QUANTA, HYPOTHESES 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 that were absolutely incompatible with the experimental criterion that, since the time of Galileo, physicists had imposed upon themselves and 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. Owing to the fundamental discontinuity that it implied and to its universality, it was initially received with surprise and almost with consternation, for 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 was imposed even before a, albeit primitive, theory of quanta could be constituted. Today, Planck’s hypothesis conceptually dominates all science: over microphysics, which was essentially constituted on its basis and is currently developing at a dizzying pace in its two major branches, atomic physics and nuclear physics; over classical physics, which, contrary to initial pessimistic forecasts, not only survived but also assumed the role of statistical physics (at our scale) of quantum microphysical phenomena and even the role of a limiting model to which, in principle, quantum definitions and laws must tend whenever, for whatever reason—even purely hypothetical and formal—the necessity of the quantum hypothesis tends to vanish; over chemistry, whose foundations now merge with those of atomic and nuclear physics; over astronomy, which owes to it the discovery and, above all, the interpretation of many of its modern astonishing results and the foundations of its current cosmological views.
It is perhaps not superfluous to affirm that what precedes does not in any way mean that the current theory of quanta has now reached, even in a relative sense, perfection; it is far from that. Only the quantum hypothesis, though still shrouded in a veil of mystery, can be considered beyond dispute and absolutely necessary.
1. **Preliminaries to the Hypothesis. Kirchhoff’s Law.** - Toward the middle of the 19th century, as is well known, heat was recognized as a generic form of energy, which in material bodies assumes the form of the kinetic energy of molecules. Experience, however, showed 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, consideration turned to radiant heat or radiation, and the question of its emission and absorption by matter arose. It was then that G. Kirchhoff formulated the first universal law concerning II.
Every radiation, independently of the particular optical theory by which one intends to interpret it, possesses a characteristic frequency ν, while every material substance, besides its particular nature—which will be symbolically indicated by (n)—possesses a temperature, which, when referred to the absolute scale, will be indicated by T. With this premise, it is logical to think that the ability of every substance to emit radiation of frequency ν, its emissive power, must in general depend on ν, T, and n, which will be indicated by Eν(ν, T, n). Similarly, the absorptive power of every substance for the same radiation (i.e., the ratio of the quantity of that radiation absorbed by the substance to the quantity that strikes it) must be thought to depend in general on ν, T, and n, and thus be indicated by Aν(ν, T, n). Consequently, it might seem logical to think that in general the ratio Eν/Aν must also depend on ν, T, n. But Kirchhoff, with a stroke of genius, intuited—and then, in a certain sense, demonstrated—that it must be independent of the nature of the substance, because if it were not, devices could be conceived to evade the second law of thermodynamics. Therefore, it must be that Eν/Aν = f(ν, T), with f(ν, T) being a universal function. This is one of the various equivalent expressions of Kirchhoff’s theorem. Observing, finally, that in the particular case of Aν = 1 (i.e., of a substance that completely absorbs every radiation that strikes it), the universal function f(ν, T) identifies with the emissive power of that same substance, namely, that of the so-called black body, one immediately understands the great interest in the exact determination of the emissive power of the black body.
One should have been able to arrive at this theoretically by exploiting the great generality of Kirchhoff’s theorem, i.e., its validity for any optical theory, even by substituting for the unknown mechanism of atoms emitting and absorbing radiation ideal Hertzian oscillators (v. ELECTROLOGY) of all frequencies; but all theorists who attempted to follow this path, among them Rayleigh, unanimously arrived at the prediction of a paradoxical emissive power which, if ever realized, would have caused a universal catastrophe. All the energy of the universe would have had to transform into radiation of extremely high, and thus lethal, frequency, and therefore the temperature would have had to tend everywhere to absolute zero. There remained nothing to do but attempt experimental determination. Thus, Pringsheim and Lummer, overcoming serious difficulties, determined the emissive power of the black body at various frequencies ν and for a series of constant temperatures between approximately 80 and 2000 degrees absolute. The results were represented by 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, Planck’s decisive intervention occurred. First, he expressed the graphical results of the experiments with a mathematical formula, and thus, on this basis, he could immediately intuit—and then demonstrate—that the actual law could in no way be interpreted according to the principles of classical physics because it was undoubtedly in contrast with the hypothesis, always accepted by classical physics, that all transfers of energy must occur in a continuous manner, a hypothesis that inexorably led to the aforementioned erroneous law; but that it was sufficient to admit that matter could emit radiant energy only discontinuously, in quanta or amounts of energy equal to the product of a constant h (now called Planck’s constant) and the frequency ν of the radiation emitted by the matter, for every difficulty to vanish at once.
