Is Einstein's general theory of relativity correct?爱因斯坦的广义相对论是正确的吗？16. How are pulsars formed?脉冲星是如何形成的？17. Is our Milky Way Galaxy special?我们的银河系特别吗？18. What is the volume, composition, and significance of the deep biosphere? 深层生物圈的规模、组成和意义是什么？19. Will humans one day have to leave the planet (or die trying)?人类有一天会不得不离开地球吗（还是会在尝试中死去）？20. Where do the heavy elements in the universe come from?宇宙中的重元素来自何处？21. Is it possible to understand the structure of compact stars and matter? 有可能了解致密恒星和物质的结构吗？22. What is the origin of the high-energy cosmic neutrinos?高能宇宙中微子的起源是什么？23. What is gravity?什么是重力？六、物理学1. Is there a diffraction limit?有衍射极限吗？2. What is the microscopic mechanism for high-temperature superconductivity?高温超导的微观机理是什么？3. What are the limits of heat transfer in matter?物质传热的极限是什么？4. What are the fundamental principles of collective motion?集体运动的基本原理是什么？5. What are the smallest building blocks of matter?什么是物质的最小组成部分？6. Will we ever travel at the speed of light?我们会以光速行驶吗？7. What is quantum uncertainty and why is it important?什么是量子不确定性，为什么它很重要？8. Will there ever be a "theory of everything"?会有“万有理论”吗？9. Why does time seem to flow in only one direction?为什么时间似乎只朝一个方向流动？10. What is dark matter?什么是暗物质？11. Can we make a real, human-size invisibility cloak?我们可以制作出真人大小的隐形斗篷吗？12.Are there any particles that behave oppositely to the properties or states of photons?是否存在与光子性质或状态相反的粒子？13. Will the Bose-Einstein condensate be widely used in the future?玻色-爱因斯坦冷凝体未来会被广泛使用吗？14. Can humans make intense lasers with incoherence comparable to sunlight? 人类能制造出与太阳光相似的非相干强激光吗？15. What is the maximum speed to which we can accelerate a particle?我们最多可以将粒子加速到多快？16.Is quantum many-body entanglement more fundamental than quantum fields?量子多体纠缠比量子场更基本吗？17. What is the optimum hardware for quantum computers?量子计算机的最佳硬件是什么？18. Can we accurately simulate the macro- and microworld?我们可以精确模拟宏观和微观世界吗？七、信息科学1. Is there an upper limit to computer processing speed?计算机处理速度是否有上限？2. Can AI replace a doctor?AI可以代替医生吗？3. Can topological quantum computing be realized?拓扑量子计算可以实现吗？4. Can DNA act as an information storage medium?DNA可以用作信息存储介质吗？八、工程与材料科学1. What is the ultimate statistical invariances of turbulence?湍流的最终统计不变性是什么？2. How can we break the current limit of energy conversion efficiencies?我们如何突破当前的能量转换效率极限？3. How can we develop manufacturing systems on Mars?我们如何在火星上开发制造系统？4. Is a future of only self-driving cars realistic?纯无人驾驶汽车的未来是否现实？九、神经科学1. What are the coding principles embedded in neuronal spike trains?神经元放电序列的编码准则是什么？2. Where does consciousness lie?意识存在于何处？3.Can human memory be stored, manipulated, and transplanted digitally?能否数字化地存储、操控和移植人类记忆？4. Why do we need sleep?为什么我们需要睡眠？5. What is addiction and how does it work?什么是成瘾？6. Why do we fall in love?为什么我们会坠入爱河？7. How did speech evolve and what parts of the brain control it?言语如何演变形成，大脑的哪些部分对其进行控制？8. How smart are nonhuman animals?除人类以外的其他动物有多聪明？9. Why are most people right-handed?为什么大多数人都是右撇子？10. Can we cure neurodegenerative diseases?我们可以治愈神经退行性疾病吗？11. Is it possible to predict the future?有可能预知未来吗？12. Can we more effectively diagnose and treat complex mental disorders?精神障碍能否有效诊断和治疗？十、生态学1. Can we stop global climate change?我们可以阻止全球气候变化吗？2. Where do we put all the excess carbon dioxide?我们能把过量的二氧化碳存到何处？3. What creates the Earth's magnetic field (and why does it move)?是什么创造了地球的磁场（为什么它会移动）？4.Will we be able to predict catastrophic weather events (tsunami, hurricanes, earthquakes) more accurately?我们是否能够更准确地预测灾害性事件（海啸、飓风、地震）？5. What happens if all the ice on the planet melts?如果地球上所有的冰融化会怎样？6. Can we create an environmentally friendly replacement for plastics?我们可以创造一种环保的塑料替代品吗？7. Can we achieve a situation where essentially every material can be recycled and reused?几乎所有材料都可以回收再利用是否可以实现？8. Will we soon see the end of monocultures like wheat, maize, rice, and soy?我们会很快看到小麦、玉米、大米和大豆等单一作物的终结吗？十一、能源科学1. Could we live in a fossil-fuel-free world?我们可以生活在一个去化石燃料的世界中吗？2. What is the future of hydrogen energy?氢能的未来是怎样的？3. Will cold fusion ever be possible?冷聚变有可能实现吗？十二、人工智能1. Will injectable, disease-fighting nanobots ever be a reality?可注射的抗病纳米机器人会成为现实吗？2. Will it be possible to create sentient robots?是否有可能创建有感知力的机器人？3. Is there a limit to human intelligence?人类智力是否有极限？4. Will artificial intelligence replace humans?人工智能会取代人类吗？5. How does group intelligence emerge?群体智能是如何出现的？6. Can robots or AIs have human creativity?机器人或AI 可以具有人类创造力吗？7.Can quantum artificial intelligence imitate the human brain?量子人工智能可以模仿人脑吗？8. Could we integrate with computers to form a human-machine hybrid species? 我们可以和计算机结合以形成人机混合物种吗？。
Quantum unique ergodicity
