Quantum theory wave mechanics

Quantum chemistry would become the (necessary) intermediary in the metamorphosis of the current scientist into the scientist of the future, who was idealized as neither a physicist nor a chemist, but a sort of hybrid of the two. Such a scientist would transcend the typical physicist or chemist and needed training in empirical chemistry, in physical chemistry, in metallurgy, in crystal structure, as well as in theoretical physics, including mechanics and electromagnetic theory, and in particular in quantum theory, wave mechanics, the structure of atoms and molecules, in thermodynamics, statistical mechanics, and finally in what Slater called chemical physics. 

The theory of chemical reactions has many facets including elaborate quantum mechanical scattering approaches that treat the kinetic energy of atoms by proper wave mechanical methods. These approaches to chemical reaction theory go far beyond the capabilities of a product like HyperChem as many of the ideas are yet to have wide-spread practical implementations. 

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. 

MaxweU-Boltzmaim particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles ate indistinguishable. Eor example, individual electrons in a soHd metal do not maintain positional proximity to specific atoms. These electrons obey Eermi-Ditac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

Only with Bohr s 1913-1923 introduction of the "old quantum theory" (itself strongly inspired by chemical periodicity patterns vide infra) and the final discovery of Schrodinger s wave mechanics in 1925 would the periodic table be supplanted as the deepest expression of current chemical understanding ([21], p 2). 

The first consistent attempt to unify quantum theory and relativity came after Schrddinger s and Heisenberg s work in 1925 and 1926 produced the rules for the quantum mechanical description of nonrelativistic systems of point particles. Mention should be made of the fact that in these developments de Broglie s hypothesis attributing wave-corpuscular properties to all matter played an important role. Central to this hypothesis are the relations between particle and wave properties E — hv and p = Ilk, which de Broglie advanced on the basis of relativistic dynamics. 

In recent years the old quantum theory, associated principally with the names of Bohr and Sommerfeld, encountered a large number of difficulties, all of which vanished before the new quantum mechanics of Heisenberg. Because of its abstruse and difficultly interpretable mathematical foundation, Heisenberg s quantum mechanics cannot be easily applied to the relatively complicated problems of the structures and properties of many-electron atoms and of molecules in particular is this true for chemical problems, which usually do not permit simple dynamical formulation in terms of nuclei and electrons, but instead require to be treated with the aid of atomic and molecular models. Accordingly, it is especially gratifying that Schrodinger s interpretation of his wave mechanics3 provides a simple and satisfactory atomic model, more closely related to the chemist s atom than to that of the old quantum theory. 

Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schrbdinger inteijected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schrodinger s wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. 

Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the Schrodinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential field. The formal mathematical postulates of quantum theory are presented in Chapter 3. 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

In this section we state the postulates of quantum mechanics in terms of the properties of linear operators. By way of an introduction to quantum theory, the basic principles have already been presented in Chapters 1 and 2. The purpose of that introduction is to provide a rationale for the quantum concepts by showing how the particle-wave duality leads to the postulate of a wave function based on the properties of a wave packet. Although this approach, based in part on historical development, helps to explain why certain quantum concepts were proposed, the basic principles of quantum mechanics cannot be obtained by any process of deduction. They must be stated as postulates to be accepted because the conclusions drawn from them agree with experiment without exception. 

Prior to Dirac s relativistic quantum theory, W. Pauli (1927) showed how spin could be incorporated into non-relativistic quantum mechanics. Since the subject of relativistic quantum mechanics is beyond the scope of this book, we present in this chapter Pauli s modification of the wave-function description so... 

The relationship between spin and the symmetry character of the wave function can be established in relativistic quantum theory. In non-relativistic quantum mechanics, however, this relationship must be regarded as a postulate. 

The first two chapters serve as an introduction to quantum theory. It is assumed that the student has already been exposed to elementary quantum mechanics and to the historical events that led to its development in an undergraduate physical chemistry course or in a course on atomic physics. Accordingly, the historical development of quantum theory is not covered. To serve as a rationale for the postulates of quantum theory, Chapter 1 discusses wave motion and wave packets and then relates particle motion to wave motion. In Chapter 2 the time-dependent and time-independent Schrodinger equations are introduced along with a discussion of wave functions for particles in a potential field. Some instructors may wish to omit the first or both of these chapters or to present abbreviated versions. 

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... 

The fundamental equivalence between Schrodinger s wave-mechanical and Heisenberg s matrix-mechanical representation of quantum theory implies that H (or Hm>) may be viewed as a differential operator or a matrix. The latter viewpoint is usually more convenient in the... 

During the last five decades, an alternative way of looking at the quantum theory of atoms, molecules, and solids in terms of the electron density in three-dimensional (3D) space, rather than the many-electron wave function in the multidimensional configuration space, has gained wide acceptance. The reasons for such popularity of the density-based quantum mechanics are the following ... 

The Occurrence of Her mite Functions in Wave Mechanics. The Hcrmite functions which wc have discussed in the Inst section occur in the wave mechanical treatment of the harmonic oscillator1). Although this is a very simple mechanical system the analysis of its properties is of great importance because of its application to the quantum theory of radiation. 

Pauling, who previously had heard Sommerfeld lecture on the old quantum theory at Caltech, now heard Sommerfeld lecture in Munich on the new wave mechanics. Pauling instantly liked the physicists new interpretation of the electron. The hydrogen atom could be thought of as... 

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