Some Unusual Topics in Quantum Mechanics

Foreword to the First Edition
Preface to the Second Edition
Preface to the First Edition
Acknowledgement
Notation
Contents
1 Position Operators of Non-relativistic Quantum Mechanics
1.1 Introduction
1.2 Spinless Relativistic Particle
1.3 Lie Algebra of the Poincare Group
1.4 Position Operators of NRQM
1.5 Projective Representations
1.6 Non-relativistic Limit
1.7 Phase Factors of the Galilean Group
1.8 Further Exercises
1.9 Notes
1.9.1 Time Translation and Time Evolution
1.9.2 The Newton-Wigner Position Operator
1.9.3 Poincare and Galilean Group Representations
References
2 A Bundle Picture of Quantum Mechanics
2.1 The Bundle Picture
2.1.1 Sections
2.1.2 Parallel Transport
2.1.3 Change of Basis or of Trivialization
2.1.4 Transport Round a Loop and Curvature
2.2 Covariant Derivative
2.2.1 Covariant Derivative of a Section
2.2.2 Covariant Derivative of an Operator
2.3 The Base X of Galilean Frames
2.3.1 The Hypothesis of Parallel Transport
2.3.2 Calculation of i
2.3.3 Galilean Bundle Curvature Is Zero
2.4 Application to Accelerated Frames
2.4.1 A Linearly Accelerated Frame
2.4.2 Rotating Frame: Centrifugal and Coriolis Forces
2.5 Further Exercises
2.6 Notes
2.6.1 Bundle Picture
2.6.2 Geometric Phase
References
3 A Beam of Particles = A Plane Wave?
3.1 A Coherent Bunch
3.2 Scattering Theory
3.2.1 Scattering States and the Moller Operator
3.2.2 Scattering Matrix
3.2.3 Properties of Ω() and S
3.3 Transition Rate
3.3.1 Irreversibility of Transitions
3.3.2 Transition Probability per Unit Time
3.4 Cross Sections
3.4.1 Non-relativistic Scattering from a Potential
3.4.2 Colliding Non-relativistic Beams
3.4.3 Relativistic Scattering
3.5 Comments on Formulas of Sect.3.2.3
3.5.1 Moller Operators
3.5.2 S-Matrix
3.5.3 Integral Equations
3.5.4 Ω(+) and S in Energy Basis
3.6 Further Exercises
3.7 Notes
References
4 Star-Product Formulation of Quantum Mechanics
4.1 Weyl Ordering and the Star Product
4.2 Derivation for Star-Product Expression
4.3 Wigner Distribution Function
4.4 Trace of
4.5 Trace of a Product: Expectation Values
4.6 Eigenvalues
4.7 Dynamics and the Moyal Bracket
4.8 Star Exponential and the Path Integral
4.9 Further Exercises
4.10 Notes
4.10.1 Weyl Correspondence and Wigner Distribution
4.10.2 Star Product and Moyal Bracket
References
5 Can There Be a Non-linear Quantum Mechanics?
5.1 Hamiltonian Equations in Quantum Mechanics
5.2 Observables and Poisson Bracket
5.3 Symmetry Transformations
5.4 Eigenvalues
5.5 Non-linear Quantum Mechanics?
5.6 Non-linear Terms: Simple Examples
5.6.1 Example 1: Extra Energy Level
5.6.2 Example 2: Asymptotic States
5.7 Further Exercises
5.8 Notes
5.8.1 Non-linear Quantum Mechanics
5.8.2 Non-linear Schrodinger Equation
References
6 Interaction = Exchange of Quanta
6.1 Non-relativistic ``Potential''
6.2 The Simplest Model of Quanta Exchange
6.2.1 The Hamiltonian
6.2.2 Bare and Dressed States
6.2.3 Single-Dressed B-particle
6.2.4 B-B Effective Interaction
6.3 Exercises
6.4 Notes
References
7 Proof of Wigner's Theorem
7.1 Rays and Symmetry Transformation
7.2 Wigner's Theorem
7.3 Bargmann Invariant
7.4 A Lemma
7.5 Proof of the Theorem
7.6 The Final Step
7.7 Further Exercises
7.8 Notes
