Quantum mechanics on noncommutative Riemann surfaces

1、microscopic world 微观世界2、macroscopic world 宏观世界3、quantum theory 量子[理]论4、quantum mechanics 量子力学5、wave mechanics 波动力学6、matrix mechanics 矩阵力学7、Planck constant 普朗克常数8、wave-particle duality 波粒二象性9、state 态10、state function 态函数11、state vector 态矢量12、superposition principle of state 态叠加原理13、orthogonal states 正交态14、antisymmetrical state 正交定理15、stationary state 对称态16、antisymmetrical state 反对称态17、stationary state 定态18、ground state 基态19、excited state 受激态20、binding state 束缚态21、unbound state 非束缚态22、degenerate state 简并态23、degenerate system 简并系24、non-deenerate state 非简并态25、non-degenerate system 非简并系26、de Broglie wave 德布罗意波27、wave function 波函数28、time-dependent wave function 含时波函数29、wave packet 波包30、probability 几率31、probability amplitude 几率幅32、probability density 几率密度33、quantum ensemble 量子系综34、wave equation 波动方程35、Schrodinger equation 薛定谔方程36、Potential well 势阱37、Potential barrien 势垒38、potential barrier penetration 势垒贯穿39、tunnel effect 隧道效应40、linear harmonic oscillator线性谐振子41、zero proint energy 零点能42、central field 辏力场43、Coulomb field 库仑场44、δ-function δ-函数45、operator 算符46、commuting operators 对易算符47、anticommuting operators 反对易算符48、complex conjugate operator 复共轭算符49、Hermitian conjugate operator 厄米共轭算符50、Hermitian operator 厄米算符51、momentum operator 动量算符52、energy operator 能量算符53、Hamiltonian operator 哈密顿算符54、angular momentum operator 角动量算符55、spin operator 自旋算符56、eigen value 本征值57、secular equation 久期方程58、observable 可观察量59、orthogonality 正交性60、completeness 完全性61、closure property 封闭性62、normalization 归一化63、orthonormalized functions 正交归一化函数64、quantum number 量子数65、principal quantum number 主量子数66、radial quantum number 径向量子数67、angular quantum number 角量子数68、magnetic quantum number 磁量子数69、uncertainty relation 测不准关系70、principle of complementarity 并协原理71、quantum Poisson bracket 量子泊松括号72、representation 表象73、coordinate representation 坐标表象74、momentum representation 动量表象75、energy representation 能量表象76、Schrodinger representation 薛定谔表象77、Heisenberg representation 海森伯表象78、interaction representation 相互作用表象79、occupation number representation 粒子数表象80、Dirac symbol 狄拉克符号81、ket vector 右矢量82、bra vector 左矢量83、basis vector 基矢量84、basis ket 基右矢85、basis bra 基左矢86、orthogonal kets 正交右矢87、orthogonal bras 正交左矢88、symmetrical kets 对称右矢89、antisymmetrical kets 反对称右矢90、Hilbert space 希耳伯空间91、perturbation theory 微扰理论92、stationary perturbation theory 定态微扰论93、time-dependent perturbation theory 含时微扰论94、Wentzel-Kramers-Brillouin method W. K. B.近似法95、elastic scattering 弹性散射96、inelastic scattering 非弹性散射97、scattering cross-section 散射截面98、partial wave method 分波法99、Born approximation 玻恩近似法100、centre-of-mass coordinates 质心坐标系101、laboratory coordinates 实验室坐标系102、transition 跃迁103、dipole transition 偶极子跃迁104、selection rule 选择定则105、spin 自旋106、electron spin 电子自旋107、spin quantum number 自旋量子数108、spin wave function 自旋波函数109、coupling 耦合110、vector-coupling coefficient 矢量耦合系数111、many-partic le system 多子体系112、exchange forece 交换力113、exchange energy 交换能114、Heitler-London approximation 海特勒-伦敦近似法115、Hartree-Fock equation 哈特里-福克方程116、self-consistent field 自洽场117、Thomas-Fermi equation 托马斯-费米方程118、second quantization 二次量子化119、identical particles全同粒子120、Pauli matrices 泡利矩阵121、Pauli equation 泡利方程122、Pauli’s exclusion principle泡利不相容原理123、Relativistic wave equation 相对论性波动方程124、Klein-Gordon equation 克莱因-戈登方程125、Dirac equation 狄拉克方程126、Dirac hole theory 狄拉克空穴理论127、negative energy state 负能态128、negative probability 负几率129、microscopic causality 微观因果性本征矢量eigenvector本征态eigenstate本征值eigenvalue本征值方程eigenvalue equation本征子空间eigensubspace (可以理解为本征矢空间)变分法variatinial method标量scalar算符operator表象representation表象变换transformation of representation表象理论theory of representation波函数wave function波恩近似Born approximation玻色子boson费米子fermion不确定关系uncertainty relation狄拉克方程Dirac equation狄拉克记号Dirac symbol定态stationary state定态微扰法time-independent perturbation定态薛定谔方程time-independent Schro(此处上面有两点)dinger equati on 动量表象momentum representation角动量表象angular mommentum representation占有数表象occupation number representation坐标（位置）表象position representation角动量算符angular mommentum operator角动量耦合coupling of angular mommentum对称性symmetry对易关系commutator厄米算符hermitian operator厄米多项式Hermite polynomial分量component光的发射emission of light光的吸收absorption of light受激发射excited emission自发发射spontaneous emission轨道角动量orbital angular momentum自旋角动量spin angular momentum轨道磁矩orbital magnetic moment归一化normalization哈密顿hamiltonion黑体辐射black body radiation康普顿散射Compton scattering基矢basis vector基态ground state基右矢basis ket ‘右矢’ket基左矢basis bra简并度degenerancy精细结构fine structure径向方程radial equation久期方程secular equation量子化quantization矩阵matrix模module模方square of module内积inner product逆算符inverse operator欧拉角Eular angles泡利矩阵Pauli matrix平均值expectation value （期望值）泡利不相容原理Pauli exclusion principle氢原子hydrogen atom球鞋函数spherical harmonics全同粒子identical partic les塞曼效应Zeeman effect上升下降算符raising and lowering operator 消灭算符destruction operator产生算符creation operator矢量空间vector space守恒定律conservation law守恒量conservation quantity投影projection投影算符projection operator微扰法pertubation method希尔伯特空间Hilbert space线性算符linear operator线性无关linear independence谐振子harmonic oscillator选择定则selection rule幺正变换unitary transformation幺正算符unitary operator宇称parity跃迁transition运动方程equation of motion正交归一性orthonormalization正交性orthogonality转动rotation自旋磁矩spin magnetic monent(以上是量子力学中的主要英语词汇，有些未涉及到的可以自由组合。

The Quantum Mechanics ofNanostructures在二十世纪初期，科学家们开始探索微观世界，发现物质不是像人们过去认为的那样实体的，而是由无数微小的基本粒子组成的。

这一发现彻底颠覆了过去的认知，引发了科学领域的革命。

随着科技的进步，人们发现微观世界的可塑性和变化性越来越重要，特别是在纳米结构研究中。

量子力学是研究微观物质的分支学科。

在纳米结构中，由于尺寸的微小和表面的高度变化，量子力学的效应变得尤为显著。

实际上，纳米尺度下的物理现象是非常不同于宏观世界的，要求我们开展全新的理论和方法。

在纳米材料的研究中，量子力学的应用广泛存在。

举个例子，纳米颗粒中的量子点对光的吸收和发射起着至关重要的作用。

在一个纳米结构中的电子不能回避任一本质上的受限束缚，其运动从单一的粒子系统变成了固体晶体系统中的波函数。

这种类似波动的运动和其他微观量子效应，如局域化和微观隧道效应，导致纳米材料的复杂性和变化性增加。

因此，我们需要量子力学的方法来描述这种动力学过程。

随着纳米科技的进步和发展，人们已经发现，许多量子力学效应对纳米结构和器件的性质和功能具有重要作用。

例如，纳米线和二维材料的表面态密度高，电荷输运现象变得更为复杂，需要借助量子力学来建模和描述。

同时，在加工成像等领域，人们也发现量子力学的应用可以提供更精确的解决方案。

使用电子和其他粒子来探测纳米结构的表面，电子显微镜等技术已经成为普遍地工具，这些技术通常需要使用量子力学的方法。

通过计算纳米电子学的电子输运，设计高效的器件也成为了可行和实现的了。

总之，量子力学已经成为了纳米领域的基础，不断发展和完善的量子力学方法可以帮助我们更好地理解纳米结构的物质行为和复杂性。

这一发现和理论将在我们未来的纳米技术，制造和生物医学领域发挥重要的作用，改变我们生活的方方面面。

量子力学专业英语词汇1、microscopic world 微观世界2、macroscopic world 宏观世界3、quantum theory 量子[理]论4、quantum mechanics 量子力学5、wave mechanics 波动力学6、matrix mechanics 矩阵力学7、Planck constant 普朗克常数8、wave-particle duality 波粒二象性9、state 态10、state function 态函数11、state vector 态矢量12、superposition principle of state 态叠加原理13、orthogonal states 正交态14、antisymmetrical state 正交定理15、stationary state 对称态16、antisymmetrical state 反对称态17、stationary state 定态18、ground state 基态19、excited state 受激态20、binding state 束缚态21、unbound state 非束缚态22、degenerate state 简并态23、degenerate system 简并系24、non-deenerate state 非简并态25、non-degenerate system 非简并系26、de Broglie wave 德布罗意波27、wave function 波函数28、time-dependent wave function 含时波函数29、wave packet 波包30、probability 几率31、probability amplitude 几率幅32、probability density 几率密度33、quantum ensemble 量子系综34、wave equation 波动方程35、Schrodinger equation 薛定谔方程36、Potential well 势阱37、Potential barrien 势垒38、potential barrier penetration 势垒贯穿39、tunnel effect 隧道效应40、linear harmonic oscillator 线性谐振子41、zero proint energy 零点能42、central field 辏力场43、Coulomb field 库仑场44、δ-function δ-函数45、operator 算符46、commuting operators 对易算符47、anticommuting operators 反对易算符48、complex conjugate operator 复共轭算符49、Hermitian conjugate operator 厄米共轭算符50、Hermitian operator 厄米算符51、momentum operator 动量算符52、energy operator 能量算符53、Hamiltonian operator 哈密顿算符54、angular momentum operator 角动量算符55、spin operator 自旋算符56、eigen value 本征值57、secular equation 久期方程58、observable 可观察量59、orthogonality 正交性60、completeness 完全性61、closure property 封闭性62、normalization 归一化63、orthonormalized functions 正交归一化函数64、quantum number 量子数65、principal quantum number 主量子数66、radial quantum number 径向量子数67、angular quantum number 角量子数68、magnetic quantum number 磁量子数69、uncertainty relation 测不准关系70、principle of complementarity 并协原理71、quantum Poisson bracket 量子泊松括号72、representation 表象73、coordinate representation 坐标表象74、momentum representation 动量表象75、energy representation 能量表象76、Schrodinger representation 薛定谔表象77、Heisenberg representation 海森伯表象78、interaction representation 相互作用表象79、occupation number representation 粒子数表象80、Dirac symbol 狄拉克符号81、ket vector 右矢量82、bra vector 左矢量83、basis vector 基矢量84、basis ket 基右矢85、basis bra 基左矢86、orthogonal kets 正交右矢87、orthogonal bras 正交左矢88、symmetrical kets 对称右矢89、antisymmetrical kets 反对称右矢90、Hilbert space 希耳伯空间91、perturbation theory 微扰理论92、stationary perturbation theory 定态微扰论93、time-dependent perturbation theory 含时微扰论94、Wentzel-Kramers-Brillouin method W. K. B.近似法95、elastic scattering 弹性散射96、inelastic scattering 非弹性散射97、scattering cross-section 散射截面98、partial wave method 分波法99、Born approximation 玻恩近似法100、centre-of-mass coordinates 质心坐标系101、laboratory coordinates 实验室坐标系102、transition 跃迁103、dipole transition 偶极子跃迁104、selection rule 选择定则105、spin 自旋106、electron spin 电子自旋107、spin quantum number 自旋量子数108、spin wave function 自旋波函数109、coupling 耦合110、vector-coupling coefficient 矢量耦合系数111、many-particle system 多子体系112、exchange forece 交换力113、exchange energy 交换能114、Heitler-London approximation 海特勒-伦敦近似法115、Hartree-Fock equation 哈特里-福克方程116、self-consistent field 自洽场117、Thomas-Fermi equation 托马斯-费米方程118、second quantization 二次量子化119、identical particles 全同粒子120、Pauli matrices 泡利矩阵121、Pauli equation 泡利方程122、Pauli’s exclusion principle泡利不相容原理123、Relativistic wave equation 相对论性波动方程124、Klein-Gordon equation 克莱因-戈登方程125、Dirac equation 狄拉克方程126、Dirac hole theory 狄拉克空穴理论127、negative energy state 负能态128、negative probability 负几率129、microscopic causality 微观因果性。