a r X i v :m a th -p h /0301035v 2 31 J u l 2003NOTE ON QUANTUM UNIQUE ERGODICITYSTEVE ZELDITCHThe purpose of this note is to record an observation about quantum unique ergodicity (QUE)which is relevant to the recent construction of H.Donnelly [D]of quasi-modes on certain non-positively curved surfaces,and to similar quasi-mode constructions known for many years as bouncing ball modes on Bunimovich stadia [BSS,H,BZ1,BZ2].Our new observation (Proposition 0.1)is the asymptotic vanishing of near off-diagonal matrix elements for eigenfunctions of QUE systems.As a corollary,we find that quantum ergodic (QE)systems possessing quasi-modes with singular limits and with a limited number of frequencies cannot be QUE.We begin by recalling that QUE (for Laplacians)concerns the matrix elements Aϕi ,ϕj of pseudodifferential operators relative to an orthonormal basis {ϕj }of eigenfunctions ∆ϕj =λ2j ϕj , ϕj ,ϕk =0.of the Laplacian ∆of a compact Riemannian manifold (M,g ).We denote the spectrum of ∆by Sp (∆).By definition,∆is QUE if Aϕj ,ϕj → S ∗M σA dL (1)where dL is the (normalized)Liouville measure on the unit (co-)tangent bundle.The term ‘unique’indicates that no subsequence of density zero of eigenfunctions need be removed when taking the limit.The main result of this note is that all off-diagonal terms of QUE systems tend to zero if the eigenvalue gaps tend to zero.This strengthens the conclusion of [Z]that almost all off-diagonal terms (with vanishing gaps)tend to zero in general QE situations.As will be seen below,it also provides evidence that Donnelly’s examples are non-QUE and establishesa localization statement of Heller-O’Connor [HO].Proposition 0.1.Suppose that ∆is QUE.Suppose that {(λi r ,λj r ),i r =j r }is a sequence of pairs of eigenvalues of√Date :February 7,2008.Research partially supported by NSF grant DMS-0071358and by the Clay Mathematics Institute .12STEVE ZELDITCHIf the eigenfunctions are real,then dνis a real (signed)measure.Our first observation is that any such weak limit must be a constant multiple of Liouville measure dL .Indeed,we first have:| A ∗Aϕi ,ϕj |≤| A ∗Aϕi ,ϕi |1/2| A ∗Aϕj ,ϕj |1/2.(3)Taking the limit along the sequence of pairs,we obtain | S ∗M |σA |2dν|≤ S ∗M |σA |2dL.(4)It follows that dν<<dL (absolutely continuous).But dL is an ergodic measure,so if dν=fdL is an invariant measure with f ∈L 1(dL ),then f is constant.Thus,dν=CdL,for some constant C.(5)We now observe that C =0if ϕi ⊥ϕj (i.e.if i =j ).This follows if we substitute A =I in(2),use orthogonality and (5).This result has implications for the possible ‘scarring’of quasi-modes of QUE systems.We first recall that a quasi-mode of order s for ∆is a sequence {ψk }of L 2-normalized functions which solves ||(∆−µk )ψk ||L 2=O (µ−s/2k ),(6)for a sequence of quasi-eigenvalues µk (see [CdV]for background).In particular a quasi-mode of order 0satisfies ||(∆−µk )ψk ||L 2=O (1).Such (relatively low-order)quasi-modes can be easily constructed for the stadium [BSS,H,BZ1,BZ2]and for Donnelly’s surfaces [D].As with eigenfunctions,we can consider the limits Aψj ,ψj → S ∗MσA dν(7)of matrix elements Aψj ,ψj of quasimodes.We say that the quasi-mode ‘scars’if the limit measure dνhas a non-zero singular component relative to dL .For instance,bouncing ball modes of stadia ‘scar’on the Lagrangean manifold with boundary formed by the bouncing ball orbits in the central rectangle,and the similar quasi-modes in [D]scar on the circles in the cylindrical part.The existence of such scarring quasi-modes suggests that these systems are not QUE.To explore this suggestion,we consider the decomposition of scarring quasi-modes into sums of true eigenfunctions.Definition:We say that a quasimode {ψk }of order 0as in (6)with ||ψk ||L 2=1has n (k )essential frequencies if,for each k ,there exists a subset Λk ⊂Sp (∆)∩[µk −δ,µk +δ]with n (k )=#Λk and constants c jk ∈C such that ψk =j :λ2j ∈Λkc kj ϕj +ηk ,with ||ηk ||L 2=o (1).(8)The following problems then seem interesting (the first is implicit in [HO]).•Bound the number n (k )of essential frequencies of a quasimode {ψk }of order 0which tends to a singular (i.e.non-Liouville)classical limit,e.g.a periodic orbit measure.NOTE ON QUANTUM UNIQUE ERGODICITY3•Bound the order s of a quasimode with a singular limit(intuitively,the diameter of the set of eigenvalues which composes its packet of eigenfunctions.)In other words,the questions are whether one can build a quasimode with a singular limit and(i)with anomalously few essential frequencies,or(ii)with anomalously low order.This softens the mathematicians’criterion of scarring as the existence of a sequence of actual modes(eigenfunctions)whose limit measureνin(2)has a singular component relative to Liouville measure[S].The following shows that quasi-modes with a uniformly bounded number of essential frequencies and singular limits do not exist for QUE systems.Corollary0.2.If there exists a quasi-mode{ψk}of order0for∆as in(8)and a constant C>0with the properties:•(i)n(k)≤C,∀k;•(ii) Aψk,ψk → S∗MσA dµwhere dµ=dL.Then∆is not QUE.Proof.We argue by contradiction.If∆were QUE,we would have(in the notation of(8): Aψk,ψk = i,j:λ2i,λ2j∈Λk c kj¯c ki Aϕi,ϕj +o(1)= j:λ2j∈Λk|c kj|2 Aϕj,ϕj + i=j:λ2i,λ2j∈Λk c kj¯c ki Aϕi,ϕj +o(1)= S∗MσA dL+o(1),by Proposition0.1.This contradicts(ii).In the last line we used that|λi−λj|→0if λ2i,λ2j∈Λk and that j:λ2j∈Λk|c kj|2=1+o(1),since||ψk||L2=1.The assumption that n(k)≤C could be weakened if we knew something about the rate of decay of the individual elements Aϕj,ϕk and| Aϕj,ϕj − S∗MσA dL|.We now consider the implications for the stadium and for Donnelly’s surface.In both cases, it is unknown how many essential frequencies are needed to build the associated bouncing ball quasi-modes.On average,intervals offixed width have a uniformly bounded number of∆-eigenvalues in dimension2,and this suggests that n(k)≤C.Our result would thenshow that such systems are automatically not QUE(as is widely believed).On the other√hand,the standard remainder estimate for Weyl’s law allows O(4STEVE ZELDITCHbounds on the concentration of eigenfunctions in the central part of stadia or in collars around hyperbolic closed geodesics Riemannian manifolds,which show that the optimal order of quasimodes with singular concentration in these regions is0.They further show that stadium eigenfunctions cannot scar on smaller sets than the entire set of bouncing ball orbits.References[BSS] A.Backer,R.Schubert,and P.Stifter,On the number of bouncing ball modes in billiards.J.Phys.A30(1997),no.19,6783–6795.