UID:
almafu_9961294318302883
Umfang:
1 online resource (xxii, 656 pages) :
,
digital, PDF file(s).
Ausgabe:
First edition.
ISBN:
9781108985062
,
1108985068
,
9781108986199
,
1108986196
,
9781108976299
,
1108976298
Inhalt:
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
Anmerkung:
Title from publisher's bibliographic system (viewed on 03 Jan 2023).
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Cover -- Half-title -- Endorsements -- Title page -- Copyright information -- Dedication -- Contents -- Preface -- Acknowledgements -- Introduction -- Part I Formalism and Basic Problems -- Introduction: Historical Background -- 0.1 Experiments Point towards Quantum Mechanics -- 0.2 Quantized States: Matrix Mechanics, and Waves for Particles: Correspondence Principle -- 0.3 Wave Functions Governing Probability, and the Schrödinger Equation -- 0.4 Bohr-Sommerfeld Quantization and the Hydrogen Atom -- 0.5 Review: States of Quantum Mechanical Systems -- 1 The Mathematics of Quantum Mechanics 1: Finite-Dimensional Hilbert Spaces -- 1.1 (Linear) Vector Spaces [mathbb(V)] -- 1.2 Operators on a Vector Space -- 1.3 Dual Space, Adjoint Operation, and Dirac Notation -- 1.4 Hermitian (Self-Adjoint) Operators and the Eigenvalue Problem -- 1.5 Traces and Tensor Products -- 1.6 Hilbert Spaces -- 2 The Mathematics of Quantum Mechanics 2: Infinite-Dimensional Hilbert Spaces -- 2.1 Hilbert Spaces and Related Notions -- 2.2 Functions as Limits of Discrete Sets of Vectors -- 2.3 Integrals as Limits of Sums -- 2.4 Distributions and the Delta Function -- 2.5 Spaces of Functions -- 2.6 Operators in Infinite Dimensions -- 2.7 Hermitian Operators and Eigenvalue Problems -- 2.8 The Operator D[sub(xx')] -- 3 The Postulates of Quantum Mechanics and the Schrödinger Equation -- 3.1 The Postulates -- 3.2 The First Postulate -- 3.3 The Second Postulate -- 3.4 The Third Postulate -- 3.5 The Fourth Postulate -- 3.6 The Fifth Postulate -- 3.7 The Sixth Postulate -- 3.8 Generalization of States to Ensembles: the Density Matrix -- 4 Two-Level Systems and Spin-1/2, Entanglement, and Computation -- 4.1 Two-Level Systems and Time Dependence -- 4.2 General Stationary Two-State System -- 4.3 Oscillations of States -- 4.4 Unitary Evolution Operator -- 4.5 Entanglement.
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4.6 Quantum Computation -- 5 Position and Momentum and Their Bases -- Canonical Quantization, and Free Particles -- 5.1 Translation Operator -- 5.2 Momentum in Classical Mechanics as a Generator of Translations -- 5.3 Canonical Quantization -- 5.4 Operators in Coordinate and Momentum Spaces -- 5.5 The Free Nonrelativistic Particle -- 6 The Heisenberg Uncertainty Principle and Relations, and Gaussian Wave Packets -- 6.1 Gaussian Wave Packets -- 6.2 Time Evolution of Gaussian Wave Packet -- 6.3 Heisenberg Uncertainty Relations -- 6.4 Minimum Uncertainty Wave Packet -- 6.5 Energy-Time Uncertainty Relation -- 7 One-Dimensional Problems in a Potential V(x) -- 7.1 Set-Up of the Problem -- 7.2 General Properties of the Solutions -- 7.3 Infinitely Deep Square Well (Particle in a Box) -- 7.4 Potential Step and Reflection and Transmission of Modes -- 7.5 Continuity Equation for Probabilities -- 7.6 Finite Square Well Potential -- 7.7 Penetration of a Potential Barrier and the Tunneling Effect -- 8 The Harmonic Oscillator -- 8.1 Classical Set-Up and Generalizations -- 8.2 Quantization in the Creation and Annihilation Operator Formalism -- 8.3 Generalization -- 8.4 Coherent States -- 8.5 Solution in the Coordinate, |x〉, Representation (Basis) -- 8.6 Alternative to |x〉 Representation: Basis Change from |n〉 Representation -- 8.7 Properties of Hermite Polynomials -- 8.8 Mathematical Digression (Appendix): Classical Orthogonal Polynomials -- 9 The Heisenberg Picture and General Picture -- Evolution Operator -- 9.1 The Evolution Operator -- 9.2 The Heisenberg Picture -- 9.3 Application to the Harmonic Oscillator -- 9.4 General Quantum Mechanical Pictures -- 9.5 The Dirac (Interaction) Picture -- 10 The Feynman Path Integral and Propagators -- 10.1 Path Integral in Phase Space -- 10.2 Gaussian Integration -- 10.3 Path Integral in Configuration Space.