八卦一下量子机器学习的历史人工智能和量子信息在讲量子机器学习之前我们先来八卦一下人工智能和量子信息。

1956，达特茅斯，十位大牛聚集于此，麦卡锡（John McCarthy）给这个活动起了个别出心裁的名字：“人工智能夏季研讨会”（Summer Research Project on Artificial Intelligence），现在被普遍认为是人工智能的起点。

AI的历史是非常曲折的，从符号派到联结派，从逻辑推理到统计学习，从经历70年代和80年代两次大规模的政府经费削减，到90年代开始提出神经网络，默默无闻直到2006年Hinton提出深层神经网络的层级预训练方法，从专注于算法到李飞飞引入ImageNet，大家开始注意到数据的重要性，大数据的土壤加上计算力的摩尔定律迎来了现在深度学习的火热。

量子信息的历史则更为悠久和艰难。

这一切都可以归结到1935年，爱因斯坦，波多尔斯基和罗森在“Can Quantum-Mechanical Description of Physical Reality be Considered Complete?”一文中提出了EPR悖论，从而引出了量子纠缠这个概念。

回溯到更早一点，1927年第五次索尔维会议，世界上最主要的物理学家聚在一起讨论新近表述的量子理论。

会议上爱因斯坦和波尔起了争执，爱因斯坦用“上帝不会掷骰子”的观点来反对海森堡的不确定性原理，而玻尔反驳道，“爱因斯坦，不要告诉上帝怎么做”。

这一论战持续了很多年，伴随着量子力学的发展，直到爱因斯坦在1955年去世。

爱因斯坦直到去世也还一直坚持这个世界没有随机性这种东西，所有的物理规律都是确定性的，给定初态和演化规律，物理学家就能推算出任意时刻系统的状态。

而量子力学生来就伴随了不确定性，一只猫在没测量前可以同时“生”和'死'，不具备一个确定的状态，只有测量后这只猫才具备“生”和'死'其中的一种状态，至于具体是哪一种状态量子力学只能告诉我们每一种态的概率，给不出一个确定的结果。

