Research on the Theory of Quanta

de Broglie, L. (1925). Recherches sur la théorie des quanta [Research on the theory of quanta]. Annales de Physique, 10th Series, Vol. III, January-February 1925. Translated from the original French.

NoteTo the Reader

This is Louis de Broglie’s 1925 doctoral thesis, one of the most revolutionary works in the history of physics. In this paper, de Broglie proposes that all matter has wave-like properties, introducing the concept of “matter waves” or “phase waves.”

For General Chemistry students, this paper provides the theoretical foundation for understanding electron orbitals. When de Broglie suggested that electrons behave as waves, he explained why electrons in atoms can only occupy certain energy levels: the electron wave must form a standing wave around the nucleus. This directly leads to the quantum mechanical model of the atom you study in chemistry.

The famous de Broglie equation, λ = h/p (wavelength equals Planck’s constant divided by momentum), emerges from this work and remains fundamental to understanding atomic structure today.

Summary

The history of optical theories shows that scientific thought has long hesitated between a dynamical conception and a wave conception of light; these two representations are therefore probably less opposed than had been supposed, and the development of quantum theory seems to confirm this conclusion. ¹

¹ What this thesis accomplishes: De Broglie’s 1925 thesis answered a fundamental question: if light waves have particle properties (photons), do particles have wave properties? His answer was yes. Every particle with momentum p has an associated wavelength λ = h/p. This explained why electrons in atoms can only have certain energies (the wave must form a standing wave), gave physical meaning to Bohr’s quantization rules, and laid the groundwork for Schrödinger’s wave equation. For Gen Chem students: this is why we describe electrons as “waves” and draw orbitals as probability clouds.

Guided by the idea of a general relationship between the notions of frequency and energy, we assume in the present work the existence of a periodic phenomenon of a nature yet to be specified, which would be linked to every isolated piece of energy and which would depend on its proper mass through the Planck-Einstein equation. The theory of relativity then leads us to associate with the uniform motion of any material point the propagation of a certain wave whose phase travels through space faster than light (Chapter I). To generalize this result to the case of non-uniform motion, we are led to assume a proportionality between the momentum four-vector of a material point and a vector characteristic of the propagation of the associated wave, whose time component is the frequency. Fermat’s principle applied to the wave then becomes identical to the principle of least action applied to the moving body. The rays of the wave are identical to the possible trajectories of the moving body (Chapter II).

The preceding statement applied to the periodic motion of an electron in Bohr’s atom allows us to recover the quantum stability conditions as expressions of the resonance of the wave along the length of the trajectory (Chapter III). This result can be extended to the case of circular motions of the nucleus and electron around their common center of gravity in the hydrogen atom (Chapter IV).

The application of these general ideas to Einstein’s light quantum leads to numerous very interesting concordances. It allows us to hope, despite the difficulties that remain, for the constitution of an optics that is both atomistic and wave-like, establishing a sort of statistical correspondence between the wave linked to the grain of light energy and Maxwell’s electromagnetic wave (Chapter V). In particular, the study of the scattering of X-rays and γ-rays by amorphous bodies serves to show how desirable such a reconciliation is today (Chapter VI).

Finally, the introduction of the notion of phase wave into statistical mechanics leads to justifying the intervention of quanta in the dynamical theory of gases and to recovering the laws of black-body radiation as expressing the distribution of energy among atoms in a gas of light quanta.

Historical Introduction

I. From the 16th to the 20th Century

Modern science was born at the end of the 16th century following the intellectual renewal brought about by the Renaissance. While positional astronomy was becoming more precise day by day, the sciences of equilibrium and motion, statics and dynamics, were slowly being established. It is well known that Newton was the first to make dynamics a homogeneous body of doctrine and, through his memorable law of universal gravitation, opened up an enormous field of applications and verifications for the new science. During the 18th and 19th centuries, a great number of mathematicians, astronomers, and physicists developed Newton’s principles, and mechanics reached such a degree of beauty and rational harmony that its character as a physical science was almost forgotten. In particular, it became possible to derive this entire science from a single principle, the principle of least action, first stated by Maupertuis, then in another manner by Hamilton, whose mathematical form is remarkably elegant and condensed.

Through its involvement in acoustics, hydrodynamics, optics, and capillarity, mechanics appeared for a moment to reign over all domains. It had somewhat more difficulty absorbing a new branch of science born in the 19th century: thermodynamics. While one of the two great principles of this science, that of the conservation of energy, readily lends itself to interpretation by mechanical conceptions, this is not the case for the second, that of the increase of entropy. The work of Clausius and Boltzmann on the analogy between thermodynamic quantities and certain quantities appearing in periodic motions, work which is now very much back in vogue, did not succeed in completely restoring agreement between the two points of view. But the admirable kinetic theory of gases of Maxwell and Boltzmann, and the more general doctrine called statistical mechanics of Boltzmann and Gibbs, showed that dynamics, when supplemented by probability considerations, allows the interpretation of the fundamental notions of thermodynamics.

From the 17th century onward, the science of light, optics, had attracted the attention of researchers. The most common phenomena (rectilinear propagation, reflection, refraction), those which today form our geometrical optics, were naturally the first to be known. Several scientists, notably Descartes and Huygens, worked to unravel their laws, and Fermat summarized them by a synthetic principle that bears his name and which, stated in our current mathematical language, recalls by its form the principle of least action. Huygens had leaned toward a wave theory of light, but Newton, sensing in the great laws of geometrical optics a profound analogy with the dynamics of the material point of which he was the creator, developed a corpuscular theory of light called the “emission theory” and even managed to account, with the help of somewhat artificial hypotheses, for phenomena now classified in wave optics (Newton’s rings).