References
8 Hilbert Space: An Introduction
8.1 Definition
8.2 Pythagoras Theorem
8.3 Bessel's Inequality
8.4 Schwarz and Triangle Inequalities
8.5 Complete Orthonormal Set
8.6 Subspaces
8.7 Direct Sum of Hilbert Spaces
8.8 Tensor Product of Hilbert Spaces
8.9 Bounded Linear Operators
8.10 Bounded Linear Functionals
8.11 Adjoint of a Bounded Operator
8.12 Projection Operators
8.13 Unitary Operators
8.14 Eigenvalues and Spectrum
8.15 Spectral Theorem
8.16 Density Matrix
8.17 Notes
References
9 What Is an ``Essentially Self-Adjoint'' Operator?
9.1 Introduction
9.2 Unbounded Operators
9.2.1 Domain and Range of an Operator
9.3 Graph of an Operator
9.4 Closed Operator
9.5 Adjoint in the General Case
9.6 Symmetric and Self-Adjoint Operators
9.6.1 Boundary Conditions for Self-Adjointness of H=-d2/dx2 on L2([a,b])
9.7 Self-Adjoint Extensions
9.8 Notes
References
10 Is There a Time-Energy Uncertainty Relation?
10.1 Introduction
10.1.1 Internal and External Time
10.1.2 Individual Measurement Versus StatisticalRelations
10.1.3 The Einstein-Bohr Argument of 1927
10.2 Mandelstam-Tamm Relation
10.2.1 A Characteristic Time for a Time-Dependent Observable
10.2.2 Half-Life of a State
10.3 Krylov-Fock Survival Probability
10.4 Wigner's Uncertainty Relation
10.4.1 Definition of t and E
10.4.2 Calculation of t
10.4.3 The Product (t)2(E)2
10.5 Notes
References
11 Fock Spaces
11.1 Introduction
11.2 Space for Many Particles
11.3 Fock Space for Bosons
11.3.1 Fock Space Fs
11.3.1.1 Symmetrizer
11.3.1.2 Annihilation Operator on General Fock Space Fn
11.3.1.3 Annihilation Operator on Fns
11.3.1.4 Creation Operator on Fn
11.3.1.5 Creation Operator on Fns
11.3.1.6 af Is Adjoint of af
11.3.2 Commutation Relations
11.4 Fock Space for Fermions
11.4.1 Fock Space Fa
11.4.1.1 Annihilation and Creation Operators on Fa
11.4.2 Anti-commutation Relations
11.5 Orthonormal Bases in Hns and Hna
11.5.1 Orthonormal Basis in Hn
11.5.2 Orthonormal Basis in Hns
11.5.3 Orthonormal Basis in Hna
11.6 Occupation Number Representation
11.6.1 Symmetric Occupation States
11.6.2 Antisymmetric Occupation States
11.7 Summary
11.7.1 Af, Af
11.7.2 Symmetrizer S(n) and Symmetric Subspaces
11.7.3 Antisymmetrizer A(n) and AntisymmetricSubspaces
11.7.4 Orthonormal Bases in Hns
11.7.5 Orthonormal Basis in Hna
11.8 Notes
References
12 Second Quantization
12.1 Tensor Product of Observables
12.2 One-Particle Observables
12.3 Two-Particle Observables
12.3.1 Bosons
12.3.2 Fermions
12.4 Symmetries
12.5 Examples
12.5.1 Non-relativistic Particles
12.5.1.1 Bosons
12.5.1.2 Fermions
12.5.2 Heisenberg Picture
12.5.2.1 The Name ``Second Quantization''
12.6 Many Species of Particles: The Standard Convention
12.7 A Briefest Outline of Feynman Diagrams
12.8 Summary
12.8.1 One-Particle Operators
12.8.2 Two-Particle Observables
12.8.2.1 Bosons
12.8.2.2 Fermions
12.8.3 Symmetries
12.9 Notes
12.9.1 Definition of the Tensor Product
12.9.2 The Standard Convention
References
13 Relativistic Configuration States and Quantum Fields
13.1 Spin Zero Particle of Mass m
13.2 Configuration Space States
13.2.1 Inner Product in Configuration Space
13.2.2 "426830A ct,x|ct,y"526930B ≠0
13.3 Fock Space for Spin Zero Particles
13.3.1 Microcausality
