The Schrödinger and Riccati equations
著者
書誌事項
The Schrödinger and Riccati equations
（Lecture notes in chemistry, 70）
Springer-Verlag, c1999
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注記
Includes bibliographical references (p. 187-218)
内容説明・目次
内容説明
The linear Schroedinger 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 Schroedinger equation.
The authors develop the Schroedinger-Riccati equation as an approach to determine solutions of the time-independent, linear Schroedinger equation.
目次
1 Introduction.- The Linear Schroedinger Equation.- 2 Derivation of the Schroedinger 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 Schroedinger Hamiltonian Operator.- 2.4.1 Atoms.- 2.4.2 Molecules.- 2.5 The Invariant Form of the Hamiltonian Operator.- 3 The Schroedinger 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 Schroedinger 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 Schroedinger Equation.- 5.1 Alternatives for the Computational Solution of the Schroedinger 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 Schroedinger Equation.- 6.1 The Equation and Its Solutions.- 6.2 Time-Evolution Operators.- 6.3 Time-Evolution Representations.- 6.3.1 The Schroedinger 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 Schroedinger Equation.- 7 The Non-Linear Schroedinger 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 Schroedinger 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 Schroedinger-Riccati Equation.- 10 The Schroedinger-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 Schroedinger-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 Schroedinger-Riccati Equation.- 11 Numerical Experience with the Schroedinger-Riccati Equation.- 11.1 1-Dimensional Schroedinger 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.
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