Wave mechanics mechanical model

Schrodinger equation valence electrons wave function wavelength, X wave mechanical model wave-particle duality of nature... 

Values of the radiative rate constant fcr can be estimated from the transition probability. A suggested relationship14 57 is given in equation (25), where nt is the index of refraction of the medium, emission frequency, and gi/ga is the ratio of the degeneracies in the lower and upper states. It is assumed that the absorption and emission spectra are mirror-image-like and that excited state distortion is small. The basic theory is based on a field wave mechanical model whereby emission is stimulated by the dipole field of the molecule itself. Theory, however, has not so far been of much predictive or diagnostic value. 

Comparing the results of the Bohr and the wave mechanical models, we find that ... 

The surprising conclusion to be drawn from this analysis is that a quantitative description of orbital angular momentum, as detailed as the wave-mechanical model, can be obtained within the old quantum theory. 

Soon after first publication Schrodinger s wave-mechanical model was extended by Madelung [39] on the basis of the obvious correspondence with the classical theory of hydrodynamics, already pointed out in Schrodinger s original papers [34] (II,p.l7). Writing the time dependence of k in terms of an action function... 

The formulation of spatially separated a and 7r interactions between a pair of atoms is grossly misleading. Critical point compressibility studies show [71] that N2 has essentially the same spherical shape as Xe. A total wave-mechanical model of a diatomic molecule, in which both nuclei and electrons are treated non-classically, is thought to be consistent with this observation. Clamped-nucleus calculations, to derive interatomic distance, should therefore be interpreted as a one-dimensional section through a spherical whole. Like electrons, wave-mechanical nuclei are not point particles. A wave equation defines a diatomic molecule as a spherical distribution of nuclear and electronic density, with a common quantum potential, and pivoted on a central hub, which contains a pith of valence electrons. This valence density is limited simultaneously by the exclusion principle and the golden ratio. 

On a plot of Z/N vs Z for all stable nuclides the field of stability is outlined very well by a profile, defined by the special points of the periodic table derived from 4. Furthermore, hem lines that divide the 264 nuclides into 11 groups of 24 intersect the convergence line, Z/N = r, at most of the points that define the periodic function. If the hem lines are extended to intersect the line Z/N = 0.58, a different set of points are projected out and found to match the periodicity, derived from the wave-mechanical model. 

From a chemical point of view the most important result is that number theory predicts two alternative periodic classifications of the elements. One of these agrees with experimental observation and the other with a wave-mechanical model of the atom. The subtle differences must be ascribed to a constructionist error that neglects the role of the environment in the wave-mechanical analysis. It is inferred that the wave-mechanical model applies in empty space Z/N = 0.58), compared to the result, observed in curved non-empty space, (Z/N = t). The fundamental difference between the two situations reduces to a difference in space-time curvature. 

The wave-mechanical model of the atom shows a more complex structure of the atom and the way electrons configure themselves in the principal energy levels. Principal energy levels are divided into sublevels, each with its own distinct set of orbitals. This more complex structure is outlined with the help of this diagram. The principal energy levels in the atom are numbered 1 through 7. 

The modern view of the periodic table explains its structure in terms of an Aufbau procedure based on the wave-mechanical model of the hydrogen atom. Although seductive at first glance, the model is totally inadequate to account for details of the observed electronic configurations of atoms, and makes no distinction between isotopes of the same element. The attractive part of the wave-mechanical model is that it predicts a periodic sequence of electronic configurations readily specified as a function of atomic number. The periodicity follows from the progressive increase of four quantum numbers n, l, mi and s, such that... 

Each of the three theories accounts for some, but not all aspects of elemental periodicity. The common ground among the three may well reveal the suspected link with space-time structure. What is required is to combine aspects of the wave-mechanical model of hydrogen, the structure of atomic nuclei and number theory. 

The results considered in this section are very important. We have seen that the wave mechanical model can be used to explain the arrangement of elements in the periodic table. This model allows us to understand that the similar chemistry exhibited by the members of a given group arises from the fact that they all have the same valence electron configuration. Only the principal quantum number of the occupied orbitals changes in going down a particular group. 

Standing wave a stationary wave as on a string of a musical instrument in the wave mechanical model, the electron in the hydrogen atom is considered to be a standing wave. 

Wave mechanical model a model for the hydrogen atom in which the electron is assumed to behave as a standing wave. (12.7)... 

The Schrodinger wave equation In 1926, Austrian physicist Erwin Schrbdinger (1887-1961) furthered the wave-particle theory proposed by de Broglie. Schrbdinger derived an equation that treated the hydrogen atom s electron as a wave. Remarkably, Schrbdinger s new model for the hydrogen atom seemed to apply equally well to atoms of other elements—an area in which Bohr s model failed. The atomic model in which electrons are treated as waves is called the wave mechanical model of the atom or, more commonly, the quantum mechanical model of the atom. Like Bohr s model,... 

Summary of the Wave Mechanical Model and Valence-Electron Configurations... 

To understand how the electron s position Is represented In the wave mechanical model... 

Orbitals In the Bohr model, the electron was assumed to move in circular orbits. In the wave mechanical model, on the other hand, the electron states are described by orbitals. Orbitals are nothing like orbits. To approximate the idea of an orbital, picture a single male firefly in a room in the center of which is suspended an open vial of a chemical which attracts fireflies. The room is extremely dark and in one corner there is a camera with an open shutter. Every time the firefly "flashes," the camera records a pinpoint of light and thus the firefly s position in the room at that particular moment. The firefly senses the attractant and, as you can imagine, it spends a lot of time at the vial or close to it. However, now and then the insect flies randomly around the room. 

O How does the wave mechanical model of the atom differ from Bohr s model ... 

In terms of the obsolete Bohr model, this meant the electron was transferred to an orbit with a larger radius. In the wave mechanical model, these higher energy states correspond to different kinds of orbitals with different shapes. 

The Bohr model was discarded because it could be applied only to hydrogen. The wave mechanical model can be applied to all atoms in basically the same form as we have just used it for hydrogen. In fact, the major triumph of this model is its ability to explain the periodic table of the elements. Recall that the elements on the periodic table are arranged in vertical groups, which contain elements that typically show similar chemical properties. For example, the halogens shown to the left are chemically similar. The wave mechanical model of the atom allows us to explain, based on electron arrangements, why these similarities occur. We will see later how this is done. 

Principal Components of the Wave Mechanical Model of the Atom... 

The concepts we have discussed in this chapter are very important. They allow us to make sense of a good deal of chemistry. When it was first observed that elements with similar properties occur periodically as the atomic number increases, chemists wondered why. Now we have an explanation. The wave mechanical model pictures the electrons in an atom as arranged in orbitals, with each orbital capable of holding two electrons. 

For reasons we will explore in the next chapter, elements with a particular type of valence configuration all show very similar chemical behavior. Thus groups of elements, such as the alkali metals, show similar chemistry because all the elements in that group have the same type of valence-electron arrangement. This concept, which explains so much chemistry, is the greatest contribution of the wave mechanical model to modern chemistry. 

As we explore this topic, and as we use theories to explain other types of chemical behavior later in the text, it is important that we distinguish the observation (steel msts) from the attempts to explain why the observed event occurs (theories). The observations remain the same over the decades, but the theories (our explanations) change as we gain a clearer understanding of how nature operates. A good example of this is the replacement of the Bohr model for atoms by the wave mechanical model. 

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