The beginning of the 19th century saw a reaction against Newton’s ideas in favor of those of Huygens. The interference experiments, the first of which were due to Young, were difficult if not impossible to interpret from the corpuscular point of view. Fresnel then developed his admirable elastic theory of the propagation of light waves, and from then on the credit of Newton’s conception continually diminished.

One of Fresnel’s great successes was to explain the rectilinear propagation of light, whose interpretation was so intuitive in the emission theory. When two theories based on ideas that seem entirely different to us account with the same elegance for the same experimental truth, one can always ask whether the opposition between the two points of view is real and is not due solely to the insufficiency of our efforts at synthesis. ² This question was not asked in Fresnel’s time, and the notion of the light corpuscle was considered naive and abandoned.

² Wave-particle duality: De Broglie is foreshadowing his key insight. For 200 years, waves and particles were considered opposites. Light was either one or the other. De Broglie will show they’re complementary descriptions of the same reality. This wave-particle duality is now a cornerstone of quantum mechanics. In Gen Chem, we see this when we describe electrons as both particles (with mass and charge) and waves (with wavelength λ = h/p). The electron’s wave nature determines atomic structure; its particle nature determines how it interacts with other particles.

The 19th century saw the birth of an entirely new branch of physics that has brought immense upheavals to our conception of the world and to our industry: the science of electricity. We need not recall here how it was established thanks to the work of Volta, Ampère, Laplace, Faraday, etc. What matters only is to say that Maxwell was able to summarize in formulas of superb mathematical concision the results of his predecessors and show how all of optics could be considered as a branch of electromagnetism. The work of Hertz and even more that of H. A. Lorentz perfected Maxwell’s theory; Lorentz also introduced the notion of the discontinuity of electricity, already developed by J. J. Thomson and so brilliantly confirmed by experiment. Certainly, the development of electromagnetic theory removed the reality from Fresnel’s elastic ether and thereby seemed to separate optics from the domain of mechanics, but many physicists, following Maxwell himself, still hoped at the end of the last century to find a mechanical explanation of the electromagnetic ether and, consequently, not only to subject optics again to dynamical explanations, but also to subject all electrical and magnetic phenomena to them at the same time.

The century thus ended illuminated by the hope of an imminent and complete synthesis of all physics.

II. The 20th Century: Relativity and Quanta

However, some shadows remained in the picture. Lord Kelvin, in 1900, announced that two dark clouds appeared menacingly on the horizon of physics. ³ One of these clouds represented the difficulties raised by the famous Michelson-Morley experiment, which seemed incompatible with the ideas then accepted. The second cloud represented the failure of statistical mechanics methods in the domain of black-body radiation; the equipartition theorem of energy, a rigorous consequence of statistical mechanics, leads in fact to a well-defined distribution of energy among the various frequencies in thermodynamic equilibrium radiation; however, this law, the Rayleigh-Jeans law, is in gross contradiction with experiment, and it is even almost absurd because it predicts an infinite value for the total energy density, which obviously has no physical meaning. ⁴

³ Lord Kelvin’s “two clouds”: In 1900, the famous physicist Lord Kelvin declared that physics was essentially complete, except for “two small clouds” on the horizon. These “clouds” turned out to be revolutionary: one led to Einstein’s theory of relativity, the other to quantum mechanics. For chemistry, the quantum cloud was transformative. It explained why atoms emit only specific colors of light (atomic spectra), why electrons don’t spiral into the nucleus, and ultimately gave us the orbital model of the atom you learn in General Chemistry.

⁴ The ultraviolet catastrophe: A black body is an idealized object that absorbs all light and, when heated, emits radiation at all wavelengths. Think of a glowing piece of iron: as it heats up, it glows red, then orange, then white. Classical physics predicted that a hot object should emit infinite energy at short wavelengths (ultraviolet and beyond), which is obviously absurd. This crisis was called the “ultraviolet catastrophe.” Planck resolved it by proposing that energy comes in discrete packets (quanta), not continuous waves. This was the birth of quantum theory.

In the early years of the 20th century, Lord Kelvin’s two clouds condensed, if I may say so, one into the theory of relativity, the other into the theory of quanta.

How the difficulties raised by the Michelson experiment were first studied by Lorentz and FitzGerald, how they were then resolved by A. Einstein through an intellectual effort perhaps without parallel, is something we shall not develop here, the question having been expounded many times in recent years by voices more authoritative than ours. We shall therefore assume in this exposition that the essential conclusions of the theory of relativity are known, at least in its restricted form, and we shall call upon them whenever necessary.

We shall, on the contrary, briefly indicate the development of quantum theory. The notion of quanta was introduced into science in 1900 by Max Planck. This scientist was then theoretically studying the question of black-body radiation and, since thermodynamic equilibrium should not depend on the nature of the emitters, he had imagined a very simple emitter called “Planck’s resonator” consisting of an electron subject to a quasi-elastic constraint and thus possessing a vibration frequency independent of its energy. If one applies to energy exchanges between such resonators and radiation the classical laws of electromagnetism and statistical mechanics, one recovers the Rayleigh law whose undeniable inaccuracy we pointed out above. To avoid this conclusion and find results more in conformity with experimental facts, Planck admitted a strange postulate: “Energy exchanges between resonators (or matter) and radiation take place only in finite quantities equal to h times the frequency, h being a new universal constant of physics.” ⁵ To each frequency there thus corresponds a sort of atom of energy, a quantum of energy.

