Wave mechanical model

In 1913 Niels Bohr proposed a system of rules that defined a specific set of discrete orbits for the electrons of an atom with a given atomic number. These rules required the electrons to exist only in these orbits, so that they did not radiate energy continuously as in classical electromagnetism. This model was extended first by Sommerfeld and then by Goudsmit and Uhlenbeck. In 1925 Heisenberg, and in 1926 Schrn dinger, proposed a matrix or wave mechanics theory that has developed into quantum mechanics, in which all of these properties are included. In this theory the state of the electron is described by a wave function from which the electron s properties can be deduced. 

Wave mechanics is based on the fundamental principle that electrons behave as waves (e.g., they can be diffracted) and that consequently a wave equation can be written for them, in the same sense that light waves, soimd waves, and so on, can be described by wave equations. The equation that serves as a mathematical model for electrons is known as the Schrodinger equation, which for a one-electron system is... 

Schrodinger s equation is widely known as a wave equation and the quantum formalism developed on the basis thereof is called wave mechanics. This terminology reflects historical developments in the theory of matter following various conjectures and experimental demonstration that matter and radiation alike, both exhibit wave-like and particle-like behaviour under appropriate conditions. The synthesis of quantum theory and a wave model was first achieved by De Broglie. By analogy with the dual character of light as revealed by the photoelectric effect and the incoherent Compton scattering... 

Only those problems that can be reduced to one-dimensional one-particle problems can be solved in closed form by the methods of wave mechanics, which excludes all systems of chemical interest. As shown before, several chemical systems can be approximated by one-dimensional model systems, such as a rotating diatomic molecule modelled in terms of a rotating particle in a fixed orbit. The trick is to find a one-dimensional potential function, V that provides an approximate model of the interaction of interest, in the Schrodinger formulation... 

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

A good deal of this work had no impact in the development of models of molecular structure and the elucidation of reaction mechanisms one reason was Perrin s own coolness to quantum wave mechanics. 108 Another, according to Oxford s Harold Thompson, who studied with Nernst and Fritz Haber, was that researchers like Lecomte "did not know enough chemistry he was a physicist." 109 Perrin, too, approached physical chemistry as a physicist, not as a chemist. He had little real interest or knowledge of organic chemistry. But what made his radiation hypothesis attractive to many chemists was his concern with transition states and the search for a scheme of pathways defining chemical kinetics. 

Quantum wave mechanics gave chemistry a new "understanding," but it was an understanding absolutely dependent on purely chemical facts already known. What enabled the theoretician to get the right answer the first time, in a set of calculations, was the experimental facts of chemistry, which, Coulson wrote, "imply certain properties of the solution of the wave equation, so that chemistry could be said to be solving the mathematicians problems and not the other way around."36 So complex are the possible interactions among valence electrons that one must either use an exact mathematical model of a... 

With the failure of the Bohr model it was found that the properties of an electron in an atom had to be described in wave-mechanical terms (p. 54). Each Bohr model energy level corresponding to... 

The use of effective mass to understand the state of the microstructure is chosen to conserve the application of the free electron model by letting the mass of the electron incorporate the electron interactions with the lattice, which are experiencing potential energy interactions. Considering the total energy, E, of an electron in a solid, based on wave mechanics, then ... 

In principle, quantum mechanics permits the calculation of molecular energies and therefore thermodynamic properties. In practice, analytic solutions of the equations of wave mechanics are not generally accessible, especially for molecules with many atoms. However, with the advances in computer technology and programming, and the development of new computational methods, it is becoming feasible to calculate energies of molecules by ab initio quantum mechanics [11]. Furthermore, molecular modeling with substantial complexity and molecular mechanics treatments for... 

It is then quite understandable why, without the necessary mathematical machinery, the relevant concepts cannot be properly grasped. On the other hand, the mathematical disguise that is characteristic of quantum-chemistry courses makes both teachers and students pay more attention to the complexities of the mathematics (the tools, the trees ) and lose the physics (the actual world, the forest ). Although mathematics is essential for a deep understanding of quantum chemistry, the underlying physical picture and its connection with mathematics are equally important. AOs, MOs and related concepts derive from SchrOdinger s wave mechanics, which is an approximation to nature. According to Simons (96), "orbital concepts are merely aspects of the best presently available model they are not real in the same sense that experimental observations are. ... 

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... 

In the modern model of the atom, based on wave mechanics, the conception of electronic orbits in the old model is replaced by the idea of the probability of the occurrence of an electron at a given point. The conclusions, however, which can be drawn from the older model remain the same in the newer conception, and it is important to remember that in this new model the essential points of Bohr s theory have not been discarded, but merely interpreted differently and very greatly refined. 

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. 

The quantity / is usually called the characteristic length of the interaction potential, and is also employed in the more realistic wave-mechanical treatment. Many authors employ the symbol, a, which is equal to /-1. The form of the molecular interaction potential can be determined in terms of a suitable model, from experimental measurements of the temperature dependence of the viscosity of a gas, whence the characteristic length can be estimated. 

This model includes wave mechanics, in which the electron in a hydrogen atom is described as a wave. 

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