Planck’s law is today generally referred, not to the specific emissive power, but to the specific density uν of the radiation inside a cavity, which is entirely equivalent. It is, denoting by k the well-known Boltzmann constant:
\[
u_v = \frac{8 \pi v^2}{c^3} \cdot \frac{h v}{e^{\frac{h v}{k T}} - 1}
\]
It is this 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 that any physical law’s equations must satisfy leads to a natural derivation of the constant h. It has the character of an action, that is, a physical quantity that is never explicitly encountered in ordinary scientific considerations, yet which dominates as a supreme regulator over the development of all physical phenomena when viewed from a more general and elevated standpoint. Indeed, there exists a principle of least action, first enunciated on the basis of not entirely clear metaphysical considerations, but later recognized as synthesizing the entirety of the most fundamental principles of physics. This principle asserts that among the infinite possible evolutions of any physical system, the natural evolution is characterized by the fact that it corresponds to the minimal variation of the system’s action. In particular, for an isolated system, this variation must be zero, and thus the action remains constant. Finally, in the case of a system undergoing periodic evolution, this constant action will be equivalent to the product of its constant energy and the period of the said evolution.
The specific value of the constant h can be deduced from each of the numerous measurements that, taken together, enabled the determination of Planck’s law. All, without any appreciable discrepancy, led to the value h = 6.544 × 10⁻²⁷ erg·sec, that is, a value of an extremely small order (equivalent to a zero followed by twenty-seven decimal places) and followed by the digits 6544.
To grasp the magnitude of the ratio of 10⁻²⁰ between the h of cosmic rays and that of the longest waves, it is helpful to use a representation. If the small q₁ of emission corresponding to long waves were represented by a segment of 1 cm, the large q₂ involved in the emission of cosmic rays would have to be represented by a segment of 10⁻²⁰ cm, equal to 10⁻¹⁵ km, that is, a length sufficient to wrap around the Earth’s equator 25 billion times.
### 4. The Action Quantum h
The requirement that radiating material systems emit their radiation in quanta of amount h applies without 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 regardless of the amplitude of vibration, or what is equivalent, of their energy state. However, the same requirement encounters serious difficulties when one considers that most vibrating systems vary their frequency with their state of excitation. Planck, with a brilliant adjustment to his original theory, not only overcame this difficulty but elevated the hypothesis of quanta to an even higher level. Observing that every periodic system possesses, in each of its energy states, its own action, he noted that to obtain the necessary energy jump of hν, it sufficed simply to impose upon systems the requirement that they must vary their action by quanta of amount h. Thus, the constant h assumed the rank of the true and unique atom of action of that quantity which, as already stated, governs the highest lines of classical theories in all fields of physics and which now assumes an even greater role in the microphysical domain.
### 5. The Hypothesis of Radiation Quanta
Yet the hypothesis of quanta was too radical and profound not to raise another conceptual difficulty. It was argued that it is quite conceivable that a system (an atom) when it possesses sufficient energy emits a given quantum, naturally of lesser amount; however, it is not understood how absorption can occur when the entire quantum is not yet present. For example, it was noted that light from certain faint stars could be collected and measured, so tenuous that a single quantum of light had to be considered as diffused over hundreds of meters of its path and thus required a finite time for absorption, during which the absorbing system could have in its vicinity only a minimal part of the quantum it was required either to absorb completely or not at all. These intuitive objections were considered so serious that for some time Planck himself inclined toward accepting the compromise that radiation should be emitted in quanta but absorbed continuously.
It was Einstein who definitively resolved the objection in one stroke: radiation had to be considered as consisting of quanta of energy hν, or, as is often said today, of photons. He justified this bold hypothesis by demonstrating that it not only did not lead to contradictions with known laws but also facilitated the still imperfect intuitive interpretations of some of them, particularly those of the photoelectric effect and of specific colors in relation to the ancient law of Dulong and Petit, both of which had remained quite mysterious. While many physicists hesitated before Einstein’s hypothesis, which (erroneously) they regarded as a return to Newton’s old emission theory, the hypothesis proved so useful that within a few years it was generally accepted, and shortly thereafter, due to the discoveries it had led to regarding new phenomena absolutely incompatible with classical views, it was considered one of the greatest and most secure cornerstones of modern physics.
### 6. The First Application to the Study of Atomic Structure
The idea that the constitution of the atom should strictly correspond to the etymology of the term by which it is designated had long since been abandoned, especially following the observation of the regular multiplicity of spectral lines that various atoms could emit (Ritz’s Combination Law), an unmistakable sign of their complex internal structural constitution. Among the various hypotheses regarding this constitution, the one originally suggested by J. Perrin and later revived and corroborated by experimental results from Rutherford had gained considerable acceptance. It supposed atoms to be like a miniature planetary system, with a number of electrons equal to the atomic number of the atom itself, orbiting around a nucleus containing almost the entire mass and a positive electric charge equivalent to that of all the electrons. However, the correlations observed between spectra and the said planetary constitution were certainly unattainable on the basis of classical laws, even in the simplest case of the hydrogen atom, consisting of a single electron orbiting a single proton.