[BL]J.Bourgain and E.Lindenstrauss,Entropy of Quantum Limits Commun.Math.Phys.233(2003), 153-171.[BZ1]N.Burq and M.Zworski,Control in the presence of a black box,arxiv preprint math.AP/0304184 (2003).[BZ2]N.Burq and M.Zworski,Bouncing ball modes and quantum chaos,arxiv preprint math.AP/0306278 (2003).[CdV]Y.Colin de Verdi`e re,Quasi-modes sur les varietes Riemanniennes.Invent.Math.43(1977),no.1, 15–52.[D]H.G.Donnelly,Quantum unique ergodicity,Proc.Amer.Math.Soc.131(2003),no.9,2945–2951. [FN] F.Faure,S.Nonnenmacher,On the maximal scarring for quantum cat map eigenstates,arxiv preprint nlin.CD/0304031(2003).[FND] F.Faure,S.Nonnenmacher,S.De Bievre Scarred eigenstates for quantum cat maps of minimal periods,arxiv preprint nlin.CD/0207060(2003).[H] E.J.Heller,Wavepacket dynamics and quantum chaology.Chaos et physique quantique(Les Houches,1989),547–664,North-Holland,Amsterdam,1991.[HO] E.J.Heller and P.W.O’Connor,Quantum localization for a strongly classical chaotic system,Phys.Rev.Lett.61(20)(1988),2288-2291.[L] E.Lindenstrauss,Invariant measures and arithmetic quantum unique ergodicity,(preprint,2003). [RS]Z.Rudnick and P.Sarnak,The behaviour of eigenstates of arithmetic hyperbolic m.Math.Phys.161(1994),no.1,195–213.[S]P.Sarnak,Arithmetic quantum chaos.The Schur lectures(1992)(Tel Aviv),183–236,Israel Math.Conf.Proc.,8,Bar-Ilan Univ.,Ramat Gan,1995.[W]S.A.Wolpert,The modulus of continuity forΓ0(m)\H semi-classical m.Math.Phys.216 (2001),no.2,313–323.[Z]S.Zelditch,Quantum transition amplitudes for ergodic and for completely integrable systems.J.Funct.Anal.94(1990),no.2,415–436.Department of Mathematics,Johns Hopkins University,Baltimore,MD21218,USA E-mail address:zelditch@。
英语作文素材 普朗克 量子
英语作文素材普朗克量子Max Planck and the Quantum Era.In the annals of physics, few figures loom as large as Max Planck. His groundbreaking contributions to the field, particularly his pivotal role in the development of quantum theory, have revolutionized our understanding of the universe at its most fundamental level.Early Life and Education.Max Planck was born on April 23, 1858, in Kiel, Germany. From a young age, he exhibited an insatiable thirst for knowledge, excelling in mathematics and physics. After completing his secondary education, Planck enrolled in the University of Munich, where he studied under the esteemed physicist Ludwig Boltzmann.Planck's Law: The Birth of Quantum Theory.In 1894, Planck embarked on an investigation to unravel the enigmatic behavior of heat radiation. The prevailing theory at the time was that thermal radiation wasdistributed continuously across all wavelengths. However, experiments consistently showed discrepancies with this law.Determined to find a solution, Planck proposed aradical idea: that energy exchanges between matter and radiation occur in discrete packets called quanta. This concept, known as Planck's law, revolutionized physics by introducing the notion of quantization, a fundamental concept in the quantum realm.The Quantum Revolution.Planck's law had far-reaching implications beyond thermal radiation. It laid the groundwork for AlbertEinstein's theory of the photoelectric effect, which confirmed the existence of photons as the quanta of light. This marked a profound shift in physics, from a classical worldview to a quantum one.Planck's Legacy.Max Planck's contributions to quantum theory are immeasurable. His groundbreaking work not only laid the foundation for quantum physics but also paved the way for numerous technological advancements in fields such as laser technology, electronics, and particle physics.In addition to his scientific achievements, Planck was also a respected professor, serving as the rector of the University of Berlin and president of the German Physical Society. His influence on subsequent generations of physicists is undeniable.Conclusion.Max Planck's legacy as a pioneer of quantum theory is secure. His groundbreaking work revolutionized our understanding of the universe and set the stage for the technological marvels we enjoy today. As we continue to explore the quantum realm, the principles established byPlanck will continue to guide our endeavors, illuminating the hidden mysteries of the microcosm.。
数学名言
布劳威尔(Luitzen Egbertus Jan Brouwer 1881-1996) 荷兰数学家、哲学家
是逻辑依赖数学，而不是数学依赖逻辑。
阿贝尔(Niels Henrik Abel 1802-1829) 挪威数学家
在我看来，一个人如果要在数学上有所进步，他必须向大师们学习，而不应向徒弟们学习。
阿尔布斯纳特(John Arbuthnot 1667-1735) 英国文学家、数学家
数学使思维产生活力，并使思维不受偏见、轻信与迷信的影响与干扰。
无论一数学定理如何正确，它不能给出一个美丽的印象便永远不算完美。
No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful.
