UID:
almafu_9961635168202883
Umfang:
1 online resource (xxix, 553 pages) :
,
digital, PDF file(s).
Ausgabe:
First edition.
ISBN:
9781108670791
,
1108670792
,
9781108555241
,
1108555241
Inhalt:
Quantum mechanics is one of the principle pillars of modern physics. It also remains a topic of great interest to mathematicians. Since its discovery it has inspired and been inspired by many topics within modern mathematics, including functional analysis and operator algebras, Lie groups, Lie algebras and their representations, principle bundles, distribution theory, and much more. Written with beginning graduate students in mathematics in mind, this book provides a thorough treatment of (nonrelativistic) quantum mechanics in a style that is leisurely, without the usual theorem-proof grammar of pure mathematics, while remaining mathematically honest. The author takes the time to fully develop the required mathematics and employs a consistent mathematical presentation to clarify the often-confusing notation of physics texts. Along the way the reader encounters several topics requiring more advanced mathematics than found in many discussions of the subject, making for a fascinating course in how mathematics and physics interact.
Anmerkung:
Title from publisher's bibliographic system (viewed on 07 Sep 2020).
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Cover -- Half-title -- Title page -- Copyright information -- Contents -- Preface -- Prolegomenon -- 1 The Harmonic Oscillator: Classical versus Quantum -- 1.1 The Classical Harmonic Oscillator -- 1.2 The Quantum Mechanical Treatment -- 1.3 What Does It All Mean? -- 1.4 Foundational Issues -- 1.5 End Notes -- 2 The Mathematical Structure of Quantum Mechanics -- 2.1 The Minimalist Rules -- 2.2 Wave Mechanics -- 2.3 Adjoints and Self-Adjoint Operators -- 2.4 The Position and Momentum Operators -- 2.5 End Notes -- 3 Observables and Expectation Values -- 3.1 Elementary Properties of Expectation Values -- 3.2 Can a Quantum Observable have a Precise Value? -- 3.3 What Happens Upon Measurement? -- 3.4 The Measurement Problem -- 3.5 End Notes -- 4 The Projection Postulate Examined -- 4.1 The Physicist's Approach -- 4.2 The Mathematician's Rigor -- 4.3 An Important Class of Operators -- 4.4 The Spectrum of a Self-Adjoint Operator -- 4.5 End Notes -- 5 Rigged Hilbert Space and the Dirac Calculus -- 5.1 Gelfand Triples and the Rigging of H -- 5.2 The Position and Momentum Operators in the Dirac Calculus -- 5.3 Products of Bras and Kets -- 5.4 Spectral Decomposition and the Dirac Calculus -- 5.5 End Notes -- 6 A Review of Classical Mechanics -- 6.1 Newtonian Mechanics and the Euler-Lagrange Equation in Cartesian Coordinates -- 6.2 Lagrangian Mechanics -- 6.3 Hamiltonian Mechanics and Poisson Brackets -- 6.4 Noether's Theorem -- 6.5 End Notes -- 7 Hamilton-Jacobi Theory [star] -- 7.1 Generalized Coordinates Reexamined -- 7.2 Canonical Transformations -- 7.3 The Hamilton-Jacobi Equation -- 7.4 Some Sample Applications -- 7.5 End Notes -- 8 Classical Mechanics Regain'd -- 8.1 The Quantum Evolution Equation -- 8.2 Commutation Relations and the Ehrenfest Theorems -- 8.3 Commuting Self-Adjoint Operators -- 8.4 The Baker-Hausdorff Formula -- 8.5 End Notes.
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9 Wave Mechanics I: Heisenberg Uncertainty -- 9.1 Statement of the Principle -- 9.2 Interpretation -- 9.3 Minimal-Uncertainty States -- 9.4 The Fourier Transform and Uncertainty -- 9.5 End Notes -- 10 Wave Mechanics II: The Fourier Transform -- 10.1 The Fourier Transform -- 10.2 Eigenvalues and Eigenfunctions: Hermite Functions -- 10.3 The Position and Momentum Representations -- 10.4 Wave Packets and Superposition -- 10.5 End Notes -- 11 Wave Mechanics III: The Quantum Oscillator -- 11.1 Ladder Operators and the Ground State -- 11.2 Higher Energy States -- 11.3 Generating Function and Completeness -- 11.4 Coherent States of the Oscillator -- 11.5 End Notes -- 12 Angular Momentum I: Basics -- 12.1 Angular Momentum Operators -- 12.2 Eigenvectors and Eigenvalues -- 12.3 Derivation -- 12.4 Further Remarks on Angular Momentum -- 12.5 End Notes -- 13 Angular Momentum II: Representations of su(2) -- 13.1 The Pauli Spin Matrices and the Lie Algebra -- 13.2 Angular Momentum and su(2)-Representations -- 13.3 The Quaternions H, S3, SU(2), SO(3), and SO(4) -- 13.4 Representations of Lie Groups SU(2) and SO(3) -- 13.5 End Notes -- 14 Angular Momentum III: The Central Force Problem -- 14.1 Orbital Angular Momentum -- 14.2 Legendre Functions and Spherical Harmonics -- 14.3 Radial Symmetry and Representations -- 14.4 A Single Particle in a General Central Potential -- 14.5 End Notes -- 15 Wave Mechanics IV: The Hydrogenic Potential -- 15.1 An Algebraic Approach to the Radial Equation -- 15.2 Power Series and Hypergeometric Functions -- 15.3 The Full Solution for Bound State Electrons -- 15.4 The Unbound Electron in the Coulomb Potential -- 15.5 End Notes -- 16 Wave Mechanics V: Hidden Symmetry Revealed -- 16.1 Quantum Numbers, Degeneracy, and Fine Structure -- 16.2 The Laplace-Runge-Lenz Vector -- 16.3 Hidden Symmetry.
