The supersymmetric Dirac equation the application to hydrogenic atoms / Allen Hirshfeld.

Includes bibliographies and index.

14.4 The Johnson-Lippmann Operator as the Generator of Supersymmetry15. Extending the Solution Space; 15.1 The -Induced Radial Supersymmetry; 15.2 The Supersymmetric Ground State in the Representation; 15.3 The General Solutions in the Representation; 16. A Different Extension of the Solution Space; 16.1 The .-Induced Radial Supersymmetry; 16.2 The Supersymmetric Ground State in the Representation; 16.3 The General Solutions in the Representation; 17. The Relation of the Solutions to Kramer's Equation; 17.1 The Eigenvalue Problem for Traceless 2 × 2 Matrices.

Preface; Contents; List of Figures; 1. Introduction; 2. The Classical Kepler Problem; 2.1 Central Forces; 2.2 The Laplace Vector; 3. Symmetry of the Classical Problem; 3.1 Lie Groups and Lie Algebras; 3.2 Some Special Lie Algebras; 3.3 Poisson Brackets; 3.4 The Inverse Square Law; 4. From Solar Systems to Atoms; 4.1 Rutherford Scattering; 4.2 Conservation of the Laplace Vector; 4.3 The Differential Cross Section; 5. The Bohr Model; 5.1 Spectroscopic Series; 5.2 The Postulates of the Model; 5.3 The Predictions of the Model; 5.4 Correction for Finite Nuclear Mass.

6. Interpretation of the Quantum Rules6.1 The Sommerfeld-Wilson Quantization Conditions; 6.2 de Broglie's Wave Interpretation; 7. Sommerfeld's Model for Non-Relativistic Electrons; 7.1 Assumptions of the Model; 7.2 Results of the Model for Non-Relativistic Hydrogen Atoms; 7.3 The Eccentricity; 8. Quantum Mechanics of Hydrogenic Atoms; 8.1 Quantization; 8.2 Quantum Mechanical Relation Between A and L; 8.3 Pauli's Hydrogenic Realization of so(4); 8.4 so(4) and the Spectrum of Hydrogenic Atoms; 9. The Schrödinger Equation and the Confluent Hypergeometric Functions.

9.1 The Schrödinger Equation and Its Solutions9.2 Laguerre Polynomials and Associated Laguerre Functions; 10. Non-Relativistic Hydrogenic Atoms with Spin; 10.1 Spin Variables, the Pauli Hamiltonian and Factorization; 10.2 A Theorem Concerning the Anticommutation of K; 10.3 Pauli Spinors; 10.4 Concerning the Operator (r); 10.5 The Key Equation: Concerning the Operator (A); 10.6 The Factorization Method; 10.7 The Definition of Eccentricity; 11. Elements of Supersymmetric Quantum Mechanics; 11.1 General Considerations; 11.2 Supersymmetry of Non-Relativistic Hydrogenic Atoms.

12. Sommerfeld's Derivation of the Relativistic Energy Level Formula12.1 Assumptions of the Model; 12.2 The Energies of the Bound States; 13. The Dirac Equation; 13.1 The Hamiltonian; 13.2 Total Angular Momentum; 13.3 The Dirac Operator; 13.4 A Complete Set of Mutually Commuting Operators; 13.5 The Dirac Spinors; 13.6 The Radial Equations in Polar Coordinates; 14. The Primary Supersymmetry of the Dirac Equation; 14.1 A Derivation of the Johnson-Lippmann Operator; 14.2 Commutation and Anticommutation Relations of the Johnson-Lippmann Operator; 14.3 Eccentricity.

The solution of the Dirac equation for an electron in a Coulomb field is systematically treated here by utilizing new insights provided by supersymmetry. It is shown that each of the concepts has its analogue in the non-relativistic case. Indeed, the non-relativistic case is developed first, in order to introduce the new concepts in a familiar context. The symmetry of the non-relativistic model is already present in the classical limit, so the classical Kepler problem is first discussed in order to bring out the role played by the Laplace vector, one of the central concepts of the whole book.

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