The Bell that Rings Light

The following sections are included:

• Acknowledgement

• Preface

• Contents

The notion that the material world is composed of “atoms” is an old one. The first record we have of it is the version held by Democritus, in opposition to Aristotle's world picture. It never dominated the scientific imagination until well after Galileo invented the telescope. Seeing the Milky Way dissolve into a myriad of individual stars did something to the scientific imagination of the Renaissance that has yet to be completely undone. In short, if the universe at large is filled with distinct stars wheeling about, then why shouldn't the universe in miniature behave thus? And suddenly Democritus was no longer about something preposterous. Isaac Newton then gave the world some laws of physics which could conceivably apply to these little objects and, behold, a new science was born. If you think I am oversimplifying all of scientific history, you are right. But just remember how far Euclid got with only three objects (point, line, plane) and five laws to govern them. Well, you can also go pretty far with just atoms and Newton's laws of motion…

The following sections are included:

• Coulomb's Law

• Light

• The Dual Nature of Matter

Now, on to the atom. The atom contains electrons and we may influence the energy of those electrons using electromagnetic radiation. Therefore, our discussion of the nature of electrons and light will be significant in the discussion of the atom. Coulomb's Law will also be invoked in the upcoming discussion…

Certain old assumptions die hard, particularly those having to do with circularity of motion. The circle is a sacrosanct figure for many cultures and, though it may be refuted as the actual description of a natural phenomenon, it is never completely abandoned as the ideal for which nature somehow strives. Bohr's circular orbits would give way to other models but the role of the circle itself continued to enlarge. Here we are going to explore, not quantum theory itself but the assumptions on which it is based. These include the circle, or more accurately the sphere, in a prominent role…

It would be tempting, and in some sense enough, to say that because we observe certain behaviors in small particles that are characteristic of waves, we are therefore justified in describing these objects by a classical wave equation. If it walks like a wave and quacks like a wave, we may as well call it a wave. So it would be nice to be able to arrive at the wave equation for a quantum particle by reasoning from a few basic premises that make sense in the context of quantum mechanics. Just as the classical wave equation for a vibrating string can be derived from a discussion of forces acting on the string, so we would equally appreciate a derivation of the same equation based on the first principles of quantum mechanics…

Because so much of what happens next rides on the assumption that an electron behaves like a wave, we thought it best to digress momentarily to show you some of the mathematical tools typically employed in discussions of ordinary waves, such as those one would see in a vibrating guitar string, for example. Several positions over time of a string plucked at its midpoint are shown in Fig. 6.1, and if you were to plot the displacement at the point x = 0.5 throughout time, you would see an oscillating pattern, like Fig. 6.2, which is why the equation describing the plucked string is one of those generically called the “wave equation”…

Before going on to something as complex as an atom, let's look at a model problem in some detail. The first one is the one-dimensional particle-in-a-box problem. This turns out to be an excellent conceptual model for conjugated dye molecules (see Chapter 21) and also a model for trapped charged particles. The problem and its solutions are similar to the vibrating string just discussed. The potential term is shown graphically and mathematically in Fig. 7.1…

It is worth spending a little time looking at how the wave equation will directly inform our approach to the hydrogen atom. We have seen the formalism applied to the particle-in-a-box problem, but hydrogen is much more complex. It is better to use the analogy with the vibrating string for our initial foray. All of the phenomena mentioned in the discussion of the vibrating string have a direct analogy to some aspect of the wave equation. If we can see how the analogy works, perhaps we will gain an insight into how to approach the puzzle of hydrogen…

The historical development of the Schrödinger equation is available for your edification in some quantum mechanics texts, and good luck to you, too. Seriously, the way people figured these things out was as an axiomatic system whose laws lead you to the results observed in nature. By axiomatic, I simply mean that the basic underlying assumption that an electron might obey a wave equation strikes me as an axiom upon which quantum mechanics rests. The alternative assumption of an electron behaving like a body orbiting another body according to Newton's laws leads to radically different conclusions. In this approach you see the shadows of Euclid and Newton, standing off in the dim distance and cheering the brave physicists on. But, unlike Euclidean geometry or Newtonian physics, the answer to be explained was far from believable…

We can now write the Schrödinger equation for any system. The problem is: how can we solve it for any system, or for even one system? It turns out that we can solve the equation exactly for a one-electron system. All other cases will require some form of approximation. We'll not try to reproduce that solution here. For now we shall concentrate only the solutions and interpret them. A word of warning; we revert back to the physicist's notation. In a subsequent chapter, we will explicitly connect the language of the physicist to that of the mathematician…

The following sections are included:

