Ten years ago, when I started writing on the physics of vibration, I had in mind a single volume. Four years ago I was reconciled to the need for two, and now must confess that the complexity of the subject has made it advisable to get the second of the projected three parts into print without waiting for the third to accompany it between the same covers. It is still my hope to do justice to the vibrations of extended systems, but the difficulties are considerable and not made easier by the vigour with which some of the central topics are being pursued at present.

Of all the encouragement I have enjoyed I particularly wish to record with the warmest thanks the help of Dr Edmund Crouch and Dr John Hannay who long ago, as research students, derived for me some solutions of Schrödinger's equation which provided a stout anchor for my thoughts: Dr Bob Butcher who has firmly guided me in my brief excursions into structural chemistry and molecular spectra: and Dr Andrew Phillips who devoted more time than he could have been expected to spare to a critical reading of much of the typescript. If the state of that typescript as delivered to the printers did not achieve even a modest standard of tidiness, the fault is entirely the consequence of copious afterthoughts on my part, and in no way to be blamed on Mrs Janet Thulborn whose patient and faithful typing deserved better respect, and has certainly earned my gratitude.

In combining Parts 1 and 2 into a single volume only minor changes have been made, apart from the correction of errors. I have not attempted to bring the treatment up to date even for such rapidly expanding fields as the study of chaos in non-linear systems, but have been content to add a small number of references. Where an argument could be corrected, clarified or extended in the space of a few sentences, these are signposted in the margin and placed at the end of the chapter. The system of marginal cross-references has met with critical approval and I have taken the opportunity of adding to their number.

My hopes of ever completing a further volume on the vibrations of extended systems have by now grown faint. There are too many exciting new ideas that are not yet ready for the type of exposition that suited the present work. Fortunately the development, by both classical and quantal methods, of the physics of simple vibrators produces a reasonably selfcontained argument, and I am grateful, as always, to Cambridge University Press for making possible this synthesis of concepts which in modern physics should be regarded always as complementary, never as antagonistic.

The writing of this book has occupied several years, and now that it is finished it is time to ask what sort of a book it is. For all its length, it turns out to be only the first volume out of two, any hope of covering the topic in a single volume having vanished as the project developed. It must be obvious, therefore, that it is not a textbook in the sense of an adjunct to a conventional course of lectures (there are not even any questions at the ends of the chapters). On the other hand, it is certainly not an advanced treatise, for many of the more difficult topics are treated at a much lower level than is to be found in the specialist works devoted to them. Moreover, I can claim no professional skill in most of them, and this is both a confession and an advertisement. For by writing about them in a way that illuminates for me the essential physical thought underlying what is often a very complicated calculation, I hope to have provided a treatment that will enlighten others in the same unlearned state. The field is very wide, ranging from applied mathematics (non-linear vibration and stability theory) to electrical engineering (oscillators), and taking in masers, nuclear magnetic resonance, neutron scattering and many other matters on the way. Nobody could hope to make himself a master of all these, and few advanced treatises dealing with one topic think fit to mention the analogies with others.

The original maser of Gordon, Zeiger and Townes, driven by a focussed stream of ammonia molecules in their antisymmetrical state, provides a conveniently explicit example on which to base a discussion of the principles underlying coherent excitation of a vibrator by stimulated emission. It was shown in chapter 18 how a quadrupole electrostatic lens served to separate symmetric from antisymmetric states, and we shall assume that separation is perfect; it is easy to extend the argument to include a proportion of molecules in the symmetric state. In addition we shall ignore any complications arising from the multitude of rotational states leading to the fine structure shown in fig. 18.8, and shall assume that only one line contributes, for example the strong 3,3 line at 23.9 GHz. Since the microwave cavity resonator, if it is to be excited by the molecules, must normally be very closely tuned to their natural frequency this assumption is realistic.

The simplest intuitive approach to the maser is by way of Einstein's treatment of radiation in terms of stimulated and spontaneous processes.† Excited molecules passing through the resonator, when it is already in an oscillatory condition, are stimulated by the field; if the resonator frequency is well matched to the molecular levels they may make a transition down to the ground state and on leaving the resonator have 2Δ0 less energy than when they entered.

After this excursion into the field of non-linear vibrators, we now return to the harmonic vibrator and take up a point which had begun to reveal itself at the end of chapter 13, where we found that under the influence of a uniform, but arbitrarily time-varying, force the vibrator never forgets its initial state. If it started in the ground state, for ever afterwards its response can be described by the movement of a compact distribution in the σ-representation of equivalent classical vibrators; only the centroid of the distribution responds to the applied force. The harmonic vibrator is thus extraordinarily resistant to randomization. To be sure, if the force is not uniform, but depends on the displacement of the oscillating particle, the result just summarized is no longer true. Nevertheless, in the most important application, where an oscillator of atomic dimensions is influenced by electromagnetic vibrations, the force due to the electric field is as nearly uniform as makes no difference, since the wavelength of electromagnetic waves at a typical atomic resonant frequency is a thousand times the size of an atom. The disturbing aspect of the resistance of a vibrator to randomization is that in all theories of black-body radiation, before and after Planck, it is assumed that material oscillators and electromagnetic vibrations in a cavity will eventually share the chaotic state that allows statistical mechanics to be applied.