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10.4 Path Integral over Coherent States (in "Harmonic Phase Space") -- 10.5 Correlation Functions and Their Generating Functional -- 11 The Classical Limit and Hamilton-Jacobi (WKB Method), the Ehrenfest Theorem -- 11.1 Ehrenfest Theorem -- 11.2 Continuity Equation for Probability -- 11.3 Review of the Hamilton-Jacobi Formalism -- 11.4 The Classical Limit and the Geometrical Optics Approximation -- 11.5 The WKB Method -- 12 Symmetries in Quantum Mechanics I: Continuous Symmetries -- 12.1 Symmetries in Classical Mechanics -- 12.2 Symmetries in Quantum Mechanics: General Formalism -- 12.3 Example 1. Translations -- 12.4 Example 2. Time Translation Invariance -- 12.5 Mathematical Background: Review of Basics of Group Theory -- 13 Symmetries in Quantum Mechanics II: Discrete Symmetries and Internal Symmetries -- 13.1 Discrete Symmetries: Symmetries under Discrete Groups -- 13.2 Parity Symmetry -- 13.3 Time Reversal Invariance, T -- 13.4 Internal Symmetries -- 13.5 Continuous Symmetry -- 13.6 Lie Groups and Algebras and Their Representations -- 14 Theory of Angular Momentum I: Operators, Algebras, Representations -- 14.1 Rotational Invariance and SO(n) -- 14.2 The Lie Groups SO(2) and SO(3) -- 14.3 The Group SU(2) and Its Isomorphism With SO(3) Mod [mathbb(Z)][sub(2)] -- 14.4 Generators and Lie Algebras -- 14.5 Quantum Mechanical Version -- 14.6 Representations -- 15 Theory of Angular Momentum II: Addition of Angular Momenta and Representations -- Oscillator Model -- 15.1 The Spinor Representation, j = 1/2 -- 15.2 Composition of Angular Momenta -- 15.3 Finding the Clebsch-Gordan Coefficients -- 15.4 Sums of Three Angular Momenta, [vec(J)][sub(1)] + [vec(J)][sub(2)] + [vec(J)][sub(3)]: Racah Coefficients -- 15.5 Schwinger's Oscillator Model.
