The Schrödinger and Riccati Equations

ISBN-13: 9783540651055 / Angielski / Miękka / 1998 / 218 str.

Serafin Fraga;Jose M. Garcia de la Vega;Eric S. Fraga

Opis

The linear Schrodinger equation is central to Quantum Chemistry. It is presented within the context of relativistic Quantum Mechanics and analysed both in time-dependent and time-independent forms. The Riccati equation is used to study the one-dimensional Schrodinger equation.
The authors develop the Schrodinger-Riccati equation as an approach to determine solutions of the time-independent, linear Schrodinger equation.

Spis treści

1 Introduction.- The Linear Schrödinger Equation.- 2 Derivation of the Schrödinger Equation.- 2.1 The Dirac Equation.- 2.2 The Breit Correction.- 2.3 The Generalized Dirac-Breit Equation.- 2.3.1 Separation of the Coordinates of the Centre of Mass.- 2.3.2 Reduction to Non-Relativistic Form.- 2.4 The Schrödinger Hamiltonian Operator.- 2.4.1 Atoms.- 2.4.2 Molecules.- 2.5 The Invariant Form of the Hamiltonian Operator.- 3 The Schrödinger Equation in Position Space.- 3.1 The Hamiltonian Operator and Its Eigenfunctions.- 3.1.1 The Hamiltonian Operator.- 3.1.2. The Eigenfunctions.- 3.2 Local Properties.- 3.2.1 Nodes and Nodal Planes.- 3.2.2 Local Energies.- 3.2.3 Coalescence and Cusp Conditions.- 3.2.4 Null Kinetic- and Potential-Energy Regions.- 3.3 Global Properties.- 3.3.1 The Potential Energy Hypersurface.- 3.3.2 The Hellmann-Feynman Theorem.- 3.3.3 The Hypervirial Theorem.- 3.3.4 The Virial Theorem.- 4 The Schrödinger Equation In Momentum Space.- 4.1 Introduction.- 4.1.1 The r-Representation versus the p-Representation.- 4.1.2 The Fourier Transform Method.- 4.2 The Transformed Equation.- 4.2.1 General Formulation.- 4.2.2 The Hydrogen Atom.- 4.3 The Transformed Functions.- 4.3.1 Hydrogenic Orbitals.- 4.4 Properties and Expectation Values.- 4.4.1 Symmetry.- 4.4.2 Momentum Density Distributions.- 4.4.3 Expectation Values.- 5 The Local Schrödinger Equation.- 5.1 Alternatives for the Computational Solution of the Schrödinger Equation.- 5.1.1 Computational Solutions of the Global Problem.- 5.1.2 Local vs. Global Computational Solutions.- 5.2 The Local Energy Methods.- 5.2.1 The Frost Equation.- 5.2.2 Reduced Local Energies.- 5.3 Local Orbital Energies.- 6 The Time-Dependent Schrödinger Equation.- 6.1 The Equation and Its Solutions.- 6.2 Time-Evolution Operators.- 6.3 Time-Evolution Representations.- 6.3.1 The Schrödinger Representation.- 6.3.2 The Heisenberg Representation.- 6.3.3 The Interaction Representation.- 6.4 The Solution of the Interaction Evolution Equation.- 6.4.1 The Perturbation Series Expansion.- 6.4.2 The Magnus Formula.- 6.5 Evolution of the System: Transition Probabilities.- The Non-Linear Schrödinger Equation.- 7 The Non-Linear Schrödinger Equation.- 7.1 Solitary Waves and Solitons.- 7.2 The 1-Dimensional NLS Equation.- 7.2.1 Conservation Laws and Conserved Quantities.- 7.2.2 Analytic Solution.- 7.3 Numerical Solution of the 1-Dimensional NLS Equation.- 7.3.1 The Finite Element Method.- 7.3.2 Mesh Refinement Techniques.- 7.4 The Generalized 1-Dimensional NLS Equations.- 7.4.1 Equation i?t + ?xx + qc|?|2? + qq+|?|4?+ i qm|?|2x? + i qu|?|2?x = 0.- 7.4.2 Equation i?t = — ?xx + U(x)? + ?(x,t)? + w?.- 7.5 The Problem of Higher Dimensions.- The Riccati Equation.- 8 The Riccati Equation and Its Solution.- 8.1 Derivation of the Riccati Equation.- 8.2 Solution of the Riccati Equation.- 8.2.1 Solution of the General Riccati Equation.- 8.2.2 Solution of the Transformed Riccati Equation.- 8.2.3 Solution of the Special Riccati Equation.- 9 Quantum-Mechanical Applications of the Riccati Equation.- 9.1 Introduction.- 9.1.1 The Sturm-Liouville Equation.- 9.1.2 The 1-Dimensional Schrödinger Equation.- 9.1.3 The Riccati Equation for the Logarithmic Derivative.- 9.2 Solution of Sturm-Liouville Equations.- 9.2.1 Equations ?”(E,x) + A(x)?’(E,x) + B(E,x)?(E,x) = 0.- 9.2.2 Equations ?”(E,x) + B(E,x)?(E,x) = 0.- The Schrödinger-Riccati Equation.- 10 The Schrödinger-Riccati Equation.- 10.1 Formulation for Spin-Free Functions.- 10.1.1 Basic Definitions and Relationships.- 10.1.2 The Riccati Local Energies.- 10.1.3 The Schrödinger-Riccati Equation.- 10.1.4 Evaluation of the Correction Function ?.- 10.1.5 Evaluation of the Energy.- 10.2 Formulation for Antisymmetric Functions.- 10.2.1 Two-electron Systems.- 10.2.2 General Case.- 10.3 Applicability of the Schrödinger-Riccati Equation.- 11 Numerical Experience with the Schrödinger-Riccati Equation.- 11.1 1-Dimensional Schrödinger Equation.- 11.2 The Groundstate of the Hydrogen Atom.- 11.2.1 Preliminary Test.- 11.2.2 Evaluation of ? and ??.- 11.3 Groundstates of Ions of the He-Isoelectronic Series.- 11.3.1 Evaluation of the Basic Quantities.- 11.3.2 Evaluation of the Quantities T(n).- 11.3.3 Evaluation of the Energy Contributions.- 12 References and Bibliography.- 12.1 References.- 12.2 Bibliography.- Appendix. Matrix Notation.

Blurb

The linear Schrödinger equation is presented in the context of relativistic Quantum Mechanics. It is analysed in time-dependent and time-independent forms and its local and global properties are inspected. The authors develop the Schrödinger-Riccati equation as an approach to determine solutions of the time-independent, linear Schrödinger equation.