⁵ Planck’s quantum hypothesis: This is the birth of quantum theory. Before Planck, physicists believed energy could have any value, like water flowing continuously. Planck proposed something radical: energy comes in discrete “packets” called quanta, with energy E = hν. The constant h = 6.626 × 10⁻³⁴ J·s is Planck’s constant, one of the fundamental constants of nature. You use this equation constantly in chemistry when calculating photon energies. If ν = 5 × 10¹⁴ Hz (visible light), then E = (6.626 × 10⁻³⁴)(5 × 10¹⁴) ≈ 3.3 × 10⁻¹⁹ J per photon.

The observational data provided Planck with the necessary basis for calculating the constant h, and the value found then (h = 6.545 × 10⁻²⁷) has hardly been modified by the innumerable subsequent determinations made by the most diverse methods. This is one of the finest examples of the power of theoretical physics.

Rapidly, quanta spread like wildfire and soon permeated all parts of physics. While their introduction removed certain difficulties relating to the specific heats of gases, it allowed Einstein first, then Nernst and Lindemann, and finally in a more perfect form Debye, Born, and von Karman to construct a satisfactory theory of the specific heats of solids and to explain why the Dulong-Petit law sanctioned by classical statistics has important exceptions and is, just like the Rayleigh law, only a limiting form valid in a certain domain.

Quanta also penetrated into a science where they would hardly have been expected: the theory of gases. Boltzmann’s method leads to leaving undetermined the value of the additive constant appearing in the expression for entropy. Planck, to account for Nernst’s theorem and obtain the exact prediction of chemical constants, admitted that quanta had to be involved, and he did so in a rather paradoxical form by attributing to the phase space element of a molecule a finite size equal to h³.

The study of the photoelectric effect raised a new enigma. ⁶ The photoelectric effect is the expulsion by matter of moving electrons under the influence of radiation.

⁶ The photoelectric effect: When light shines on a metal surface, electrons are ejected. Classical physics predicted that brighter light (more energy) should give electrons more kinetic energy. But experiments showed something strange: brightness only affected how many electrons were ejected, not their energy. Only higher frequency light (bluer light) gave electrons more kinetic energy. Red light, no matter how bright, couldn’t eject electrons from certain metals at all. Einstein explained this in 1905 by proposing that light itself comes in packets (photons) with energy E = hν. Each electron absorbs exactly one photon. This earned Einstein the Nobel Prize in 1921, and it’s why solar cells and digital cameras work.

Experiment shows, a paradoxical fact, that the energy of the expelled electrons depends on the frequency of the exciting radiation and not on its intensity. Einstein, in 1905, accounted for this strange phenomenon by admitting that radiation can only be absorbed in quanta; consequently, if the electron absorbs energy hν and if it must expend work w to leave the matter, its final kinetic energy will be hν − w. This law has been well verified. With his profound intuition, Einstein felt that there was reason to return in some way to the corpuscular conception of light and put forward the hypothesis that all radiation of frequency ν is divided into energy atoms of value hν. This hypothesis of light quanta (Lichtquanten), in opposition to all the facts of wave optics, was judged too simplistic and rejected by most physicists. While Lorentz, Jeans, and others raised formidable objections against it, Einstein replied by showing how the study of fluctuations in black-body radiation also led to the conception of a discontinuity of radiant energy. The international physics congress held in Brussels in 1911 under the auspices of M. Solvay devoted itself entirely to the question of quanta, and it was following this congress that Henri Poincaré published, shortly before his death, a series of articles on quanta, showing the necessity of accepting Planck’s ideas.

In 1913, Niels Bohr’s theory of the atom appeared. ⁷ He admitted, with Rutherford and Van Den Broek, that the atom is formed of a positive nucleus surrounded by a cloud of electrons, the nucleus carrying N elementary positive charges of 4.77 × 10⁻¹⁰ e.s.u., and the number of electrons being N so that the whole is neutral. N is the atomic number equal to the order number of the element in Mendeleev’s periodic series. To be able to predict optical frequencies, particularly for hydrogen whose one-electron atom is especially simple, Bohr makes two hypotheses: 1° Among the infinity of trajectories that an electron rotating around the nucleus can describe, only certain ones are stable, and the stability condition involves Planck’s constant. We shall specify in Chapter III the nature of these conditions; 2° When an intra-atomic electron passes from one stable trajectory to another, there is emission or absorption of a quantum of energy of frequency ν. The emitted or absorbed frequency ν is thus related to the variation δε of the total energy of the atom by the relation |δε| = hν.

⁷ Bohr’s atomic model: Bohr combined Rutherford’s nuclear model with Planck’s quantum ideas to solve a major problem: classical physics predicted that orbiting electrons should continuously radiate energy and spiral into the nucleus in about 10⁻¹¹ seconds. Atoms shouldn’t be stable! Bohr proposed that electrons can only occupy certain quantized orbits where they don’t radiate. When electrons jump between orbits, they emit or absorb photons with energy exactly equal to the energy difference: ΔE = hν. This explained why hydrogen emits only specific colors of light (its line spectrum). The energy levels you learn in Gen Chem (E_n = −13.6 eV/n² for hydrogen) come directly from Bohr’s model.

⁸ What you see in the lab: When you look at a hydrogen discharge tube through a spectroscope, you see distinct colored lines: red (656 nm, n = 3 → 2), cyan (486 nm, n = 4 → 2), and violet lines. These are the Balmer series, transitions ending at n = 2. Bohr’s model, and de Broglie’s wave interpretation of it, predict exactly these wavelengths. The Lyman series (UV, ending at n = 1) and Paschen series (IR, ending at n = 3) complete the picture.

