Quanti, Hypotheses and Theory of

QUANTA, HYPOTHESIS AND THEORY OF. - The hypothesis of quanta is the first, in order of time 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. RELATIVITY).

The hypothesis of quanta was proposed in 1900 by the great theoretical physicist Max Planck. By reason of the fundamental discontinuity that it implied and of its universality, it was initially received with surprise and almost with dismay, because 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 imposed itself even before it had been possible to arrive at the constitution of even a primitive theory of quanta. Today Planck's hypothesis conceptually dominates all of science: over microphysics, which was essentially constituted on the basis of it, and is currently in vertiginous development in its two major branches: atomic physics and nuclear physics; over classical physics which, contrary to the first pessimistic predictions, was able not only to survive, but also to assume the role of statistical physics (on our scale) of microphysical quantum phenomena and even the role of limiting model toward which, in principle, the definitions and the 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; over chemistry, whose foundations are now merged with those of atomic and nuclear physics; over astronomy, which owes to it the discovery and above all the interpretation of many of its astonishing modern results and the foundations of its current cosmological views.

It is perhaps not superfluous to affirm that the foregoing does not at all mean that the current theory of quanta has by now reached even a relative perfection; it is far from that. Only the quantum hypothesis, though still wrapped in a halo of mystery, can be considered beyond dispute and absolutely necessary.

1. Prodromes of the hypothesis. Kirchhoff's law. - Toward the middle of the 19th century, as is well known, heat was recognized as having the generic nature of energy, which in material bodies takes 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 in particular in cosmic space; thus one came to the consideration of radiant heat or radiation and the question arose of its emission and its absorption by matter. It was then that G. Kirchhoff gave its first universal law.

Every radiation, independently of the particular optical theory by which one intends to interpret it, possesses its own characteristic frequency v, while every material, besides its own nature which will be indicated symbolically by (n), possesses a temperature which, referring to the absolute scale, will be indicated by T. This premised, 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 indicated by Eᵥ (v, T, n). Analogously the absorbing power of every substance for the same radiation (that is, the ratio of the quantity of that radiation which the substance absorbs to the quantity that strikes it) must be thought to depend in general on v, T and n, and therefore be indicated by Aᵥ (v, T, n). Consequently it would seem logical to think that in general also the ratio Eᵥ/Aᵥ must likewise depend on v, T, n. But Kirchhoff ingeniously intuited, and then in a certain sense demonstrated, that it must be independent of the nature of the substance, because if this were not so one could devise arrangements capable of eluding the 2nd law of thermodynamics. Therefore it must 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. Observing finally that in the particular case of Aᵥ = t (that is, of a substance that completely absorbs every radiation that strikes it) the universal function f (v, T) identifies itself with the emissive power of that same substance, that is, of the so-called black body, one understands at once what high interest the exact determination of the emissive power of the black body has.

It should have been possible to arrive at it theoretically, exploiting the great generality of Kirchhoff's theorem, that is, its validity for any optical theory, even by thinking of replacing the unknown mechanism of the atoms emitting and absorbing radiations with ideal Hertzian oscillators (v. ELECTRICITY) of all frequencies; but all the theoreticians who attempted to follow this path, among them Rayleigh, arrived in agreement at the prediction of a paradoxical emissive power which, if it had ever been realized, would have provoked a universal catastrophe. All the energy of the universe would have had to be transformed into radiation of very high and therefore deadly frequency, and therefore the temperature would have had to tend everywhere to absolute zero. There remained therefore only to attempt the experimental determination. Thus Pringsheim and Lummer, having overcome serious difficulties, determined the emissive power of the black body at various frequencies v, and for a series of constant temperatures between about 80 and 2000 degrees absolute. The results were represented with a corresponding series of curves (isotherms of black radiation) which fully confirmed the absolute unreliability of the theoretical predictions.

2. Planck's law and hypothesis

At this point there was the decisive intervention of Planck. In the first place he expressed with a mathematical formula the graphical results of the experimenters, and thus on the basis of it he was able immediately to intuit ingeniously, and then to demonstrate, that the actual law was in no way interpretable on the basis of the principles of classical physics because it was undoubtedly in contrast with the hypothesis, always admitted by the latter, that all transfers of energy must always occur in a continuous manner, a hypothesis that inexorably led to the aforementioned erroneous law; that, however, it was sufficient to admit that matter could not emit radiant energy except discontinuously, by quanta and of an energy amount 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 vanish at once.

