Quantum mechanics/ (Record no. 176805)
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| 000 -LEADER | |
|---|---|
| fixed length control field | 00331nam a2200133Ia 4500 |
| 020 ## - INTERNATIONAL STANDARD BOOK NUMBER | |
| International Standard Book Number | 3540277064 |
| 040 ## - CATALOGING SOURCE | |
| Transcribing agency | CUS |
| 082 ## - DEWEY DECIMAL CLASSIFICATION NUMBER | |
| Classification number | 530.12 |
| Item number | BAS/Q |
| 100 ## - MAIN ENTRY--PERSONAL NAME | |
| Personal name | Basdevant, Jean-Louis |
| 245 #0 - TITLE STATEMENT | |
| Title | Quantum mechanics/ |
| Statement of responsibility, etc. | Jean-Louis Basdevant |
| 260 ## - PUBLICATION, DISTRIBUTION, ETC. (IMPRINT) | |
| Place of publication, distribution, etc. | Berlin: |
| Name of publisher, distributor, etc. | Springer, |
| Date of publication, distribution, etc. | 2005. |
| 300 ## - PHYSICAL DESCRIPTION | |
| Extent | 511 p. |
| 505 ## - FORMATTED CONTENTS NOTE | |
| Formatted contents note | <br/>1. Quantum Phenomena 1<br/>1.1 The Franck and Hertz Experiment 3<br/>1.2 Interference of Matter Waves 5<br/>1.2.1 The Young Double-Slit Experiment 6<br/>1.2.2 Interference of Atoms in a Double-Slit Experiment.... 7<br/>1.2.3 Probabilistic Aspect of Quantum Interference 8<br/>1.3 The Experiment of Davisson and Germer 10<br/>1.3.1 Diffraction of X Rays by a Crystal 10<br/>1.3.2 Electron Diffraction 12<br/>1.4 Summary of a Few Important Ideas 15<br/>Further Reading 15<br/>Exercises 16<br/>2. The Wave Function and the Schrodinger Equation 17<br/>2.1 The Wave Function 18<br/>2.1.1 Description of the State of a Particle 18<br/>2.1.2 Position Measurement of the Particle 19<br/>2.2 Interference and the Superposition Principle 20<br/>2.2.1 De Broglie Waves 20<br/>2.2.2 The Superposition Principle 21<br/>2.2.3 The Wave Equation in Vacuum 22<br/>2.3 Free Wave Packets 24<br/>2.3.1 * Definition of a Wave Packet 24<br/>2.3.2 Fourier Transformation 24<br/>2.3.3 Structure of the Wave Packet 25<br/>2.3.4 Propagation of a Wave Packet: the Group Velocity ... 26<br/>2.3.5 Propagation of a Wave Packet:<br/>Average Position and Spreading 27<br/>2.4 Momentum Measurements and Uncertainty Relations 28<br/>2.4.1 The Momentum Probability Distribution 29<br/>2.4.2 Heisenberg Uncertainty Relations 30<br/>2.5 The Schrodinger Equation 31XII Contents<br/>2.5.1 Equation of Motion 32<br/>2.5.2 Particle in a Potential: Uncertainty Relations 32<br/>2.5.3 Stability of Matter 33<br/>2.6 Momentum Measurement in a Time-of-Flight Experiment ... 34<br/>Further Reading 36<br/>Exercises<br/>3. Physical Quantities and Measurements • • 39<br/>3.1 Measurements in Quantum Mechanics 40<br/>3.1.1 The Measurement Procedure 40<br/>3.1.2 Experimental Facts 41<br/>3.1.3 Reinterpretation of Position<br/>and Momentum Measurements 41<br/>3.2 Physical Quantities and Observables 42<br/>3.2.1 Expectation Value of a Physical Quantity 42<br/>3.2.2 Position and Momentum Observables 43<br/>3.2.3 Other Observables: the Correspondence Principle 44<br/>3.2.4 Commutation of Observables 44<br/>3.3 Possible Results of a Measurement 45<br/>3.3.1 Eigenfunctions and Eigenvalues of an Observable 45<br/>3.3.2 Results of a Measurement<br/>and Reduction of the Wave Packet 46<br/>3.3.3 Individual Versus Multiple Measurements 47<br/>3.3.4 Relation to Heisenberg Uncertainty Relations 47<br/>3.3.5 Measurement and Coherence of Quantum Mechanics.. 