• Schur's Lemma

It is the general consensus among mathematicians I know that “The Spherical Harmonics” would be a great name for a rock group, or possibly a barbershop quartet with somewhat portly members. Language conjures an image, no matter how technical the language may be. All of the discussion we have been having about groups and symmetries is part of the corner of mathematics called algebra, or sometimes out of honesty, abstract algebra. It is indeed a direct descendent of that Arabic and late medieval activity whose name is derived from the word al-jabr, meaning to restore. In its abstract form, it bears little resemblance to either the medieval subject or its renaissance counterpart as taught in high school, which we all have studied and whose exposition can be found in books entitled “College Algebra”. That title is a tip-off that the contents of the book restore (al-jabr) what was supposed to have been learned in high school. In contrast, the most abstract renderings of modern algebra can be found in texts ingenuously titled “Basic Algebra 1”. It is some matter of debate whether language is doing its job in these cases…

The following sections are included:

• A Guided Tour of the Radial Part of Schrödinger's Equation

Gentle reader, you have tolerated much to get to this point. Together we have made forays into almost every corner of mathematics in search of an explanation of the marvelous light of hydrogen. Will it pay? I assure you it will pay, and handsomely. But do not hope for perfection, because that is not what science gives. Only mathematics can ever be perfect. In this chapter, we will explore the quantum number. The language will be that of the mathematician. You might want a translator, if you are more comfortable with the chemical or physical dialect. Later chapters will do this for you…

The following sections are included:

• Introduction

One way to deal with the specific geometry of a molecule is to return to Coulomb's law, that is, to look at electron–electron repulsion. VSEPR is such an electrostatic theory of bonding. As with Lewis dot structures, it ignores specific orbitals. The observed geometry reflects the attempt to minimize electron–electron repulsion by maximizing the distance between electrons. Bond angles are determined solely by the number of valence electrons around a central atom. It is instructive to use examples…

I have personally met students who are majoring in chemistry or physics or math and believe that those pictures of orbitals you see in every chemistry or physics book are shapes that have actually been measured or observed in some way. They are not. There is no scientific instrument that can directly measure the shape of an electron orbital. The reason is not merely that scientists just don't try hard enough. The reason is Heisenberg's Uncertainty Principle…

Our quest for a bonding theory leads now to molecular orbital theory (MO theory). This is an extension of the atomic orbital theory we have just learned. If it is “correct” for atoms, we must also be able to apply it to molecules, right= We'll start with an “easy” case. Consider the simplest molecule, , a one-electron system; the molecular analogy to the hydrogen atom. Begin with a thought experiment. Imagine that we could measure the probability density, ψ², that is, the probability per unit volume of finding the electron in . If we construct a plot of ψ² as a function of the distance from the hydrogen atoms, we would see the plot shown in Fig. 18.1…

We have listed this as a separate bonding model, but in reality this is not a different theory. We can make VB theory take the same mathematical form as MO theory, if we wish. However, this defeats its major benefit: a physical picture of bonding in large (especially organic) molecules. As always, we begin with something simple. In this case, we start with BeH₂. We already know that the H–Be–H bond angle is 180°;. Be has two valence electrons; a 1s²2s² configuration. Since all of the electrons are paired, we can only imagine forming two bonds (actually, we can't imagine forming any bonds without changing the electron configuration) by unpairing the 2s electrons, so that the electron configuration becomes 1s²2s2p…

Hydrogen bonds occur in the most polar molecules. That is, those molecules containing the elements with the largest electronegativity: N, O and F. These are not truly bonds in the sense that we have been using that term. Hydrogen has only a single electron involved in bonding. It can closely approach an electronegative atom without any electron–electron repulsion. This interaction is what we call a hydrogen bond. The bonds are a result of the skewed electron distribution in these molecules. One end is essentially positive and the other is essentially negative. For example, look at HF, where dimers are dominant…

So, we have completed our journey through basic quantum theory and the application to chemical structure. Is there some way that all of this material can be tied together in an application? Of course, and we'll use extremes: a simple quantum model (the one-dimensional particle-in-a-box) and a very large, complex dye molecule. I'm sure that the particle-in-a-box, especially in one-dimension, seemed useless at the time we developed it. It turns out that it is an excellent model for conjugated (we'll see what that means in a moment) molecules. In short, the dye molecules constitute a case study in quantum mechanics and case studies are an excellent way to explore what we have learned in science or any other area of knowledge…

In the preface, the purpose of this text was described from the perspective of my co-author, a mathematician who, prior to making that career choice, considered chemistry as a career. Here, I will sum up our goals and hopes for you, from the perspective of a chemist, who prior to making that choice considered mathematics (and, subsequently, engineering) as a possible major. If all of this sounds like two (at one time) very confused people, well, it should! The main purpose in writing this book was to combine a very rigorous mathematical approach with a more practical chemical approach and to bring you to a point where you will feel comfortable exploring quantum mechanics in more depth in your advanced courses. Our somewhat confused backgrounds were just what the National Science Foundation ordered, when we began teaching an interdisciplinary course (Integrated Mathematics and Physical Sciences) for incoming first year students…

The following sections are included:

• Index