Modern Science Made Easy
By one of the leading physicists of the twentieth century, George Gamow's One, Two, Three…Infinity is one of the most memorable popular books on physics, mathematics, and science generally ever written, famous for having, directly or indirectly, launched the academic and/or scientific careers of many young people whose first real encounter with the wonders and mysteries of mathematics and science was through reading this book as a teenager. Untypically for popular science books, this one is enhanced by the author's own delightful sketches. Reviewers were enthusiastic when One, Two, Three…Infinity was published in 1947.

In the Author's Own Words:

Critical Acclaim for One, Two, Three…Infinity:

"A stimulating and provocative book for the science-minded layman." — Kirkus Reviews

"This is a layman's book as readable as a historical novel, but every chapter bears the solid imprint of authoritative research." — San Francisco Chronice

"George Gamow succeeds where others fail because of his remarkable ability to combine technical accuracy, choice of material, dignity of expression, and readability." — Saturday Review of Literature

M. PLANCK AND LIGHT QUANTA

The roots of Max Planck's revolutionary statement that light can be emitted and absorbed only in the form of certain discrete energy packages goes back to much earlier studies of Ludwig Boltzmann, James Clerk Maxwell, Josiah Willard Gibbs, and others on the statistical description of the thermal properties of material bodies. The Kinetic Theory of Heat considered heat to be the result of random motion of the numerous individual molecules of which all material bodies are formed. Since it would be impossible (and also purposeless) to follow the motion of each single individual molecule participating in thermal motion, the mathematical description of heat phenomena must necessarily use statistical method. Just as the government economist does not bother to know exactly how many acres are seeded by fanner John Doe or how many pigs he has, a physicist does not care about the position or velocity of a particular molecule of a gas which is formed by a very large number of individual molecules. All that counts here, and what is important for the economy of a country or the observed macroscopic behavior of a gas, are the averages taken over a large number of farmers or molecules.

One of the basic laws of Statistical Mechanics, which is the study of the average values of physical properties for very large assemblies of individual particles involved in random motion, is the so-called Equipartition Theorem, which can be derived mathematically from the Newtonian laws of Mechanics. It states that: The total energy contained in the assembly of a large number of individual particles exchanging energy among themselves through mutual collisions is shared equally (on the average) by all the particles. If all particles are identical, as for example in a pure gas such as oxygen or neon, all particles will have on the average equal velocities and equal kinetic energies. Writing E for the total energy available in the system, and N for the total number of particles, we can say that the average energy per particle is E/N. If we have a collection of several kinds of particles, as in a mixture of two or more different gases, the more massive molecules will have the lesser velocities, so that their kinetic energies (proportional to the mass and the square of the velocity) will be on the average the same as those of the lighter molecules.

Consider, for example, a mixture of hydrogen and oxygen. Oxygen molecules, which are 16 times more massive than those of hydrogen, will have average velocity [squareroot of 16] = 4 times smaller than the latter.

While the equipartition law governs the average distribution of energy among the members of a large assembly of particles, the velocities and energies of individual particles may deviate from the averages, a phenomenon known as statistical fluctuations. The fluctuations can also be treated mathematically, resulting in curves showing the relative number of particles having velocities greater or less than the average for any given temperature. These curves, first calculated by J. Clerk Maxwell and carrying his name, are shown in Fig. 1 for three different temperatures of the gas. The use of the statistical method in the study of thermal motion of molecules was very successful in explaining the thermal properties of material bodies, especially in the case of gases; in application to gases the theory is much simplified by the fact that gaseous molecules fly freely through space instead of being packed closely together as in liquids and solids.

Statistical Mechanics and Thermal Radiation

Toward the end of the nineteenth century Lord Rayleigh and Sir James Jeans attempted to extend the statistical method, so helpful in understanding thermal properties of material bodies, to the problems of thermal radiation. All heated material bodies emit electromagnetic waves of different wavelengths. When the temperature is comparatively low — the boiling point of water, for example — the predominant wavelength of the emitted radiation is rather large. These waves do not affect the retina of our eye (that is, they are invisible) but are absorbed by our skin, giving the sensation of warmth, and one speaks therefore of heat or infrared radiation. When the temperature rises to about 600°C (characteristic of the heating units of an electric range) a faint red light is seen. At 2000°C (as in the filament of an electric bulb) a bright white light which contains all the wavelengths of the visible radiation spectrum from red to violet is emitted. At the still higher temperature of an electric arc, 4000°C, a considerable amount of invisible ultraviolet radiation is emitted, the intensity of which rapidly increases as the temperature rises still higher. At each given temperature there is one predominant vibration frequency for which the intensity is the highest, and as the temperature rises this predominant frequency becomes higher and higher. The situation is represented graphically in Fig. 2, which gives the distribution of intensity in the spectra corresponding to three different temperatures.

Comparing the curves in Figs. 1 and 2, we notice a remarkable qualitative similarity. While in the first case the increase of temperature moves the maximum of the curve to higher molecular velocities, in the second case the maximum moves to higher radiation frequencies. This similarity prompted Rayleigh and Jeans to apply to thermal radiation the same Equipartition Principle that had turned out to be so successful in the case of gas; that is, to assume that the total available energy of radiation is distributed equally among all possible vibration frequencies. This attempt led, however, to catastrophic results! The trouble was that, in spite of all similarities between a gas formed by individual molecules and thermal radiation formed by electromagnetic vibrations, there exists one drastic difference: while the number of gas molecules in a given enclosure is always finite even though usually very large, the number of possible electromagnetic vibrations in the same enclosure is always infinite. To understand this statement, one must remember that the wave-motion pattern in a cubical enclosure, let us say, is formed by the superposition of various standing waves having their nodes on the walls of the enclosure.

The situation can be visualized more easily in a simpler case of one-dimensional wave motion, as of a string fastened at its two ends. Since the ends of the string cannot move, the only possible vibrations are those shown in Fig. 3 and correspond in musical terminology to the fundamental tone and various overtones of the vibrating string. There may be one half-wave on the entire length of the string, two half-waves, three half-waves, ten half-waves, ... a...