Dynamical Symmetry 1st Edition by Carl Wulfman ISBN 9789814291378 9814291374

CHAPTER 1 Introduction

1.1 On Geometric Symmetry and Invariance in the Sciences

1.2 Fock’s Discovery

1.3 Keplerian Symmetry

1.4 Dynamical Symmetry

1.5 Dynamical Symmetries Responsible for Degeneracies and Their Physical Consequences

1.6 Dynamical Symmetries When Energies Can Vary

1.7 The Need for Critical Reexamination of Concepts of Physical Symmetry. Lie’s Discoveries

Appendix A. Historical Note

References

CHAPTER 2 Physical Symmetry and Geometrical Symmetry

2.1 Geometrical Interpretation of the Invariance Group of an Equation; Symmetry Groups

2.2 On Geometric Interpretations of Equations

2.3 Geometric Interpretations of Some Transformations in the Euclidean Plane

2.4 The Group of Linear Transformations of Two Variables

2.5 Physical Interpretation of Rotations

2.6 Intrinsic Symmetry of an Equation

2.7 Non-Euclidean Geometries

2.8 Invariance Group of a Geometry

2.9 Symmetry in Euclidean Spaces

2.10 Symmetry in the Spacetime of Special Relativity

2.11 Geometrical Interpretations of Nonlinear Transformations: Stereographic Projections

2.12 Continuous Groups that Leave Euclidean and Pseudo-Euclidean Metrics Invariant

2.13 Geometry, Symmetry, and Invariance

Appendix A: Stereographic Projection of Circles

References

CHAPTER 3 On Symmetries Associated With Hamiltonian Dynamics

3.1 Invariance of a Differential Equation

3.2 Hamilton’s Equations

3.3 Transformations that Convert Hamilton’s Equations into Hamilton’s Equations; Symplectic Grou

3.4 Invariance Transformations of Hamilton’s Equations of Motion

3.5 Geometrization of Hamiltonian Mechanics

3.6 Symmetry in Two-Dimensional Symplectic Space

3.7 Symmetry in Two-Dimensional Hamiltonian Phase Space

Definition of Two-dimensional Hamiltonian Phase–Space:

Definition of Symmetry in Two-Dimensional Hamiltonian Phase–Space:

3.8 Symmetries Defined by Linear Symplectic Transformations

3.9 Nonlinear Transformations in Two-Dimensional Phase Space

3.10 Dynamical Symmetries as Intrinsic Symmetries of Differential Equations and as Geometric Symmetr

Exercises:

References

CHAPTER 4 One-Parameter Transformation Groups

4.1 Introduction

4.2 Finite Transformations of a Continuous Group Define Infinitesimal Transformations and Vector Fie

4.3 Spaces in which Transformations will be Assumed to Act on

4.4 The Defining Equations of One-Parameter Groups of Infinitesimal Transformations. Group Generator

4.5 The Differential Equations that Define Infinitesimal Transformations Define Finite Transformatio

4.6 The Operator of Finite Transformations

4.7 Changing Variables in Group Generators

4.8 The Rectification Theorem

4.9 Conversion of Non-autonomous ODEs to Autonomous ODEs

4.10 N-th Order ODEs as Sets of First-Order ODEs

4.11 Conclusion

Appendix A. Homeomorphisms, Diffeomorphisms, and Topology

Exercises

References

CHAPTER 5 Everywhere-Local Invariance

5.1 Invariance under the Action of One-Parameter Lie Transformation Groups

5.2 Transformation of Infinitesimal Displacements

5.3 Transformations and Invariance of Work, Pfaffians, and Metrics

5.4 Point Transformations of Derivatives

5.5 Contact Transformations

5.6 Invariance of an Ordinary Differential Equation of First-Order under Point Transformations; Exte

5.7 Invariance of Second-Order Ordinary Differential Equations under Point Transformations; Harmonic

5.8 The Commutator of Two Operators

5.9 Invariance of Sets of ODEs. Constants of Motion

5.10 Conclusion

Appendix A. Relation between Symmetries and Intergrating Factors

Appendix B. Proof That the Commutator of Lie Generators is Invariant under Diffeomorphisms

Appendix C. Isolating and Non-isolating Integrals of Motion

Exercises

References

CHAPTER 6 Lie Transformation Groups and Lie Algebras

6.1 Relation of Many-Parameter Lie Transformation Groups to Lie Algebras

6.2 The Differential Equations That Define Many-Parameter Groups

6.3 Real Lie Algebras

6.4 Relations between Commutation Relations and the Action of Transformation Groups: Some Examples

6.5 Transitivity

6.6 Complex Lie Algebras

6.7 The Cartan–Killing Form; Labeling and Shift Operators

6.8 Casimir Operators

6.9 Groups That Vary the Parameters of Transformation Groups

6.10 Lie Symmetries Induced from Observations

6.11 Conclusion

Appendix A. Definition of Lie Groups by Partial Differential Equations

Exercises.