The magnificent fortune of Bohr’s theory over the past ten years is well known. It immediately allowed the prediction of the spectral series of hydrogen and ionized helium; ⁸ the study of X-ray spectra and the famous Moseley law relating the atomic number to the spectral markers of the Röntgen domain considerably extended its field of application. Sommerfeld, Epstein, Schwarzschild, Bohr himself, and others have perfected the theory, stated more general quantization conditions, explained the Stark and Zeeman effects, interpreted optical spectra in their details, etc. But the profound significance of quanta remained unknown. The study of the X-ray photoelectric effect by Maurice de Broglie, that of the γ-ray photoelectric effect due to Rutherford and Ellis, have increasingly accentuated the corpuscular character of these radiations, the energy quantum hν seeming each day more to constitute a true atom of light. But the old objections against this view remained, and even in the domain of X-rays, the wave theory achieved fine successes: prediction of Laue interference phenomena and scattering phenomena (work of Debye, W. L. Bragg, etc.). However, quite recently, scattering in turn has been subjected to the corpuscular point of view by H. A. Compton: his theoretical and experimental work has shown that an electron scattering radiation must undergo a certain impulse as in a collision; naturally, the energy of the radiation quantum is thereby diminished, and consequently the scattered radiation presents a variable frequency depending on the direction of scattering and lower than the frequency of the incident radiation.

In short, the moment seemed to have come to attempt an effort with the aim of unifying the corpuscular and wave points of view and of deepening somewhat the true meaning of quanta. This is what we have recently done, and the present thesis has as its principal object to present a more complete exposition of the new ideas we have proposed, of the successes to which they have led us, and also of the very numerous gaps they contain.⁽¹⁾

Chapter I: The Phase Wave

I. The Quantum Relation and Relativity

One of the most important new conceptions introduced by relativity is that of the inertia of energy. According to Einstein, energy must be considered as having mass, and all mass represents energy. Mass and energy are always related to each other by the general relation: ⁹

⁹ Mass-energy equivalence: Einstein’s famous E = mc² (1905) tells us that mass and energy are interchangeable. The c² factor (9 × 10¹⁶ m²/s²) is enormous, which is why nuclear reactions release such tremendous energy from tiny mass changes. For chemistry, this equation explains where the energy comes from in nuclear reactions and why the mass of a nucleus is slightly less than the sum of its protons and neutrons (the “mass defect” becomes binding energy).

\[\text{energy} = \text{mass} \times c^2\]

c being the constant called “speed of light” but which we prefer to name “limiting speed of energy” for reasons explained below. Since there is always proportionality between mass and energy, matter and energy should be considered as two synonymous terms designating the same physical reality.

Atomic theory first, then electronic theory, have taught us to consider matter as essentially discontinuous, and this leads us to admit that all forms of energy, contrary to the old ideas about light, are if not entirely concentrated in small portions of space, at least condensed around certain singular points.

The principle of the inertia of energy attributes to a body whose proper mass (that is, measured by an observer attached to it) is m₀ a proper energy m₀c². ¹⁰

¹⁰ Proper mass (rest mass): The “proper mass” m₀ is what we now call rest mass, the mass measured when the object is at rest. For an electron, m₀ = 9.109 × 10⁻³¹ kg. In relativity, a moving object appears to have greater mass: m = m₀/√(1 − v²/c²). This relativistic mass increases without limit as v approaches c, which is why nothing with mass can reach light speed. In Gen Chem, we usually use the rest mass since atomic electrons move slowly enough that relativistic effects are small (except for heavy atoms).

¹¹ Relativistic kinetic energy: At everyday speeds, kinetic energy is ½mv², as you learned in physics. But at speeds approaching the speed of light, mass effectively increases and the formula becomes more complex. For chemistry, don’t worry: electrons in atoms move at about 1% of the speed of light (roughly 2 × 10⁶ m/s in hydrogen), so relativistic corrections are small. They become important only for heavy atoms like gold, where inner electrons move at about 60% of light speed, causing effects like gold’s color.

If the body is in uniform motion with a velocity v = βc relative to an observer whom we shall call for simplicity the stationary observer, its mass will have for him the value m₀/√(1 − β²) in accordance with a well-known result of relativistic dynamics, and consequently its energy will be m₀c²/√(1 − β²). Since kinetic energy can be defined as the increase that the energy of a body undergoes for the stationary observer when it passes from rest to velocity v = βc, one finds for its value the following expression: ¹¹

\[E_{\text{kin}} = \frac{m_0 c^2}{\sqrt{1 - \beta^2}} - m_0 c^2 = m_0 c^2 \left( \frac{1}{\sqrt{1 - \beta^2}} - 1 \right)\]

which naturally for small values of β leads to the classical form:

\[E_{\text{kin}} = \frac{1}{2} m_0 v^2\]

Having recalled this, let us seek in what form we can introduce quanta into relativistic dynamics. It seems to us that the fundamental idea of quantum theory is the impossibility of considering an isolated quantity of energy without associating a certain frequency with it. This connection is expressed by what I shall call the quantum relation: ¹²

¹² The quantum relation: E = hν is the Planck-Einstein relation you use constantly in chemistry. De Broglie’s insight was to turn this around: if light (waves) has particle properties with E = hν, perhaps particles (matter) have wave properties too. This simple but profound idea led to the de Broglie wavelength and ultimately to quantum mechanics. Every time you use λ = h/p or discuss electron orbitals as standing waves, you’re using de Broglie’s extension of this relation.

\[\text{energy} = h \times \text{frequency}\]

h being Planck’s constant.

The progressive development of quantum theory has several times highlighted mechanical action, and attempts have been made many times to give the quantum relation a statement involving action instead of energy. Certainly, the constant h has the dimensions of an action, namely ML²T⁻¹, and this is not due to chance since the theory of relativity teaches us to classify action among the principal “invariants” of physics. But action is a quantity of a very abstract character, and following numerous meditations on light quanta and the photoelectric effect, we have been led to take the energetic statement as our basis, while then seeking why action plays such a great role in many questions.