More informationQuantum Computing for Computer ScientistsThe multidisciplinaryfield of quantum computing strives to exploit someof the uncanny aspects of quantum mechanics to expand our computa-tional horizons.Quantum Computing for Computer Scientists takes read-ers on a tour of this fascinating area of cutting-edge research.Writtenin an accessible yet rigorous fashion,this book employs ideas and tech-niques familiar to every student of computer science.The reader is notexpected to have any advanced mathematics or physics background.Af-ter presenting the necessary prerequisites,the material is organized tolook at different aspects of quantum computing from the specific stand-point of computer science.There are chapters on computer architecture,algorithms,programming languages,theoretical computer science,cryp-tography,information theory,and hardware.The text has step-by-stepexamples,more than two hundred exercises with solutions,and program-ming drills that bring the ideas of quantum computing alive for today’scomputer science students and researchers.Noson S.Yanofsky,PhD,is an Associate Professor in the Departmentof Computer and Information Science at Brooklyn College,City Univer-sity of New York and at the PhD Program in Computer Science at TheGraduate Center of CUNY.Mirco A.Mannucci,PhD,is the founder and CEO of HoloMathics,LLC,a research and development company with a focus on innovative mathe-matical modeling.He also serves as Adjunct Professor of Computer Sci-ence at George Mason University and the University of Maryland.QUANTUM COMPUTING FORCOMPUTER SCIENTISTSNoson S.YanofskyBrooklyn College,City University of New YorkandMirco A.MannucciHoloMathics,LLCMore informationMore informationcambridge university pressCambridge,New York,Melbourne,Madrid,Cape Town,Singapore,S˜ao Paulo,DelhiCambridge University Press32Avenue of the Americas,New York,NY10013-2473,USAInformation on this title:/9780521879965C Noson S.Yanofsky and Mirco A.Mannucci2008This publication is in copyright.Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published2008Printed in the United States of AmericaA catalog record for this publication is available from the British Library.Library of Congress Cataloging in Publication dataYanofsky,Noson S.,1967–Quantum computing for computer scientists/Noson S.Yanofsky andMirco A.Mannucci.p.cm.Includes bibliographical references and index.ISBN978-0-521-87996-5(hardback)1.Quantum computers.I.Mannucci,Mirco A.,1960–II.Title.QA76.889.Y352008004.1–dc222008020507ISBN978-0-521-879965hardbackCambridge University Press has no responsibility forthe persistence or accuracy of URLs for external orthird-party Internet Web sites referred to in this publicationand does not guarantee that any content on suchWeb sites is,or will remain,accurate or appropriate.More informationDedicated toMoishe and Sharon Yanofskyandto the memory ofLuigi and Antonietta MannucciWisdom is one thing:to know the tho u ght by which all things are directed thro u gh allthings.˜Heraclitu s of Ephe s u s(535–475B C E)a s quoted in Dio g ene s Laertiu s’sLives and Opinions of Eminent PhilosophersBook IX,1. More informationMore informationContentsPreface xi1Complex Numbers71.1Basic Definitions81.2The Algebra of Complex Numbers101.3The Geometry of Complex Numbers152Complex Vector Spaces292.1C n as the Primary Example302.2Definitions,Properties,and Examples342.3Basis and Dimension452.4Inner Products and Hilbert Spaces532.5Eigenvalues and Eigenvectors602.6Hermitian and Unitary Matrices622.7Tensor Product of Vector Spaces663The Leap from Classical to Quantum743.1Classical Deterministic Systems743.2Probabilistic Systems793.3Quantum Systems883.4Assembling Systems974Basic Quantum Theory1034.1Quantum States1034.2Observables1154.3Measuring1264.4Dynamics1294.5Assembling Quantum Systems1325Architecture1385.1Bits and Qubits138viiMore informationviii Contents5.2Classical Gates1445.3Reversible Gates1515.4Quantum Gates1586Algorithms1706.1Deutsch’s Algorithm1716.2The Deutsch–Jozsa Algorithm1796.3Simon’s Periodicity Algorithm1876.4Grover’s Search Algorithm1956.5Shor’s Factoring Algorithm2047Programming Languages2207.1Programming in a Quantum World2207.2Quantum Assembly Programming2217.3Toward Higher-Level Quantum Programming2307.4Quantum Computation Before Quantum Computers2378Theoretical Computer Science2398.1Deterministic and Nondeterministic Computations2398.2Probabilistic Computations2468.3Quantum Computations2519Cryptography2629.1Classical Cryptography2629.2Quantum Key Exchange I:The BB84Protocol2689.3Quantum Key Exchange II:The B92Protocol2739.4Quantum Key Exchange III:The EPR Protocol2759.5Quantum Teleportation27710Information Theory28410.1Classical Information and Shannon Entropy28410.2Quantum Information and von Neumann Entropy28810.3Classical and Quantum Data Compression29510.4Error-Correcting Codes30211Hardware30511.1Quantum Hardware:Goals and Challenges30611.2Implementing a Quantum Computer I:Ion Traps31111.3Implementing a Quantum Computer II:Linear Optics31311.4Implementing a Quantum Computer III:NMRand Superconductors31511.5Future of Quantum Ware316Appendix A Historical Bibliography of Quantum Computing319 by Jill CirasellaA.1Reading Scientific Articles319A.2Models of Computation320More informationContents ixA.3Quantum Gates321A.4Quantum Algorithms and Implementations321A.5Quantum Cryptography323A.6Quantum Information323A.7More Milestones?324Appendix B Answers to Selected Exercises325Appendix C Quantum Computing Experiments with MATLAB351C.1Playing with Matlab351C.2Complex Numbers and Matrices351C.3Quantum Computations354Appendix D Keeping Abreast of Quantum News:QuantumComputing on the Web and in the Literature357by Jill CirasellaD.1Keeping Abreast of Popular News357D.2Keeping Abreast of Scientific Literature358D.3The Best Way to Stay Abreast?359Appendix E Selected Topics for Student Presentations360E.1Complex Numbers361E.2Complex Vector Spaces362E.3The Leap from Classical to Quantum363E.4Basic Quantum Theory364E.5Architecture365E.6Algorithms366E.7Programming Languages368E.8Theoretical Computer Science369E.9Cryptography370E.10Information Theory370E.11Hardware371Bibliography373Index381More informationPrefaceQuantum computing is a fascinating newfield at the intersection of computer sci-ence,mathematics,and physics,which strives to harness some of the uncanny as-pects of quantum mechanics to broaden our computational horizons.This bookpresents some of the most exciting and interesting topics in quantum computing.Along the way,there will be some amazing facts about the universe in which we liveand about the very notions of information and computation.The text you hold in your hands has a distinctflavor from most of the other cur-rently available books on quantum computing.First and foremost,we do not assumethat our reader has much of a mathematics or physics background.This book shouldbe readable by anyone who is in or beyond their second year in a computer scienceprogram.We have written this book specifically with computer scientists in mind,and tailored it accordingly:we assume a bare minimum of mathematical sophistica-tion,afirst course in discrete structures,and a healthy level of curiosity.Because thistext was written specifically for computer people,in addition to the many exercisesthroughout the text,we added many programming drills.These are a hands-on,funway of learning the material presented and getting a real feel for the subject.The calculus-phobic reader will be happy to learn that derivatives and integrals are virtually absent from our text.Quite simply,we avoid differentiation,integra-tion,and all higher mathematics by carefully selecting only those topics that arecritical to a basic introduction to quantum computing.Because we are focusing onthe fundamentals of quantum computing,we can restrict ourselves to thefinite-dimensional mathematics that is required.This turns out to be not much more thanmanipulating vectors and matrices with complex entries.Surprisingly enough,thelion’s share of quantum computing can be done without the intricacies of advancedmathematics.Nevertheless,we hasten to stress that this is a technical textbook.We are not writing a popular science book,nor do we substitute hand waving for rigor or math-ematical precision.Most other texts in thefield present a primer on quantum mechanics in all its glory.Many assume some knowledge of classical mechanics.We do not make theseassumptions.We only discuss what is needed for a basic understanding of quantumxiMore informationxii Prefacecomputing as afield of research in its own right,although we cite sources for learningmore about advanced topics.There are some who consider quantum computing to be solely within the do-main of physics.Others think of the subject as purely mathematical.We stress thecomputer science aspect of quantum computing.It is not our intention for this book to be the definitive treatment of quantum computing.There are a few topics that we do not even touch,and there are severalothers that we approach briefly,not exhaustively.As of this writing,the bible ofquantum computing is Nielsen and Chuang’s magnificent Quantum Computing andQuantum Information(2000).Their book contains almost everything known aboutquantum computing at the time of its publication.We would like to think of ourbook as a usefulfirst step that can prepare the reader for that text.FEATURESThis book is almost entirely self-contained.We do not demand that the reader comearmed with a large toolbox of skills.Even the subject of complex numbers,which istaught in high school,is given a fairly comprehensive review.The book contains many solved problems and easy-to-understand descriptions.We do not merely present the theory;rather,we explain it and go through severalexamples.The book also contains many exercises,which we strongly recommendthe serious reader should attempt to solve.There is no substitute for rolling up one’ssleeves and doing some work!We have also incorporated plenty of programming drills throughout our text.These are hands-on exercises that can be carried out on your laptop to gain a betterunderstanding of the concepts presented here(they are also a great way of hav-ing fun).We hasten to point out that we are entirely language-agnostic.The stu-dent should write the programs in the language that feels most comfortable.Weare also paradigm-agnostic.If declarative programming is your favorite method,gofor it.If object-oriented programming is your game,use that.The programmingdrills build on one another.Functions created in one programming drill will be usedand modified in later drills.Furthermore,in Appendix C,we show how to makelittle quantum computing emulators with MATLAB or how to use a ready-madeone.(Our choice of MATLAB was dictated by the fact that it makes very easy-to-build,quick-and-dirty prototypes,thanks to its vast amount of built-in mathematicaltools.)This text appears to be thefirst to handle quantum programming languages in a significant way.Until now,there have been only research papers and a few surveyson the topic.Chapter7describes the basics of this expandingfield:perhaps some ofour readers will be inspired to contribute to quantum programming!This book also contains several appendices that are important for further study:Appendix A takes readers on a tour of major papers in quantum computing.This bibliographical essay was written by Jill Cirasella,Computational SciencesSpecialist at the Brooklyn College Library.In addition to having a master’s de-gree in library and information science,Jill has a master’s degree in logic,forwhich she wrote a thesis on classical and quantum graph algorithms.This dualbackground uniquely qualifies her to suggest and describe further readings.More informationPreface xiii Appendix B contains the answers to some of the exercises in the text.Othersolutions will also be found on the book’s Web page.We strongly urge studentsto do the exercises on their own and then check their answers against ours.Appendix C uses MATLAB,the popular mathematical environment and an es-tablished industry standard,to show how to carry out most of the mathematicaloperations described in this book.MATLAB has scores of routines for manip-ulating complex matrices:we briefly review the most useful ones and show howthe reader can quickly perform a few quantum computing experiments with al-most no effort,using the freely available MATLAB quantum emulator Quack.Appendix D,also by Jill Cirasella,describes how to use online resources to keepup with developments in quantum computing.Quantum computing is a fast-movingfield,and this appendix offers guidelines and tips forfinding relevantarticles and announcements.Appendix E is a list of possible topics for student presentations.We give briefdescriptions of different topics that a student might present before a class of hispeers.We also provide some hints about where to start looking for materials topresent.ORGANIZATIONThe book begins with two chapters of mathematical preliminaries.Chapter1con-tains the basics of complex numbers,and Chapter2deals with complex vectorspaces.Although much of Chapter1is currently taught in high school,we feel thata review is in order.Much of Chapter2will be known by students who have had acourse in linear algebra.We deliberately did not relegate these chapters to an ap-pendix at the end of the book because the mathematics is necessary to understandwhat is really going on.A reader who knows the material can safely skip thefirsttwo chapters.She might want to skim over these chapters and then return to themas a reference,using the index and the table of contents tofind specific topics.Chapter3is a gentle introduction to some of the ideas that will be encountered throughout the rest of the ing simple models and simple matrix multipli-cation,we demonstrate some of the fundamental concepts of quantum mechanics,which are then formally developed in Chapter4.From there,Chapter5presentssome of the basic architecture of quantum computing.Here one willfind the notionsof a qubit(a quantum generalization of a bit)and the quantum analog of logic gates.Once Chapter5is understood,readers can safely proceed to their choice of Chapters6through11.Each chapter takes its title from a typical course offered in acomputer science department.The chapters look at that subfield of quantum com-puting from the perspective of the given course.These chapters are almost totallyindependent of one another.We urge the readers to study the particular chapterthat corresponds to their favorite course.Learn topics that you likefirst.From thereproceed to other chapters.Figure0.1summarizes the dependencies of the chapters.One of the hardest topics tackled in this text is that of considering two quan-tum systems and combining them,or“entangled”quantum systems.This is donemathematically in Section2.7.It is further motivated in Section3.4and formallypresented in Section4.5.The reader might want to look at these sections together.xivPrefaceFigure 0.1.Chapter dependencies.There are many ways this book can be used as a text for a course.We urge instructors to find their own way.May we humbly suggest the following three plans of action:(1)A class that provides some depth might involve the following:Go through Chapters 1,2,3,4,and 5.Armed with that background,study the entirety of Chapter 6(“Algorithms”)in depth.One can spend at least a third of a semester on that chapter.After wrestling a bit with quantum algorithms,the student will get a good feel for the entire enterprise.(2)If breadth is preferred,pick and choose one or two sections from each of the advanced chapters.Such a course might look like this:(1),2,3,4.1,4.4,5,6.1,7.1,9.1,10.1,10.2,and 11.This will permit the student to see the broad outline of quantum computing and then pursue his or her own path.(3)For a more advanced class (a class in which linear algebra and some mathe-matical sophistication is assumed),we recommend that students be told to read Chapters 1,2,and 3on their own.A nice course can then commence with Chapter 4and plow through most of the remainder of the book.If this is being used as a text in a classroom setting,we strongly recommend that the students make presentations.There are selected topics mentioned in Appendix E.There is no substitute for student participation!Although we have tried to include many topics in this text,inevitably some oth-ers had to be left out.Here are a few that we omitted because of space considera-tions:many of the more complicated proofs in Chapter 8,results about oracle computation,the details of the (quantum)Fourier transforms,and the latest hardware implementations.We give references for further study on these,as well as other subjects,throughout the text.More informationMore informationPreface xvANCILLARIESWe are going to maintain a Web page for the text at/∼noson/qctext.html/The Web page will containperiodic updates to the book,links to interesting books and articles on quantum computing,some answers to certain exercises not solved in Appendix B,anderrata.The reader is encouraged to send any and all corrections tonoson@Help us make this textbook better!ACKNOLWEDGMENTSBoth of us had the great privilege of writing our doctoral theses under the gentleguidance of the recently deceased Alex Heller.Professor Heller wrote the follow-ing1about his teacher Samuel“Sammy”Eilenberg and Sammy’s mathematics:As I perceived it,then,Sammy considered that the highest value in mathematicswas to be found,not in specious depth nor in the overcoming of overwhelmingdifficulty,but rather in providing the definitive clarity that would illuminate itsunderlying order.This never-ending struggle to bring out the underlying order of mathematical structures was always Professor Heller’s everlasting goal,and he did his best to passit on to his students.We have gained greatly from his clarity of vision and his viewof mathematics,but we also saw,embodied in a man,the classical and sober ideal ofcontemplative life at its very best.We both remain eternally grateful to him.While at the City University of New York,we also had the privilege of inter-acting with one of the world’s foremost logicians,Professor Rohit Parikh,a manwhose seminal contributions to thefield are only matched by his enduring com-mitment to promote younger researchers’work.Besides opening fascinating vis-tas to us,Professor Parikh encouraged us more than once to follow new directionsof thought.His continued professional and personal guidance are greatly appre-ciated.We both received our Ph.D.’s from the Department of Mathematics in The Graduate Center of the City University of New York.We thank them for providingus with a warm and friendly environment in which to study and learn real mathemat-ics.Thefirst author also thanks the entire Brooklyn College family and,in partic-ular,the Computer and Information Science Department for being supportive andvery helpful in this endeavor.1See page1349of Bass et al.(1998).More informationxvi PrefaceSeveral faculty members of Brooklyn College and The Graduate Center were kind enough to read and comment on parts of this book:Michael Anshel,DavidArnow,Jill Cirasella,Dayton Clark,Eva Cogan,Jim Cox,Scott Dexter,EdgarFeldman,Fred Gardiner,Murray Gross,Chaya Gurwitz,Keith Harrow,JunHu,Yedidyah Langsam,Peter Lesser,Philipp Rothmaler,Chris Steinsvold,AlexSverdlov,Aaron Tenenbaum,Micha Tomkiewicz,Al Vasquez,Gerald Weiss,andPaula Whitlock.Their comments have made this a better text.Thank you all!We were fortunate to have had many students of Brooklyn College and The Graduate Center read and comment on earlier drafts:Shira Abraham,RachelAdler,Ali Assarpour,Aleksander Barkan,Sayeef Bazli,Cheuk Man Chan,WeiChen,Evgenia Dandurova,Phillip Dreizen,C.S.Fahie,Miriam Gutherc,RaveHarpaz,David Herzog,Alex Hoffnung,Matthew P.Johnson,Joel Kammet,SerdarKara,Karen Kletter,Janusz Kusyk,Tiziana Ligorio,Matt Meyer,James Ng,SeverinNgnosse,Eric Pacuit,Jason Schanker,Roman Shenderovsky,Aleksandr Shnayder-man,Rose B.Sigler,Shai Silver,Justin Stallard,Justin Tojeira,John Ma Sang Tsang,Sadia Zahoor,Mark Zelcer,and Xiaowen Zhang.We are indebted to them.Many other people looked over parts or all of the text:Scott Aaronson,Ste-fano Bettelli,Adam Brandenburger,Juan B.Climent,Anita Colvard,Leon Ehren-preis,Michael Greenebaum,Miriam Klein,Eli Kravits,Raphael Magarik,JohnMaiorana,Domenico Napoletani,Vaughan Pratt,Suri Raber,Peter Selinger,EvanSiegel,Thomas Tradler,and Jennifer Whitehead.Their criticism and helpful ideasare deeply appreciated.Thanks to Peter Rohde for creating and making available to everyone his MAT-LAB q-emulator Quack and also for letting us use it in our appendix.We had a gooddeal of fun playing with it,and we hope our readers will too.Besides writing two wonderful appendices,our friendly neighborhood librar-ian,Jill Cirasella,was always just an e-mail away with helpful advice and support.Thanks,Jill!A very special thanks goes to our editor at Cambridge University Press,HeatherBergman,for believing in our project right from the start,for guiding us through thisbook,and for providing endless support in all matters.This book would not existwithout her.Thanks,Heather!We had the good fortune to have a truly stellar editor check much of the text many times.Karen Kletter is a great friend and did a magnificent job.We also ap-preciate that she refrained from killing us every time we handed her altered draftsthat she had previously edited.But,of course,all errors are our own!This book could not have been written without the help of my daughter,Hadas-sah.She added meaning,purpose,and joy.N.S.Y.My dear wife,Rose,and our two wondrous and tireless cats,Ursula and Buster, contributed in no small measure to melting my stress away during the long andpainful hours of writing and editing:to them my gratitude and love.(Ursula is ascientist cat and will read this book.Buster will just shred it with his powerful claws.)M.A.M.。

Quantum MechanicsQuantum mechanics is a fascinating and complex field of physics that has revolutionized our understanding of the fundamental laws governing the universe. At its core, quantum mechanics deals with the behavior of particles at the smallest scales, where classical physics no longer applies. This branch of physics has given rise to a multitude of mind-bending concepts that challenge ourintuition and push the boundaries of what we thought was possible. One of the most fundamental principles of quantum mechanics is the concept of superposition. This idea states that a particle can exist in multiple states simultaneously until it is measured, at which point it collapses into a single state. This concept is famously illustrated by Schr?dinger's thought experiment involving a cat in a box, which is both alive and dead until the box is opened. The implications of superposition are profound and have led to the development of technologies such as quantum computing, which harnesses the power of superposition to perform calculations at speeds far beyond what classical computers can achieve. Another key concept in quantum mechanics is entanglement, which Einstein famously referred to as "spooky action at a distance." Entanglement occurs when two particles become linked in such a way that the state of one particle is instantaneously correlated with the state of the other, regardless of the distance between them. This phenomenon has been experimentally verified and has profound implications for the nature of reality, suggesting that particles can communicate with each otherfaster than the speed of light. The uncertainty principle, formulated by Werner Heisenberg, is another cornerstone of quantum mechanics that states that it is impossible to simultaneously know both the exact position and momentum of a particle. This principle introduces a fundamental indeterminacy into the behavior of particles at the quantum level, challenging our classical notions of determinism and causality. The uncertainty principle has profound implications for our understanding of the nature of reality and has inspired philosophical debates about the nature of free will and the limits of human knowledge. Quantum mechanics also introduces the concept of wave-particle duality, which suggeststhat particles such as electrons and photons exhibit both wave-like and particle-like behavior depending on how they are observed. This duality is exemplified bythe famous double-slit experiment, where particles exhibit interference patterns characteristic of waves when not observed, but behave like particles when observed. This phenomenon challenges our classical intuitions about the nature of particles and has led to the development of quantum field theory, which describes particles as excitations of underlying quantum fields. Despite its incredible success in explaining the behavior of particles at the quantum level, quantum mechanics remains a deeply mysterious and enigmatic theory. The probabilistic nature of quantum phenomena, where the outcome of a measurement is inherently uncertain, has led to debates about the role of consciousness in collapsing the wave function and determining the outcome of experiments. This has sparked discussions about the nature of reality and the limits of human knowledge, with some arguing that quantum mechanics reveals a fundamentally subjective aspect of the universe. In conclusion, quantum mechanics is a rich and multifaceted field that challenges our classical intuitions and pushes the boundaries of what we thought was possible. From the concept of superposition to the phenomenon of entanglement, quantum mechanics has revolutionized our understanding of the fundamental laws governing the universe. While the theory remains deeply mysterious and enigmatic, its implications for technology and our understanding of reality are profound. As we continue to explore the implications of quantum mechanics, we are sure to uncover even more mind-bending phenomena that will reshape our understanding of the universe.。