In mathematics the art of proposing a question must be held of higher value than solving it.
我看到了，但我简直不能相信它！ 《康托尔给狄特金的信》
卡当(Elie Cartan 1869-1951) 法国数学家
玻尔(Niels Bohr 1885-1962) 丹麦物理学家
专家是一个可以制造所有错误的人，而这些错误只存在于一个非常狭少的领域。
遇事不决量子力学 英语
遇事不决量子力学英语Quantum Mechanics in Decision-MakingIn the face of complex and uncertain situations, traditional decision-making approaches often fall short. However, the principles of quantum mechanics, a field of physics that explores the behavior of matter and energy at the subatomic level, can provide valuable insights and a new perspective on problem-solving. By understanding and applying the fundamental concepts of quantum mechanics, individuals and organizations can navigate challenging scenarios with greater clarity and effectiveness.One of the key principles of quantum mechanics is the idea of superposition, which suggests that particles can exist in multiple states simultaneously until they are observed or measured. This concept can be applied to decision-making, where the decision-maker may be faced with multiple possible courses of action, each with its own set of potential outcomes. Rather than prematurely collapsing these possibilities into a single decision, the decision-maker can embrace the superposition and consider the various alternatives in a more open and flexible manner.Another important aspect of quantum mechanics is the principle of uncertainty, which states that the more precisely one property of a particle is measured, the less precisely another property can be known. This principle can be applied to decision-making, where the decision-maker may be faced with incomplete or uncertain information. Instead of trying to eliminate all uncertainty, the decision-maker can acknowledge and work within the constraints of this uncertainty, focusing on making the best possible decision based on the available information.Furthermore, quantum mechanics introduces the concept of entanglement, where two or more particles can become inextricably linked, such that the state of one particle affects the state of the other, even if they are physically separated. This idea can be applied to decision-making in complex systems, where the actions of one individual or organization can have far-reaching and unpredictable consequences for others. By recognizing the interconnectedness of the various elements within a system, decision-makers can better anticipate and navigate the potential ripple effects of their choices.Another key aspect of quantum mechanics that can inform decision-making is the idea of probability. In quantum mechanics, the behavior of particles is described in terms of probability distributions, rather than deterministic outcomes. This probabilistic approach can be applied to decision-making, where the decision-maker canconsider the likelihood of different outcomes and adjust their strategies accordingly.Additionally, quantum mechanics emphasizes the importance of observation and measurement in shaping the behavior of particles. Similarly, in decision-making, the act of observing and gathering information can influence the outcomes of a situation. By being mindful of how their own observations and interventions can impact the decision-making process, decision-makers can strive to maintain a more objective and impartial perspective.Finally, the concept of quantum entanglement can also be applied to the decision-making process itself. Just as particles can become entangled, the various factors and considerations involved in a decision can become deeply interconnected. By recognizing and embracing this entanglement, decision-makers can adopt a more holistic and integrated approach, considering the complex web of relationships and dependencies that shape the outcome.In conclusion, the principles of quantum mechanics offer a unique and compelling framework for navigating complex decision-making scenarios. By embracing the concepts of superposition, uncertainty, entanglement, and probability, individuals and organizations can develop a more nuanced and adaptable approach to problem-solving. By applying these quantum-inspired strategies, decision-makers can navigate the challenges of the modern world with greater clarity, resilience, and effectiveness.。
量子是一种玄学方法英语
量子是一种玄学方法英语Quantum physics is a branch of science that has captivated the minds of scientists and non-scientists alike. It is a field filled with strange and counterintuitive phenomena that challenge our understanding of how the world works. Quantum mechanics, in particular, is known for its mind-bending concepts such as superposition, entanglement, and wave-particle duality. This branch of science is often referred to as a "mysterious" and "magical" method due to its puzzling and unpredictable nature.Quantum mechanics is based on the principles that govern the behavior of particles at the atomic and subatomic levels. Unlike classical physics, which deals with the macroscopic world, quantum mechanics focuses on the quantum realm, where particles exhibit wave-like properties and can exist in multiple states simultaneously until measured.One of the key features of quantum mechanics is superposition. This concept states that particles can exist in multiple states or locations at the same time until obser ved. Schrödinger's famous thought experiment, in which a cat inside a box is simultaneously alive and dead until the box is opened, illustrates this phenomenon. This mind-boggling idea challenges our intuition and raises questions about the nature of reality. Another intriguing aspect of quantum mechanics is entanglement. When two particles become entangled, their properties becomeinterdependent, regardless of the distance between them. This means that measuring the state of one particle instantaneously determines the state of the other, no matter how far apart they are. Einstein famously called this phenomenon "spooky action at a distance." The concept of entanglement has led to the development of quantum teleportation and quantum cryptography, which have the potential to revolutionize communication and computing.Furthermore, quantum mechanics challenges the classical concept of particles having definite properties. According to wave-particle duality, particles can behave as both waves and particles depending on the experimental setup. This means that particles can exhibit characteristics of both particles and waves simultaneously, adding to the mystery of quantum mechanics.Despite its success in explaining the behavior of atoms and subatomic particles, quantum mechanics is still not fully understood. It has been described as a "magical" and "mysterious" method due to its ability to produce unexpected and counterintuitive results. The probabilistic nature of quantum mechanics, where predictions are made based on the likelihood of outcomes rather than definitive results, adds to its enigmatic nature.The potential applications of quantum mechanics are vast. Quantum computers, currently in their infancy, have the potential to performcomplex calculations exponentially faster than classical computers. Quantum cryptography promises unbreakable encryption, ensuring secure communication in a world where digital security is crucial. Furthermore, quantum sensors have the ability to detect incredibly small changes in physical quantities, making them invaluable in fields like medicine, defense, and environmental monitoring.In conclusion, quantum mechanics is a field that continues to perplex and fascinate scientists and laypeople alike. Its counterintuitive concepts, such as superposition, entanglement, and wave-particle duality, make it appear as a mysterious and magical method. Despite its challenges, quantum mechanics holds immense potential for technological advancements and deeper understanding of the fundamental workings of the universe. As we continue to explore and unravel the mysteries of quantum physics, we embark on a thrilling journey into the unknown.。