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16.4 Momentum Representation and SO(4) Symmetry -- 16.5 End Notes -- 17 Wave Mechanics VI: Hidden Symmetry Solved -- 17.1 Fock's Treatment of the Momentum Space Equation -- 17.2 Group Characters and Representations -- 17.3 The Momentum Space Equation and Characters -- 17.4 The Infinitesimal Generators of SO(4) [star] -- 17.5 End Notes -- 18 Angular Momentum IV: Addition Rules and Spin -- 18.1 Coupled Angular Momenta -- 18.2 The Selection Rules -- 18.3 Spin-1/2 Systems -- 18.4 Rotations of Wave Functions and Spin-1/2 Particles -- 18.5 End Notes -- 19 Wave Mechanics VII: Pauli's Spinor Theory -- 19.1 Tensor Products and Internal Degrees of Freedom -- 19.2 Action of the Double Cover of SO(3) on L[sup(2)](R[sup(3)]) ⊗ V[sup(1/2)] -- 19.3 The Spin and Magnetic Moment of the Electron -- 19.4 The Hydrogenic Potential with Spin -- 19.5 End Notes -- 20 Clifford Algebras and Spin Representations [star] -- 20.1 Clifford Algebras -- 20.2 Low-Dimensional Algebras -- 20.3 The Groups Pin and Spin -- 20.4 Spin Representations and Spinors -- 20.5 End Notes -- 21 Many-Particle Quantum Systems -- 21.1 Multi-Particle States and Tensor Products -- 21.2 The Axiom for Multi-Component Systems -- 21.3 Coupled Angular Momenta, Again -- 21.4 A Mathematical Interlude: Bases for Tensor Products -- 21.5 End Notes -- 22 The EPR Argument and Bell's Inequalities -- 22.1 The EPR Criticism of Quantum Mechanics -- 22.2 The Coupled Spin-1/2 System in Quantum Mechanics -- 22.3 Bell Inequalities, Realism, and Nonlocality -- 22.4 The GHZ Scheme for Spin Triplets -- 22.5 End Notes -- 23 Ensembles and Density Operators -- 23.1 The Spin-1/2 System Revisited -- 23.2 Density Operators I: Finite-Dimensional Setting -- 23.3 Matrix Calculations in the Spin-1/2 System -- 23.4 Density Operators II: Infinite-Dimensional Setting -- 23.5 End Notes -- 24 Bosons and Fermions -- 24.1 Bosons and Fermions.
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24.2 N Indistinguishable Quanta -- 24.3 Algebraic Structure of the Tensor Algebra -- 24.4 Analytic Structure of the Tensor Algebra -- 24.5 End Notes -- 25 The Fock Space for Indistinguishable Quanta -- 25.1 Notation for Fock Space -- 25.2 Annihilation and Creation Operators -- 25.3 Bosonic Fock Space -- 25.4 Fermionic Fock Space -- 25.5 End Notes -- 26 An Introduction to Quantum Statistical Mechanics -- 26.1 Statistical Mechanics -- 26.2 The Most Probable Configuration via the Maximum-Term Method -- 26.3 The Mechanics of Maximizing Q(n) -- 26.4 The Fundamental Law and the Canonical Ensemble -- 26.5 End Notes -- 27 Quantum Dynamics -- 27.1 The Schr[ddot(o)]dinger Picture -- 27.2 Deriving the Schr[ddot(o)]dinger Equation: Stone's Theorem -- 27.3 The Heisenberg Picture and the Heisenberg Equation -- 27.4 Synthesis: The Dirac, or Interaction, Picture -- 27.5 End Notes -- 28 Unitary Representations and Conservation Laws -- 28.1 Euclidean Symmetries -- 28.2 Conservation Laws -- 28.3 Phase Transformations -- 28.4 Local Phase Symmetry Requires a Mediating Field -- 28.5 End Notes -- 29 The Feynman Formulation of Quantum Mechanics -- 29.1 A New Look at Quantum Mechanics -- 29.2 Feynman's Propagator Calculus -- 29.3 Evaluating Path Integrals: Quadratic Lagrangians -- 29.4 The Free Particle Propagator -- 29.5 End Notes -- 30 A Mathematical Interlude: Gaussian Integrals -- 30.1 Fresnel and Gaussian Integrals -- 30.2 The Complex Gaussian Integral in N Dimensions -- 30.3 Generalizing F[sub(N)](a, c, d) -- 30.4 Hilbert Matrices and Cauchy Determinants -- 30.5 End Notes -- 31 Evaluating Path Integrals I -- 31.1 The Piecewise Linear Family -- 31.2 The Constant Force Propagator -- 31.3 The Filtration Measures for the Polynomial Family -- 31.4 From the Dirac Calculus to the Path Integral -- 31.5 End Notes -- 32 Evaluating Path Integrals II -- 32.1 The Proposal.
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32.2 The Harmonic Oscillator and Fourier Sums -- 32.3 The Harmonic Oscillator and Polynomial Sums [star] -- 32.4 The Forced Harmonic Oscillator -- 32.5 End Notes -- Epilogue -- Resources for Individual Exploration -- Bibliography -- Index.
Weitere Ausg.:
ISBN 9781108429764
Weitere Ausg.:
ISBN 1108429769
Sprache:
Englisch
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