Planck's law, rather than to the specific emissive power, is today generally referred to the specific density uᵥ of the radiation within a cavity, a thing altogether equivalent. It is, designating by k the well-known 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, provoked 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 that is never encountered explicitly in ordinary scientific considerations, but which instead reigns as supreme regulator over the unfolding of all physical phenomena, when they are considered from the most general and elevated point of view. There exists in fact a principle of least action, first enunciated on the basis of not entirely clear considerations of a metaphysical nature, but later recognized as synthesizing the whole body of the most fundamental principles of physics. This principle states that among the infinite possible evolutions of every physical system, the natural evolution is characterized by the fact that to it corresponds the least variation of the action of the system. In particular, for an isolated system the variation must be zero and therefore the action remains constant. In the case, finally, of a system with periodic evolution, this constant action will be 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, permitted 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 extremely small order (equivalent to zero point twenty-six zeros followed by the digits 6544).

The extreme smallness of h also affects the dimensions of the energy quanta hv, which, however, given the very great variability of the frequencies v, can be extremely different from one another and produce equally different effects. To fix ideas on this essential point, one should consider that the frequencies v pass with rapid increase from the minimum ones of long electromagnetic waves (v = ~ 10⁴ n/sec.) to those of medium, short, ultra-short 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.), and that therefore the corresponding hv stand to one another as the numbers 1:5, 10¹⁰:10²⁰.

To grasp the magnitude of the ratio 10²⁰ between the hv of cosmic rays and those of the longest waves, it is helpful to use a representation. If the small quantum of emission corresponding to long waves were represented by a little segment of 1 cm., the large quantum that was brought into play in the emission of cosmic rays would have to be represented by a segment of 10²⁰ cm., equal to 10¹⁵ km, that is, of such a length as to suffice to wrap around the Earth's equator 25 billion times.

4. The quantum of action h

The imposition on radiating material systems to emit their radiation in quanta of amount hv applies without any difficulty to the ideal case of Hertzian oscillators, because they have the property, common to all harmonic oscillators, of always vibrating with the same frequency regardless of the amplitude of the vibration, or, what amounts to the same thing, of their energy state. But the same imposition encounters serious difficulties when one considers that the majority of vibrating systems vary their frequency with their state of excitation. Planck, with an ingenious refinement of his original theory, not only eliminated that difficulty, but raised the quantum hypothesis to an even higher plane. Since every periodic system possesses, in each of its energy states, its own action, he observed that to obtain the necessary energy jump hv, it was enough simply to require systems to vary their action in quanta of amount h. Thus the constant h rose to the rank of the true and sole atom of action of that quantity which, as already said, governs the highest lines of the classical theories in all fields of physics and which now assumes an even greater role in the microphysical domain.

5. The hypothesis of quanta of radiation

But the quantum hypothesis was too radical and profound not to raise yet another conceptual difficulty. It was observed that it is fairly easily conceivable that a system (an atom), when it possesses sufficient energy, emits a given quantum of it, naturally of smaller amount; it is not understood, however, how absorption can occur when the whole quantum is not yet present. There was cited, for example, the fact that one could collect and measure the light coming from certain faintly shining stars, so tenuous that one of their quanta of light had to be considered spread over hundreds of metres of its path and therefore to require a finite time of absorption, during which the absorbing system could have in its vicinity only a minimal part of the quantum that it was strictly required either to absorb completely or not to absorb at all. And these intuitive objections were considered so serious that, for some time, Planck himself leaned toward accepting the compromise that radiation should be emitted in quanta but absorbed in a continuous manner.

It was Einstein who definitively resolved the objection at a stroke: radiation had to be considered as made up of quanta of energy hv, or, as is often said today, of photons. He justified the bold hypothesis by demonstrating that it not only led to no conflicts with known laws, but facilitated the still defective intuitive interpretations of some of them, in particular 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 quite mysterious. While many physicists were hesitating before the Einsteinian hypothesis, which they (erroneously) regarded as a return to the old Newtonian theory of emission, the hypothesis itself proved so useful that, after a few years, it was generally accepted and, shortly thereafter, because of the discoveries to which it had led of new phenomena absolutely incompatible with classical views, it 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 must correspond rigorously to the etymology of the word by which it is designated had long since faded, especially following the observation of the regular multiplicity of the spectral lines that the various atoms could emit (Ritz combination law), an indisputable sign of their internally complex structural constitution. Among the various hypotheses concerning this constitution, one had been well received which was originally suggested by J. Perrin, then taken up and corroborated with experimental results by Rutherford. It supposed atoms to be like a microscopic planetary system with a number of electrons equal to the atomic number of the atom itself, gravitating around a nucleus containing almost the entire mass and an electric charge equivalent to that of all the electrons, but positive. However, the interpretation of the observed spectral correlations with the aforementioned planetary constitution was certainly unattainable on the basis of classical laws, even for the simplest case of the hydrogen atom, consisting of a single electron gravitating around a single proton.