中科院博士研究生英语精读教材翻译及原文整理解读
第1课知识的悖论The Paradox of KnowledgeThe greatest achievement of humankind in its long evolution from ancient hominoid ancestors to its present status is the acquisition and accumulation of a vast body of knowledge about itself, the world, and the universe. The products of this knowledge are all those things that, in the aggregate, we call "civilization," including language, science, literature, art, all the physical mechanisms, instruments, and structures we use, and the physical infrastructures on which society relies. Most of us assume that in modern society knowledge of all kinds is continually increasing and the aggregation of new information into the corpus of our social or collective knowledge is steadily reducing the area of ignorance about ourselves, the world, and the universe. But continuing reminders of the numerous areas of our present ignorance invite a critical analysis of this assumption.In the popular view, intellectual evolution is similar to, although much more rapid than, somatic evolution. Biological evolution is often described by the statement that "ontogeny recapitulates phylogeny"--meaning that the individual embryo, in its development from a fertilized ovum into a human baby, passes through successive stages in which it resembles ancestral forms of the human species. The popular view is that humankind has progressed from a state of innocent ignorance, comparable to that of an infant, and gradually has acquired more and more knowledge, much as a child learns in passing through the several grades of the educational system. Implicit in this view is an assumption that phylogeny resembles ontogeny, so that there will ultimately be a stage in which the accumulation of knowledge is essentially complete, at least in specific fields, as if society had graduated with all the advanced degrees that signify mastery of important subjects.Such views have, in fact, been expressed by some eminent scientists. In 1894 the great American physicist Albert Michelson said in a talk at the University of Chicago:While it is never safe to affirm that the future of Physical Science has no marvels in store even more astonishing than those of the past, it seems probable that most of the grand underlying principles have been firmly established and that further advances are to be sought chiefly in the rigorous application of these principles to all the phenomena which come under our notice .... The future truths of Physical Science ate to be looked for in the sixth place of decimals.In the century since Michelson's talk, scientists have discovered much more than the refinement of measurements in the sixth decimal place, and none is willing to make a similar statement today. However, many still cling to the notion that such a state of knowledge remains a possibility to be attained sooner or later. Stephen Hawking, thegreat English scientist, in his immensely popular book A Brief History of Time (1988), concludes with the speculation that we may "discover a complete theory" that "would be the ultimate triumph of human reason--for then we would know the mind of God." Paul Davies, an Australian physicist, echoes that view by suggesting that the human mind may be able to grasp some of the secrets encompassed by the title of his book The Mind of God (1992). Other contemporary scientists write of "theories of everything," meaning theories that explain all observable physical phenomena, and Nobel Laureate Steven Weinberg, one of the founders of the current standard model of physical theory, writes of his Dreams of a Final Theory (1992).Despite the eminence and obvious yearning of these and many other contemporary scientists, there is nothing in the history of science to suggest that any addition of data or theories to the body of scientific knowledge will ever provide answers to all questions in any field. On the contrary, the history of science indicates that increasing knowledge brings awareness of new areas of ignorance and of new questions to be answered.Astronomy is the most ancient of the sciences, and its development is a model of other fields of knowledge. People have been observing the stars and other celestial bodies since the dawn of recorded history. As early as 3000 B.C. the Babylonians recognized a number of the constellations. In the sixth century B.C., Pythagoras proposed the notion of a spherical Earth and of a universe with objects in it chat moved in accordance with natural laws. Later Greek philosophers taught that the sky was a hollow globe surrounding the Earth, that it was supported on an axis running through the Earth, and chat stars were inlaid on its inner surface, which rotated westward daily. In the second century A.D., Ptolemy propounded a theory of a geocentric (Earth-centered) universe in which the sun, planets, and stars moved in circular orbits of cycles and epicycles around the Earth, although the Earth was not at the precise center of these orbits. While somewhat awkward, the Ptolemaic system could produce reasonably reliable predictions of planetary positions, which were, however, good for only a few years and which developed substantial discrepancies from actual observations over a long period of time. Nevertheless, since there was no evidence then apparent to astronomers that the Earth itself moves, the Ptolemaic system remained unchallenged for more than 13 centuries.In the sixteenth century Nocolaus Copernicus, who is said to have mastered all the knowledge of his day in mathematics, astronomy, medicine, and theology, became dissatisfied with the Ptolemaic system. He found that a heliocentric system was both mathematically possible and aesthetically more pleasing, and wrote a full exposition of his hypothesis, which was not published until 1543, shortly after his death. Early inthe seventeenth century, Johannes Kepler became imperial mathematician of the Holy Roman Empire upon the death of Tycho Brahe, and he acquired a collection of meticulous naked-eye observations of the positions of celestial bodies chat had been made by Brahe. On the basis of these data, Kepler calculated that both Ptolemy and Copernicus were in error in assuming chat planets traveled in circular orbits, and in 1609 he published a book demonstrating mathematically chat the planets travel around the sun in elliptical orbits. Kepler's laws of planetary motion are still regarded as basically valid.In the first decade of the seventeenth century Galileo Galilei learned of the invention of the telescope and began to build such instruments, becoming the first person to use a telescope for astronomical observations, and thus discovering craters on the moon, phases of Venus, and the satellites of Jupiter. His observations convinced him of the validity of the Copernican system and resulted in the well-known conflict between Galileo and church authorities. In January 1642 Galileo died, and in December of chat year Isaac Newton was born. Modern science derives largely from the work of these two men.Newton's contributions to science are numerous. He laid the foundations for modem physical optics, formulated the basic laws of motion and the law of universal gravitation, and devised the infinitesimal calculus. Newton's laws of motion and gravitation are still used for calculations of such matters as trajectories of spacecraft