The Dreams That Stuff is Made of

by Stephen Hawking

  LECTURE 20

  Substitution of the matrix calculus by the general operational calculus for improved treatment of aperiodic motions—Concluding remarks.

  In the case of aperiodic straight-line motion, another procedure must therefore be adopted, which Wiener and I have recently developed. Only an outline of the fundamental ideas can be given here. An Hermitian form can be associated with every matrix, as already shown; likewise a linear transformation of the form used above,

  (1)

  Then the product of two matrices corresponds to the successive application of two such transformations:

  These together give

  (2)

  where

  As seen, the matrix enters here not as a “quantity” or “system of quantities,” but as an operator which, from an infinite system of quantities y 1, y 2 . . . , yields another system x 1, x2 . . . . The precise physical significance of these quantities is still very obscure. A calculus of operators can, therefore, be substituted for the matrix calculus and this method becomes fruitful if applied in the following way: An infinite system of quantities x1, x2 . . . may define a function with a continuous range of arguments; for instance these quantities may be taken as the coefficients of a Fourier series. It is advantageous to operate with this function instead of with the coefficients, because we then have the whole machinery of the calculus at our disposal, and differential or integral equations replace infinite sets of simultaneous equations in an infinite number of variables. These equations possess solutions under certain conditions even when the original representation in series collapses. Of course Fourier series will not be used here, but general trigonometric series of the form

  (3)

  The coefficients xn are determined from the function x (t ) by taking averages,

  (4)

  Instead of the matrix q = (q (mn)) we make use of the function of two variables,

  (5)

  and of the derived “average operator,”

  (6)

  It is then easy to show that operator products, formed by the successive application of operators, correspond to matrix products. An explicit representation of operators is not, however, necessary. It is sufficient to consider linear operators in general, that is such operators for which the simple formulaq (x (t ) + y (t )) = q x (t ) + q y (t )

  holds. Thus multiplication by a function of t, differentiation and integration with respect to t , for example, are all operators. Of special importance is the differential operator D = d /d t .

  Under certain conditions a matrix can be associated with an operator. The sequence of energy levels of this matrix is ordered not with respect to indices m , n, but relatively to the energy values themselves. The elements of the matrix which correspond to the operator q are defined by

  (7)

  In many cases this matrix does not exist although the sum of elements in a row

  (8)

  does. For the operator D for instance,

  q (V , W ) therefore does not exist as a continuous function. If W has discrete values, q (V , W ) is the diagonal matrix (Wn δnm ). The sum of the elements in a row

  however, always exists. From this example it is seen how the operational method permits a treatment of singular cases where the matrix representation breaks down.

  A more exhaustive treatment of the method is as yet unwarranted. Suffice to say that it has been possible to show that, in the case of the harmonic oscillator, the operational calculus gives the same result as the matrix calculus. Moreover, treatment of uniform rectilinear motion is possible, a case where the matrix calculus breaks down completely. Investigations on the theorems of angular momentum, on hyperbolic orbits in the hydrogen atom and on similar problems are in progress.

  In closing I should like to add a few general remarks. The first concerns the question of whether it is possible to visualize the laws of physics as formulated in this new manner and whether the processes in the atom can be conceived to exist in space and time. A definite answer will only be possible when we can see all the consequences of the new theory, perhaps only when new principles have been discovered. But it already seems certain that the usual conceptions of space and time are not rigorously compatible with the character of the new laws.

  Consider for instance the hydrogen atom. The classical theory not only gives the orbits of the electron but seems to assign a meaning to the position of the electron at each instant. In the new theory the energy and moment of momentum of a state can be given, but it appears to be impossible to give any further description of this state as a geometrical orbit and even more impossible to fix the position of the electron at any instant. Space points and time points in the ordinary sense do not exist. These conceptions can only be introduced subsequently in limiting cases.

  On the other hand it seems to me that we have a right to use the terms “orbit” or even “ellipse,” “hyperbola,” etc. in the new theory, if we agree to interpret them rationally and to understand by them the quantum processes which go over in the limit to the orbits, ellipses, hyperbolas, etc. of the classical theory. This not only gives a convenient terminology but expresses the following fact: The world of our imagination is narrower and more special in its logical structure than the world of physical things. Our imagination is restricted to a limiting case of possible physical processes. This philosophical point of view is not new: it has always been the guiding thought of physicists since Copernicus, and it came so clearly to the fore in the theory of relativity that philosophy was compelled to take a definite stand towards it. In the quantum theory this guiding principle assumes an even more predominant rôle, but in this case it is supported by such an enormous weight of evidence that a flat denial seems much more difficult than it was when the theory of relativity came up for consideration.