It was then (1913) that N. Bohr communicated his impressive discovery. If one supposed that the action A of the proton-electron planetary system had to be quantized with A = 1h, 2h,... nh,..., then only discrete orbits of the electron of order 1, 2... n,... would have been possible on which it would have had to have well-determined frequencies v₁, v₂,... vₙ,... while the entire system would have possessed the energies E₁, E₂,... Eₙ,... It then turned out that hypothetical jumps of the electron from any orbit of order n to that of order 2 had to correspond to energy jumps of amount Eₙ-E₂ with the corresponding emission of radiation which, by Planck's law of emission, obviously had to satisfy the condition Eₙ-E₂ = hv₂, ... Similarly, if it was supposed that the jumps of the electron 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 had to be obtained. Now, the frequencies v₂,... corresponded for the various n to those of the lines actually emitted by the hydrogen atom in the visible field and formulated empirically by the well-known Balmer law, while the frequencies v₁,... corresponded to those emitted by the same atom in the ultraviolet field and formulated by the Lyman law.

In conclusion, the quantization of the planetary hydrogen atom according to the views of Planck, Einstein and Bohr had rationally thrown the first bridge between the mysterious ultra-microscopic world inside atoms and that on our own scale.

Atomic theory developed until about 1923 on these same foundations, achieving notable successes, but also encountering, for the interpretation of certain categories of phenomena, serious difficulties, ever more intractable owing to the fact that, like those which had later led to the hypothesis of quanta, they appeared ever less eliminable without some new revolutionary innovation. In other words, after so many successes, a new state of crisis had been reached.

7. Quantum mechanics and wave mechanics

In an attempt to resolve this crisis, two apparently equivalent paths were envisaged almost simultaneously. Both begin with a critique of the previous theory, followed by the proposal of a constructive procedure that avoided its drawbacks.

W. Heisenberg (1925) thought, rightly, that the previous theory had been harmed by having introduced, for the sole purpose of giving the theory itself an intuitive appearance, concepts that have no counterpart in experimental reality, among them that of electronic orbits. Consequently he developed a theory in which nothing enters but notions perfectly defined and controllable qualitatively and quantitatively by means of experiment. But since this obliges us to consider the possible values of many physical quantities as a set of infinite values that are not continuous but discrete, representable only by means of infinite matrices, Heisenberg dispensed outright also with the notion of continuity and developed the whole quantum theory in the difficult formalism of the mechanical theory of matrices. The results thus obtained represent a notable improvement over those provided by the primitive theory.

The proposal and development of the second path were preceded by a notable theoretical development which it is useful and historically dutiful to recall.

The conceptually most disturbing effect of the quantization of magnetic waves was that of having led to the consideration of photons, mysterious physical entities which, with their different behaviors in numerous experiments, manifest sometimes corpuscular, sometimes wave-like characters, so much as to induce one to recognize in them a singular wave-corpuscle dualism that classical physics not only cannot interpret but cannot even conceive. It was then that L. de Broglie (1924) asked himself whether an analogous dualism should not also arise from the quantization of matter, giving rise to a quid which, besides manifesting in certain circumstances the usual material characters, should in others manifest characters of a wave-like type. He believed he could answer this interesting question affirmatively by generalizing the particular fact of having succeeded in associating with the uniform translational motion of velocity v of a free material point considered relativistically (v. RELATIVITÀ) the propagation of a wave of velocity V = c²/v and of frequency v linked to the kinetic energy T of the material point through a relation hv = T, of evident quantum nature owing to the intervention of the quantum of action h.

The material point would thus no longer be subject to the mandatory classical condition of having to move in a straight line, but to that of moving along a ray of that wave, following its vicissitudes in the event that the wave should undergo phenomena of diffraction. In other words, in passing from the classical conception to that of de Broglie, the motion of the material point comes to be 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 only sketched and limited to the abstract case of the material point, provoked among others the celebrated experiments of Davisson and Germer which led to the first true diffraction fringes of corpuscular rays, which by now are obtained in various ways and also employed technically to check the microscopic structure of metals.

The theory of de Broglie, unlike that of Planck and Einstein, was immediately received with favor and induced many to hope for the possibility of an intuitive interpretation of quantum phenomena by means of these mysterious waves which were called pilot, phantom, etc. But the intuitive generalization thus heralded never came. The further refinements of the theory, fundamentally due to E. Schrödinger and to P.A.M. Dirac, all comprised under the generic denomination of wave mechanics, are so abstract that, despite their formal utility, they cannot be considered particularly here. They constitute the formal demonstration of the reliability of the theory of quanta even beyond the limits of our representative faculties.

8. Philosophical repercussions of the theory of quanta

Evidently the quantum revolution could not fail to have appreciable philosophical repercussions, as the atomic hypothesis did. But more than these immediate repercussions, what impresses and raises objections are the indeterministic and anticausalistic affirmations of a notable number of physicists. Because the current formalism always provides only systems of solutions each affected by indeterminacies of an amount connected to the constant h, they believe they can affirm 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 a principle of indeterminacy, which they seek to justify also by other means.

Other physicists, on the other hand, among them the pioneers of the theory of quanta, Planck and Einstein, think that there exist no sufficient reasons to exclude so radically and definitively every possibility of determination.

Paolo Straneo