and satellites and orbits of planets. In 1846, relying on such calculations as a guide to observation, astronomers discovered the planet Neptune.While calculations based on Newton's laws are accurate, they are dismayingly complex when three or more bodies are involved. In 1915, Einstein announced his theory of general relativity, which led to a set of differential equations for planetary orbits identical to those based on Newtonian calculations, except for those relating to the planet Mercury. The elliptical orbit of Mercury rotates through the years, but so slowly that the change of position is less than one minute of arc each century. The equations of general relativity precisely accounted for this precession; Newtonian equations did not.Einstein's equations also explained the red shift in the light from distant stars and the deflection of starlight as it passed near the sun. However, Einstein assumed chat the universe was static, and, in order to permit a meaningful solution to the equations of relativity, in 1917 he added another term, called a "cosmological constant," to the equations. Although the existence and significance of a cosmological constant is still being debated, Einstein later declared chat this was a major mistake, as Edwin Hubble established in the 1920s chat the universe is expanding and galaxies are receding fromone another at a speed proportionate to their distance.Another important development in astronomy grew out of Newton's experimentation in optics, beginning with his demonstration chat sunlight could be broken up by a prism into a spectrum of different colors, which led to the science of spectroscopy. In the twentieth century, spectroscopy was applied to astronomy to gun information about the chemical and physical condition of celestial bodies chat was not disclosed by visual observation. In the 1920s, precise photographic photometry was introduced to astronomy and quantitative spectrochemical analysis became common. Also during the 1920s, scientists like Heisenberg, de Broglie, Schrodinger, and Dirac developed quantum mechanics, a branch of physics dealing with subatomic particles of matter and quanta of energy. Astronomers began to recognize that the properties of celestial bodies, including planets, could be well understood only in terms of physics, and the field began to be referred to as "astrophysics."These developments created an explosive expansion in our knowledge of astronomy. During the first five thousand years or more of observing the heavens, observation was confined to the narrow band of visible light. In the last half of this century astronomical observations have been made across the spectrum of electromagnetic radiation, including radio waves, infrared, ultraviolet, X-rays, and gamma rays, and from satellites beyond the atmosphere. It is no exaggeration to say chat since the end of World War II more astronomical data have been gathered than during all of the thousands of years of preceding human history.However, despite all improvements in instrumentation, increasing sophistication of analysis and calculation augmented by the massive power of computers, and the huge aggregation of data, or knowledge, we still cannot predict future movements of planets and other elements of even the solar system with a high degree of certainty. Ivars Peterson, a highly trained science writer and an editor of Science News, writes in his book Newton's Clock (1993) that a surprisingly subtle chaos pervades the solar system. He states:In one way or another the problem of the solar system's stability has fascinated and tormented asrtonomers and mathematicians for more than 200 years. Somewhat to the embarrassment of contemporary experts, it remains one of the most perplexing, unsolved issues in celestial mechanics. Each step toward resolving this and related questions has only exposed additional uncertainties and even deeper mysteries.Similar problems pervade astronomy. The two major theories of cosmology, general relativity and quantum mechanics, cannot be stated in the same mathematical language, and thus are inconsistent with one another, as the Ptolemaic and Copernicantheories were in the sixteenth century, although both contemporary theories continue to be used, but for different calculations. Oxford mathematician Roger Penrose, in The Emperors New Mind (1989), contends that this inconsistency requires a change in quantum theory to provide a new theory he calls "correct quantum gravity."Furthermore, the observations astronomers make with new technologies disclose a total mass in the universe that is less than about 10 percent of the total mass that mathematical calculations require the universe to contain on the basis of its observed rate of expansion. If the universe contains no more mass than we have been able to observe directly, then according to all current theories it should have expanded in the past, and be expanding now, much more rapidly than the rate actually observed. It is therefore believed that 90 percent or more of the mass in the universe is some sort of "dark matter" that has not yet been observed and the nature of which is unknown. Current theories favor either WIMPs (weakly interacting massive particles) or MACHOs (massive compact halo objects). Other similar mysteries abound and increase in number as our ability to observe improves.The progress of biological and life sciences has been similar to that of the physical sciences, except that it has occurred several centuries later. The theory of biological evolution first came to the attention of scientists with the publication of Darwin's Origin of Species in 1859. But Darwin lacked any explanation of the causes of variation and inheritance of characteristics. These were provided by Gregor Mendel, who laid the mathematical foundation of genetics with the publication of papers in 1865 and 1866.Medicine, according to Lewis Thomas, is the youngest science, having become truly scientific only in the 1930s. Recent and ongoing research has created uncertainty about even such basic concepts as when and how life begins and when death occurs, and we are spending billions in an attempt to learn how much it may be possible to know about human genetics. Modern medicine has demonstrably improved both our life expectancies and our health, and further improvements continue to be made as research progresses. But new questions arise even more rapidly than our research resources grow, as the host of problems related to the Human Genome Project illustrates.From even such an abbreviated and incomplete survey of science as this, it appears that increasing knowledge does not result in a commensurate decrease in ignorance, but, on the contrary, exposes new lacunae in our comprehension and confronts us with unforeseen questions disclosing areas of ignorance of which we were not previously aware.Thus the concept of science as an expanding body of knowledge that will eventually encompass or dispel all significant areas of ignorance is an illusion. Scientists and philosophers are now observing that it is naive to regard science as a process that begins with observations that are organized into theories and are then subsequently tested by experiments. The late Karl Popper, a leading philosopher of science, wrote in The Growth of Scientific Knowledge (1960) chat science starts from problems, not from observations, and chat every worthwhile new theory raises new problems. Thus there is no danger that science will come to an end because it has completed its task, clanks to the "infinity of our ignorance."At least since Thomas Kuhn published The Structure of Scientific