The quantum relation would probably not have much meaning if energy could be distributed continuously in space, but we have just seen that this is probably not the case. One can therefore conceive that, by virtue of a great law of Nature, to each piece of energy of proper mass m₀ there is linked a periodic phenomenon of frequency ν₀ such that: ¹³

¹³ De Broglie’s fundamental hypothesis: This is the central idea of the entire thesis. De Broglie proposes that every particle with mass m₀ has an intrinsic “vibration” with frequency ν₀ = m₀c²/h. For an electron, this works out to about 1.2 × 10²⁰ Hz, an incredibly high frequency. This internal oscillation, when combined with the particle’s motion, produces the famous de Broglie wavelength λ = h/p. The physical nature of this “vibration” was unclear to de Broglie, but the mathematical consequences were revolutionary: it explained why electrons in atoms can only have certain energies (quantization) and laid the foundation for Schrödinger’s wave mechanics.

\[h\nu_0 = m_0 c^2\]

ν₀ being measured, of course, in the system attached to the piece of energy. This hypothesis is the basis of our system: it is worth, like all hypotheses, what the consequences that can be deduced from it are worth.

Should we suppose the periodic phenomenon localized inside the piece of energy? This is by no means necessary, and it will result from paragraph III that it is probably spread over an extended portion of space. Besides, what should we understand by the interior of a piece of energy? The electron is for us the type of the isolated piece of energy, the one we believe, perhaps wrongly, to know best; now according to accepted conceptions, the energy of the electron is spread throughout all space with a very strong condensation in a region of very small dimensions whose properties are moreover very poorly known to us. What characterizes the electron as an atom of energy is not the small place it occupies in space, I repeat that it occupies all of it, but the fact that it is indivisible, not subdivisible, that it forms a unit.⁽²⁾

Having admitted the existence of a frequency linked to the piece of energy, let us seek how this frequency manifests itself to the stationary observer mentioned above. The Lorentz-Einstein time transformation tells us that a periodic phenomenon linked to the moving body appears slowed to the stationary observer in the ratio of 1 to √(1 − β²); this is the famous slowing down of clocks. Therefore the frequency observed by the stationary observer will be ¹⁴

¹⁴ Time dilation: In special relativity, a moving clock appears to run slower to a stationary observer. For a clock moving at velocity v, time slows by the factor √(1 − v²/c²). At ordinary speeds this effect is negligible, but it matters for GPS satellites and particle accelerators. De Broglie uses time dilation here to show how the internal “vibration” frequency of a moving particle appears different to a stationary observer. This seeming contradiction (the frequency both slows down and speeds up depending on how you look at it) is resolved by the phase wave concept.

\[\nu_1 = \nu_0 \sqrt{1 - \beta^2} = \frac{m_0 c^2}{h} \sqrt{1 - \beta^2}\]

On the other hand, since the energy of the moving body for the same observer is equal to m₀c²/√(1 − β²), the corresponding frequency according to the quantum relation is ν = (1/h) × m₀c²/√(1 − β²). The two frequencies ν₁ and ν are essentially different since the factor √(1 − β²) does not appear in them in the same way. There is here a difficulty that puzzled me for a long time; I managed to resolve it by proving the following theorem that I shall call the theorem of the harmony of phases:

“The periodic phenomenon linked to the moving body and whose frequency for the stationary observer is equal to ν₁ = (m₀c²/h)√(1 − β²) appears to him constantly in phase with a wave of frequency ν = (1/h) × m₀c²/√(1 − β²) propagating in the same direction as the moving body with velocity V = c/β.” ¹⁵

¹⁵ Theorem of harmony of phases: This theorem resolves a paradox. Due to time dilation, a moving particle’s internal oscillation appears slowed down to a stationary observer. But the quantum relation E = hν says a faster particle (higher energy) should have a higher frequency! De Broglie resolves this by showing that the particle remains “in step” with an associated wave traveling at velocity V = c/β (faster than light). The particle “surfs” this wave, always at a crest. This is not a physical wave carrying energy (nothing travels faster than light), but rather a phase wave describing the spatial distribution of the quantum phase. This concept is the precursor to the quantum mechanical wavefunction Ψ.

The proof is very simple. Suppose that at time t = 0, there is agreement of phase between the periodic phenomenon linked to the moving body and the wave defined above. At time t, the moving body has traversed since the initial instant a distance equal to x = *β**ct* and the phase of the periodic phenomenon has varied by ν₁t = (m₀c²/h)√(1 − β²) × (x/βc). The phase of the portion of wave that covers the moving body has varied by:

\[\nu \left( t - \frac{\beta x}{c} \right) = \frac{m_0 c^2}{h} \cdot \frac{1}{\sqrt{1 - \beta^2}} \left( \frac{x}{\beta c} - \frac{\beta x}{c} \right) = \frac{m_0 c^2}{h} \sqrt{1 - \beta^2} \cdot \frac{x}{\beta c}\]

As we announced, the agreement of phases persists.

It is possible to give another proof of this theorem, identical at bottom, but perhaps more striking. If t₀ represents the time for an observer attached to the moving body (proper time of the moving body), the Lorentz transformation gives:

\[t_0 = \frac{1}{\sqrt{1 - \beta^2}} \left( t - \frac{\beta x}{c} \right)\]

The periodic phenomenon that we imagine is represented for the same observer by a sinusoidal function of ν₀t₀. For the stationary observer, it is represented by the same sinusoidal function of ν₀/√(1 − β²) × (t − *β**x/c), a function which represents a wave of frequency ν₀/√(1 − β²) propagating with velocity c/β* in the same direction as the moving body.

