G. A. Arteca

Large Order Perturbation Theory and Summation Methods in Quantum Mechanics

The problems usually found in Theoretical Chemistry and Physical Chemistry involve the use of quantum mechanical models which, as a general rule, do not have exact solutions. Due to this very compelling reason a formidable effort has been devoted to develop approximate methods to solve the Schrodinger equation from the very birth of the Quantum Mecha nics.

Rayleigh-Schrodinger Perturbation Theory (RSPT)

We devote this Section to present RSPT in a way appropriate to our specific needs. There are several alternative manners to introduce this formalism which can be found in the standard literature /1–3/.

Application of the FM to the Anharmonic Oscillator

This paragraph is devoted to the application of the FM developed in §.44 to a widely studied power series with zero convergence radius: the RSPT for anharmonic oscillators. As discussed before, the anharmonic oscillator models are closely related with a λϕ2M field theory (Appendix B). The model is also found when describing molecular vibrations, diffusion processes, laser theories, etc. (Appendix A). We have resorted to this model time and again in preceding chapters but right now we are able to obtain highly accurate results.

FM and Vibrational Potentials of Diatomic Molecules

The electronic energy of diatomic molecules as a function of the distance between the atoms is usually expanded around any of the following points: the united atom, the separated atoms or equilibrium. This last possibility is only valid for bound molecular states.

Multidimensional Systems: The Problem of the Zeeman Effect in Hydrogen

The aim of this chapter is to present an up-to-date, general overview of a problem of current interest: the properties of matter placed in strong magnetic fields. Moreover, we shall discuss here different theoretical methods that have been developed and applied to study several phenomena appearing under the effect of external magnetic fields.

Application of the FM to the Stark Effect in Hydrogen

The Stark effect in hydrogen has deserved a remarkable attention since the birth of Quantum Mechanics /1–9/.

Divergence of the Perturbation Series

The RSPT (Chapter III) allows one to get an approximation to the eigenvalues (En) of a given Hamiltonian operator through a series in powers of a real parameter λ. However, the usefulness of the power series is conditioned by a fundamental question: its convergence.

Application of the FM to Models with Confining Potentials

Quantum mechanical models with confining potentials have deserved remarkable attention in recent years, because they provide convenient phenomenological potentials to explain the hadronic spectroscopy within the context of quark theory. The aim of this chapter is to employ the FM to calculate the eigenvalues of these models by purely numerical approaches and also to derive accurate analytical expressions. We do not discuss here the applicability of the model, since this question is beyond the scope of this book. However, the interested reader may resort to Appendix F for additional details about the importance and usefulness of the confining models in elementary particle physics.

Application of the VFM to One-Dimensional Systems with Boundary Conditions for Finite Values of the Coordinates

Qiantum Systems subjected to finite BC are valuable in several fields of Chemistry and Physics. Thus, a large number of phenomena require the use of bounded quantum models in order to rationalize them in a natural manner.

Geometrical Connection Between the VFM and the JWKB Method

Previous paragraphs were devoted to discussing several functional energy representations of physical systems, through the generalization of semiclassical relationships, and the Heisenberg inequalities or the de Broglie hypothesis. It has been shown that all these approximations lead to eigenvalues depending on quantum numbers and parameters contained within the Hamiltonian, similarly to those obtained via the JWKB method and the variational theorem /1–13/ (see Chapter VI).