Particle in a central potential. Hydrogen atom model

Peter Woit

Ana Marı́a Cetto, Luis de la Peña

Chapter 12 uses the results presented in the previous chapter to deal with systems consisting of two particles interacting via a central potential. It therefore begins by transforming the Hamiltonian of the two-particle system into one that separates the center-of-mass motion from the relative motion. When the potential is spherically symmetric, the relative motion is conveniently described in spherical coordinates to take advantage of the conservation of angular momentum and the solution of the corresponding eigenvalue problem. The chapter contains a detailed discussion of the rigid rotor and the free particle, as well as the selection rules for angular momentum, in preparation for the study of the most important problem of atomic physics, the hydrogen atom. The H spectrum is discussed in detail, and the introduction of minimal coupling to the electromagnetic field allows for a discussion of the normal Zeeman effect as well as the Aharonov-Bohm effect.

P. C. W. Davies, David S. Betts

Waldemar Wolney Filho

In this work we are presenting a modified Coulomb potential function to describe the interaction between two micro-scopic electric charges. In particular, concerning the interaction between the proton and the electron in the hydrogen atom. The modified potential function is the product of the classical Coulomb potential and an oscillatory function dependent on a quantized phase factor. The oscillatory function picks up only selected points along the Coulomb potential, creating potential wells and barriers around the nucleus of the atom. The new potential reveals us new features of the hydrogen atom. Searching for a manner to determine the phase factor, we are using the concept of the de Broglie particle wavelike behavior and the quantum analogue of the virial theorem for describing the bound motion of a particle in a central force field. This procedure is a kind of feedback action, where we are making use of well established concepts of the quantum mechanics aiming to determine the phase factor of the new interaction potential.

D. Baye, M. Dufour, Benjamin Fuks E

David K. Ferry

The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. In general, this is a multi-dimensional problem, and not one of the simpler one-dimensional problems with which we have been concerned in the previous chapters. Once we begin to treat multiple dimensions, then state degeneracy begins to arise more frequently, and the most common problem treated is that of the hydrogen atom vs. the helium atom. These extra dimensions provide more degrees of freedom and more complexity. In the hydrogen atom, there is only a single electron, and the problem is solvable. On the other hand, adding a second electron for the helium atom makes the problem extremely difficult to solve, as it has become a three-body problem in which the interactions are difficult to handle.