48<br/>3.4 Energy Eigenfunctions and Stationary States 48<br/>3.4.1 Isolated Systems: Stationary States 49<br/>3.4.2 Energy Eigenstates and Time Evolution oO<br/>3.5 The Probability Current<br/>3.6 Crossing Potential Barriers ^2<br/>3.6.1 The Eigenstates of the Hamiltonian 52<br/>3.6.2 Boundary Conditions at the Discontinuities<br/>of the Potential<br/>3.6.3 Reflection and Transmission on a Potential Step 54<br/>3.6.4 Potential Barrier and Tunnel Effect 56<br/>3.7 Summary of Chapters 2 and 3<br/>Further Reading<br/>. 60<br/>Exercises<br/>4. Quantization of Energy in Simple Systems 53<br/>4.1 Bound States and Scattering States 53<br/>4.1.1 Stationary States of the Schrodinger Equation 54<br/>4.1.2 Bound States<br/>4.1.3 Scattering States<br/>4.2 The One Dimensional Harmonic Oscillator 56Contents XIII<br/>4.2.1 Definition and Classical Motion 66<br/>4.2.2 The Quantum Harmonic Oscillator 67<br/>4.2.3 Examples 69<br/>4.3 Square-Well Potentials 70<br/>4.3.1 Relevance of Square Potentials 70<br/>4.3.2 Bound States in a One-Dimensional<br/>Square-Well Potential 71<br/>4.3.3 Infinite Square Well 73<br/>4.3.4 Particle in a Three-Dimensional Box 74<br/>4.4 Periodic Boundary Conditions 75<br/>4.4.1 A One-Dimensional Example 75<br/>4.4.2 Extension to Three Dimensions 77<br/>4.4.3 Introduction of Phase Space 78<br/>4.5 The Double Well Problem and the Ammonia Molecule 78<br/>4.5.1 Model of the NH3 Molecule 79<br/>4.5.2 Wave Functions 79<br/>4.5.3 Energy Levels 81<br/>4.5.4 The Tunnel Effect and the Inversion Phenomenon .... 82<br/>4.6 Other Applications of the Double Well 84<br/>Further Reading 86<br/>Exercises 87<br/>5. Principles of Quantum Mechanics 89<br/>5.1 Hilbert Space 90<br/>5.1.1 The State Vector 90<br/>5.1.2 Scalar Products and the Dirac Notations 90<br/>5.1.3 Examples 91<br/>5.1.4 Bras and Kets, Brackets 92<br/>5.2 Operators in Hilbert Space 92<br/>5.2.1 Matrix Elements of an Operator 92<br/>5.2.2 Adjoint Operators and Hermitian Operators 93<br/>5.2.3 Eigenvectors and Eigenvalues 94<br/>5.2.4 Summary: Syntax Rules in Dirac's Formalism 95<br/>5.3 The Spectral Theorem 95<br/>5.3.1 Hilbertian Bases 95<br/>5.3.2 Projectors and Closure Relation 96<br/>5.3.3 The Spectral Decomposition of an Operator 96<br/>5.3.4 Matrix Representations 97<br/>5.4 Measurement of Physical Quantities 99<br/>5.5 The Principles of Quantum Mechanics 100<br/>5.6 Structure of Hilbert Space 104<br/>5.6.1 Tensor Products of Spaces 104<br/>5.6.2 The Appropriate Hilbert Space 105<br/>5.6.3 Properties of Tensor Products 105<br/>5.6.4 Operators in a Tensor Product Space 106XIV Contents<br/>5.6.5 Simple Examples 106<br/>5.7 Reversible Evolution and the Measurement Proce.ss 107<br/>Further Reading 110<br/>Exercises Ill<br/>6. Two-State Systems, Principle of the Maser 115<br/>6.1 Two-Dimensional Hilbert Space 115<br/>6.2 A