Contemporary MathematicsDudley Ernest LittlewoodAlun O MorrisDudley Ernest(D.E.)Littlewood was born in London on September7th1903, the only son of Harry Bramley Littlewood,a Solicitor’s Clerk,and Ada Piper. He must have shown early promise,for on completion of his school education at Tottenham County School,he obtained a highly coveted place at Trinity College, Cambridge supported by a major open entrance scholarship in addition to a state scholarship.Significantly,due to later developments,fellow students in Cambridge were Philip Hall and W.V.D.Hodge;all three graduated as Wranglers in1925.His Director of Studies in Trinity was J.E.Littlewood(no relation)who in a testimonial perceptively commented that‘his has always been an unusual type of mind’. D.E.Littlewood did commence on research in analysis in the strong research school headed by J.E.Littlewood with S.Pollard as his research supervisor.However,he did not take to the rigour of research in analysis and so,after a year he withdrew -there were alsofinancial reasons.For the following two years he took up a number of temporary positions as a schoolteacher,including a short period at Christ’s Hospital School.Again,this was not really to his taste,he even claimed in the farewell dinner on his retirement that he had been dismissed from one position due to problems with class discipline. However,in1928he had in view of his subsequent career,the great good fortune to be appointed as a temporary part-time Lecturer at University College,Swansea a constituent college of the University of Wales.This was where A.R.Richardson, one of a few algebraists in the U.K.at that time,was Professor of Mathematics and whom Littlewood found was‘bursting with problems’.Thus commenced in the early years a fruitful collaboration and then a long career of highly imaginative innovative research.Littlewood was to stay in Swansea until1947,except for a short period in1930as a full-time assistant in Dundee.Richardson must have immediately appreciated his potential for he ensured his return through his appointment to an Assistant Lectureship in1930and promotion to a Lectureship in1934.In1930,Littlewood had married Muriel Dyson and in1935their only child Malcolm was born.Although the family were very happy in Swansea,Littlewood was keen to return to Cambridge.In1947his wish was fulfilled through his ap-pointment as a University Lecturer,but he had no college attachment.This meant that he had no tutoring duties,but also that he had no office.However,his ear-lier student colleague,Hodge,came to the rescue and allowed Littlewood to use a second room he had in the Arts School where,for example,Littlewood met upc 0000(copyright holder)12ALUN O MORRISwith his postgraduate students J.A.(Sandy)Green and G.E.(Tim)Wall.But he and his family were not happy in Cambridge and so he was extremely pleased to return to Wales in1948tofill the vacant Chair of Mathematics at the University College of North Wales,Bangor-another constituent college of the University of Wales.This is where he spent the following22years up to his retirement in1970. He lived with his family in a neighbouring coastal resort Llandudno some twenty miles away from where he travelled by train to work on three days every week.He died at home on6th October1979a few weeks after breaking a leg.Although Littlewood claimed that some of the early problems which Richardson presented to him did notfire his imagination,a recent historical note[2]suggests that his paper[6]was equally significant with those of O.Ore and J.H.M.Wed-derburn in extending the concept of quotient rings to noncommutative rings-he considered amongst other things what is now called thefirst Weyl algebra.Sig-nificantly the motivation came from the direction of mathematical physics and in particular Dirac’s classic book on quantum mechanics,an interest that persisted throughout his life.Also,Patrick du Val credited Littlewood[5]as being thefirst to identify the quaternion groups T and I with the vertices of the regular polytopes {3,4,3}and{3,3,5}.But Richardson’s greatest favour to Littlewood was to point him in the direction of the work of the German mathematicians F.G.Frobenius and I.Schur and also Alfred Young who curiously Littlewood never met.It was this which lead to his most significant contributions.Thefirst of these appeared in what is now regarded as a classic paper,the joint paper with A.R. Richardson,[9].This is where they re-discovered Schur functions,or as they named them,S-functions-they were disappointed that these already appeared in the work of I.Schur and indeed even prior to him in C.G.Jacobi work almost a century earlier.Also,this is where immanants appeared for thefirst time and where the connection of Schur functions with invariant matrices was noticed.But as H.W. Turnbull[14]indicated in his Obituary to A.R.Richardson,There was atfirst perhaps a feeling of disappointment for thecollaborators that Schur had anticipated them to some extent,but the new insight,that these immanants and S-functions(asthey came to be called in honour of Schur)were related to in-variant matrices was indeed worth the loss of precedence;and asLittlewood has said,‘It was quite marvellous to see the intricatetheory taking shape under our hands’.But the most celebrated of all the results in this paper is their so-called Littlewood-Richardson Rule for multiplying Schur functions whose significance seem to grow with the years.(A recent Google search showed up2080hits).In fact,at that stage this was basically a conjecture,although a‘proof’appeared in Littlewood’s classic book[7]attributed to G.de B.Robinson.Rigorous proofs did not appear until the late1970’s,by now there are numerous proofs from various directions but the most satisfying in many respects is the one given by I.G.Macdonald in his book[10]in that it‘rescued’the one presented by Littlewood and kept theflavour.This is not the place to give a full account of Littlewood’s subsequent work. More details about his research and other contributions may be found in the Obit-uary[1].This will relate his massive contribution to the use of Schur functions in invariant theory which has not been properly acknowledged,his many strikingD E LITTLEWOOD3 Schur function identities,the introduction to plethysms of Schur functions,his pa-per on modular representations of the symmetric groups and even his contributions to theoretical physics,especially those in his latter years.What will,however,be attempted will be to give the background to his dis-covery of the Hall-Littlewood symmetric functions or polynomials as these were his last contribution in this area and were the subject of this conference.It was in late1956that he presented me with my Ph D problem of determining what he called the spin representations of the symmetric groups suggesting that I should approach this by restricting the spin representation of the orthogonal groups to the symmetric groups.Some progress was made;but sometime in the summer of1958Littlewood met up with Sandy Green who was then a lecturer at the University of Manchester.Green asked Littlewood what he was working on and Littlewood presumably answered(for that was his attitude by that time)that by then that he did little himself and worked mainly through his research students. So Littlewood told him about my work-Green immediately responded by asking him whether he knew about I.Schur’s monumental tour-de-force where he already had determined the irreducible projective representations of the symmetric groups in his1911paper[13].Green was rather pleased with himself in that he had been able to catch Littlewood out in this way for Littlewood thought that he was quite expert on the works of Frobenius and Schur.It was in this paper,of course,that Schur had introduced what are now referred to as Schur Q-functions which play the same role for projective representations as the usual Schur functions play for the ordinary representations of the symmetric groups.Littlewood was already aware of Green’s own remarkable tour-de-force in1955where he had solved the problem of determining the irreducible characters of the general linear groups[3].(Indeed, Littlewood had given another research student,Ivor Morris,the impossible problem of simplifying Green’s work!).In that paper,Green made crucial use of the so-called Hall polynomials introduced by Phillip Hall in some unpublished work.There,a family of symmetric functions Pλ(x,t)had been defined in an indirect way.(See the account on Philip Hall’s work for more details).However,Littlewood in typical fashion very quickly responded by defining a new class of symmetric functions which he claimed are the same as those due to Hall of which Schur and Schur Q-functions are special cases.In October1958he presented his new results to his research students in three lectures.It even meant a totally new and uniform way of getting to Schur and Schur Q-functions.He immediately wrote the work up for publication and submitted it on10th December 1958to the Proceedings of the London Mathematical Society.Although he did not share this information with his research students,he must have been more than disappointed in early1959when his paper was summarily rejected.This must have confirmed his feeling by then that his work was not properly appreciated or understood.The referee was clearly ignorant on the background and certainly had no inkling of what had truly been achieved.The paper was,of course,hard to follow and may have lacked rigour,especially in its original form.But Littlewood had sent Green a copy and Green immediately realised what had been achieved -indeed Littlewood had solved a conjecture made by Ian Macdonald who was a colleague of Green at Manchester at that time and who had been introduced to Hall’s work.In due course,Green was astonished to hear about the rejection and remarkably he was somehow able to persuade the Society to recall the paper.The4ALUN O MORRISrevised version,with a few minor changes,was submitted on20th May1960and was immediately accepted[8]-this is reprinted below.In fact,as mentioned above,Littlewood claimed that his symmetric functions Qλ(x,t)‘are the same as those defined by Hall in some unpublished...’,in fact they are different but are related by the relation Qλ(x,t)=bλ(t)Pλ(x,t).The initial proof of this was complicated and depended on some heavy calculations involving the characters of the general linear groups and their orthogonality relations obtained by Green[4].A more direct proof was given by the author[11]in1963.In fact, Littlewood got his due recognition when Ian Macdonald re-christened them Hall-Littlewood functions in his book[10]in1979.Indeed,it was through this book that not only did these functions get their due recognition,but indeed a great deal of Littlewood’s earlier work was expounded in a more rigorous and understandable way.As mentioned earlier,Littlewood did not publish further in this area.As quoted in his Obituary,in a letter to W.Ledermann he said‘I was writing(in the1950’s) lots of mathematical papers of which nobody took any notice and I wondered what was the point of it all.So I stopped writing them except in connection with work for research students’.Not only am I grateful that he wrote this one,but the numerous references in the last quarter century makes it clear that others would feel in the same way.It is sad that he did not live to see the world-wide recognition of his work by now.In fact,he will be one of the few recognised by his inclusion in the New Oxford Dictionary of National Biography.Again,for a more detailed appraisal of Littlewood,both of his work and of him as a person,I refer to the Obituary[1].As mentioned there,he was a very shy per-son who mixed very little with others.There is no record of how he interacted with his fellow students as an undergraduate.Although,their work eventually coincided through the Hall-Littlewood polynomials named after them,it seems that Little-wood never subsequently interacted with Philip Hall.Both of them were lonely figures in algebra in the U.K.in the1930’s,in Hall’s Obituary[12]it is claimed that‘For many years he was the only algebraist working in England’.There were others in the U.K.of course,but they,like Littlewood were in Wales or in Scotland where H.W.Turnbull and A.C.Aitken were pre-eminent.Philip Hall however was also‘not gregarious,and cared little for large gatherings or formal occasions; he was reticent rather than shy’.With both of them sharing these qualities,it is perhaps not unsurprising that they never met.But the main reason for this was their totally different approach or attitude to mathematics,and Littlewood possibly fell into the category‘He was not always so helpful to others with whom he was in serious disagreement’.Littlewood was also totally isolated from other contemporaries.He did corre-spond with others such as van der Waerden,A.Young and A.C.Aitken.He never met G.de B.Robinson but had met H.W.Turnbull,but only because he was an External Examiner in the University of Wales.Littlewood never attended math-ematical meetings or conferences.However in the summer of1958,he was asked by the Senate of his University to officially represent them at the meeting of the International Congress of Mathematics in Edinburgh.It is claimed that he spent his whole time there asking‘Have you seen Aitken?’-it is not known whether they actually did meet.D E LITTLEWOOD5After my appointment to a lectureship at Aberystwyth,I invited Littlewood on three occasions to visit and to give seminars,but he refused on each occasion. Thefirst time,his reason for not coming was that his dog was ill;the second time that his mother-in-law was ill;the third time that his wife was unwell.My belief is that his wife was not too keen to see him travel.There is no evidence to confirm that she was a real invalid-she did outlive him by a number of years.While at Bangor,he claimed that the only time that he found for research was on his train journeys on three days per week between Llandudno and Bangor for at home he had too many household chores to deal with.Littlewood may have complained about the lack of interest in his work,but possibly he was largely responsible for this in that he did not realise that publication alone is not sufficient and other avenues must also be taken to‘sell’one’s work.References[1] C.C.H.Barker and A.O.Morris,Obituary-Dudley Ernest Littlewood,Bull.London Math.Soc.,15(1983),56-69.[2]S.C.Coutinho,Quotient rings of noncommutative rings in thefirst half of the20th century,Arch.Hist.Exact Sci.,58(2004),255-281.[3]J.A.Green,The characters of thefinite general linear groups,Trans.Amer.Math.Soc.80,(1955),402-447.[4]J.A.Green,Les polynomes de Hall et les caract`e res des groupes GL(n,q),Colloque d’alg`e bresup´e rieure,(Brussels,1956),206-217.[5] D.E.Littlewood,The groups of the regular solids in n-dimensions,Proc.London Math.Soc.(2)32,(1931),10-20.[6] D.E.Littlewood,On the classification of algebras,Proc.London Math.Soc.(2)35,(1933),200-240.[7] D. E.Littlewood,The theory of group characters and matrix representations of groups(Clarendon Press,Oxford)1940.[8] D.E.Littlewood,On certain symmetric functions,Proc.London Math.Soc.(3)11,(1961),485-498.[9] D.E.Littlewood and A.R.Richardson,Group characters and algebra,Phil.Trans.Roy.Soc.London Ser.A233(1934),99-141.[10]I.G.Macdonald,Symmetric functions and Hall polynomials(Clarendon Press,Oxford)1979.[11] A.O.Morris,The multiplication of Hall functions,Proc.London Math.Soc.(3)13,(1963),733-742.[12]J.E.Roseblade,J.G.Thompson and J.A.Green,Obituary-Philip Hall,Bull.LondonMath.Soc.16(1984),603-626[13]I.Schur,¨Uber die Darstellung der symmmetrischen und der alternierenden Gruppe durchgebrochene lineare Substitutionen.J.reine angew.Math.139,(1911),155-250.[14]H.W.Turnbull,Obituary-Archibald Read Richardson,J.London Math.Soc.,31(1956),376-384.Institute of Mathematics and Physics,University of Wales,Aberystwyth,Ceredi-gion SY232HAE-mail address:alun@。

