Introduction

Everyday experience is a helpful guide as we attempt to model the physical world around us. For example, interpreting wave motion is aided by noting the movement of a bobber on the ripples of a pond. Anticipating trajectories resulting from colliding bodies may be facilitated from observation of a pool table cue ball. Understanding action at a distance from a force field is furthered by participating in a game of catch. With regard to the subject at hand, observing ocean water crashing into a rock jetty, listening to sound echoing through a mountain valley, or locating a penny on the bottom of a fountain provides a basis of comprehension for wave–matter interaction.

By contrast attempts to phenomenologically model a chemical process in which matter undergoes a transformation of constitution present daunting challenges. Ignoring for a moment the fact that reaction rates commonly occur in a time frame that render them humanly imperceptible, a more fundamental impediment is the scale of matter dimensionally. We cannot 'see' the system interacting, we may only verify the occurrence of chemistry from observing affects on its surroundings: heat evolution, a phase change, a flash of light, a smell, a puff of smoke. In fact, the only 'illuminating' probe at these dimensions is a light wave. Suppose then that we are somehow magically able to ride a light beam, as if it allows us to don nanoscopes for the purpose of observing matter in action during a chemical process. Even with this advantage, we would quickly find that attempts to utilize rules for cue balls or other projectile motions do not apply. We would also learn the light itself is not just a casual observer in this environment, but is an intimate part of the system dynamics.

Ultimately would come to the realization that an entirely new set of principles and guidelines, far outside the box of those ingrained from familiar observation, are required to correctly model and predict events. The approach taken here will then be to review our understanding of the behavior of macroscopic matter. Particular focus will be given to instances where the governing rules hold fast, and where there is a disconnect. This is facilitated by reviewing experimental evidence that could not be explained away by accepted guidelines. We will then introduce the necessary modifications to allow matter's description at the atomic and molecular level.

While navigating this course, the advantage taken from tangible connection with everyday experience must unfortunately be abandoned. We will in fact find it necessary to incorporate some facets of our common understanding of wave behavior into the model of matter. This seems counterintuitive. Even to the most casual observer, there are obvious differences between matter and waves. Though each takes a variety of distinguishable forms, matter and waves are ultimately differentiated by a single criterion. Waves uniquely have the capability of occupying the same space at the same time. For example consider four individuals simultaneously conducting two separate conversations as depicted in figure 1.1. Everyday experience tells us that opposing pairs can communicate, even though sounds from their voices are somewhere intersecting. In addition, light reflecting off any one of them can be detected by the remaining three, even though these waves must also inhabit the same space along their journey. This property, known as superposition, allows waves to exhibit constructive and destructive interference, which for sound results in phenomena such as piano chords and devices like noise-cancelling headphones.

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By distinct contrast, one of the two fundamental characteristics of matter is that it 'takes up space' (the other, according to any physical science primer, being that it has mass. This is technically saying the same thing actually, but I digress...). Of course the implication of matter occupying space is that it must exclude other matter from that space. Despite this very fundamental of differences, matter and waves are intimately related. Most waves are actually disturbances of matter. Sound, water, and string waves cannot move through space without matter acting as a medium to enable their propagation. In fact for quite some time it was believed waves and matter had an inextricable relationship. With Huygens' introduction of the wave theory of light, it presupposed the requirement of a support medium. The search ensued for the so-called aether, until its existence was debunked as a result of experiments performed by Michelson and Morley.

In contrast to other wave forms, electromagnetic radiation creates a disturbance in space. This effect is described in physics as being generated by a field, or interacting fields as is the case here. In fact, electromagnetic radiation prefers an absence of matter. As evidenced by gazing upward towards the nighttime sky, electromagnetic radiation continuously self-generates and propagates through space when unimpeded by matter. When electromagnetic radiation of a particular type does encounter matter it may pass through, be diffracted, refracted, reflected, scattered, or absorbed. Matter and its interaction with light and other portions of the electromagnetic spectrum is the basis of a wide variety of qualitative and quantitative analytical chemistry techniques.

The arrival of spectroscopy as a characterizing tool for the composition of substances was paralleled by the fundamental questions light–matter interaction posed. By the late nineteenth and early twentieth centuries, instrumentation had achieved levels of resolution revealing information that could not be explained by contemporary theories. Attempts to formalize the mechanism of light–matter energy transfer blurred their fundamental distinction. As an initial explanation, light waves were re-imagined to possess matter-like characteristics. Eventually viewpoints shifted to treating matter from a wave perspective. In this way many experimental inconsistencies could be resolved. Ultimately, we should be resigned to the fact that both light or matter can individually exhibit wave- or particle-like characteristics. The circumstances dictating their behavior ultimately depend on the situation, but invariably occur at the atomic and molecular level. Our purpose in subsequent sections is therefore to take a nanoscopic view of matter behaving as a wave in order to gain insight into its macroscopic properties.