Familiar Example: the Polarization of Light 116<br/>6.2.1 Polarization States of a Photon 116<br/>6.2.2 Measurement of Photon Polarizations 118<br/>6.2.3 Successive Measurements and "Quantum Logic" 119<br/>6.3 The Model of the Ammonia Molecule 120<br/>6.3.1 Restriction to a Two-Dimensional Hilbert Space 120<br/>6.3.2 The Basis {|^s),|Vm>} 121<br/>6.3.3 The Basis {|V-'r)> \^l)} 123<br/>6.4 The Ammonia Molecule in an Electric Field 123<br/>6.4.1 The Coupling of NH3 to an Electric Field 124<br/>6.4.2 Energy Levels in a Fixed Electric Field 125<br/>6.4.3 Force Exerted on the Molecule<br/>by an Inhomogeneous Field 127<br/>6.5 Oscillating Fields and Stimulated Emission 129<br/>6.6 Principle and Applications of Masers 131<br/>6.6.1 Amplifier 131<br/>6.6.2 Oscillator 132<br/>6.6.3 Atomic Clocks 132<br/>Further Reading 132<br/>Exercises 133<br/>7. Commutation of Observables 135<br/>7.1 Commutation Relations 136<br/>7.2 Uncertainty Relations 137<br/>7.3 Ehrenfest's Theorem 138<br/>7.3.1 Evolution of the Expectation Value of an Observable.. 138<br/>7.3.2 Particle in a Potential V(r) 139<br/>7.3.3 Constants of Motion 140<br/>7.4 Commuting Observables 142<br/>7.4.1 Existence of a Common Eigenbasis<br/>for Commuting Observables 142<br/>7.4.2 Complete Set of Commuting Observables(CSCO).... 142<br/>7.4.3 Completely Prepared Quantum State 143<br/>7.4.4 Symmetries of the Hamiltonian<br/>and Search of Its Eigenstates 145<br/>7.5 Algebraic Solution of the Harmonic-Oscillator Problem 148<br/>7.5.1 Reduced Variables 148<br/>7.5.2 Annihilation and Creation Operators a and 148Contents XV<br/>7.5.3 Eigenvalues of the Number Operator N 149<br/>7.5.4 Eigenstates 150<br/>Further Reading 151<br/>Exercis&s 152<br/>8. The Stern-Gerlach Experiment 157<br/>8.1 Principle of the Experiment 157<br/>8.1.1 Classical Analysis 157<br/>8.1.2 Experimental Results 159<br/>8.2 The Quantum Description of the Problem 161<br/>8.3 The Observables and find 163<br/>8.4 Discussion 165<br/>8.4.1 Incompatibility of Measurements Along Different Axes 165<br/>8.4.2 Classical Versus Quantum Analysis 166<br/>8.4.3 Measurement Along an Arbitrary Axis 167<br/>8.5 Complete Description of the Atom 168<br/>8.5.1 Hilbert Space 168<br/>8.5.2 Representation of States and Observables 169<br/>8.5.3 Energy of the Atom in a Magnetic Field 170<br/>8.6 Evolution of the Atom in a Magnetic Field 170<br/>8.6.1 Schrodinger Equation 170<br/>8.6.2 Evolution in a Uniform Magnetic Field 171<br/>8.6.3 Explanation of the Stern-Gerlach Experiment 173<br/>8.7 Conclusion 175<br/>Further Reading 175<br/>Exercises 176<br/>9. Approximation Methods «<br/>9.1 Perturbation Theory 177<br/>9.1.1 Definition of the Problem 177<br/>9.1.2 Power Expansion of Energies and Eigenstates 178<br/>9.1.3 First-Order Perturbation in the Nondegenerate Case .. 179<br/>9.1.4 First-Order Perturbation in the Degenerate Case 179<br/>9.1.5 First-Order Perturbation to the Eigenstates 180<br/>9.1.6 Second-Order Perturbation to the Energy Levels 181<br/>9.1.7 Examples 181<br/>9.1.8 Remarks on the Convergence of Perturbation Theory ..182<br/>9.2 The Variational Method 183<br/>9.2.1 The Ground State 183<br/>9.2.2 Other Levels 184<br/>9.2.3 Examples of Applications of the Variational Method .. 