Revolutions (1962), it has been generally recognized that observations are the result of theories (called paradigms by Kuhn and other philosophers), for without theories of relevance and irrelevance there would be no basis for determining what observations to make. Since no one can know everything, to be fully informed on any subject (a claim sometimes made by those in authority) is simply to reach a judgment that additional data are not important enough to be worth the trouble of securing or considering.To carry the analysis another step, it must be recognized that theories are the result of questions and questions are the product of perceived ignorance. Thus it is chat ignorance gives rise to inquiry chat produces knowledge, which, in turn, discloses new areas of ignorance. This is the paradox of knowledge: As knowledge increases so does ignorance, and ignorance may increase more than its related knowledge.My own metaphor to illustrate the relationship of knowledge and ignorance is based on a line from Matthew Arnold: "For we are here as on a darkling plain...." The dark chat surrounds us, chat, indeed, envelops our world, is ignorance. Knowledge is the illumination shed by whatever candles (or more technologically advanced light sources) we can provide. As we light more and more figurative candles, the area of illumination enlarges; but the area beyond illumination increases geometrically. We know chat there is much we don't know; but we cannot know how much there is chat we don't know. Thus knowledge is finite, but ignorance is infinite, and the finite cannot ever encompass the infinite.This is a revised version of an article originally published in COSMOS 1994. Copyright 1995 by Lee Loevinger.Lee Loevinger is a Washington lawyer and former assistant attorney general of the United States who writes frequently for scientific c publications. He has participated for many years as a member, co-chair, or liaison with the National Conference of Lawyers and Scientists, and he is a founder and former chair of the Science andTechnology Section of the American Bar Association. Office address: Hogan and Hartson, 555 Thirteenth St. NW, Washington, DC 20004.人类从古类人猿进化到当前的状态这个长久的进化过程中的最大成就是有关于人类自身、世界以及宇宙众多知识的获得和积聚。
Quantum Mechanics
Quantum MechanicsQuantum Mechanics is a fascinating and complex field of study that has revolutionized our understanding of the fundamental building blocks of the universe. It is a branch of physics that deals with the behavior of very small particles, such as atoms and subatomic particles, and how they interact with each other. The principles of quantum mechanics have led to the development of technologies such as lasers, transistors, and MRI machines, and have also had a profound impact on our understanding of the nature of reality. One of the key principles of quantum mechanics is the concept of superposition, which states that particles can exist in multiple states at the same time. This idea was famously illustrated by the thought experiment known as Schr?dinger's cat, in which a cat in a box is both alive and dead until the box is opened and the cat's state is observed. This idea challenges our everyday understanding of the world, where objects are either in one state or another, but it has been supported by numerous experimental observations. Another important concept in quantum mechanics is the idea of quantum entanglement, which Einstein famously referred to as "spooky action at a distance." This phenomenon occurs when two particles become linked in such a way that the state of one particle is instantly correlated with the state of the other, regardless of the distance between them. This idea has been demonstrated in experiments and has led to the development of technologies such as quantum cryptography, which promises to revolutionize the field of secure communication. However, despite the incredible success of quantum mechanics in explaining the behavior of particles at the smallest scales, it also presents significant challenges to our understanding of the nature of reality. For example, the famous double-slit experiment demonstrates that particles can behave as both waves and particles, depending on how they are observed. This has led to the development of various interpretations of quantum mechanics, such as the Copenhagen interpretation, the many-worlds interpretation, and the pilot-wave theory, each of which offers a different perspective on the nature of reality at the quantum level. Furthermore, the principles of quantum mechanics also have profound philosophical implications. For example, the idea that particles canexist in multiple states at the same time challenges our everyday understanding ofthe world and raises questions about the nature of consciousness and observation. Additionally, the concept of quantum entanglement raises questions about thenature of causality and the relationship between particles at a fundamental level. In conclusion, quantum mechanics is a field of study that has revolutionized our understanding of the fundamental building blocks of the universe. It has led tothe development of technologies that have transformed the modern world and hasalso raised profound questions about the nature of reality and the relationship between consciousness and the physical world. While it presents significant challenges to our understanding, it also offers incredible opportunities forfurther exploration and discovery. Overall, quantum mechanics is a field that continues to inspire and intrigue scientists and philosophers alike.。
霍金博士：科学的无尽探求
霍金博士：科学的无尽探求Steven Hawking, known as one of the greatest theoretical physicists of our time, dedicated his life to unraveling the mysteries of the universe. His groundbreaking research, coupled with his unique perspective on the cosmos, has inspired generations of scientists and ignited a passion for scientific exploration. In this article, we delve into the remarkable contributions and unwavering curiosity of Dr. Hawking.I. Early Life and EducationSteven William Hawking was born on January 8, 1942, in Oxford, England. Despite being diagnosed with a rare form of motor neuron disease (amyotrophic lateral sclerosis or ALS) at the age of 21, he didn't allow his physical limitations to hinder his intellectual pursuits. Hawking completed his undergraduate studies at the University of Oxford, where he showcased his brilliance in mathematics and physics.II. The Theory of EverythingHawking's most renowned work revolves around his quest for a "Theory of Everything." This theory aims to unify the fundamental forces of the universe, including gravity, electromagnetism, and the strong and weak nuclear forces. Although he faced numerous setbacks due to his physical condition, Hawking's persistent dedication led to his groundbreaking discovery of Hawking radiation. This theory implies that black holes slowly lose mass over time, eventually evaporating and disappearing.III. Popularization of ScienceApart from his astonishing scientific contributions, Hawking also played a vital role in popularizing scientific concepts for the general public. Through his bestselling book, "A Brief History of Time," he successfully bridged the gap between complex theoretical physics and mainstream readers. The book captivated millions and allowed them to explore the origins of the universe, the nature of time, and other mind-boggling phenomena.IV. Contributions to CosmologyHawking's work in cosmology fundamentally transformed our understanding of the universe. His research on the Big Bang theory and the existence of singularities revolutionized the field. By applying quantum mechanics