References

CHAPTER 7 Dynamical Symmetry in Hamiltonian Mechanics

7.1 General Invariance Properties of Newtonian Mechanics

7.2 Relationship of Phase Space to Abstract Symplectic Space

7.3 Hamilton’s Equations in PQ Space. Constants of Motion

7.4 Poisson Bracket Operators

7.5 Hamiltonian Dynamical Symmetries in PQ Space

7.6 Hamilton’s Equations in Classical PQET Space; Conservation Laws Arising From Galilei Invarianc

7.7 Time-dependent Constants of Motion; Dynamical Groups That Act Transitively

7.8 The Symplectic Groups Sp(2n,r)

7.9 Generalizations of Symplectic Groups That Have an Infinite Number of One-parameter Groups

Appendix A. Lagrange’s Equations and the Definition of Phase Space

Appendix B. The Variable Conjugate to Time in PQET Space

Exercises

References

CHAPTER 8 Symmetries of Classical Keplerian Motion

8.1 Newtonian Mechanics of Planetary Motion

8.2 Hamiltonian Formulation of Keplerian Motions in Phase Space

8.3 Symmetry Coordinates For Keplerian Motions

8.4 Geometrical Symmetries of Bound Keplerian Systems in Phase Space

8.6 The SO(4,1) Dynamical Symmetry

8.7 Concluding Remarks

Exercises:

References

CHAPTER 9 Dynamical Symmetry in Schrodinger Quantum Mechanics

9.1 Superposition Invariance

9.2 The Correspondence Principle

9.3 Correspondence Between Quantum Mechanical Operators and Functions of Classical Dynamical Variabl

9.4 Lie Algebraic Extension of the Correspondence Principle

9.5 Some Properties of Invariance Transformations of Partial Differential Equations Relevant to Quan

9.6 Determination of Generators and Lie Algebra of Invariance Transformations of ((−1/2)∂2/∂x2

9.7 Eigenfunctions of the Constants of Motion of a Free-Particle

9.8 Dynamical Symmetries of the Schr¨odinger Equations of a Harmonic Oscillator

9.9 Use of the Oscillator Group in Pertubation Calculations

9.10 Concluding Observations

Exercises

References

CHAPTER 10 Spectrum-Generating Lie Algebras and Groups Admitted by Schr¨odinger Equations

Introduction

10.1 Lie Algebras That Generate Continuous Spectra

10.2 Lie Algebras That Generate Discrete Spectra

10.3 Dynamical Groups of N-Dimensional Harmonic Oscillators

10.4 Linearization of Energy Spectra by Time Dilatation; Spectrum-Generating Dynamical Group of Rigi

10.5 The Angular Momentum Shift Algebra; Dynamical Group of the Laplace Equation

10.6 Dynamical Groups of Systems with Both Discrete and Continuous Spectra

10.7 Dynamical Group of the Bound States of Morse Oscillators

10.8 Dynamical Group of the Bound States of Hydrogen-Like Atoms

10.9 Matrix Representations of Generators and Group Operators

10.10 Invariant Scalar Products

10.11 Direct-Products: SO(3)⊗SO(3) and the Coupling of Angular Momenta

10.12 Degeneracy Groups of Non-interacting Systems; Completions of Direct-Products

10.13 Dynamical Groups of Time-dependent Schrodinger Equations of Compound Systems; Many-Electron At

Appendix A. References Providing Invariance Groups of Schrodinger Equations

Appendix B. References to Work Dealing with Dynamical Symmetries in Nuclear Shell Theory

Exercises

References

CHAPTER 11 Dynamical Symmetry of Regularized Hydrogen-like Atoms

Introduction

11.1 Position-space Realization of the Dynamical Symmetries

11.2 The Momentum-space Representation

11.3 The Hyperspherical Harmonics Yklm

11.4 Bases Provided by Eigenfunctions of J12, J34, J56

Appendix A. Matrix Elements of SO(4,2) Generators3,16

Appendix B. N-Shift Operators For the Hyperspherical Harmonics

References

CHAPTER 12 Uncovering Approximate Dynamical Symmetries. Examples From Atomic and Molecular Physics

12.1 Introduction

12.2 The Stark Effect; One-Electron Diatomics

12.3 Correlation Diagrams and Level Crossings: General Remarks

12.4 Coupling SO(4)1 ⊗ SO(4)2 to Produce SO(4)12

12.5 Coupling SO(4)1 ⊗ SO(4)2 to Produce SO(4)1−2

12.6 Configuration Mixing in Doubly Excited States of Helium-like Atoms

12.7 Configuration Mixing Arising From Interactions Within Valence Shells of Second and Third Row At

12.8 Origin of the Period-Doubling Displayed in Periodic Charts

12.9 Molecular Orbitals in Momentum-Space; The Hyperspherical Basis

12.10 The Sturmian Ansatz of Avery, Aquilanti and Goscinski

Exercises

References

CHAPTER 13 Rovibronic Systems

13.1 Introduction

13.2 Algebraic Treatment of Anharmonic Oscillators With a Finite Number of Bound States

13.3 U(2) ⊗ U(2) Model of Vibron Coupling

13.4 Spectrum Generating Groups of Rigid Body Rotations

13.5 The U(4) Vibron Model of Rotating Vibrating Diatomics

13.6 The U(4) ⊗ U(4) Model of Rotating Vibrating Triatomics

13.7 Concluding Remarks

Exercises

References

CHAPTER 14 Dynamical Symmetry of Maxwell’s Equations

14.1 The Poincare Symmetry of Maxwell’s Equations

14.2 The Conformal and Inversion Symmetries of Maxwell’s Equations; Their Physical Interpretation

14.3 Alteration of Wavelengths and Frequencies by a Special Conformal Transformation: Interpretation

Conclusion

Exercises

References

Index