185<br/>Exercises 187XVI Contents<br/>10. Angular Momentum 189<br/>10.1 Orbital Angular Momentum and the Coniiniitation Relations 190<br/>10.2 Eigenvalues of Angular Monicntuni 190<br/>10.2.1 The Observables and J. and the Basis States |J, ni) 191<br/>10.2.2 The Operators .J± 192<br/>10.2.3 Action of J± on the States \j. ni) 192<br/>10.2.4 Quantization of and m 193<br/>10.2.5 Measurement of and Jy .. 195<br/>10.3 Orbital Angular Momentum 196<br/>10.3.1 The Quantum Numbers in and(are Integers 196<br/>10.3.2 Spherical Coordinates 197<br/>10.3.3 Eigenfunctions of and L.: the Spherical Harmonics 198<br/>10.3.4 Examples of Spherical Harmonics 199<br/>10.3.5 Example: Rotational Energy of a Diatomic Molecule .. 200<br/>10.4 Angular Momentum and Magnetic Moment 201<br/>10.4.1 Orbital Angular Momentum and Magnetic Moment... 202<br/>10.4.2 Generalization to Other Angular Momenta 203<br/>10.4.3 What Should we Think<br/>about Half-Integer Values ofjand 7n ? 204<br/>Further Reading 204<br/>Exercises 205<br/>11. Initial Description of Atoms 207<br/>11.1 The Two-Body Problem; Relative Motion 208<br/>11.2 Motion in a Central Potential 210<br/>11.2.1 Spherical Coordinates 210<br/>11.2.2 Eigenfunctions Common to H, and Lz 211<br/>11.3 The Hydrogen Atom 215<br/>11.3.1 Orders of Magnitude:<br/>Appropriate Units in Atomic Physics 215<br/>11.3.2 The Dimensionless Radial Equation 216<br/>11.3.3 Spectrum of Hydrogen 219<br/>11.3.4 Stationary States of the Hydrogen Atom 220<br/>-11.3.5 Dimensions and Orders of Magnitude 221<br/>11.3.6 Time Evolution of States of Low Energies 223<br/>11.4 Hydrogen-Like Atoms 224<br/>11.5 Muonic Atoms 224<br/>11.6 Spectra of Alkali Atoms 226<br/>Further Reading 227<br/>Exercises 228Contents XVII<br/>12. Spin 1/2 and Magnetic Resonance 231<br/>12.1 The Hilbert Space of Spin 1/2 232<br/>12.1.1 Spin Observables 233<br/>12.1.2 Representation in a Particular Basis 233<br/>12.1.3 Matrix Representation 234<br/>12.1.4 Arbitrary Spin State 234<br/>12.2 Complete Description of a Spin-1/2 Particle 235<br/>12.2.1 Hilbert Space 235<br/>12.2.2 Representation of States and Observables 235<br/>12.3 Spin Magnetic Moment 236<br/>12.3.1 The Stern-Gerlach Experiment 236<br/>12.3.2 Anomalous Zeeman Effect 237<br/>12.3.3 Magnetic Moment of Elementary Particles 237<br/>12.4 Uncorrelated Space and Spin Variables 238<br/>12.5 Magnetic Resonance 239<br/>12.5.1 Larmor Precession in a Fixed Magnetic Field Rq 239<br/>12.5.2 Superposition of a Fixed Field and a Rotating Field .. 240<br/>12.5.3 Rabi's Experiment 242<br/>12.5.4 Applications of Magnetic Resonance 244<br/>12.5.5 Rotation of a Spin 1/2 Particle by 27r 245<br/>Further Reading 246<br/>Exercises 247<br/>13. Addition of Angular Momenta,<br/>Fine and Hyperfine Structure of Atomic Spectra 249<br/>13.1 Addition of Angular Momenta 249<br/>13.1.1 The Total-Angular Momentum Operator 249<br/>13.1.2 Factorized and Coupled Bases 250<br/>13.1.3 A Simple Case: the Addition of Two Spins of 1/2 251<br/>13.1.4 Addition of Two Arbitrary