to cosmology, he developed the concept of the inflationary model, proposing that the universe rapidly expanded in the moments following the Big Bang.V. Perseverance and LegacyDespite facing immense physical challenges, Hawking's determination and resilience shine as a testament to the power of the human spirit. He continued his scientific pursuits, communicated his ideas through his speech-generating device, and fearlessly pushed the boundaries of our knowledge. Hawking's legacy extends beyond the realm of science, inspiring individuals worldwide to embrace their curiosities and persevere in the face of adversity.VI. Recognition and AwardsThroughout his illustrious career, Hawking received numerous accolades for his contributions to science and humanity. He was honored with the Albert Einstein Award, the Presidential Medal of Freedom, and was even made a Commander of the Order of the British Empire (CBE). Moreover, he held esteemed positions at the University of Cambridge, including the Lucasian Professorship of Mathematics, a position once held by Sir Isaac Newton.VII. Philanthropy and ActivismHawking was not only dedicated to scientific research but also advocated for various social and humanitarian causes. His advocacy for stem cell research, raising awareness about the impact of climate change, and championing the rights of individuals with disabilities showcased his commitment to making a positive difference in the world.VIII. ConclusionThe extraordinary life and intellectual contributions of Dr. Steven Hawking have left an indelible mark on the scientific community and the world as a whole. Through his unwavering pursuit of knowledge, he encouraged others to question the known and embrace the unknown. As we continue to explore the wonders of the universe, let us strive to embody the spirit of curiosity and perseverance that defined the late Dr. Hawking.。
菲赫金哥尔茨微积分学教程英文版
The Feynman-Kac formula is a fundamental result in stochastic calculus. It connects the solution of the heat equation to the path integral formulation of quantum mechanics. This is an introduction to the Feynman-Kac formula with applications to stochastic differential equations and mathematical finance.The first part of the book gives a brief introduction to the theory of stochastic processes and introduces the basic concepts of Brownian motion and the Gaussian measure. It then presents the Feynman-Kac formula for the heat equation and discusses its connection to the Schrödinger equation in quantum mechanics.The second part of the book covers applications of the Feynman-Kac formula to stochastic differential equations. It discusses the connection between the Feynman-Kac formula and the backward stochastic differential equation, and shows how the formula can be used to solve certain types of PDEs.The third part of the book focuses on applications of the Feynman-Kac formula to mathematical finance. It discusses the connection between the formula and the Black-Scholes equation, and shows how it can be used to price financial derivatives and model risk-neutral valuation.Overall, this book provides a comprehensive introduction to the Feynman-Kac formula and its applications. It is suitable for graduate students and researchers in mathematics, physics, and finance who are interested in stochasticcalculus and its applications.。
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arXiv:quant-ph/0406213v1 29 Jun 2004
I. INTRODUCTION
Stochastic Schr¨odinger equations and their solutions, quantum trajectories, have been extensively studied in the last 15 years (cf. [Car], [GaZ]). They provide insight into the behaviour of open quantum systems and they are invaluable for Monte Carlo simulations of the time evolution of such systems, in particular for the numerical determination of equilibrium states.
ϑt((t1, i1), . . . , (tn, in)) = e(t−tn)L0 Jin e(tn−tn−1)L0 · · · e(t2−t1)L0 Ji1 et1L0 (ρ) .
(2.3)
The probability density for these clicks to occur is equal to the trace of ϑt in (2.3). We shall denote the normalized density matrix ϑt/tr (ϑt) by Θt.
In a previous paper [Ku¨M] we have considered the ergodic properties of the observed output of open quantum systems. We found that quantum systems with finite dimensional Hilbert spaces and unique equilibrium states lead to ergodic observations. Strangely enough, the techniques needed to prove our present result seem to be entirely different from the ones used in that paper. Here we make strong use of martingales, which have been introduced to this context in [Bel]. As in [Ku¨M] we concentrate our discussion on jump processes in continuous time using the formulation of Davies and Srivinas [Dav] [SrD]. But the result also holds for diffusive Schr¨odinger equations and for quantum evolutions in discrete time, as they occur in repeated measurement situations like the micromaser [WBKM].
In this paper we prove that for any finite-dimensional quantum system and for any initial state the time average of a single quantum trajectory converges to some equilibrium state with probability one. This result holds true despite the fact that the quantum trajectory itself may stay away from equilibrium forever.
tr L(ρ) = 0 for all ρ .
பைடு நூலகம்
(2.1)
2
An unraveling of ρ is induced by a decomposition of the generator
k
L = L0 + Ji ,
i=1
(2.2)
where Ji(ρ) = ViρVi∗ would be a natural choice. In general, any decomposition can be treated for which etL0, t ≥ 0, and Ji, i = 1, . . . , k, are completely positive.
II. THE MAIN RESULT
The state of an open quantum system is described by a density matrix ρ on a finite dimensional Hilbert space H, obeying a Master equation ρ˙ = Lρ, where L is a generator of Lindblad form [Lin],
If the time evolution of an open quantum system approaches equilibrium in the time mean, then on any single trajectory of any of its unravelings the time averaged state approaches the same equilibrium state with probability 1. In the case of multiple equilibrium states the quantum trajectory converges in the mean to a random choice from these states.
The paper is organizied as follows. We formulate our result in Section II and introduce the necessary martingales in Section III. In Section IV the proof of the theorem is given. It is extended to the diffusive and discrete time cases in Sections V and VI, respectively.
denotes the change of the state of the system under the condition that during a time interval of length t no clicks are recorded. The operator ρ → Ji(ρ) on the state space describes the change of state conditioned on the occurrence of a click of detector i.
D-64 289 Darmstadt Germany, kuemmerer@mathematik.tu-darmstadt.de 2Mathematisch Instituut, Radboud Universiteit Nijmegen, Toernooiveld 1,
6525 ED Nijmegen, The Netherlands, maassen@math.kun.nl (Dated: June 25, 2004)
In the simple case that there exists only one equilibrium state, the above result implies that the path average converges to this particular state, almost surely and independently of the starting point chosen. So in one sense the quantum trajectory is ergodic in this case: The path average of any observable of the quantum system equals its expectation in the equilibrium state. However, when looked upon as a classical stochastic process with values in the space of all quantum states, the quantum trajectory need not be ergodic, even in this simple and well-behaved case: There may be disjoint regions in the space of all quantum states between which no transitions are possible.
k
L(ρ) = i[H, ρ] +
Vi ρVi∗
−
1 2
(Vi∗Viρ
−
ρVi∗Vi)
.
i=1
Here H, V1, . . . , Vk are linear operators on H, H being self-adjoint. Conservation of normalisation of ρ is expressed by the relation