Asymptotic Methods in Quantum Mechanics

S. H. Patil,

K. T. Tang

E-Book (pdf), 174 Seiten

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Quantum mechanics and the Schrodinger equation are the basis for the de scription of the properties of atoms, molecules, and nuc...

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Beschreibung

Quantum mechanics and the Schrodinger equation are the basis for the de scription of the properties of atoms, molecules, and nuclei. The development of reliable, meaningful solutions for the energy eigenfunctions of these many is a formidable problem. The usual approach for obtaining particle systems the eigenfunctions is based on their variational extremum property of the expectation values of the energy. However the complexity of these variational solutions does not allow a transparent, compact description of the physical structure. There are some properties of the wave functions in some specific, spatial domains, which depend on the general structure of the Schrodinger equation and the electromagnetic potential. These properties provide very useful guidelines in developing simple and accurate solutions for the wave functions of these systems, and provide significant insight into their physical structure. This point, though of considerable importance, has not received adequate attention. Here we present a description of the local properties of the wave functions of a collection of particles, in particular the asymptotic properties when one of the particles is far away from the others. The asymptotic behaviour of this wave function depends primarily on the separation energy of the outmost particle. The universal significance of the asymptotic behaviour of the wave functions should be appreciated at both research and pedagogic levels. This is the main aim of our presentation here.
Inhalt
1. Introduction.- 2. General Properties of Wave Functions.- 2.1 Asymptotic Form of Wave Functions.- 2.2 Asymptotic Perturbed Wave Function.- 2.3 Wave Function for rij ? 0.- 2.4 Wave Function for rij and rik ? 0.- 2.5 Local Satisfaction of Schrödinger Equation.- 2.6 Variational Stationary Property.- 2.7 Variational Approach to Perturbations.- 2.8 Generalised Virial Theorem.- 2.9 A Simple Example.- 3. Two- and Three-Electron Atoms and Ions.- 3.1 A Simple Wave Function.- 3.1.1 Energy.- 3.1.2 Multipolar Potential Perturbation.- 3.1.3 Third Order Energy Shifts.- 3.2 Wave Functions Satisfying Cusp, Coalescence and Asymptotic Conditions.- 3.2.1 Behaviour for ri ? 0.- 3.2.2 Correlation Function.- 3.2.3 Results for the Unperturbed Ground State.- 3.2.4 Multipolar Polarizabilities and Hyperpolarizabilities.- 3.2.5 Wave Functions for Excited States.- 3.3 Three-Electron Wave Functions.- 4. Polarizabilities and Dispersion Coefficients.- 4.1 Polarizabilities.- 4.1.1 Perturbative Expression.- 4.1.2 Hyperpolarizabilities.- 4.1.3 Dynamic Polarizabilities.- 4.2 Dispersion Coefficients.- 4.2.1 Relation to Dynamic Polarizabilities.- 4.2.2 Three-Body Dispersion Coefficients.- 4.3 Alkali Isoelectronic Sequences.- 4.3.1 The Wave Function.- 4.3.2 Polarizabilities.- 4.3.3 Hyperpolarizabilities and Dispersion Coefficients.- 4.4 Asymptotic Polarizabilities and Dispersion Coefficients.- 4.4.1 Asymptotic Polarizabilities.- 4.4.2 Polarizabilities of He and Ne Systems.- 4.4.3 Asymptotic Behaviour of the Effective Energy.- 4.4.4 Dispersion Coefficients for H, He and Ne.- 5. Asymptotically Correct Thomas-Fermi Model Density.- 5.1 Thomas-Fermi Model.- 5.1.1 Statistical Approach.- 5.1.2 WKB Approach.- 5.2 Solution for the Thomas-Fermi Density.- 5.3 Asymptotic Density.- 5.4 Modified Density.- 5.5 Applications.- 5.5.1 Expectation Values ?r2n?.- 5.5.2 Multipolar Polarizabilities.- 5.5.3 Dispersion Coefficients.- 6. Molecules and Molecular Ions with One and Two Electrons.- 6.1 Wave Functions for One-Electron Molecular Ions.- 6.1.1 Cusp Condition.- 6.1.2 Asymptotic Behaviour.- 6.2 Energies for One-Electron Molecular Ions.- 6.3 Wave Function for H2 and He2++.- 6.3.1 Molecular Orbital Type of Wave Function.- 6.3.2 Atomic Orbital Type of Wave Function.- 6.3.3 General Wave Function.- 6.3.4 Correlation Function.- 6.4 Results for the Ground State.- 6.4.1 Ground State Energies.- 6.4.2 Discussion.- 7. Interaction of an Electron with Ions, Atoms, and Molecules.- 7.1 Atomic Rydberg States.- 7.1.1 Perturbation Approach for Anti-symmetric Wave Functions.- 7.1.2 The Perturbed Hamiltonian.- 7.1.3 Asymptotic Core Density and Density Matrix.- 7.1.4 Penetration Energy.- 7.1.5 Exchange Energy.- 7.1.6 Second Order Terms.- 7.1.7 Total Energy Shift.- 7.1.8 Results.- 7.2 Electron-Atom and Electron-Molecule Scattering at High Energies.- 7.2.1 Perturbation Series for the Scattering Amplitude.- 7.2.2 Scattering Amplitude at High Energies.- 7.2.3 Electron-Atom Scattering.- 7.2.4 Electron-Molecule Scattering.- 8. Exchange Energy of Diatomic Systems.- 8.1 Exchange Energy of Dimer Ions.- 8.1.1 Exchange Energy of the H2+ Molecular Ion by Surface Integral Method.- 8.1.2 Exchange Energy of Multielectron Dimer Ions.- 8.2 Exchange Energy of Diatomic Molecules.- 8.2.1 Exchange Energy of the H2 Molecule.- 8.2.2 Exchange Energy of Multielectron Diatomic Molecules.- 9. Inter-atomic and Inter-ionic Potentials.- 9.1 Exchange Energy and Exchange Integral in the Heitler-London Theory.- 9.2 Generalized Heitler-London Theory.- 9.2.1 Unsymmetrized (Polarization) Perturbation Method.- 9.2.2 Symmetry Imposed Generalised Heitler-London Equation.- 9.2.3 Asymptotic Exchange Energy and Polarization Approximation.- 9.3 Inter-atomic and Inter-ionic Potentials.- 9.3.1 The 3?u State Potential of the H2 Molecule.- 9.3.2 The 2?u State Potentials of Alkali Dimer Cations.- 9.3.3 The Potential of Rare Gas Dimers.- 10. Proton and Neutron Densities in Nuclei.- 10.1 Semi-phenomenological Density.- 10.2 Determination of the Parameters.- 10.3 Results.- References.