Angular Momenta 254<br/>13.1.5 One-Electron Atoms, Spectroscopic Notations 258<br/>13.2 Fine Structure of Monovalent Atoms 258<br/>13.3 Hyperfine Structure; the 21cm Line of Hydrogen 261<br/>13.3.1 Interaction Energy 261<br/>13.3.2 Perturbation Theory 262<br/>13.3.3 Diagonalization of 263<br/>13.3.4 The Effect of an External Magnetic Field 265<br/>13.3.5 The 21cm Line in Astrophysics 265<br/>Further Reading 268<br/>Exercises 269XVIII Contents<br/>14. Entangled States, EPR Paradox and Bell's Inequality 2^3<br/>Written in collaboration with Philippe Grangier<br/>14.1 The EPR Paradox and Bell's Inequality 274<br/>14.1.1 "God Does Not Play Dice" 274<br/>14.1.2 The EPR Argument 275<br/>14.1.3 Bell's Inequality 278<br/>14.1.4 Experimental Tests 281<br/>14.2 Quantum Cryptography • • • 282<br/>14.2.1 The Communication Between Alice and Bob 282<br/>14.2.2 The Quantum No cloning Theorem 285<br/>14.2.3 Present Experimental Setups 286<br/>14.3 The Quantum Computer 287<br/>14.3.1 The Quantum Bits, or "'Q-Bits" 287<br/>14.3.2 The Algorithm of Peter Shor 288<br/>14.3.3 Principle of a Quantum Computer 289<br/>14.3.4 Decoherence 290<br/>Further Reading 290<br/>901<br/>Exercises<br/>15. The Lagrangian and Hamiltonian Formalisms,<br/>Lorentz Force in Quantum Mechanics 293<br/>15.1 Lagrangian Formalism and the Least-Action Principle 294<br/>15.1.1 Least Action Principle 294<br/>15.1.2 Lagrange Equations 295<br/>15.1.3 Energy 297<br/>15.2 Canonical Formalism of Hamilton 297<br/>15.2.1 Conjugate Momenta 297<br/>15.2.2 Canonical Equations 298<br/>15.2.3 Poisson Brackets 299<br/>15.3 Analytical Mechanics and Quantum Mechanics 300<br/>15.4 Classical Charged Particles in an Electromagnetic Field 301<br/>15.5 Lorentz Force in Quantum Mechanics 302<br/>15.5.1 Hamiltonian ^^2<br/>1*5.5.2 Gauge Invariance 303<br/>15.5.3 The Hydrogen Atom Without Spin<br/>in a Uniform Magnetic Field • • 304<br/>15.5.4 Spin-1/2 Particle in an Electromagnetic Field 305<br/>Further Reading<br/>Exercises<br/>16. Identical Particles and the Pauli Principle 309<br/>16.1 Indistinguishability of Two Identical Particles 310<br/>16.1.1 Identical Particles in Classical Physics 310<br/>16.1.2 The Quantum Problem 310<br/>16.2 Two-Particle Systems; the Exchange Operator 312Contents XIX<br/>16.2.1 The Hilbert Space for the Two Particle System 312<br/>16.2.2 The Exchange Operator<br/>Between Two Identical Particles 312<br/>16.2.3 Symmetry of the States 313<br/>16.3 The Pauli Principle 314<br/>16.3.1 The Case of Two Particles 314<br/>16.3.2 Independent Fermions and Exclusion Principle 315<br/>16.3.3 The Case of N Identical Particles 316<br/>16.3.4 Time Evolution 317<br/>16.4 Physical Consequences of the Pauli Principle 317<br/>16.4.1 Exchange Force Between Two Fermions 318<br/>16.4.2 The Ground State<br/>of N Identical Independent Particles 318<br/>16.4.3 Behavior of Fermion and Boson Sj'stems<br/>at Low Temperature 320<br/>16.4.4 Stimulated Emission and the Laser Effect 322<br/>16.4.5 Uncertainty Relations for a System of N Fermions.... 