量子力学英语
量子力学是一种描述微观世界的理论，在实验中已被证明具有极高的准确性。

虽然量子力学的概念和数学语言非常抽象，但它已成为许多现代科技和工程领域的基础。

因此，对于学习和研究量子力学的人来说，掌握一些相关的英语词汇和表达方式是非常重要的。

以下是一些量子力学英语词汇和表达方式的例子：
1. Quantum mechanics - 量子力学
2. Wave function - 波函数
3. Superposition principle - 叠加原理
4. Uncertainty principle - 不确定性原理
5. Entanglement - 纠缠态
6. Quantum state - 量子态
7. Measurement - 测量
8. Eigenstate - 本征态
9. Operator - 算符
10. Hamiltonian - 哈密顿量
11. Schrdinger equation - 薛定谔方程
12. Commutation - 对易关系
13. Hermitian operator - 厄米算符
14. Unitary operator - 酉算符
15. Quantum field theory - 量子场论
通过学习这些量子力学英语词汇和表达方式，可以更好地理解和
交流量子力学相关的概念和研究成果。

凝聚态物理材料物理专业考博量子物理领域英文高频词汇1. Quantum Mechanics - 量子力学2. Wavefunction - 波函数3. Hamiltonian - 哈密顿量4. Schrödinger Equation - 薛定谔方程5. Quantum Field Theory - 量子场论6. Quantum Entanglement - 量子纠缠7. Uncertainty Principle - 不确定性原理8. Quantum Tunneling - 量子隧穿9. Quantum Superposition - 量子叠加10. Quantum Decoherence - 量子退相干11. Spin - 自旋12. Quantum Computing - 量子计算13. Quantum Teleportation - 量子纠缠传输14. Quantum Interference - 量子干涉15. Quantum Information - 量子信息16. Quantum Optics - 量子光学17. Quantum Dots - 量子点18. Quantum Hall Effect - 量子霍尔效应19. Bose-Einstein Condensate - 玻色-爱因斯坦凝聚态20. Fermi-Dirac Statistics - 费米-狄拉克统计中文翻译：1. Quantum Mechanics - 量子力学2. Wavefunction - 波函数3. Hamiltonian - 哈密顿量4. Schrödinger Equation - 薛定谔方程5. Quantum Field Theory - 量子场论6. Quantum Entanglement - 量子纠缠7. Uncertainty Principle - 不确定性原理8. Quantum Tunneling - 量子隧穿9. Quantum Superposition - 量子叠加10. Quantum Decoherence - 量子退相干11. Spin - 自旋12. Quantum Computing - 量子计算13. Quantum Teleportation - 量子纠缠传输14. Quantum Interference - 量子干涉15. Quantum Information - 量子信息16. Quantum Optics - 量子光学17. Quantum Dots - 量子点18. Quantum Hall Effect - 量子霍尔效应19. Bose-Einstein Condensate - 玻色-爱因斯坦凝聚态20. Fermi-Dirac Statistics - 费米-狄拉克统计。

高二英语科学家名称单选题20题1.Who is known for inventing papermaking?A.ConfuciusB.GalileoC.Cai LunD.Newton答案：C。

蔡伦发明了造纸术。

孔子是思想家教育家，伽利略是天文学家和物理学家，牛顿是物理学家和数学家。

2.Which ancient scientist is famous for his contributions to astronomy?A.Zhang Hengo TzuC.PlatoD.Aristotle答案：A。

张衡对天文学有很大贡献。

老子是思想家，柏拉图和亚里士多德是哲学家。

3.Who is renowned for his medical achievements in ancient times?A.HippocratesB.SocratesC.PythagorasD.Euclid答案：A。

希波克拉底在古代以医学成就闻名。

苏格拉底是哲学家，毕达哥拉斯是数学家，欧几里得是数学家。

4.Which ancient scientist is associated with the invention of the seismograph?A.Bi ShengB.Zhuge LiangC.Zhang HengD.Sima Qian答案：C。

张衡发明了地动仪。

毕昇发明活字印刷术，诸葛亮是政治家军事家，司马迁是史学家。

5.Who is known for his contributions to mathematics in ancient Greece?A.EuclidB.Alexander the GreatC.SolonD.Herodotus答案：A。

欧几里得对古希腊数学有贡献。

亚历山大大帝是军事家，梭伦是政治家，希罗多德是历史学家。

6.Who is known for his theory of relativity?A.NewtonB.EinsteinC.DarwinD.Franklin答案：B。

a r X i v :g r -q c /9911060v 1 17 N o v 1999KIMS-1999-11-12gr-qc/9911060Quantum mechanical time contradicts theuncertainty principleHitoshi Kitada Department of Mathematical Sciences University of Tokyo Komaba,Meguro,Tokyo 153-8914,Japan e-mail:kitada@kims.ms.u-tokyo.ac.jp http://kims.ms.u-tokyo.ac.jp/November 17,1999Abstract.The a priori time in conventional quantum mechanics is shown to contra-dict the uncertainty principle.A possible solution is given.In classical Newtonian mechanics,one can define mean velocity v by v =x/t of a particle that starts from the origin at time t =0and arrives at position x at time t ,if we assume that the coordinates of space and time are given in an a priori sense.This definition of velocity and hence that of momentum do not produce any problems,which assures that in classical regime there is no problem in the notion of space-time.Also in classical relativistic view,this would be valid insofar as we discuss the motion of a particle in the coordinates of the observer’s.Let us consider quantum mechanical case where the space-time coordinates are given a priori .Then the mean velocity of a particle that starts from a point around the origin at time 0and arrives at a point around x at time t should be defined as v =x/t .Thelonger the time length t is,the more exact this value will be,if the errors of the positions at time 0and t are the same extent,say δ>0,for all t .This is a definition of the velocity,so this must hold in exact sense if the definition works at all.Thuswe have a precise value of (mean)momentum p =mv at a large time t (1)with m being the mass of the particle.Note that the mean momentum approaches the momentum at time t when t →∞as the interaction of the particle with other particles vanishes as t →∞.However in quantum mechanics,the uncertainty principle prohibits the position and momentum from taking exact values simultaneously.For illustration we consider a nor-malized state ψsuch that ψ =1in one dimensional case.Then the expectation values of the position and momentum operators Q =x and P =¯h dxon the state ψare given by q =(Qψ,ψ),p =(P ψ,ψ)1respectively,and their variances are∆q = (Q −q )ψ ,∆p = (P −p )ψ .Then their product satisfies the inequality∆q ·∆p = (Q −q )ψ (P −p )ψ ≥|((Q −q )ψ,(P −p )ψ)|=|(Qψ,P ψ)−qp |≥|Im((Qψ,P ψ)−qp )|=|Im(Qψ,P ψ)|=12¯h ly∆q ·∆p ≥¯h 2δ(3)because we have assumed the error ∆q of the coordinate x of the particle at time t is less than δ>0.But the argument (1)above tells that ∆p →0when t →∞,contradicting(3).This observation shows that,if given a pair of a priori space and time coordinates,quantum mechanics becomes contradictory.A possible solution would be to regard the independent quantities,space and mo-mentum operators,as the fundamental quantities of quantum mechanics.As time t can be introduced as a ratio x/v on the basis of the notion of space and momentum in this view †,time is a redundant notion that should not be given a role independent of space and momentum.It might be thought that in this view we lose the relation v =x/t that is necessary for the notion of time to be valid,if space and momentum operators are completely independent as we have seen.However there can be found a relation like x/t =v as an approximate relation that holds to the extent that the relation does not contradict the uncertainty principle ([1],[2]).The quantum jumps that are assumed as an axiom on observation in usual quantum mechanics may arise from the classical nature of time that determines the position and momentum in precise sense simultaneously.This nature of time may urge one to think jumps must occur and consequently one has to observe definite eigenstates.In actuality what one is able to observe is scattering process,but not the eigenstates as thefinal states of the ly jumps and eigenstates are ghosts arising based on the passed classical notion of time.Or in more exact words,the usual quantum mechanical theory is an overdetermined system that involves too many independent variables:space, momentum,and time,and in that framework time is not free from the classical image that velocity is defined by v=x/t.References[1]H.Kitada,Theory of local times,Il Nuovo Cimento109B,N.3(1994),281-302.(/abs/astro-ph/9309051,http://kims.ms.u-tokyo.ac.jp/time_I.tex).[2]H.Kitada,Quantum Mechanics and Relativity—Their Unification by Local Time,in“Spectral and Scattering Theory,”Edited by A.G.Ramm,Plenum Publishers,New York,pp.39-66,1998.(/abs/gr-qc/9612043,http://kims.ms.u-tokyo.ac.jp/ISAAC.tex,time_IV.tex).3。

算子代数书籍全文共四篇示例，供读者参考第一篇示例：算子代数(Book on Operator Algebra)算子代数作为数学领域里一个重要的分支，涉及到代数学、泛函分析、拓扑学等多个学科的交叉领域，是现代数学的一个基础且重要的研究方向。

算子代数的研究内容主要是以算子（线性变换）为基本对象的代数结构，在量子力学、控制理论、信号处理等领域有广泛的应用。

在算子代数的研究中，书籍是学习和研究最重要的工具之一。

下面我们将介绍一些经典且值得读者深入研究的算子代数书籍。

我们推荐David Blackadar的著作《Operator Algebras：Theory of C*-Algebras and von Neumann Algebras》。

这本书是算子代数领域的经典之作，内容涵盖了C*-代数和von Neumann代数的理论及相关应用。

书中以清晰的逻辑结构和详细的数学推导为特点，适合有一定数学基础的读者阅读。

除了以上几本书籍外，还有许多其他优秀的算子代数书籍，如Bruce Blackadar的《K-Theory for Operator Algebras》、John B. Conway的《A Course in Operator Theory》等，都值得读者关注和研究。

