From Classical Physics to Quantum Theory

Early attempts to understand atoms and molecules met with only limited success. By assuming that molecules behave like rebounding balls, physicists were able to predict and explain some macroscopic phenomena, such as the pressure exerted by a gas. However, their model did not account for the stability of molecules that is, it could not explain the forces that hold atoms together. It took a long time to realize— and an even longer time to accept— that the properties of atoms and molecules are not governed by the same laws that work so well for larger objects. 

The new era in physics started in 1900 with a young German physicist named Max Planck. While analyzing the data on the radiation emitted by solids heated to various temperatures, Planck discovered that atoms and molecules emit energy only in certain discrete quantities, or quanta. Physicists had always assumed that energy is continuous, which meant that any amount of energy could be released in a radiation process, so Planck s quantum theory turned physics upside down. Indeed, the flurry of research that ensued altered our concept of nature forever. 

An important property of a wave traveling through space is its speed (m), which is given by the product of its wavelength and its frequency  

The inherent sensibihty of Equation (7.1) becomes apparent if we analyze the physical dimensions involved in the three terms. The wavelength (A) expresses the length of a wave, or distance/wave. The frequency (v) indicates the number of these waves that pass any reference point per unit of time, or waves/time. Thus, the product of these terms results in dimensions of distance/time, which is speed  

Wavelength is nsnally expressed in units of meters, centimeters, or nanometers, and frequency is measured in hertz (Hz), where 

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We begin by discussing the transition from classical physics to quantum theory. In particular, we become familiar with properties of waves and electromagnetic radiation and Planck s formulation of the quantum theory. (7.1)... 

In what I broadly regard as structure (essentially quantum theory), the equation that epitomizes the transition from classical mechanics to quantum mechanics, is the de Broglie relation, k = hip, for it summarizes the central concept of duality. Stemming from duality is the aspect of reality that distinguishes quantum mechanics from classical mechanics, namely superposition y = y/A + y/R with its implication of the roles of constructive and destructive interference. Then of course, there is the means of calculating wavefixnctions, the Schrodinger equation. For simplicity I will write down its time-independent form, Hip = Eip, but it is just as important for a physical chemist to be familiar with its time-dependent form and its ramifications for spectroscopy and reaction. 

The purpose of this first computer laboratory is to review some of the fundamental concepts from classical physics, to understand what constitutes a solution to a problem in classical physics, and to introduce students to numerical solutions for the Newtonian equations of motion. QuickBASIC programs have been written which use PC graphics to display the trajeaory of an electron in the Thomson plum pudding model of the atom, the Bohr atom, and a classical model for the hydrogen-molecule ion. This early review of classical physics helps students appreciate more fully how fundamentally different quantum theory is. The material in this exercise is frequently used as a leaure demonstration to support a classroom lecture on the precursors to the quantum theory of atomic and molecular structure. 

Our ambition is to provide a modern introduction to the field of relativistic quantum chemistry, aimed at the advanced student and the practicing nonspecialist researcher. The material has been divided into five parts. Parts I and II provide the necessary background from classical physics, relativistic quantum mechanics, and group theory. Part III covers the application of these principles to fully relativistic methods for quantum chemistry within a four-component framework. Part IV deals with the main... 

Modern quantum theory, also called quantum physics or quantum mechanics, replaced Bohr s theory in 1926. Quantization arises naturally by using quantum mechanics. It is not assumed or imposed beforehand as a condition, as was done by Bohr. As we will soon see, the circular orbits that are so prominent in Bohr s model of the hydrogen atom are absent in the model based on quantum mechanics. Despite the fact that Bohr s model of the hydrogen atom is wrong, it was an important scientific development because it prompted a paradigm shift—the quantum leap—from classical physics to the new quantum physics. 

To understand the behavior of electrons in atoms and molecules requires the use of quantum mechanics. This theory predicts the allowed quantized energy levels of a system and has other features that are very different from classical physics. Electrons are described by a wavefunction, which contains all the information we can know about their behavior. The classical notion of a definite trajectory (e.g. the motion of a planet around the Sun) is not valid at a microscopic level. The quantum theory predicts only probability distributions, which are given by the square of the wavefunction and which show where electrons are more or less likely to be found. 

This book provides an up-to-date overview on (nonrelativistic) quantum theory from the point of view of potential misunderstandings and misconceptions which have led to various criticisms and "extensions" of quantum theory during the past decades. It does not, however, contain any reference to a relativistic formulation of quantum chemistry but focuses on the relation between classical physics and quantum physics. A more readable and instructive introduction to quantum mechanics from Omnes point of view can be found in another book by the same author [95]. 

Maxwell s equations describe the propagation of electromagnetic radiation as waves within the framework of classical physics however, they do not describe emission phenomena. The search for the law that defines the energy distribution of radiation from a small hole in a large isothermal cavity gave rise to quantum theory. The function that describes the frequency distribution of blackbody radiation was the first result of that new theory (Planck, 1900,1901). 

We turn now to an analysis of English chemists who provided the first systematic interpretations of chemical reaction mechanisms in which the molecule was modeled as a dynamic system of positive nuclei and negative electrons. While their approach was informed by physical ideas and theories, it was unarguably a chemical approach, consistent with classical nineteenth-century chemistry, from which it developed, and with quantum chemistry, which it helped to construct. 

The basic theories of physics - classical mechanics and electromagnetism, relativity theory, quantum mechanics, statistical mechanics, quantum electrodynamics - support the theoretical apparatus which is used in molecular sciences. Quantum mechanics plays a particular role in theoretical chemistry, providing the basis for the valence theories which allow to interpret the structure of molecules and for the spectroscopic models employed in the determination of structural information from spectral patterns. Indeed, Quantum Chemistry often appears synonymous with Theoretical Chemistry it will, therefore, constitute a major part of this book series. However, the scope of the series will also include other areas of theoretical chemistry, such as mathematical chemistry (which involves the use of algebra and topology in the analysis of molecular structures and reactions) molecular mechanics, molecular dynamics and chemical thermodynamics, which play an important role in rationalizing the geometric and electronic structures of molecular assemblies and polymers, clusters and crystals surface, interface, solvent and solid-state effects excited-state dynamics, reactive collisions, and chemical reactions. 

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