323<br/>16.4.6 Complex Atoms and Atomic Shells 324<br/>Further Reading 326<br/>Exercises 327<br/>17. The Evolution of Systems 331<br/>Written in collaboration with Gilbert Grynberg<br/>17.1 Time-Dependent Perturbation Theory 332<br/>17.1.1 Transition Probabilities 332<br/>17.1.2 Evolution Equations 332<br/>17.1.3 Perturbative Solution 333<br/>17.1.4 First-Order Solution: the Born Approximation 334<br/>17.1.5 Particular Cases 334<br/>17.1.6 Perturbative and Exact Solutions 335<br/>17.2 Interaction of an Atom with an Electromagnetic Wave 336<br/>17.2.1 The Electric-Dipole Approximation 336<br/>17.2.2 Justification of the Electric Dipole Interaction 337<br/>17.2.3 Absorption of Energy by an Atom 338<br/>17.2.4 Selection Rules 339<br/>17.2.5 Spontaneous Emission 339<br/>17.2.6 Control of Atomic Motion by Light 341<br/>17.3 Decay of a System 343<br/>17.3.1 The Radioactivity of ^'^Fe 343<br/>17.3.2 The Fermi Golden Rule 345<br/>17.3.3 Orders of Magnitude 346<br/>17.3.4 Behavior for Long Times 347<br/>17.4 The Time-Energy Uncertainty Relation 350<br/>17.4.1 Isolated Systems and Intrinsic Interpretations 350<br/>17.4.2 Interpretation of Landau and Peierls 351XX Contents<br/>17.4.3 The Einstein Bohr Controver.sy 352<br/>Further Reading 353<br/>Exercises 353<br/>18. Scattering Processes 357<br/>18.1 Concept of Cross Section 358<br/>18.1.1 Definition of Cross Section 358<br/>18.1.2 Classical Calculation 359<br/>18.1.3 Examples 360<br/>18.2 Quantum Calculation in the Born Approximation 361<br/>18.2.1 Asymptotic States 361<br/>18.2.2 Transition Probability 362<br/>18.2.3 Scattering Cross Section 363<br/>18.2.4 Validity of the Born Approximation 364<br/>18.2.5 Example: the Yukawa Potential 365<br/>18.2.6 Range of a Potential in Quantum Mechanics 366<br/>18.3 Exploration of Composite Systems 367<br/>18.3.1 Scattering Off a Bound State and the Form Factor ... 367<br/>18.3.2 Scattering by a Charge Distribution 368<br/>18.4 General Scattering Theory 372<br/>18.4.1 Scattering States 372<br/>18.4.2 The Scattering Amplitude 373<br/>18.4.3 The Integral Equation for Scattering 374<br/>18.5 Scattering at Low Energy 375<br/>18.5.1 The Scattering Length 375<br/>18.5.2 Explicit Calculation of a Scattering Length 376<br/>18.5.3 The Case of Identical Particles 377<br/>Further Reading 378<br/>Exercises 378<br/>19. Qualitative Physics on a Macroscopic Scale 381<br/>Written in collaboration with Alfred Vidal-Madjar<br/>19.1 Confined Particles and Ground State Energy 382<br/>19.1.1 The Quantum Pressure 382<br/>19.1.2 Hydrogen Atom 383<br/>19.1.3 7V-Fermion Systems and Complex Atoms 383<br/>19.1.4 Molecules, Liquids and Solids .. 384<br/>19.1.5 Hardness of a Solid 385<br/>19.2 Gravitational Versus Electrostatic Forces 386<br/>19.2.1 Screening of Electrostatic Interactions 386<br/>19.2.2 Additivity of Gravitational Interactions 387<br/>19.2.3 Ground State of a Gravity-Dominated Object 388<br/>19.2.4 Liquefaction of a Solid and the Height of Mountains .. 