QUANTI, HYPOTHESIS AND THEORY OF. — The hypothesis of quanta is the first, in chronological order and importance, of the two singular hypotheses that physics, at the beginning of this century, had to adopt in order to eliminate two profound contradictions between the predictions of its classical theories and the results of its experiments; contradictions absolutely incompatible with the experimental criterion that, since the time of Galileo, the physicist had imposed upon himself and had always rigorously followed (for the second of these hypotheses, V. RELATIVITÀ).
The hypothesis of quanta was proposed in 1900 by the great theoretical physicist Max Planck. Because of the fundamental discontinuity it implied and of its universality, it was initially received with surprise and almost with dismay, since it was by no means certain that its adoption would not bring about the collapse of all the customary theories of physics. But within a few years the hypothesis proved so useful, not only in resolving the difficulties for which it had been devised, but also in perfecting the interpretation of numerous phenomena in various fields of physics, that its universal adoption became imperative even before it had been possible to arrive at the construction of even a primitive theory of quanta. Today Planck’s hypothesis conceptually dominates the whole of science: microphysics, which was essentially constituted on its basis and is currently undergoing vertiginous development in its two principal branches, atomic physics and nuclear physics; classical physics, which, contrary to the first pessimistic predictions, was able not only to survive but also to assume the role of statistical physics (at our scale) of quantum microphysical phenomena, and even the role of limiting model toward which, in principle, definitions and quantum laws themselves must tend whenever, for any reason, even a purely hypothetical and formal one, the necessity of the quantum hypothesis tended to vanish; chemistry, whose foundations are now merged with those of atomic and nuclear physics; and astronomy, which owes to it the discovery and, above all, the interpretation of many of its astonishing modern results, as well as the foundations of its present cosmological views.
It is perhaps not superfluous to state that the foregoing by no means signifies that the present theory of quanta has by now attained even a relative perfection; it is far from having done so. Only the quantum hypothesis, although it is still shrouded in an aura of mystery, may be considered beyond dispute and absolutely necessary.
1. Precursors of the hypothesis. Kirchhoff’s law. — Toward the middle of the nineteenth century, as is well known, heat was recognized as having the general nature of energy, which in material bodies assumes the form of the kinetic energy of molecules. Experience showed, however, that one had to admit the possibility of the existence of heat, at least in transit, even where molecules did not exist, and particularly in cosmic space; thus consideration came to be given to radiant heat, or radiation, and the question arose of its emission and absorption by matter. It was then that G. Kirchhoff formulated its first universal law.
Every radiation, independently of the particular optical theory with which one intends to interpret it, possesses its own characteristic frequency v, while every substance, in addition to its own nature, which will be indicated symbolically by (n), possesses a temperature which, with reference to the absolute scale, will be indicated by T. Given this, it is logical to think that the power of every substance to emit radiation of frequency v—its emissive power—must in general depend on v, T, and n, which will be denoted by Eᵥ (v, T, n). Similarly, the absorptive power of every substance for the same radiation (that is, the ratio of the quantity of that radiation absorbed by the substance to the quantity impinging upon it) must in general be thought to depend on v, T, and n, and therefore be denoted by Aᵥ (v, T, n). Consequently, it would seem logical to think that in general the ratio Eᵥ/Aᵥ must likewise depend on v, T, and n. But Kirchhoff brilliantly intuited, and then in a certain sense demonstrated, that it must be independent of the nature of the substance, because otherwise devices could be devised capable of circumventing the second principle of thermodynamics. It must therefore be Eᵥ/Aᵥ = f (v, T), with f (v, T) a universal function. This is one of the various equivalent expressions of Kirchhoff’s theorem. Finally, observing that in the particular case of Aᵥ = t (that is, of a substance that completely absorbs every radiation impinging upon it), the universal function f (v, T) is identified with the emissive power of that same substance, namely, of the so-called black body, one immediately understands the great importance of determining exactly the emissive power of the black body.
