Wave phenomena quantum waves

The mathematical treatment of the Rutherford-Bohr atom was especially productive in Denmark and Germany. It led directly to quantum mechanics, which treated electrons as particles. Electrons, however, like light, were part of electromagnetic radiation, and radiation was generally understood to be a wave phenomenon. In 1924, the French physicist Prince Louis de Broglie (1892-1987), influenced by Einstein s work on the photoelectric effect, showed that electrons had both wave and particle aspects. Wave mechanics, an alternative approach to quantum physics, was soon developed, based on the wave equation formulated in 1926 by the Austrian-born Erwin Schrodinger (1887-1961). Quantum mechanics and wave mechanics turned out to be complementary and both were fruitful for an understanding of valence. 

The above arguments show that type II wave chaos is a genuine wave phenomenon in classical wave systems. In the context of quantum mechanics, however, type II quantum chaos is only an approximation. This is because classical walls or dynamic boundaries do not exist in quantum mechanics. The dynamical degrees of freedom of the walls, or boundaries, have to be quantized too, resulting in a higher-dimensional, but purely quantum, system, usually of type I. This fact leads us to a promising... 

A paradox which stimulated the early development of the quantum theory concerned the indeterminate nature of light. Light usually behaves as a wave phenomenon but occasionally it betrays a particle-like aspect, a schizoid tendency known as the wave-particle duality. We consider first the wave aspect of light. 

The size-evolution of the physical properties from atom to bulk might also be related in part to the variation of the surface-to-volume ratio. In addition to these classical effects, however, the quantum mechanical properties of the electrons play an equally important role. These so-called quantum-size effects can be understood most simply by realizing that a conduction electron in a metal has both particle-and wave-like properties, according to the famous particle-wave duality of quantum mechanics. Treated as a wave-phenomenon, the electron in a metal has a wavelength of one to a few nanometers. The wave-character of the electron will... 

The essential and peculiar feature of the quantum mechanical model of the atom lies in its description of electrons as waves rather than particles. It is far more inmitive to think of electrons as particles, perhaps resembling tiny marbles, than to envision them as waves. But just as we ve seen for light, experimental observations led to the idea that electrons can exhibit wave-like behavior. The first evidence of the wave nature of electrons came through diffraction experiments in 1927. Diffraction was already a well-understood phenomenon of waves, so the observation of electron diffraction strongly suggested the need for a wave-based treatment of the electron. 

It was a characteristic feature of Bohr s classical quantum mechanics that it could never be generalized to give good quantitative results for systems containing more than one electron. The extension from N = 1 to N = 2, 3,. . . came first with modern wave mechanics and Heisenberg s discovery in 1926 of the exchange phenomenon in the He-atom, which, with the identity... 

Just like any spectroscopic event EPR is a quantum-mechanical phenomenon, therefore its description requires formalisms from quantum mechanics. The energy levels of a static molecular system (e.g., a metalloprotein in a static magnetic field) are described by the time-independent Schrodinger wave equation,... 

Quantum mechanical tunneling. Tunneling is the phenomenon by which a particle transfers through a reaction barrier due to its wave-like property.Figure 1 graphically illustrates this for a carbon-hydrogen-carbon double-well system Hydrogen... 

Our discussion so far has not touched upon the origin of the stability of the H2 molecule. Reading from the articles ofthe early workers, one obtains the impression that most of them attributed the stability to resonance between the 150(1) 156(2) form of the wave function and the l5o(2)l56(l) form. This phenomenon was new to physicists and chemists at the time and was frequently invoked in explaining quantum effects. Today s workers prefer explanations that use classical language. 

The triple bond structure appears in the third place with spherical AOs and standard tableaux functions, but is not among the first four with HLSP ffinctions. This is actually misleading due to the arbitrary cutoff at four ffinctions in the table. The HLSP function triple bond has a coefficient of 0.09182, only slightly smaller that function 4 in the table. The appearance of the triple bond structure in this wave function is the quantum mechanical manifestation of the V back-bonding phenomenon invoked in qualitative arguments concerning bonding. We thereby have a quantitative approach to the concept. 

In order to describe microscopic systems, then, a different mechanics was required. One promising candidate was wave mechanics, since standing waves are also a quantized phenomenon. Interestingly, as first proposed by de Broglie, matter can indeed be shown to have wavelike properties. However, it also has particle-Uke properties, and to properly account for this dichotomy a new mechanics, quanmm mechanics, was developed. This chapter provides an overview of the fundamental features of quantum mechanics, and describes in a formal way the fundamental equations that are used in the construction of computational models. In some sense, this chapter is historical. However, in order to appreciate the differences between modem computational models, and the range over which they may be expected to be applicable, it is important to understand the foundation on which all of them are built. Following this exposition. Chapter 5 overviews the approximations inherent... 

The concepts of hybridisation and resonance are the cornerstones of VB theory. Unfortunately, they are often misunderstood and have consequently suffered from much unjust criticism. Hybridisation is not a phenomenon, nor a physical process. It is essentially a mathematical manipulation of atomic wave functions which is often necessary if we are to describe electron-pair bonds in terms of orbital overlap. This manipulation is justified by a theorem of quantum mechanics which states that, given a set of n respectable wave functions for a chemical system which turn out to be inconvenient or unsuitable, it is permissible to transform these into a new set of n functions which are linear combinations of the old ones, subject to the constraint that the functions are all mutually orthogonal, i.e. the overlap integral J p/ip dT between any pair of functions ip, and op, (i = j) is always zero. This theorem is exploited in a great many theoretical arguments it forms the basis for the construction of molecular orbitals as linear combinations of atomic orbitals (see below and Section 7.1). 

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