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16 Applications of Angular Momentum Theory: Tensor Operators, Wave Functions and the Schrödinger Equation, Free Particles -- 16.1 Tensor Operators -- 16.2 Wigner-Eckhart Theorem -- 16.3 Rotations and Wave Functions -- 16.4 Wave Function Transformations under Rotations -- 16.5 Free Particle in Spherical Coordinates -- 17 Spin and [vec(L)] + [vec(S)] -- 17.1 Motivation for Spin and Interaction with Magnetic Field -- 17.2 Spin Properties -- 17.3 Particle with Spin 1/2 -- 17.4 Rotation of Spinors with s = 1/2 -- 17.5 Sum of Orbital Angular Momentum and Spin, [vec(L)] + [vec(S)] -- 17.6 Time-Reversal Operator on States with Spin -- 18 The Hydrogen Atom -- 18.1 Two-Body Problem: Reducing to Central Potential -- 18.2 Hydrogenoid Atom: Set-Up of Problem -- 18.3 Solution: Sommerfeld Polynomial Method -- 18.4 Confluent Hypergeometric Function and Quantization of Energy -- 18.5 Orthogonal Polynomials and Standard Averages over Wave Functions -- 19 General Central Potential and Three-Dimensional (Isotropic) Harmonic Oscillator -- 19.1 General Set-Up -- 19.2 Types of Potentials -- 19.3 Diatomic Molecule -- 19.4 Free Particle -- 19.5 Spherical Square Well -- 19.6 Three-Dimensional Isotropic Harmonic Oscillator: Set-Up -- 19.7 Isotropic Three-Dimensional Harmonic Oscillator in Spherical Coordinates -- 19.8 Isotropic Three-Dimensional Harmonic Oscillator in Cylindrical Coordinates -- 20 Systems of Identical Particles -- 20.1 Identical Particles: Bosons and Fermions -- 20.2 Observables under Permutation -- 20.3 Generalization to N Particles -- 20.4 Canonical Commutation Relations -- 20.5 Spin-Statistics Theorem -- 20.6 Particles with Spin -- 21 Application of Identical Particles: He Atom (Two-Electron System) and H[sub(2)] Molecule -- 21.1 Helium-Like Atoms -- 21.2 Ground State of the Helium (or Helium-Like) Atom.
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21.3 Approximation 3: Variational Method, "Light Version" -- 21.4 H[sub(2)] Molecule and Its Ground State -- 22 Quantum Mechanics Interacting with Classical Electromagnetism -- 22.1 Classical Electromagnetism plus Particle -- 22.2 Quantum Particle plus Classical Electromagnetism -- 22.3 Application to Superconductors -- 22.4 Interaction with a Plane Wave -- 22.5 Spin-Magnetic-Field and Spin-Orbit Interaction -- 23 Aharonov-Bohm Effect and Berry Phase in Quantum Mechanics -- 23.1 Gauge Transformation in Electromagnetism -- 23.2 The Aharonov-Bohm Phase δ -- 23.3 Berry Phase -- 23.4 Example: Atoms, Nuclei plus Electrons -- 23.5 Spin-Magnetic Field Interaction, Berry Curvature, and Berry Phaseas Geometric Phase -- 23.6 Nonabelian Generalization -- 23.7 Aharonov-Bohm Phase in Berry Form -- 24 Motion in a Magnetic Field, Hall Effect and Landau Levels -- 24.1 Spin in a Magnetic Field -- 24.2 Particle with Spin 1/2 in a Time-Dependent Magnetic Field -- 24.3 Particle with or without Spin in a Magnetic Field: Landau Levels -- 24.4 The Integer Quantum Hall Effect (IQHE) -- 24.5 Alternative Derivation of the IQHE -- 24.6 An Atom in a Magnetic Field and the Landé g-Factor -- 25 The WKB -- a Semiclassical Approximation -- 25.1 Review and Generalization -- 25.2 Approximation and Connection Formulas at Turning Points -- 25.3 Application: Potential Barrier -- 25.4 The WKB Approximation in the Path Integral -- 26 Bohr-Sommerfeld Quantization -- 26.1 Bohr-Sommerfeld Quantization Condition -- 26.2 Example 1: Parity-Even Linear Potential -- 26.3 Example 2: Harmonic Oscillator -- 26.4 Example 3: Motion in a Central Potential -- 26.5 Example: Coulomb Potential (Hydrogenoid Atom) -- 27 Dirac Quantization Condition and Magnetic Monopoles -- 27.1 Dirac Monopoles from Maxwell Duality -- 27.2 Dirac Quantization Condition from Semiclassical Nonrelativistic Considerations.
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27.3 Contradiction with the Gauge Field.
Weitere Ausg.:
ISBN 9781108838733
Sprache:
Englisch
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