It was then (1913) that N. Bohr announced his remarkable discovery. If one supposed that the action A of the proton-electron planetary system should be quantized with A = 1h, 2h, ..., nh, ..., then only discrete electron orbits of order 1, 2, ..., n, ... would be possible, on which it would have to possess well-defined frequencies ν₁, ν₂, ..., νₙ, ..., while the entire system would possess energies E₁, E₂, ..., Eₙ, ... It was then verified that hypothetical jumps of the electron from any orbit of order n to that of order 2 would correspond to energy jumps of amount Eₙ - E₂ with the corresponding emission of radiation that, by Planck’s emission law, had to satisfy the condition Eₙ - E₂ = hν₂, ... Similarly, if one supposed that the electron jumps should occur from any orbit of order n to the orbit of order 1, one would obtain frequencies satisfying the condition Eₙ - E₁ = hν₁, ... Now, the frequencies ν₂ₙ, corresponding to the various n, matched those of the lines actually emitted by the hydrogen atom in the visible region and empirically formulated by Balmer’s well-known law, while the frequencies ν₁ₙ corresponded to those emitted by the same atom in the ultraviolet region and formulated by Lyman’s law.
In conclusion, the quantization of the planetary atom of hydrogen according to the views of Planck, Einstein, and Bohr had rationally thrown the first bridge between the ultra-microscopic mysterious world of the interior of atoms and that of our scale.
Atomic theory developed along these same lines until about 1923, achieving notable successes but also encountering, in the interpretation of certain categories of phenomena, serious difficulties that became increasingly troublesome due to the fact that, as those which had later led to the hypothesis of quanta, they appeared ever less removable 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
To attempt to resolve this crisis, two apparently equivalent approaches were proposed almost simultaneously. Both begin with a critique of the previous theory, followed by the proposal of a constructive method that avoids its drawbacks.W. Heisenberg (1925) rightly thought that the previous theory had been harmed by the introduction, solely for the purpose of giving the theory an intuitive appearance, of concepts that have no correspondence with experimental reality, including that of electronic orbits. Consequently, he developed a theory in which only perfectly defined and controllable notions, both qualitatively and quantitatively, through experience, intervene. But since this obliges one to consider the possible values of many physical quantities as sets of infinite non-continuous, but discrete, values, representable only by means of infinite matrices, Heisenberg forthwith eliminated the notion of continuity as well and developed the entire quantum theory within the difficult formalism of matrix mechanics. The results thus obtained represent a notable improvement over those provided by the primitive theory.
The proposal and development of the second approach were preceded by a notable theoretical development that is useful and historically necessary to recall.
The most conceptually disturbing effect of the quantization of magnetic waves was that of having led to the consideration of photons, mysterious physical entities that, with their differing behaviors in numerous experiments, manifest characteristics that are sometimes corpuscular and sometimes wave-like, to the point of inducing recognition of a singular wave-particle dualism that classical physics not only fails to interpret but even to conceive. It was then that L. de Broglie (1924) asked whether an analogous dualism might not also arise from the quantization of matter, giving rise to something that, in addition to manifesting customary material characteristics in some circumstances, should, in others, manifest characteristics of a wave-like type. He believed he could answer this question affirmatively by generalizing the particular fact that he had succeeded in associating with the uniform translational motion of velocity v of a free material point considered relativistically (v. RELATIVITY) the propagation of a wave of velocity V = c²/v and of frequency ν linked to the kinetic energy T of the material point through a relation hν = T, which is clearly of a quantum nature due to the intervention of the quantum of action h.
The material point would thus no longer be subject to the strict 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 case that 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 comes to be generalized in a manner analogous to the motion of light when one passes from geometric optics to wave optics.
It is interesting to recall that this theory, although still only outlined 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, now obtained in various ways and even employed technically to control the microscopic structure of metals.
De Broglie’s theory, unlike that of Planck and Einstein, was immediately favorably received 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, ghost, etc. But the intuitive generalization thus envisaged was never achieved. The further refinements of the fundamental theory due to E. Schrödinger and P.A.M. Dirac, all included under the generic denomination of wave mechanics, are so abstract that, despite their formal utility, they cannot be considered here in detail. They constitute the formal demonstration of the reliability of quantum theory even beyond the limits of our representational faculties.
8. Philosophical repercussions of quantum theory
Evidently, the quantum revolution could not fail to have significant philosophical repercussions, as did the atomic hypothesis. But more than these immediate repercussions, the interminist and anti-causal assertions of a considerable number of physicists have provoked objections and raised doubts. They argue that since the current formalism always provides only systems of solutions each affected by indeterminacies whose magnitude is connected to the constant h, it can be affirmed that such 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 attempt to justify by other means as well.Other physicists, however, including the pioneers of quantum theory Planck and Einstein, believe that there are not sufficient reasons for thus radically and definitively excluding all possibility of determination.