390<br/>19.3 White Dwarfs, Neutron Stars<br/>and the Gravitational Catastrophe 392Contents XXI<br/>19.3.1 White Dwarfs and the Chandrasekhar* Mass 392<br/>19.3.2 Neutron Stars 394<br/>Further Reading 396<br/>20. Early History of Quantum Mechanics 397<br/>20.1 The Origin of Quantum Concepts 397<br/>20.1.1 Planck's Radiation Law 397<br/>20.1.2 Photons 398<br/>20.2 The Atomic Spectrum 398<br/>20.2.1 Empirical Regularities of Atomic Spectra 398<br/>20.2.2 The Structure of Atoms 399<br/>20.2.3 The Bohr Atom 399<br/>20.2.4 The Old Theory of Quanta 400<br/>20.3 Spin 400<br/>20.4 Heisenberg's Matrices 401<br/>20.5 Wave Mechanics 403<br/>20.6 The Mathematical Formalization 404<br/>20.7 Some Important Steps in More Recent Years 405<br/>Further Reading 406<br/>Appendix A. Concepts of Probability Theory 407<br/>1 Fundamental Concepts 407<br/>2 Examples of Probability Laws 408<br/>2.1 Discrete Laws 408<br/>2.2 Continuous Probability Laws<br/>in One or Several Variables 408<br/>3 Random Variables 409<br/>3.1 Definition 409<br/>3.2 Conditional Probabilities 410<br/>3.3 Independent Random Variables 411<br/>3.4 Binomial Law and the Gaussian Approximation 411<br/>4 Moments of Probability Distributions 412<br/>4.1 Mean Value or Expectation Value 412<br/>4.2 Variance and Mean Square Deviation 412<br/>4.3 ^ Bienayme-Tchebycheff Inequality 413<br/>4.4 Experimental Verification of a Probability Law 413<br/>Exercises 414<br/>Appendix B. Dirac Distribution, Fourier Transformation .... 417<br/>1 Dirac Distribution, or <5 "Function" 417<br/>1.1 Definition of <5(.t) 417<br/>1.2 Examples of Functions Which Tend to (5(a:) 418<br/>1.3 Properties of S{x) 419<br/>2 Distributions 420<br/>2.1 The Space S 420XXII Contents<br/>2.2 Linear Fimctionals 420<br/>2.3 Derivative of a Di.stributioii 421<br/>2.4 Convolution Product 422<br/>3 Fourier Transformation 422<br/>3.1 Definition 422<br/>3.2 Fourier Transform of a Gaussian 423<br/>3.3 Inversion of the Fourier Transformation 423<br/>3.4 Parseval-Plancherel Theorem 424<br/>3.5 Fourier Transform of a Distribution 425<br/>3.6 Uncertainty Relation 426<br/>Exercises 427<br/>Appendix C. Operators in Infinite-Dimensional Spaces 429<br/>1 Matrix Elements of an Operator 429<br/>2 Continuous Bases 430<br/>Appendix D. The Density Operator 435<br/>1 Pure States 436<br/>1.1 A Mathematical Tool: the Tiace of an Operator 436<br/>1.2 The Density Operator of Pure States 437<br/>1.3 Alternative Formulation of Quantum Mechanics<br/>for Pure States 438<br/>2 Statistical Mixtures 439<br/>2.1 A Particular Case: an Unpolarized Spin-1/2 System .. 439<br/>2.2 The Density Operator for Statistical Mixtures 440<br/>3 Examples of Density Operators 441<br/>3.1 The Micro-Canonical and Canonical Ensembles 441<br/>3.2 The Wigner Distribution of a Spinless Point Particle.. 442<br/>4 Entangled Systems 444<br/>4.1 Reduced Density Operator 444<br/>4.2 Evolution of a Reduced Density Operator 444<br/>4.3 Entanglement and Measurement 445<br/>Further Reading 446<br/>Exerdses 446<br/> |
| 942 ## - ADDED ENTRY ELEMENTS (KOHA) | |
| Koha item type | General Books |
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| Science Library, Sikkim University | Science Library, Sikkim University | Science Library General Section | 29/08/2016 | 530.12 BAS/Q | P31807 | 29/08/2016 | General Books Science Library | Books For SU Science Library |