This should theoretically have been attainable by exploiting the great generality of Kirchhoff’s theorem, that is, its validity for any optical theory, even by replacing the unknown mechanism of the atoms emitting and absorbing radiation with ideal Hertzian oscillators (v. ELETTRIGOGIA) of all frequencies; but all the theorists who attempted to follow this path, among them Rayleigh, unanimously arrived at the prediction of a paradoxical emissive power which, had it ever been realized, would have caused a universal catastrophe. All the energy of the universe would have had to be transformed into radiation of extremely high and consequently deadly frequency, and therefore the temperature would everywhere have had to tend toward absolute zero. There thus remained no alternative but to attempt an experimental determination. Thus Pringsheim and Lummer, after overcoming serious difficulties, determined the emissive power of the black body at the various frequencies v and for a series of constant temperatures between approximately 80 and 2000 absolute degrees. The results were represented by a corresponding series of curves (isotherms of black-body radiation), which fully confirmed the utter unreliability of the theoretical predictions.
2. Planck’s law and hypothesis
At this point Planck’s decisive intervention took place. First of all, he expressed the experimenters’ graphical results by means of a mathematical formula, and thus, on its basis, he was immediately able brilliantly to intuit, and then demonstrate, that the actual law could in no way be interpreted on the basis of the principles of classical physics, because it was undoubtedly in conflict with the hypothesis, always admitted by the latter, that all transfers of energy must always occur continuously—a hypothesis that inexorably led to the erroneous law mentioned above; but that it was sufficient to admit that matter could emit radiant energy only discontinuously, in quanta and in amounts of energy equal to the product of a constant h (now called Planck’s constant) and the frequency v of the radiation emitted by matter, for every difficulty to disappear at once.Planck’s law, instead of being referred to the specific emissive power, is today generally referred to the specific density uᵥ of radiation within a cavity, which is entirely equivalent. It is, denoting by k the familiar Boltzmann constant:
uᵥ = frac{8 π v²}{c³} · frac{h v}{e^{frac{k v}{k v}} - 1}
This is the simple formula that, at the beginning of the century, brought about the greatest scientific revolution of all time.
3. Nature and value of the constant h
The condition of physical homogeneity, which the equations expressing any physical law must satisfy, uniquely determines the nature of the constant h. It has the character of an action, that is, of a physical quantity which is never encountered explicitly in ordinary scientific considerations, but which instead dominates, as a supreme regulator, the course of all physical phenomena when they are considered from the most general and elevated point of view. There indeed exists a principle of least action, initially enunciated on the basis of not entirely clear considerations of a metaphysical nature, but subsequently recognized as synthesizing the ensemble of the most fundamental principles of physics. This principle states that among the infinitely many possible evolutions of every physical system, the natural evolution is characterized by the fact that it corresponds to the minimum variation of the action of the system. In particular, for an isolated system the variation must be zero and hence the action must remain constant. Finally, in the case of a system undergoing periodic evolution, this constant action is equivalent to the product of its constant energy and the period of the said evolution.The particular value of the constant h can be deduced from each of the numerous measurements which, taken together, made possible the determination of Planck’s law. All of them, without any appreciable discrepancy, led to the value h = 6.544 · 10⁻²⁷ erg. sec., that is, to a value of an extremely small order (equivalent to zero, followed by twenty-six zeros, followed by the digits 6544).
The extreme smallness of h also affects the dimensions of the quanta of energy hᵥ, which, however, given the enormous variability of the frequencies v, may be extremely different from one another and produce effects that are likewise extremely different. To clarify this essential point, consider that the frequencies v increase rapidly from the minimum frequencies of long electromagnetic waves (v = ~ 10⁴ n/sec.) to those of medium, short, ultrashort and luminous waves (v = ~ 5.10¹⁴ n/sec.), and then to those of Röntgen, γ and cosmic rays (the latter exceeding v = 10²⁴ n/sec.); consequently, the corresponding hv stand to one another as the numbers 1:5, 10¹⁰:10²⁰.
To understand the magnitude of the ratio 10²⁰ between the hv of cosmic rays and those of the longest waves, it is useful to employ a representation. If the small quantum of emission corresponding to the long waves were represented by a segment 1 cm. long, the large quantum brought into play in the emission of cosmic rays would have to be represented by a segment 10²⁰ cm. long, equal to 10¹⁵ km, that is, of a length sufficient to wrap around the terrestrial equator 25 billion times.