It is now indispensable to reflect on the nature of the wave whose existence we have just conceived. The fact that its velocity V = c/β is necessarily greater than c (β being always less than 1, otherwise the mass would be infinite or imaginary) shows us that it cannot be a question of a wave transporting energy. Our theorem moreover tells us that it represents the distribution in space of the phases of a phenomenon; it is a “phase wave.” ¹⁶

¹⁶ Phase wave: De Broglie’s phase wave travels faster than light, which seems impossible. But this wave doesn’t carry energy or information. Think of it like the “wave” that appears to move along a line of falling dominoes, or the spot of a laser pointer swept across the Moon. The phase wave describes where the quantum phase has a particular value at each instant. What actually moves with the particle is not a single wave but a wave packet, whose group velocity equals the particle velocity. The phase wave concept evolved into the quantum mechanical wavefunction Ψ, where |Ψ|² gives the probability of finding the particle.

To clarify this last point well, we shall present a mechanical comparison, somewhat crude, but which speaks to the imagination. Let us suppose a horizontal circular platform of very large radius; to this platform are suspended identical systems formed of a spiral spring to which a weight is attached. The number of systems thus suspended per unit area of the platform, their density, decreases very rapidly as one moves away from the center of the platform in such a way that there is a condensation of systems around this center. All the spring-weight systems being identical all have the same period; let us make them oscillate with the same amplitude and the same phase. The surface passing through the centers of gravity of all the weights will be a plane that will rise and fall with an alternating motion. The ensemble thus obtained presents a very crude analogy with the isolated piece of energy as we conceive it.

The description we have just given is appropriate for an observer attached to the platform. If another observer sees the platform moving with a uniform translational motion with velocity v = βc, each weight will appear to him as a small clock undergoing Einstein’s slowing; moreover, the platform and the distribution of oscillating systems will no longer be isotropic around the center because of the Lorentz contraction. But the fundamental fact for us (paragraph 3 will make us understand it better) is the phase difference of the motions of the different weights. If, at a given moment of his time, our stationary observer considers the geometric locus of the centers of gravity of the various weights, he obtains a cylindrical surface in the horizontal sense whose vertical sections parallel to the velocity of the platform are sinusoids. It corresponds in the particular case considered to our phase wave; according to the general theorem, this surface is animated by a velocity V parallel to that of the platform, and the vibration frequency of a point of fixed abscissa that constantly rests on it is equal to the proper oscillation frequency of the springs multiplied by 1/√(1 − β²).

One sees clearly from this example (and this is our excuse for having insisted so long) how the phase wave corresponds to the transport of phase and not at all to that of energy.

The preceding results seem to us to be of extreme importance because, with the help of a hypothesis strongly suggested by the very notion of quantum, they establish a link between the motion of a moving body and the propagation of a wave, and thus allow us to foresee the possibility of a synthesis of the antagonistic theories on the nature of radiation. Already we can note that the rectilinear propagation of the phase wave is linked to the rectilinear motion of the moving body; Fermat’s principle applied to the phase wave determines the form of its rays, which are straight lines, while Maupertuis’s principle applied to the moving body determines its rectilinear trajectory, which is one of the rays of the wave. In Chapter II, we shall attempt to generalize this coincidence.

II. Phase Velocity and Group Velocity

We must now prove an important relation existing between the velocity of the moving body and that of the phase wave. If waves of very close frequencies propagate in the same direction Ox with velocities V that we shall call phase propagation velocities, these waves will give by their superposition beat phenomena if the velocity V varies with frequency ν. These phenomena have been studied notably by Lord Rayleigh in the case of dispersive media.

Let us consider two waves of neighboring frequencies ν and ν′ = ν + δν and velocities V and V′ = V + (dV/dν)δν; their superposition is expressed analytically by the following equation obtained by neglecting δν compared to ν in the second term:

\[\sin 2\pi\left(\nu t - \frac{\nu x}{V} + \varphi\right) + \sin 2\pi \left(\nu' t - \frac{\nu' x}{V'} + \varphi'\right)\]

\[= 2 \sin 2\pi \left( \nu t - \frac{\nu x}{V} + \psi \right) \cos 2\pi \left[ \frac{\delta\nu}{2} t - x \frac{d\left(\frac{\nu}{V}\right)}{d\nu} \frac{\delta\nu}{2} + \psi' \right]\]

We thus have a resultant sinusoidal wave whose amplitude is modulated at frequency δν, for the sign of the cosine matters little. This is a well-known result. If we denote by U the propagation velocity of the beat, or group velocity, we find:

¹⁷

¹⁷ Group velocity: A wave packet is made of many waves with slightly different frequencies. Individual wave crests move at the phase velocity V (faster than light for matter waves), but the envelope of the packet moves at the group velocity U. De Broglie proves that U = v, the particle velocity. This is crucial: the group velocity is where the “stuff” is. You can see this with water waves: individual ripples seem to pass through a wave packet, but the packet itself moves at a different speed. For electrons, the group velocity (= particle velocity) is what carries the energy and momentum, while the phase velocity is just a mathematical feature of the wave description.

\[\frac{1}{U} = \frac{d\left(\frac{\nu}{V}\right)}{d\nu}\]

Let us return to phase waves. If we attribute to the moving body a velocity v = βc without giving β a completely determined value but only requiring it to be between β and β + δβ, the frequencies of the corresponding waves fill a small interval ν, ν + δν.