As a prelude to the tale of matter and waves and its timeline, we must begin two centuries beforehand, to properly acknowledge antecedent milestones. The eighteenth century is appropriately known as the 'Age of Enlightenment' or 'Age of Reason.' With apology to the scientific and engineering accomplishments of the period, an incredible array of mathematical techniques and advancements were introduced by Gauss, Euler, Fourier, LaPlace, Maclaurin, LaGrange, Taylor, Leibnitz, Bernoulli, Legendre, Newton and others. At the time much of this probably seemed no more than academic indulgence with little or no connection to the real world. However a century later these techniques were essential to formulation of thermodynamics and electrodynamics by individuals including Ampere, Faraday, Maxwell, Clausius, Joule, Helmholtz, Boltzmann, Thompson (Lord Kelvin), and Gibbs.

These achievements marked a seminal moment in the annals of scientific accomplishment. After a lengthy gestation, a coming of age was signaled. With roots in human curiosity, fear, and superstition followed by a lengthy infancy of straightforward phenomenological modeling, science now embraced a new, fundamental purpose. Interpreting the workings of the everyday world was no longer the be-all and end-all. Scientists pushed the envelope of human experience to dimensions beyond what could be seen by a telescope or microscope. The frontiers of science were inextricably dependent on abstract mathematical techniques, culminating in more versatile, robust, and predictive scientific models. Theory now blazed a trail for experimental investigation.

Throughout this time, both the interpretation of matter as well as the properties and behavior of waves were thought to be on firm theoretical ground. However, evidence which emerged in the late nineteenth and early twentieth centuries blurred the lines between matter and waves. After much consternation, reflection, and debate among scientists, a blended behavior of light and matter emerged known as wave–particle duality. Much of the same mathematics, that by then had served theoreticians so well for over a century, again proved crucial and indispensable.

Quantum mechanics, the consummation of wave–matter interaction, marked a paradigm shift in more ways than the radical departure of its physics. It did not congeal from the singular ruminations or epiphany of any one individual. Quantum mechanics was an evolution of thought and philosophy coalesced from decades of work, culminating from efforts of an unprecedentedly-large collection of vital contributors. Previous landmark events in science could almost invariably be attributed to efforts of a single individual. The global effort that quantum mechanics represented was a testament to evolving human connectivity. By the twentieth century, scientists were taking full advantage of information sharing from advances in communication and experiencing an increased mobility attributable to ease of travel.

Quantum mechanics refined our understanding of matter to the point that its profound impact now demarcates the advent of 'modern physics.' Within a brief period of time it reverberated across chemistry and molecular biology as well. Subsequent to its introduction several of the leading scientists of the day, most familiarly Einstein, took on worldwide celebrity status. Nobel prizes were awarded to a variety of its principal contributors over a broad span of the twentieth century. These include prizes in physics to Planck in 1918, Einstein in 1921, Bohr in 1922, de Broglie in 1929, Heisenberg in 1932, Schrödinger in 1933, Pauli in 1945, and Born in 1954. Awards for contributions of quantum mechanics in chemistry were given to Pauling in 1954, Mullikan in 1966, and Pople and Kohn in 1998. Many other recipients in both fields were either guided in their experiments or directly impacted in their theoretical developments by quantum mechanics. It is somewhat unsettling to read the press release accompanying Mullikan's 1966 prize which points out the overwhelmingly complex nature of the discipline, essentially stating that quantum mechanics was inaccessible to the layperson. One of the main goals of this work is to help allay such predispositions or trepidations.

To punctuate the human interest aspect, no other image heralds the arrival of quantum mechanics or underscores the collective effort behind it quite like figure 1.2, a photograph of participants in the 1927 Solvay Conference. These invitation-only events feature varying focus topics that to this day they are intermittently held in Brussels, having been instituted by Belgian industrialist Ernest Solvay in 1911. The 1927 meeting, fifth in the series, featured lectures and discussions focused on the title subject: 'Electrons and Photons,' and a conference theme parallel to the topics of this book. Lewis once wrote, 'Science has its cathedrals, built by the efforts of few architects and of many workers.' The 1927 Solvay Conference validated one's standing as an architect to the sanctum of quantum mechanics. Essentially everyone who was anyone relevant to its development was present, a contingent in some ways analogous to the 1992 US men's Olympic basketball 'Dream Team.' Even those with no more than a passing knowledge of science will recognize several names. Those with a passion for it should particularly appreciate the special nature of the moment.

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