4. The quantum of action h
The imposition on radiating material systems that they emit their radiation in quanta of amount hv applies without any difficulty to the ideal case of Hertzian oscillators, because they possess the property, common to all harmonic oscillators, of always vibrating at the same frequency independently of the amplitude of the vibration, or, what amounts to the same thing, of their energetic state. But the same imposition encounters serious difficulties when one considers that most vibrating systems vary their frequency with their state of excitation. Planck, with a brilliant modification of his original theory, not only eliminated this difficulty, but carried the hypothesis of the quanta into an even higher sphere. Since every periodic system possesses, in each of its energetic states, an action of its own, he observed that, in order to obtain the necessary energy jump hv, it was simply sufficient to impose on the systems the requirement that they vary their action by quanta of amount h. Thus the constant h rose to the rank of the true and unique atom of action, of that quantity which, as already stated, regulates the highest lines of classical theories in all fields of physics and which now assumes an even greater role in the microphysical field.5. The hypothesis of radiation quanta
But the hypothesis of the quanta was too radical and profound not to raise another conceptual difficulty as well. It was observed that it is fairly easy to conceive that a system (an atom), when it possesses sufficient energy, emits a given quantum, naturally of a smaller amount; it is not clear, however, how absorption can take place when the entire quantum is not yet present. For example, the fact was cited that one could collect and measure the light coming from certain faint stars, so tenuous that a quantum of their light had to be regarded as dispersed over hundreds of metres of its path and therefore as requiring a finite absorption time, during which the absorbing system could have near it only a minimal part of the quantum which it was categorically required either to absorb in its entirety or not to absorb at all. And these intuitive objections were considered so serious that, for some time, Planck himself inclined toward accepting the intermediate position that radiation should be emitted in quanta but absorbed continuously.It was Einstein who definitively and at once resolved the objection: radiation had to be regarded as consisting of quanta of energy hv, or, as is often said today, of photons. He justified the daring hypothesis by demonstrating that it not only led to no contradictions with known laws, but also facilitated the still defective intuitive interpretations of some of them, particularly those of the photoelectric effect and of specific heats with reference to the old law of Dulong and Petit, both of which had always remained highly mysterious. While many physicists hesitated before the Einsteinian hypothesis, which they (erroneously) regarded as a return to the old Newtonian emission theory, the hypothesis itself proved so useful that, after a few years, it was generally accepted and, shortly thereafter, owing to the discoveries to which it had led of new phenomena absolutely incompatible with classical views, was regarded as one of the greatest and most secure cornerstones of modern physics.
6. The first application to the study of the constitution of atoms
The idea that the constitution of the atom should correspond rigorously to the etymology of the word by which it is designated had long since disappeared, especially following the observation of the regular multiplicity of the spectral lines which the various atoms could emit (Ritz’s Combination Law), an indisputable sign of their internally complex structural constitution. Among the various hypotheses concerning this constitution, one originally suggested by J. Perrin, and subsequently taken up and corroborated by Rutherford with experimental results, had been well received. It assumed that atoms were similar to a microscopic planetary system, with a number of electrons equal to the atomic number of the atom itself, revolving around a nucleus containing almost the entire mass and bearing an electric charge equivalent to that of all the electrons, but positive. Yet the interpretation of the observed spectral correlations on the basis of the aforesaid planetary constitution was certainly unattainable by means of classical laws, even in the simplest case of the hydrogen atom, consisting of a single electron revolving around a single proton.It was then (1913) that N. Bohr communicated his striking discovery. If it was assumed that the action A of the proton–electron planetary system had to be quantized, with A = 1h, 2h,... nh,..., only discrete orbits of the electron of order 1, 2... n,... would be possible.
on which it would have had well-defined frequencies v₁, v₂,... vₙ,... while the system as a whole would have possessed the energies E₁, E₂,... Eₙ,... It then followed that hypothetical jumps of the electron from any orbit of order n to that of order 2 had to correspond to energy changes of magnitude Eₙ-E₂, with the corresponding emission of radiation which, according to Planck’s law of emission, obviously had to satisfy the condition Eₙ-E₂ = hv₂, ... Similarly, if it was supposed that the electron’s jumps had to occur from any orbit of order n to the orbit of order 1, a frequency satisfying the condition Eₙ-E₁ = hv₁, ... would have to be obtained. Now, the frequencies v₂,... corresponded, for the various values of n, to those of the lines actually emitted by the hydrogen atom in the visible spectrum and empirically formulated by the well-known Balmer law, whereas the frequencies v₁,... corresponded to those emitted by the same atom in the ultraviolet spectrum and formulated by the Lyman law.
In conclusion, the quantization of the planetary atom of hydrogen according to the views of Planck, Einstein, and Bohr had rationally laid the first bridge between the mysterious ultra-microscopic world within atoms and the world at our scale.