We shall establish the following theorem which will be useful to us later: “The group velocity of phase waves is equal to the velocity of the moving body.” Indeed, this group velocity is determined by the formula given above in which V and ν can be considered as functions of β since we have:

\[V = \frac{c}{\beta} \qquad \nu = \frac{1}{h} \frac{m_0 c^2}{\sqrt{1 - \beta^2}}\]

One can write:

\[U = \frac{\frac{d\nu}{d\beta}}{\frac{d\left(\frac{\nu}{V}\right)}{d\beta}}\]

Now

\[\frac{d\nu}{d\beta} = \frac{m_0 c^2}{h} \cdot \frac{\beta}{(1 - \beta^2)^{3/2}}\]

\[\frac{d\left(\frac{\nu}{V}\right)}{d\beta} = \frac{m_0 c}{h} \cdot \frac{d\left(\frac{\beta}{\sqrt{1 - \beta^2}}\right)}{d\beta} = \frac{m_0 c}{h} \frac{1}{(1 - \beta^2)^{3/2}}\]

Therefore:

\[U = \beta c = v\]

The group velocity of phase waves is indeed equal to the velocity of the moving body. This result calls for a remark: in the wave theory of dispersion, if we except the absorption zones, the velocity of energy is equal to the group velocity. Here, although placed from a quite different point of view, we find an analogous result, for the velocity of the moving body is nothing other than the velocity of the displacement of energy.

III. The Phase Wave in Space-Time

Minkowski was the first to show that a simple geometric representation of the relations of space and time introduced by Einstein could be obtained by considering a 4-dimensional Euclidean manifold called the Universe or Space-time. For this purpose, he took 3 rectangular coordinate axes of space and a fourth axis normal to the first 3 on which the times multiplied by c√−1 were plotted. Today the real quantity ct is more willingly plotted on the fourth axis, but then the planes passing through this axis and normal to space have a pseudo-Euclidean hyperbolic geometry whose fundamental invariant is c²dt² − dx² − dy² − dz².

Let us therefore consider space-time referred to the 4 rectangular axes of the so-called “stationary” observer. We shall take the x-axis as the rectilinear trajectory of the moving body, and we shall represent on our paper the plane otx containing the time axis and the said trajectory. Under these conditions, the world line of the moving body is represented by a straight line inclined at less than 45° to the time axis; this line is moreover the time axis for the observer attached to the moving body. We represent on our figure the 2 time axes intersecting at the origin, which does not restrict generality.

If the velocity of the moving body for the stationary observer is βc, the slope of Ot′ has the value 1/β. The line Ox′, trace on the plane tox of the space of the moving observer at time O, is symmetric to Ot′ with respect to the bisector OD; it is easy to demonstrate this analytically by means of the Lorentz transformation, but it results immediately from the fact that the limiting velocity of energy c has the same value for all reference systems. The slope of Ox′ is therefore β. If the space surrounding the moving body is the seat of a periodic phenomenon, the state of space will become the same again for the moving observer each time a time (1/c)OA = (1/c)AB equal to the proper period T₀ = 1/ν₀ = h/(m₀c²) of the phenomenon has elapsed.

The lines parallel to Ox′ are therefore the traces of these “equiphase spaces” of the moving observer on the plane xot. The points… a′, o, a… represent in projection their intersections with the space of the stationary observer at instant 0; these intersections of 2 three-dimensional spaces are two-dimensional surfaces and even planes because all the spaces considered here are Euclidean. When time elapses for the stationary observer, the section of space-time which for him is space is represented by a line parallel to ox moving with uniform motion toward increasing t. One easily sees that the equiphase planes… a′, o, a… move in the space of the stationary observer with velocity c/β. Indeed, if the line ox₁ in the figure represents the space of the stationary observer at time t = 1, we have aa₀ = c. The phase which at t = 0 was at a is now at a₁; for the stationary observer, it has therefore moved in his space by the length a₀a₁ in the direction ox during unit time. One can therefore say that its velocity is:

\[V = a_0 a_1 = aa_0 \cot(\widehat{xox'}) = \frac{c}{\beta}\]

The set of equiphase planes constitutes what we have called the phase wave.

It remains to examine the question of frequencies. Let us make a small simplified figure.

Lines 1 and 2 represent two successive equiphase spaces of the attached observer. AB is, as we said, equal to c times the proper period T₀ = h/(m₀c²).

AC, the projection of AB on the axis Ot, is equal to

\[cT_1 = c T_0 \frac{1}{\sqrt{1 - \beta^2}}\]

This results from a simple application of trigonometric relations; however, we shall remark that in applying trigonometry to figures in the plane xot, one must always keep in mind the anisotropy peculiar to this plane. Triangle ABC gives us:

\[\overline{AB}^2 = \overline{AC}^2 - \overline{CB}^2 = \overline{AC}^2 (1 - \tan^2 \widehat{CAB}) = \overline{AC}^2 (1 - \beta^2)\]

\[\overline{AC} = \frac{\overline{AB}}{\sqrt{1 - \beta^2}}\]

Q.E.D.

The frequency 1/T₁ is that which the periodic phenomenon appears to have for the stationary observer who follows it with his eyes in its displacement. It is:

\[\nu_1 = \nu_0 \sqrt{1 - \beta^2} = \frac{m_0 c^2}{h} \sqrt{1 - \beta^2}\]

The period of the waves at a point in space for the stationary observer is given not by (1/c)AC but by (1/c)AD. Let us calculate.

In the small triangle BCD, we find the relation

\[\frac{\overline{CB}}{\overline{DC}} = \frac{1}{\beta} \quad \text{whence} \quad \overline{DC} = \beta\overline{CB} = \beta^2\overline{AC}\]

But AD = AC − DC = AC(1 − β²). The new period T is therefore equal to:

\[T = \frac{1}{c} \overline{AC} (1 - \beta^2) = T_0 \sqrt{1 - \beta^2}\]

and the frequency ν of the waves is expressed by:

\[\nu = \frac{1}{T} = \frac{\nu_0}{\sqrt{1 - \beta^2}} = \frac{m_0 c^2}{h\sqrt{1 - \beta^2}}\]

We thus recover all the results obtained analytically in the first paragraph, but now we see better how they are connected with the general conception of space-time and why the phase shift of the periodic motions occurring at different points in space depends on the way simultaneity is defined by the theory of relativity.