  Only a further extension of the theory, which in all likelihood will be very laborious, will show whether the principles given above are really sufficient to explain atomic structure. Even if we are inclined to put faith in this possibility, it must be remembered that this is only the first step toward the solution of the riddles of the quantum theory. Our theory gives the possible states of the system, but no indication of whether a system is in a given state. It gives at most the probability of the jumps. However, the statement that a system is at a given time and place in a certain state probably has a meaning, which the present state of our theory does not allow us to formulate. This is also the case with regard to the problem of light-quanta. Here the Compton effect and the related experiments of Bothe and Geiger, Compton and Simon have shown that both the energy and the momentum of light travel as a projectile from atom to atom. But the existence of interference, that is the fact that light added to light can produce darkness, is just as certain. It cannot yet be seen how these two views can be reconciled or whether a matrix representation of the electromagnetic field will lead further. An attempt to treat the statistics of cavity radiation by the new method has resulted in the elimination of serious contradictions in the classical theory. Many puzzling questions remain which fall outside the scope of these lectures.

  In the further development of the new quantum theory, the physicist cannot dispense with the aid of the mathematician. The close alliance between mathematics and physics which has reigned during the best periods of both sciences will, I hope, return and banish the mystic cloud in which physics has of late been enshrouded. The activity of the mathematician must however not carry him as far as in the theory of relativity, where the clarity of his reasoning has come to be hidden by the erection of a structure of pure speculation so vast that it is impossible to view it in its entirety. A single crystal can be clear, nevertheless a mass of fragments of this crystal is opaque. Even the theoretical physicist must be guided by the ideal of the closest possible contact with the world of facts. Only then do the formulas live and beget new life.

  EXCERPTS FROM THIRTY YEARS THAT SHOOK PHYSICS: THE STORY OF QUANTUM THEORY (CHAPTERS I AND IV)

  BY


  GEORGE GAMOW

  Illustrations by the Author

  Courtesy of Dover Publications

  CHAPTER I

  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 farmer 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 √16 = 4 times smaller than the latter.ou

  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.

  FIG. 1 Maxwell’s distribution: the number of molecules having different velocities v is plotted against the velocities for three different temperatures, 100°, 400°, and 1600° K. Since the number of molecules in the container remains constant, the areas under the three curves are the same. The average velocities of the molecules increase proportionally to the square root of the absolute temperature.

  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 wavemotion 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.

  FIG. 2 The observed distribution of radiation intensities for different frequencies ν is plotted against the frequencies. Since the radiation energy content per unit volume increases as the fourth power of the absolute temperature T, the areas under the curves increase. The frequency corresponding to maximum intensity increases proportionally to the absolute temperature.

  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 hundred, a thousand, a million, a billion . . . any number of half-waves. The corresponding vibration frequencies of various overtones will be double, triple . . . tenfold, a hundredfold, a millionfold, a billionfold ... etc., of the basic tone.

  FIG. 3 The basic tone and higher overtones in the case of the one-dimensional continuum–for example, a violin string.

  In the case of standing waves within a three-dimensional container, such as a cube, the situation will be similar though somewhat more complicated, leading to unlimited numbers of different vibrations with shorter and shorter wavelengths and correspondingly higher and higher frequencies. Thus, if E is the total amount of radiant energy available in the container, the Equipartition Principle will lead to the conclusion that each individual vibration will be allotted E/∞, an infinitely small amount of energy! The paradoxicalness of this conclusion is evident, but we can point it even more sharply by the following discussion.

  Suppose we have a cu
bical container, known as “Jeans’ cube,” the inner walls of which are made of ideal mirrors reflecting 100 per cent of the light falling on them. Of course, such mirrors do not exist and cannot be manufactured; even the best mirror absorbs a small fraction of the incident light. But we can use the notion of such ideal mirrors in theoretical discussions as the limiting case of very good mirrors. Such reasoning, whereby one thinks what would be the result of an experiment in which ideal mirrors, frictionless surfaces, weightless bars, etc., are employed, is known as a “thought experiment” (Gedankenexperiment is the original term), and is often used in various branches - of theoretical physics. If we make in a wall of Jeans’ cube a small window and shine in some light, closing the ideal shutter after that operation, the light will stay in for an indefinite time, being reflected to and fro from the ideal mirror walls. When we open the shutter sometime later we will observe a flash of the escaping light. The situation here is identical in principle to pumping some gas into a closed container and letting it out again later. Hydrogen gas in a glass container can stay indefinitely, representing an ideal case. But hydrogen will not stay long in a container made of palladium metal, since hydrogen molecules are known to diffuse rather easily through this material. Nor can one use a glass container for keeping hydrofluoric acid, which reacts chemically with glass walls. Thus, Jeans’ cube with the ideal mirror walls is after all not such a fantastic thing!