13.4 Space and Time-Reversal Covariance
13.5 Observables
13.5.1 Pauli-Jordan Function
13.5.2 Momentum and Energy
13.6 Charged Spin Zero Particles
13.6.1 Microcausality
13.6.2 Charge Conjugation
13.7 Spin 1/2 Particles
13.7.1 Introduction to SL(2,C)
13.7.1.1 ``Direct'' Boost
13.7.2 Proper Orthochronous Lorentz Group
13.8 Covariant States
13.8.1 Momentum States
13.8.2 Wigner Rotation
13.8.3 Dirac Equation in Momentum Space
13.8.4 Parity P
13.8.5 Configuration Space States
13.8.6 Dirac Equations in Configuration Space
13.9 Fock Space for Spin 12 particles
13.9.1 The Anti-particle
13.9.2 Microcausality
13.10 Notes
13.10.1 Relativistic Invariance
13.10.2 Dirac Equation
References
14 Minimum Uncertainty States
14.1 Uncertainty Inequality
14.2 Condition for Equality
14.2.1 Example
14.3 Coherent States
14.3.1 Annihilation and Creation Operators
14.3.2 Eigenstates of af
14.3.3 Minimum Uncertainty States
14.4 Notes
References
15 Path Integrals in General Coordinates
15.1 Introduction
15.2 Feynman Path Integral
15.2.1 The Short-Time Propagator (STP)
15.2.2 The VPM determinant
15.3 Path Integral as a Scheme of Quantization
15.3.1 A Free Particle in Two Dimensions
15.4 Time Re-parametrization
15.5 Pseudo Time Scaling of Paths
15.6 Coulomb Potential
15.7 Schrodinger Equations for Riemannian Configuration Space
15.7.1 Lagrangian STP
15.7.2 The Canonical STP
15.8 Notes
15.8.1 K-S Transformation
15.8.2 General Reviews
References
16 A Brief Pre-history of Matrix Mechanics and O(4) Symmetry of the Hydrogen Atom
16.1 Niels Bohr(1913)
16.2 Albert Einstein (1917)
16.3 Ladenberg (1919)
16.3.1 Drude's Classical Theory of Dispersion
16.3.2 Loss of Energy: Classical Radiation
16.3.3 Loss of Energy: Quantum Theory
16.3.4 Kramers (1924)
16.4 Bohr, Kramers, Slater (1924)
16.5 Thomas, Reiche, and Kuhn (1925)
16.5.1 Remark About the Kuhn Formula
16.6 Heisenberg
16.6.1 Heisenberg's Proof of the Thomas-Kuhn Sum Rule, July 1925
16.7 Hydrogen Atom and O(4)
16.7.1 Runge-Lenz Vector in Classical Mechanics
16.7.2 Runge-Lenz Vector in Quantum Mechanics
16.7.3 A Useful Result
16.8 Lie Algebra of O(4)
16.8.1 Lie Algebra of O(3) and SU(2)
16.8.1.1 Important Note
16.9 Energy Spectrum via O(4)
16.10 Note
References
17 Non-locality in Quantum Mechanics and Bell's Inequality
17.1 Introduction
17.1.1 Probabilities in Quantum Mechanics
17.1.2 EPR Incompleteness Argument
17.1.3 David Bohm's Reformulation of the EPRArgument
17.1.4 Hidden Variables and Bell's Inequality
17.2 Bell's Hidden Variable Model for a Spin 1/2 Particle
17.2.1 Impossibility Proof of Hidden Variable Theories
17.3 The Bohm Version of EPR: Two Spins
17.3.1 Measuring Spins in Two Different Directions
17.4 Bell's Inequality
17.4.1 Wigner's Proof
17.4.2 BCHSH Generalization
17.5 Bell's Inequality Violation (BIV)
17.5.1 Entangled Photon Pairs
17.5.1.1 State of Entangled Photons
17.5.1.2 Polarization Vectors in Classical Optics
17.5.1.3 Bell Inequality for Photons
17.5.1.4 Polarizers
17.5.1.5 Aspect Experiment with Variable Axes
17.5.2 Creation of Entangled Photon Pairs by SPDC
17.5.3 Summary of Experimental Evidence for BIV
17.5.4 Are Inequalities Really Necessary?
17.6 Epilogue
17.7 Notes
17.8 General Reviews
References
Index