By the time a student of physics is ready to tackle quantum mechanics he has become familiar with the classical harmonic vibrator through many examples, and knows the crucial role it played in the development of Planck's ideas. It is natural then to concentrate on the mathematical aspects of the harmonic oscillator equation in quantum mechanics, the solution of Schrödinger's equation, normalization of the wave-function, calculation of mean values and of matrix elements, leaving the physics to look after itself. At a more advanced level the harmonic vibrator provides the entry into field theories, second quantization etc., and as a general rule it tends to be viewed more as a vehicle for instruction in more interesting matters than as a physical system having its own considerable interest and importance. Here we shall seek to redress the balance and study the vibrator as a thing in itself, without losing sight of the variety of physical problems to which the results can be applied. One especially important set of applications, however, will get little attention at this stage – the vibrations of compound bodies and of extended physical systems provide such wealth of interest as to justify a volume to themselves; and it is to such matters that the whole of the third part of this work will be devoted. As soon as one begins to contemplate the harmonic vibrator one becomes aware of the exceptional nature of its behaviour, in that it conforms more closely than any other system to the classical rules.

THE UNIVERSE AS A MACHINE

The seventeenth century was the century when science assumed its modern form and the scientific spirit infected Europe. It was the time when Aristotle's view of nature was rejected and Galileo's great book of the universe was adopted. The new science was nourished by an optimism that mankind could discover the laws of nature.

One of the most significant and influential figures in seventeenth-century natural philosophy was René Descartes. Early in his life, Descartes rebelled against the traditions in which he had been thoroughly educated. He sought new foundations for knowledge, foundations which could underpin certainty in our knowledge of nature. Convinced of the indubitable logic of mathematics, Descartes came to identify mathematics with physics.

TOWARD AN IDEA OF ENERGY

The law of conservation of energy is one of the most fundamental laws of physics. No matter what you do, energy is always conserved. So why do people tell us to conserve energy? Evidentally the phrase “conserve energy” has one meaning to a scientist and quite a different meaning to other people, for example, to the president of a utility company or to a politician. What then, exactly, is energy?

The notion of energy is one of the few elements of mechanics not handed down to us from Isaac Newton. The idea was not clearly grasped until the middle of the nineteenth century. Nevertheless, we can find its germ even earlier than Newton.

IF THE EARTH MOVES: ARISTOTELIAN OBJECTIONS

In 1543 Nicolaus Copernicus's book De Revolutionibus Orbium Coelestium (On the Revolutions of the Heavenly Spheres) appeared in print. Copernicus, mindful of his personal safety, had waited until his deathbed to publish his ideas. Within the pages of De Revolutionibus Copernicus set the earth spinning on its axis and revolving around the sun. In his attempt to return the heavens to their simple beauty, Copernicus had to make the sun, not the earth, the center of the universe, and in doing so he tore the heart out of the Aristotelian world. Without the solid, immovable earth at the center of the universe, there could be no Aristotelian laws of motion. And without these laws, there were none at all, for Copernicus had no laws to replace those he destroyed.

TEMPERATURE AND PRESSURE

Everybody talks about the weather, and that usually means the temperature, an inescapable part of our environment. Yet Newton's laws of mechanics tell us nothing about temperature. Is there any connection between mechanics and temperature?

In Chapter 10 we saw a connection. If you drop a block from above a table, its potential energy first turns into kinetic energy, and then is transformed into thermal energy when the block hits the table. After a while the only evidence that those events occurred is a slight warming of the surroundings, that is, a small increase in temperature.

What really happens is that the kinetic energy of the falling block is turned into the energy of motion of atoms and molecules. The energy is still there, but the motions are in random directions, not the organized motion of a whole block of matter.

THE QUEST FOR PRECISION

Not long after Copernicus published his revolutionary book, Tycho Brahe (1546–1601) provided a multitude of new observations that, despite his own intentions, provided crucial support for the Copernican hypothesis.

FREEWAYS IN THE SKY

Not many years ago, the only conceivable use of the beautiful celestial mechanics developed over hundreds of years was to compute the positions of bodies in the heavens. Today that situation has changed radically.

COORDINATE SYSTEMS

Galileo discovered through the law of inertia that there is no single preferred reference frame. To use this discovery most efficiently, we need to discuss some geometrical ideas. The first kind of geometrical construction we need is a way of describing where things are.

If the world were only one-dimensional, everything would be on a single line. To describe where something is on that line we would first pick a point on the line as a point of reference, the origin. Then we would pick a direction along the line as the positive direction – let's say to the right of the starting point. And having made those choices we would only need to give one number – call it the x coordinate – to specify the location of a point.

COOLING OFF

How do you make something colder? Making something hotter is easy. For example, if you need to warm yourself on a chilly night, you can build a fire with little or no technology. But to cool yourself on a hot day is quite another matter.