第二篇示例：算子代数是一门研究线性算子及其代数性质的数学分支，是数学中一个重要且广泛应用的领域。

算子代数的概念最早可以追溯到20世纪初，随着数学理论的深入探索和发展，算子代数的研究范围不断拓展，应用领域也越来越广泛，成为现代数学中一个不可或缺的领域。

在算子代数的学习和研究中，一本好的教材或参考书籍是必不可少的。

一本优秀的算子代数书籍应该具备清晰的逻辑结构、深入浅出的讲解、充分的实例和习题以及最新的研究成果和发展动态。

今天我们就来推荐一些值得一读的算子代数书籍，希望能够帮助学习者更好地掌握这一领域的知识。

首先推荐的是《Introduction to Operator Algebras》这本书。

1、microscopic world 微观世界2、macroscopic world 宏观世界3、quantum theory 量子[理]论4、quantum mechanics 量子力学5、wave mechanics 波动力学6、matrix mechanics 矩阵力学7、Planck constant 普朗克常数8、wave-particle duality 波粒二象性9、state 态10、state function 态函数11、state vector 态矢量12、superposition principle of state 态叠加原理13、orthogonal states 正交态14、antisymmetrical state 正交定理15、stationary state 对称态16、antisymmetrical state 反对称态17、stationary state 定态18、ground state 基态19、excited state 受激态20、binding state 束缚态21、unbound state 非束缚态22、degenerate state 简并态23、degenerate system 简并系24、non-deenerate state 非简并态25、non-degenerate system 非简并系26、de Broglie wave 德布罗意波27、wave function 波函数28、time-dependent wave function 含时波函数29、wave packet 波包30、probability 几率31、probability amplitude 几率幅32、probability density 几率密度33、quantum ensemble 量子系综34、wave equation 波动方程35、Schrodinger equation 薛定谔方程36、Potential well 势阱37、Potential barrien 势垒38、potential barrier penetration 势垒贯穿39、tunnel effect 隧道效应40、linear harmonic oscillator 线性谐振子41、zero proint energy 零点能42、central field 辏力场43、Coulomb field 库仑场44、δ-function δ-函数45、operator 算符46、commuting operators 对易算符47、anticommuting operators 反对易算符48、complex conjugate operator 复共轭算符49、Hermitian conjugate operator 厄米共轭算符50、Hermitian operator 厄米算符51、momentum operator 动量算符52、energy operator 能量算符53、Hamiltonian operator 哈密顿算符54、angular momentum operator 角动量算符55、spin operator 自旋算符56、eigen value 本征值57、secular equation 久期方程58、observable 可观察量59、orthogonality 正交性60、completeness 完全性61、closure property 封闭性62、normalization 归一化63、orthonormalized functions 正交归一化函数64、quantum number 量子数65、principal quantum number 主量子数66、radial quantum number 径向量子数67、angular quantum number 角量子数68、magnetic quantum number 磁量子数69、uncertainty relation 测不准关系70、principle of complementarity 并协原理71、quantum Poisson bracket 量子泊松括号72、representation 表象73、coordinate representation 坐标表象74、momentum representation 动量表象75、energy representation 能量表象76、Schrodinger representation 薛定谔表象77、Heisenberg representation 海森伯表象78、interaction representation 相互作用表象79、occupation number representation 粒子数表象80、Dirac symbol 狄拉克符号81、ket vector 右矢量82、bra vector 左矢量83、basis vector 基矢量84、basis ket 基右矢85、basis bra 基左矢86、orthogonal kets 正交右矢87、orthogonal bras 正交左矢88、symmetrical kets 对称右矢89、antisymmetrical kets 反对称右矢90、Hilbert space 希耳伯空间91、perturbation theory 微扰理论92、stationary perturbation theory 定态微扰论93、time-dependent perturbation theory 含时微扰论94、Wentzel-Kramers-Brillouin method W. K. B.近似法95、elastic scattering 弹性散射96、inelastic scattering 非弹性散射97、scattering cross-section 散射截面98、partial wave method 分波法99、Born approximation 玻恩近似法100、centre-of-mass coordinates 质心坐标系101、laboratory coordinates 实验室坐标系102、transition 跃迁103、dipole transition 偶极子跃迁104、selection rule 选择定则105、spin 自旋106、electron spin 电子自旋107、spin quantum number 自旋量子数108、spin wave function 自旋波函数109、coupling 耦合110、vector-coupling coefficient 矢量耦合系数111、many-particle system 多子体系112、exchange forece 交换力113、exchange energy 交换能114、Heitler-London approximation 海特勒-伦敦近似法115、Hartree-Fock equation 哈特里-福克方程116、self-consistent field 自洽场117、Thomas-Fermi equation 托马斯-费米方程118、second quantization 二次量子化119、identical particles 全同粒子120、Pauli matrices 泡利矩阵121、Pauli equation 泡利方程122、Pauli’s exclusion principle泡利不相容原理123、Relativistic wave equation 相对论性波动方程124、Klein-Gordon equation 克莱因-戈登方程125、Dirac equation 狄拉克方程126、Dirac hole theory 狄拉克空穴理论127、negative energy state 负能态128、negative probability 负几率129、microscopic causality 微观因果性本征矢量eigenvector本征态eigenstate本征值eigenvalue本征值方程eigenvalue equation本征子空间eigensubspace (可以理解为本征矢空间)变分法variatinial method标量scalar算符operator表象representation表象变换transformation of representation表象理论theory of representation波函数wave function波恩近似Born approximation玻色子boson费米子fermion不确定关系uncertainty relation狄拉克方程Dirac equation狄拉克记号Dirac symbol定态stationary state定态微扰法time-independent perturbation定态薛定谔方程time-independent Schro(此处上面有两点)dinger equation 动量表象momentum representation角动量表象angular mommentum representation占有数表象occupation number representation坐标（位置）表象position representation角动量算符angular mommentum operator角动量耦合coupling of angular mommentum对称性symmetry对易关系commutator厄米算符hermitian operator厄米多项式Hermite polynomial分量component光的发射emission of light光的吸收absorption of light受激发射excited emission自发发射spontaneous emission轨道角动量orbital angular momentum自旋角动量spin angular momentum轨道磁矩orbital magnetic moment归一化normalization哈密顿hamiltonion黑体辐射black body radiation康普顿散射Compton scattering基矢basis vector基态ground state基右矢basis ket ‘右矢’ket基左矢basis bra简并度degenerancy精细结构fine structure径向方程radial equation久期方程secular equation量子化quantization矩阵matrix模module模方square of module内积inner product逆算符inverse operator欧拉角Eular angles泡利矩阵Pauli matrix平均值expectation value （期望值）泡利不相容原理Pauli exclusion principle氢原子hydrogen atom球鞋函数spherical harmonics全同粒子identical particles塞曼效应Zeeman effect上升下降算符raising and lowering operator 消灭算符destruction operator产生算符creation operator矢量空间vector space守恒定律conservation law守恒量conservation quantity投影projection投影算符projection operator微扰法pertubation method希尔伯特空间Hilbert space线性算符linear operator线性无关linear independence谐振子harmonic oscillator选择定则selection rule幺正变换unitary transformation幺正算符unitary operator宇称parity跃迁transition运动方程equation of motion正交归一性orthonormalization正交性orthogonality转动rotation自旋磁矩spin magnetic monent(以上是量子力学中的主要英语词汇，有些未涉及到的可以自由组合。

塔诺基量子力学第三卷英文回答：Quantum mechanics, as described in the third volume of the Tannocky series, is a fascinating and complex field of study. It deals with the behavior of particles at the quantum level, where classical physics no longer applies.In this realm, particles can exist in multiple states simultaneously and can exhibit wave-particle duality.One of the fundamental principles of quantum mechanicsis superposition. This concept states that a particle can exist in multiple states or positions at the same time. For example, imagine a particle that can be in either the "up"or "down" state. In classical physics, the particle wouldbe in one state or the other. However, in quantum mechanics, the particle can exist in a superposition of both states simultaneously. It is only when a measurement is made that the particle "chooses" one of the states.Another important principle in quantum mechanics is entanglement. This phenomenon occurs when two particles become linked in such a way that the state of one particle is dependent on the state of the other, regardless of the distance between them. This concept is often described using the famous thought experiment of two entangled particles, known as "spooky action at a distance." For example, if two entangled particles are in a superposition of "up" and "down" states, measuring one particle will instantaneously determine the state of the other particle, even if they are light-years apart.Quantum mechanics also introduces the concept of uncertainty, as described by Heisenberg's uncertainty principle. This principle states that certain pairs of physical properties, such as position and momentum, cannot both be precisely known at the same time. The more precisely one property is known, the less precisely the other can be known. This inherent uncertainty at the quantum level is a fundamental aspect of the nature of reality.Overall, quantum mechanics is a highly mathematical and abstract field, but its principles have been successfully applied in various technological advancements. For example, quantum computers utilize the principles of superposition and entanglement to perform calculations at a much faster rate than classical computers. Quantum cryptography also relies on the principles of quantum mechanics to ensure secure communication.中文回答：量子力学是塔诺基系列第三卷中描述的一个引人入胜且复杂的研究领域。

量子力学专业英语词汇1、microscopic world 微观世界2、macroscopic world 宏观世界3、quantum theory 量子[理]论4、quantum mechanics 量子力学5、wave mechanics 波动力学6、matrix mechanics 矩阵力学7、Planck constant 普朗克常数8、wave-particle duality 波粒二象性9、state 态10、state function 态函数11、state vector 态矢量12、superposition principle of state 态叠加原理13、orthogonal states 正交态14、antisymmetrical state 正交定理15、stationary state 对称态16、antisymmetrical state 反对称态17、stationary state 定态18、ground state 基态19、excited state 受激态20、binding state 束缚态21、unbound state 非束缚态22、degenerate state 简并态23、degenerate system 简并系24、non-deenerate state 非简并态25、non-degenerate system 非简并系26、de Broglie wave 德布罗意波27、wave function 波函数28、time-dependent wave function 含时波函数29、wave packet 波包30、probability 几率31、probability amplitude 几率幅32、probability density 几率密度33、quantum ensemble 量子系综34、wave equation 波动方程35、Schrodinger equation 薛定谔方程36、Potential well 势阱37、Potential barrien 势垒38、potential barrier penetration 势垒贯穿39、tunnel effect 隧道效应40、linear harmonic oscillator 线性谐振子41、zero proint energy 零点能42、central field 辏力场43、Coulomb field 库仑场44、δ-function δ-函数45、operator 算符46、commuting operators 对易算符47、anticommuting operators 反对易算符48、complex conjugate operator 复共轭算符49、Hermitian conjugate operator 厄米共轭算符50、Hermitian operator 厄米算符51、momentum operator 动量算符52、energy operator 能量算符53、Hamiltonian operator 哈密顿算符54、angular momentum operator 角动量算符55、spin operator 自旋算符56、eigen value 本征值57、secular equation 久期方程58、observable 可观察量59、orthogonality 正交性60、completeness 完全性61、closure property 封闭性62、normalization 归一化63、orthonormalized functions 正交归一化函数64、quantum number 量子数65、principal quantum number 主量子数66、radial quantum number 径向量子数67、angular quantum number 角量子数68、magnetic quantum number 磁量子数69、uncertainty relation 测不准关系70、principle of complementarity 并协原理71、quantum Poisson bracket 量子泊松括号72、representation 表象73、coordinate representation 坐标表象74、momentum representation 动量表象75、energy representation 能量表象76、Schrodinger representation 薛定谔表象77、Heisenberg representation 海森伯表象78、interaction representation 相互作用表象79、occupation number representation 粒子数表象80、Dirac symbol 狄拉克符号81、ket vector 右矢量82、bra vector 左矢量83、basis vector 基矢量84、basis ket 基右矢85、basis bra 基左矢86、orthogonal kets 正交右矢87、orthogonal bras 正交左矢88、symmetrical kets 对称右矢89、antisymmetrical kets 反对称右矢90、Hilbert space 希耳伯空间91、perturbation theory 微扰理论92、stationary perturbation theory 定态微扰论93、time-dependent perturbation theory 含时微扰论94、Wentzel-Kramers-Brillouin method W. K. B.近似法95、elastic scattering 弹性散射96、inelastic scattering 非弹性散射97、scattering cross-section 散射截面98、partial wave method 分波法99、Born approximation 玻恩近似法100、centre-of-mass coordinates 质心坐标系101、laboratory coordinates 实验室坐标系102、transition 跃迁103、dipole transition 偶极子跃迁104、selection rule 选择定则105、spin 自旋106、electron spin 电子自旋107、spin quantum number 自旋量子数108、spin wave function 自旋波函数109、coupling 耦合110、vector-coupling coefficient 矢量耦合系数111、many-particle system 多子体系112、exchange forece 交换力113、exchange energy 交换能114、Heitler-London approximation 海特勒-伦敦近似法115、Hartree-Fock equation 哈特里-福克方程116、self-consistent field 自洽场117、Thomas-Fermi equation 托马斯-费米方程118、second quantization 二次量子化119、identical particles 全同粒子120、Pauli matrices 泡利矩阵121、Pauli equation 泡利方程122、Pauli’s exclusion principle泡利不相容原理123、Relativistic wave equation 相对论性波动方程124、Klein-Gordon equation 克莱因-戈登方程125、Dirac equation 狄拉克方程126、Dirac hole theory 狄拉克空穴理论127、negative energy state 负能态128、negative probability 负几率129、microscopic causality 微观因果性。

