Atom, atomic mass quantum mechanical model

The idea of electrons existing in definite energy states was fine, but another way had to be devised to describe the location of the electron about the nucleus. The solution to this problem produced the modern model of the atom, often called the quantum mechanical model. In this new model of the hydrogen atom, electrons do not travel in circular orbits but exist in orbitals with three-dimensional shapes that are inconsistent with circular paths. The modern model of the atom treats the electron not as a particle with a definite mass and velocity, but as a wave with the properties of waves. The mathematics of the quantum mechanical model are much more complex, but the results are a great improvement over the Bohr model and are in better agreement with what we know about nature. In the quantum mechanical model of the atom, the location of an electron about the nucleus is described in terms of probability, not paths, and these volumes where the probability of finding the electron is high are called orbitals. 

In Chapter 6 we learned that although an electron is a particle with a known mass, it exhibits wavelike properties. The quantum mechanical model of the atom, which gives rise to the fainiliar shapes of s and p atomic orbitals, treats electrons in atoms as waves, rather than particles. Therefore, rather than use arrows to denote the locations and spins of electrons, we will adopt a convention whereby a singly occupied orbital will appear as a light color and a doubly occupied... 

The accuracy of the CSP approximation is, as test calculations for model. systems show, typically very similar to that of the TDSCF. The reason for this is that for atomic scale masses, the classical mean potentials are very similar to the quantum mechanical ones. CSP may deviate significantly from TDSCF in cases where, e.g., the dynamics is strongly influenced by classically forbidden regions of phase space. However, for simple tunneling cases it seems not hard to fix CSP, by running the classical trajectories slightly above the barrier. In any case, for typical systems the classical estimate for the mean potential functions works extremely well. 

An important simplification in the analysis of He scattering is the large mass mismatch between He and most other atoms that are the constituents of solid surfaces. Therefore, energy transfer between the He atom and the surface is very much limited and elastic He diffraction or inelastic He scattering can be modelled rather easily and quantum mechanically. The situation is very different when heavier atoms or molecules are scattered from a surface. In this case energy exchange between projectile and surface will be facile, and in most cases only classical mechanics can be used to model the interaction. Most of the... 

The simplest model of a rotating diatomic molecule is a rigid rotor or dumbbell model in which the two atoms of mass and m2 are considered to be joined by a rigid, weightless rod. The allowed energy levels for a rigid rotor may be shown by quantum mechanics to be... 

Here U N) is the interaction potential energy for the complete system at a specific configuration, uniformly the same quantity that has been discussed above. U(TV) is an ejfective potential designed to be used in classical-limit partition function calculations, e.g. Eq. (3.17), p. 40, in order to include quantum mechanical effects approximately. We will call this /(TV) the quadratic Feynman-Hibbs (QFH) model. In Eq. (3.67), Mj is the mass of atom j, and V is the Laplacian of the... 

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