Atomic theory developed until about 1923 on these same foundations, achieving notable successes but also encountering, in interpreting certain categories of phenomena, serious difficulties, which became increasingly troubling because, like those that had eventually led to the hypothesis of the q., they appeared less and less capable of being eliminated without some new revolutionary innovation. In other words, after so many successes, it had returned to a new state of crisis.
7. Quantum mechanics and wave mechanics
In an attempt to resolve this crisis, two apparently equivalent paths were proposed almost simultaneously. Both began with a critique of the preceding theory, followed by the proposal of a constructive procedure that avoided its shortcomings.W. Heisenberg (1925) rightly thought that the preceding theory had been harmed by introducing, solely in order to give the theory itself an intuitive appearance, concepts that had no correspondence in experimental reality, among them that of electronic orbits. Consequently, he developed a theory involving only notions that were perfectly defined and qualitatively and quantitatively controllable through experience. But since experience compelled one to consider the possible values of many physical quantities as a set of infinitely many values that were not continuous but discrete, representable only by means of infinite matrices, Heisenberg also dispensed altogether with the notion of continuity and developed the entire quantum theory in the difficult formalism of matrix mechanics. The results thus obtained represented a considerable improvement over those supplied by the primitive theory.
The proposal and development of the second path were preceded by a notable theoretical development that it is useful, and historically necessary, to recall.
The conceptually most disturbing effect of the quantization of magnetic waves was that it led to the consideration of photons, mysterious physical entities which, through their differing behavior in numerous experiments, display at times corpuscular and at times wave-like characteristics, so much so as to compel recognition of a singular wave-particle dualism in them—something that classical physics not only cannot interpret but cannot even conceive. It was then that L. de Broglie (1924) asked whether an analogous dualism should not also arise from the quantization of matter, giving rise to a something that, besides displaying customary material characteristics in some circumstances, should in others display wave-like characteristics. He believed he could answer this interesting question affirmatively by generalizing the particular fact that he had succeeded in associating with the uniform translational motion, at velocity v, of a free material point considered relativistically (v. RELATIVISMO) the propagation of a wave with velocity V = c²/v and frequency v, linked to the kinetic energy T of the material point through a relation hv = T, whose quantum character was evident from the involvement of the quantum of action h.
The material point would thus no longer be subject to the strict classical condition that it must move in a straight line, but rather to that of moving along a ray of that wave, following its behavior if the wave were to undergo diffraction phenomena. In other words, in passing from the classical conception to that of de Broglie, the motion of the material point becomes generalized in a manner analogous to the motion of light when one passes from geometrical optics to wave optics.
It is interesting to recall that this theory, although still only sketched out and limited to the abstract case of the material point, led, among other things, to the celebrated experiments of Davisson and Germer, which produced the first actual diffraction fringes of corpuscular rays. Such fringes are now obtained in various ways and are also used technically to examine the microscopic structure of metals.
De Broglie’s theory, unlike those of Planck and Einstein, was immediately received favorably and led many to hope for the possibility of an intuitive interpretation of quantum phenomena by means of these mysterious waves, which were called pilote, fantasma, etc. But the anticipated intuitive generalization never came. The subsequent refinements of the theory, fundamentally due to E. Schrödinger and P.A.M. Dirac, all encompassed under the general designation of wave mechanics, are so abstract that, despite their formal usefulness, they cannot be considered here in detail. They constitute the formal demonstration of the reliability of the theory of the q. even beyond the limits of our representational faculties.
8. Philosophical repercussions of the theory of the q. - Evidently, the quantum revolution could not fail to have significant philosophical repercussions, just as the atomic hypothesis had had. But more than these immediate repercussions, the indeterministic and anti-causalist assertions of a considerable number of physicists are troubling and give rise to objections. Because the current formalism always provides only systems of solutions, each affected by uncertainties whose magnitude is connected with the constant h, they believe they can assert that this uncertainty is an inevitable consequence of the quantum hypothesis and not of the formalism employed, and that therefore, for microscopic phenomena, the principle of causality does not hold, but rather a principle of indeterminacy, which they also seek to justify by other means.
Other physicists, however, among them the pioneers of the theory of the q., Planck and Einstein, believe that there are insufficient grounds for excluding so radically and definitively every possibility of determination.