The principles of quantum mechanics量子力学的原理量子力学是一种研究微观粒子运动的物理学理论，由于其所涉及的尺度非常小，因此与我们日常生活中接触的物理学有着很大的不同。

量子力学早在20世纪初就开始引起了科学界的关注，随着时间的推移，人们对于量子力学的认识也愈发深刻。

本文将介绍量子力学的一些原理。

波粒二象性理论波粒二象性理论是指，物质既可以表现为粒子，也可以表现为波动。

一个量子粒子不仅具有质量和能量，同时也会表现出波动性。

例如，电子具有波的性质，具体表现在它的运动方式上。

当一个电子被观察时，它的位置被确定，电子就表现为一个点粒子；而当电子不被观察时，它呈现出波动性质。

不确定性原理不确定性原理是量子力学中最基本的原理之一。

它指出了在量子世界之中存在一种不确定性。

当我们试图观测一个量子粒子的位置和动量时，我们能够获得的信息是有限的。

这是因为观察者与粒子的相互作用会改变粒子的状态，而这种改变并不是可以被预测的。

因此，我们对于粒子位置和动量的观测结果永远是不完整的。

量子纠缠量子纠缠是指，在某些特殊的情况下，两个或更多的量子粒子之间会相互纠缠。

这种纠缠是一种独特的量子力学现象。

“纠缠”意味着这些量子粒子之间会以一种非常奇特的方式联系在一起，即使它们之间的距离非常远。

当一个粒子发生改变时，它的纠缠伙伴粒子也会立即相应地改变。

量子隧穿效应量子隧穿效应是一种量子力学现象，它表明了粒子能够穿越它在经典力学中是不可能的势垒。

在经典的物理学里，当一个粒子碰到了一个由电子或原子构成的势垒时，它就会被反弹回去。

但在量子力学的框架下，粒子有可能穿越势垒，同时保持着它的波动性。

量子态和量子算符在量子力学中，粒子的状态用量子态来描述。

它也可以用一个数学对象，量子算符，来表示。

通过计算，我们可以得到粒子处于某种状态的概率。

量子态和量子算符也可以被用来表示初态与末态之间的相互作用。

结语在量子力学的世界，经典物理学的规律已经不再适用。

十个高冷物理学名词物理学是一门研究自然界中基本物质和其相互关系的科学。

在物理学中，有许多高深的名词和概念，这些名词常常令人困惑，也是物理学家们日常工作的核心。

下面介绍十个高冷的物理学名词。

1.相对论（Relativity）相对论是爱因斯坦相对论的简称，它是现代物理学的重要基石之一。

相对论有狭义相对论和广义相对论两种形式，它们描述了时空的性质，推翻了牛顿力学的观念，提出了质量和能量之间的等价性。

2.量子力学（Quantum Mechanics）量子力学是研究微观粒子行为的物理学分支。

它引入了量子概念，描述了微观粒子的不确定性和波粒二象性。

量子力学在原子、分子和粒子物理学中具有重要应用。

3.黑洞（Black Hole）黑洞是一种极其致密的天体，其引力场强大到连光都无法逃离。

黑洞形成于恒星坍塌过程中，由大量质量集中在极小的空间内而形成。

它的奇异性和引力效应使得黑洞成为宇宙学研究中的热点。

4.弦理论（String Theory）弦理论是一种试图统一引力和量子力学的理论。

它认为物质的基本单位并非点粒子，而是无限细小的弦状物体。

弦理论在解释宇宙起源、黑洞的性质和粒子物理学等方面有重要意义。

5.量子纠缠（Quantum Entanglement）量子纠缠是一种奇特的量子现象。

当两个或多个粒子纠缠在一起时，它们之间的状态无论如何分离，都会存在相互关联。

这种纠缠关系超出了经典物理学的范畴，被广泛应用于量子通信和量子计算等领域。

6.宇宙背景辐射（Cosmic Microwave Background）宇宙背景辐射是宇宙大爆炸后余辉的热辐射。

它是宇宙中的微弱微波信号，被认为是宇宙起源的重要证据之一。

通过研究宇宙背景辐射，科学家可以了解宇宙的演化和结构。

7.量子计算（Quantum Computing）量子计算是利用量子力学原理进行计算的一种新型计算方法。

相比传统计算机，量子计算机具有更高的计算速度和处理能力，被视为未来计算科学的重要方向。

第37卷第3期2021 年3月商丘师范学院学报JOURNAL OF SHANGQIU NORMAL UNIVERSITYVol. 37 No. 3March，2021量子力学中测不准关系、方程和时空等基本问题的新探索张一方(云南大学物理系，云南昆明650091)摘要：量子力学中的某些问题仍应该探索.首先讨论了量子力学的基础和各种解释，并提出量子力学的非线 性混沌-孤子解释.其次研究了测不准关系，探讨了最普适的测不准关系及其数学形式.第三讨论不可逆性和统计 性,并提出熵的算符表示•第四研究了量子力学方程.最后探讨了量子理论中的二象性.关键词：量子力学；解释；量子场论；时空;对称性；测不准关系中图分类号:〇572.2 文献标识码：A文章编号：1672 - 3600(2021)03 - 0023 - 06New research of basic problems on uncertainty relation, equations and time - space,etc. ,in quantum mechanicsCHANG Yifang(Department of Physics,Yunnan University,Kunming650091 ,China)Abstract：Some problems of quantum mechanics should still be researched. First, the foundations and various interpretations of quantum mechanics are discussed, and the nonlinear chaos -soliton interpretation of quantum mechanics is proposed. Next,the uncertainty relations are investigated,and a very general uncertainty relation and its mathematical form are searched. Thirdly, the irreversibility and statistics are discussed, and an operator representation of entropy is proposed. Fourthly, the equations of quantum mechanics are investigated. Finally, the duality in quantum theory are searched.Key words ：quantum mechanics ； interpretation ； quantum field theory ； space - time ； symmetry ； uncertainty relationDirac指出“哈密顿量对于量子理论才真正是十分重要的”“只能通过哈密顿量或其概念的某种推广”发展理论.其基本 程序是由相对论不变的作用量积分得到拉氏量，再导出哈密顿量,得到量子理论.而“将来的量子理论”“一定有某种东西与哈 密顿理论对应已知量子力学最初的两种形式:Schrodinger波动力学主要起源于波动性;而Heisenberg矩阵力学主要起源于不连续性.其 中能量体现粒子性，波函数体现波动性.经典波动方程就是质量为0的Klein-Gord〇n(K G)方程，而Dirac方程是K G方程的一 阶推广.基于对量子力学结构的逻辑分析，笔者认为它只有一个基本原理:波粒二象性.统计性是其相应的数学特性.而其他 原理都是由此导出的物理或数学结果量子场论只是把二象性推广到场.量子力学的发展是基于长、短波时分别是Rayleigh-Jeans公式和W i e n公式，这已经暗含其主要适用于中能，而高能（短 波）是Wien公式，8卩Maxwell- Boltzmann(MB)分布、G a m m a分布•量子力学必然联系于光子、电磁相互作用，例如黑体辐射、光 电效应、氢原子等.反之，目前量子力学、量子场论也主要适用于电磁相互作用[U,而对强、弱、引力相互作用则理论必须发展. 量子力学中波包瞬间塌缩是超光速的.笔者提出粒子物理中的基本原理是必须区分已经检验的实验事实和优美的理论假说.由此提出粒子理论中的7个重大问 题，并且讨论了相应的量子理论某些可能的发展本文对测不准关系、量子方程等量子力学和量子理论的基本问题进行了 某些新探索.1量子力学的基础和各种解释由于量子力学的基本性和复杂性，对其的解释和探索一直是理论物理议论纷纷的热点之一.Jammer对量子力学中的基本 问题和各种解释进行了全面的经典论述[5].量子力学最著名的解释是哥本哈根的几率解释，它的两个基本原理是定域（干涉）原理和波谱分解原理.在量子力学中几率守恒，几率密度（即粒子数平均密度）守恒及总几率都不变.几率守恒是物质不灭定收稿日期:2020 - 01 -22;修回日期:2020 - 09 - 21基金项目：国家自然科学基金资助项目（11664044)作者简介:张一方（1947—），男，云南昆明人，云南大学教授，主要从事理论物理的研究24商丘师范学院学报2021 年律在微观世界的精确表现，它与幺正条件紧密联系.1970年Ballentine系统讨论了量子力学的统计解释[61.对几率解释提出不同观点的众多理论中最著名的是de Broglie- B o h m非线性理论和隐变量解释.基于Everett的多世界理 论[7’8],1971年Dewitt等提到多世界解释和隐变量理论[9]. 1972年Van Fraassen提出消除波包塌缩的模态解释（model interpretation).以后发展为著名的Kochen- Dieks- Healey理论.1986年Cramer提出量子力学的相互作用解释_l t l],其与Be丨丨不等式的检验和非局域性一致.1987年BaUentine提出量子力学的主要解释是:统计系综解释，新哥本哈根解释,R.B.G r i f f i t h s 一致性历史解释，多世界解释和量子势5种.1992年H o m e和Whitaker系统讨论了量子力学的现代系综解释[|2].从1954年起 nde集中批评波粒二象性,而提出一种被Born称为“唯粒子论”的解释方案，但Jammer认为这是“统计系综解释的一种特 殊版本”[51.此外，还有量子力学的去相干理论等.1992年Omn e s提出量子力学的新解释[131，并提出其认识论[13_141.其中的关键概念是“退相干”（de™herenCe).通过纠缠 态（entangled state)在受到环境的作用时，会发生退相干效应这种机制，从量子力学的基本原理出发，就可以统一描述宏观世 界和微观世界的物理学.1995年Rob Clifton及Bub.Goldstein理论的基本方法是区分为理论（动力学、数学）态（theoretical state)和事件（值、物理）态（sta t e of affairs).近年Jeffery B u b等提出量子力学信息解释.这联系于量子信息论.T h o m在《结构稳 定性与形态发生学》中提出波函数作为按一定频率改变拓扑类型的超曲面上的形态就是局域曲率[15].赵国求等具体提出相 互作用是在与量子力学的曲率解释[~7]，其中波长联系于粒子环流半径，曲率波包取代质点，波函数是曲率波，曲率的大小表 示粒子性，曲率在时空中的变化表示波动性.并且特征曲率/； = 1/4* =p/77.T h o m和赵国求等的量子力学曲率解释是把波函 数理解为曲率函数.笔者认为这其实是量子力学和广义相对论结合的结果，其中质量大小决定曲率大小，就是广义相对论中 质量决定空间的弯曲程度.根据公式p =1 2 ,密度越大，几率越大，曲率越大.这样就可以联系于量子引力.结合黎曼几何，还可以结合笔者计算电子磁矩的方法[2].薛定谔猫的佯谬是微观不确定、统计性与宏观确定性的关系.这可以联系于非线性,可能发展出非线性量子理论 Monroe等[~用囚禁在Paul阱中的9B e+实现介观尺度上的猫态，质心运动相干态波包与内部态纠缠.观测结果肯定了量子态 叠加原理的正确性，展示了量子力学中的非定域性.量子性最早类比于波动性中的驻波.如波动性有所改变，量子性也许有所 不同.量子数h可能可变或者连续.这就联系于泛量子论12^251.猫态联系于宏观量子现象，对应2个或多个世界.这些世界不 能交流，则拓扑分离，是平行世界.这些世界互相纠缠，则可以包括生与死，阴和阳等不同状态.而状态变化可能与参量达到混 沌值等有关，出现幽灵.非线性理论的多次迭代导致混沌，但对应一种统计性及分布函数.这类似多个粒子、多次事件测不准，然而具有统计性及 分布函数.它联系于电子云的概率分布和相应的统计解释.由此可以提出量子力学的非线性混沌-孤子解释.笔者证明在各 种具有孤子解的非线性方程中都可以得到混沌，而只有某些具有混沌解的非线性方程有孤子解.两种解的条件是不同的，某 些参数是某个常数时得到孤子，而这些参数在一定区域变化时出现分岔-混沌.这种混沌-孤子双解可以对应于量子理论中 的波-粒二象性，由此联系于非线性波动力学的双重解，并且存在若干新的意义％’271.某些实验证明，改变测量方式完全可以 将实验结果从波动条纹改变为粒子行为.Yanhim Shih(1983)可以确定粒子性和（或）波动性.Mandel等测量在两点的两个光子 的联合几率作为分离函数，证明在信号干涉中存在非经典效应[281.在实验设置1中单个光子不出现干涉.在实验设置2的情 况1中光子也不出现干涉，但在情况2中光子出现干涉.所以不仅单个光子，而且作为一个整体的一对纠缠光子也不出现干 涉.并且由非线性解释可以联系于流体力学解释.通常/!—0时，量子力学化为经典力学，这即对应原理.如此量子场论应该化为经典场论.但这与大量子数；并不普遍 等效.Liboff对二者的不同作了明确区分例如在大原子中可能是形式对应原理，如G細m a分布.Blasone提出一个 量子系统近似等价于两个经典系统[301.这对应de Broglie-B o h m非线性量子力学的双重解.J.von Nemnami在《量子力学的数学基础》中指出Gilbert空间的点表示物理系统的状态，物理系统的可观测量由Hilbert 空间的线性算子表示，而能量算子的本征值和本征函数就是该系统的能级及相应的定态.结论必须在4条公理假定成立的条 件下.其中E(A + B + C + •••)= E(A) + E(B) + £(C)+ •••(1)五是线性算符,/I、S、C等无相互作用•而它们可以互相纠缠.因此，目前的量子理论本质上应该是线性的[U8].这一般可能对应于自由粒子，即无相互作用，如无引力相互作用时就是 欧氏空间.而电磁相互作用也是线性理论，只有附加流时才是非线性.但光子-光子相互作用时就是非线性光学.这应该联系 于电磁广义相对论^32].—般的电磁理论仅涉及单个光子.而其余的三种相互作用场必然产生非线性.张永德认为相互作用必定导致量子理论的非线性，相对论量子场论和考虑相互作用的非相对论量子力学的基本方程组 都是非线性的;量子理论的量子化条件是非齐次二次型非线性的i33],因此整体而言，量子理论本质上是非线性的.2测不准关系1934年K.R.Popper已经指出[34]，从量子力学的基本原理可以导出测不准关系，因此它不是具有独立地位的原理. Margemni指出只应当限于在多次测量的统计意义上理解测不准关系M l.对于时间-能量的测不准关系，争论更大.1933年 Pauli就指出，不可能引进一个时间算符，因此根本不存在能量和时间的对易关系[36].粒子在运动,有速度^和动量p,其位置就不确定;反之j确定，则不运动.特别对于波.这似乎可以联系于布朗运动.测 不准应该基于无相互作用或某种特殊的波，如线性波等.前提改变，如是非线性波、孤波等时,测不准应该有所不同.基于此可 以具体推导.第3期张一方:量子力学中测不准关系、方程和时空等基本问题的新探索25H.Dehmelt开创了俘获单个电子和原子的研究，由此获得1989年诺贝尔物理奖.20世纪80年代中期Maryland大学和 Munich大学的实验发现,打开探测器时光表现为粒子，关闭探测器时单个光子也表现出波动.1990年艾戈勒用扫描隧道显微 镜（S T M)移动氙原子排列成IBM.氙原子固定，则土 = 10_s c m,Ap = mAv —h/A x.(2)氙质量^ = 131^^.=丨.229父1051\^¥/<:2,所以加=/1/^4*=48丨.33(；111/8.氙原子每秒运动4.81[11,此时测不准关系是 否成立？1993年I B M研究可以直接看到单个电子的波动性（波函数）.这些结果与测不准的关系值得研究.量子纠缠态和远距离移物已经被实验证实.其难点在于测不准原理.但该原理在一定条件下也可能被突破.波的测不准关系= 1表示波包长度和波长之间的测不准.这是波的性质，与量子理论无关.同时，粒子波动性及其方 程导致能量的分立态（即量子化）.B o h m指出测不准原理是由三个基本假定:波粒二象性，几率，能量-动量的不可分性，互相 结合导出的[371.三者的统一体是整个量子论的基础.原子中的测不准关系4£…山=7J ,BPA{- ^j)A t = h ,(3)n所以<)=忐，(4)通常是K测不准，更一般是A(R^j)A t = 1 .(5)其中是Rydberg常数.对Z、n—定的原子，如Z= 1，n= 1的氢原子只能是山—》，或者厶K不等于0,其测不准.或者 测不准，公式不成立.这就是李兹组合原理与A t的关系，B PAvAt = 1. (7)则A—〇〇，必须承认其在时空中无法描述.而目前光谱已经相当精确.n大时更易精确检验.目前形式的波动性如果不成立[2],测不准原理就应该有所发展.例如对非线性理论[2_~.进一步，测不准关系可以推广到 各种测不准量卸,,4*,.的关系.卸相当于=(卸）2/2m，因为4£乂S九，所以(A p)2At &2mh ；(8)反之 Ap 多A/A c，则 = (4/>)2/2m 彡 /i2/2m(4*)2 ,所以AE(A x)2 3：h2/2m.(9)彡/i2除以式(8)可得(止〇2//^為/i/2m，或者多办/2m •(10)这与我们得到的光速测不准公式[3M9]是一致的.然而，测不准关系与守恒定律存在不一致之处.多A ,如果A是测量能量或能量变化4£所需的时间间隔，则光子稳 定山—0，光速不变.但如此一切稳定粒子运动速度都应该不变.场方程及其孤子解原则上可以描述粒子的轨道.它可能联系于de Broglie-Bohm的波导理论.这样孤子与测不准关系不 一致.更一般是非线性量子理论[2#]与测不准原理的关系.彼此算符化的量构成共轭量.不可对易的两个算符量不能同时测定，这就是最普适的测不准关系.在此讨论其一般的数 学形式，设[W，/V] =A f/V+順=iG.其特例是C=0时是反对易关系•C就是测不准度.彼此用算符表示：M表象中）；(ID(12)反之，W(M表象中）_dM于是AMAN ^G.(13) 3量子理论、不可逆性和统计性量子理论的根本困难，Prigogine等认为在于无法处理不可逆过程.它应该结合广义熵（信息），引人微观熵、量子熵等.量 子力学是统计性的，而统计性可以描述不可逆性，所以量子力学应该描述不可逆性.量子力学方程和分子运动论的统计方程相似，所以其具有统计性.统计性又源于摩擦和信息缺失，因为经典和量子各是 任意值和离散值.布朗运动，基态能A/2,绝对零度不能达到等都表明微观领域的永动.布朗运动在一定条件下也许可以放大为宏观布朗运 动效应，虽然放大时可能已经输入能量•根据熵的定义dS = AE/T,(14)26商丘师范学院学报2021 年引入熵必然引人温度7\而根据S = - k\gP = - 2k\g f//，必变化导致S变化.熵增大对应于波包必然扩散.由式（14)得A/r；是组合常数.这类似屮 对定态dS- - 2k(p d* + p,dy+ p Az - £d() = 2i—lV VA e x p U p j/n).此时就是d S对应于四维动量.lV dt Eip ,(15)(16)(17)(18)贝!l dS =-2k—d t.(19)lVS和 <都可逆或都不可逆，二者变化成正比.封闭系统熵增大，能量守恒是定态，所以时间也有方向性.考虑时间箭头和统计性时，发展能量为熵.Prigogint■引人微观熵算符，这可以结合粒子物理中温度和熵的引人方法.他的 超算符导致不可逆性.这可能对应于非线性算符.更一般,可重整化的量子理论应该结合半群，引人温度等，导出不可逆性.假设熵与量K共轭，AS = 4£/r，4£也=(A S)(f4<),所以T不变时[=7!AF/AY = dF/TAt = [F,S]= [F,H]/T.(20)由此表示熵的算符为S _ izi AT dt •(21)设动童通量密度7^ =服2W的共扼量是X，如果L盖，则A T^AX ^v ,T^X - XT^= iv .(22)结合测不准关系P…V,X - XP^V U = y(23)所以X = X/^,T^=-l V d{x/v J.(24)而〜又是算符m dxv量子力学和统计力学都基于平均值.二者应该类比发展.密度p方程是刘维定理，其对应Heisenberg方程，而不是 Schrodinger方程.量子力学类似于统计力学发展为量子统计;统计力学类似于量子力学发展为动力学、方程.特别是量子统计 应该包括托马斯-费米方程.量子力学、量子场论和统计性都是对多个粒子事件，对系综成立.应该用统计学的方法全面整理、表述、修改量子力学、量 子场论，然后再推广、发展量子论.4量子力学方程的研究Dim e提出[w] “不应认为量子力学的现在形式是最后的形式”，它只是“迄今为止人们能够给出的最好的理论”，可能将来“会得到一个改进了的量子力学，使其回到决定论.”但这必须放弃某些现在认为没有问题的基本思想:4〇].笔者在探讨微观相 对论的基础上，提出对极小时空,光速应存在统计起伏.特别在高维柱形卷曲空间中光速是可变的和量子化的.由此讨论修 改、发展相对论和量子论的可能的某些方法，并且定量提出存在势和相互作用时几种新的量子力学方程4M2].Heisenberg方程只是Poisson括号变形的经典方程.Schrodinger方程(25)只是/>,的算符表示方程，而它都可以化为方程Pi= 4念(11^).(26)这和一般的算符方程a dP,= lV T~ ^dX i(27)又有所不同.如/(必）=_ :=〇，则 dj l i//)= + md^//= 〇，即~= 0 »(28)也就是Klein- Gord〇n(K G)方程•这是K G方程和Dime方程之间的又一种关系.对此再求导，= 〇,(29)第3期张一方:量子力学中测不准关系、方程和时空等基本问题的新探索27艮P( m2i/f) +m(m2i//)=0.(30)这是0'的D irac方程，是求导二次的结果.这是波函数的标度变换.广义函数类似算符，前者可用于多重产生，粒子理论等;后者已用于量子理论,并将用于多重产生等.Poisson括号都是对 易关系，应该可以推广为反对易关系等广义形式.5量子理论中的二象性问题衍射中运动的非全同性导致整体的统计性.量子理论的各种不足可能基于原来就无法完全一致的波粒二象性:2],因此波 尔才发展出互补原理.Y u tak a等的实验证明[431单个光子具有粒子性（在缝隙处不劈裂为两半）和波动性（具有隧穿效应，显示 出自干涉，最后反射和折射各占50% ,符合粒子波的几率性）.单个光子在远处平板上看不到干涉条纹.二象性把连续、不连续统一起来.波动有周期性就会有一种不连续，对应着粒子.光子和粒子的衍射、折射、反射、色散等 反映连续性;而光子和粒子的发射、吸收等瞬时过程则是不连续的.最小能量、心是不连续的，而频率。