Chapter II: Maupertuis’s Principle and Fermat’s Principle

I. Purpose of This Chapter

In this chapter we wish to try to generalize the results of Chapter I for the case of a moving body whose motion is not rectilinear and uniform. Non-uniform motion presupposes the existence of a force field to which the moving body is subject. In the present state of our knowledge, there seem to be only two kinds of fields: gravitational fields and electromagnetic fields. The generalized theory of relativity interprets the gravitational field as due to a curvature of space-time. In the present thesis, we shall systematically leave aside everything concerning gravitation, intending to return to it in another work. For us, therefore, at present, a force field will be an electromagnetic field and the dynamics of non-uniform motion will be the study of the motion of a body carrying an electric charge in an electromagnetic field.

We must expect to encounter rather great difficulties in this chapter because the theory of relativity, a very sure guide when dealing with uniform motions, is still rather hesitant in its conclusions about non-uniform motion. During Mr. Einstein’s recent stay in Paris, Mr. Painlevé raised amusing objections against relativity; Mr. Langevin was able to dismiss them without difficulty because they all involved accelerations whereas the Lorentz-Einstein transformation applies only to uniform motions. The arguments of the illustrious mathematician have nonetheless proved once more that the application of Einsteinian ideas becomes very delicate as soon as accelerations are involved, and in this respect they are very instructive. The method that allowed us to study the phase wave in Chapter I will no longer be of any use to us here.

The phase wave that accompanies the motion of a moving body, if our conceptions are admitted, has properties that depend on the nature of this moving body since its frequency, for example, is determined by the total energy. It therefore seems natural to suppose that if a force field acts on the motion of a moving body, it will also act on the propagation of its phase wave. Guided by the idea of a profound identity between the principle of least action and that of Fermat, I was led from the beginning of my research on this subject to admit that for a given value of the total energy of the moving body, and hence of the frequency of its phase wave, the dynamically possible trajectories of the one coincided with the possible rays of the other. This led me to a very satisfactory result that will be presented in Chapter III, namely the interpretation of the intra-atomic stability conditions established by Bohr. Unfortunately, rather arbitrary hypotheses were needed concerning the values of the propagation velocities Vof the phase wave at each point of the field. On the contrary, we shall use here a method that seems to us much more general and more satisfactory. We shall study on the one hand the mechanical principle of least action in its Hamiltonian and Maupertuisian forms in classical dynamics and in that of relativity, and on the other hand, from a very general point of view, the propagation of waves and Fermat’s principle. We shall then be led to conceive a synthesis of these two studies, a synthesis that can be debated but whose theoretical elegance is incontestable. We shall at the same time obtain the solution of the problem posed.

II. The Two Principles of Least Action in Classical Dynamics

In classical dynamics, the principle of least action in its Hamiltonian form is stated as follows: ¹⁸

¹⁸ Principle of least action: This is one of the most profound principles in physics. It states that nature “chooses” paths that make the quantity called action (energy × time) stationary. A ball rolling down a hill, a planet orbiting the Sun, even light bending around a massive object, all follow paths that minimize (or make stationary) the action. De Broglie shows that Fermat’s principle for waves (light takes the path of shortest time) is mathematically equivalent to the principle of least action for particles. This deep connection between wave optics and particle mechanics hints that waves and particles might be two aspects of the same reality.

“The equations of dynamics can be deduced from the fact that the integral ∫_t1^t₂ ℒdt taken between the limits of time for given initial and final values of the parameters q_i that determine the state of the system has a stationary value.” By definition, ℒ is called the Lagrange function and is assumed to depend on the variables q_i and q̇_i = dq_i/dt.

We therefore have:

\[\delta \int_{t_1}^{t_2} \mathscr{L} dt = 0\]

From this, by a well-known method of the calculus of variations, we deduce the equations called Lagrange’s equations:

\[\frac{d}{dt}\left(\frac{\partial \mathscr{L}}{\partial \dot{q}_i}\right) = \frac{\partial \mathscr{L}}{\partial q_i}\]

equal in number to that of the variables q_i.

It remains to define the function ℒ. Classical dynamics sets:

\[\mathscr{L} = E_{\text{kin}} - E_{\text{pot}}\]

the difference of the kinetic and potential energies. We shall see later that relativistic dynamics uses a different value of ℒ.

Let us now pass to the Maupertuisian form of the principle of least action. For this, let us first note that Lagrange’s equations in the general form given above admit a first integral called “energy of the system” and equal to:

\[W = -\mathscr{L} + \sum_i \frac{\partial \mathscr{L}}{\partial \dot{q}_i} \dot{q}_i\]

provided, however, that the function ℒ does not depend explicitly on time, which we shall always assume in what follows. For we then have:

\[\frac{dW}{dt} = -\sum_i \frac{\partial \mathscr{L}}{\partial q_i}\dot{q}_i - \sum_i \frac{\partial \mathscr{L}}{\partial \dot{q}_i}\ddot{q}_i + \sum_i \frac{\partial \mathscr{L}}{\partial q_i}\dot{q}_i + \sum_i \frac{d}{dt}\left(\frac{\partial \mathscr{L}}{\partial \dot{q}_i}\right)\dot{q}_i\]

\[= \sum_i \dot{q}_i \left[\frac{d}{dt}\left(\frac{\partial \mathscr{L}}{\partial \dot{q}_i}\right) - \frac{\partial \mathscr{L}}{\partial q_i}\right]\]

a quantity that is zero by Lagrange’s equations. Therefore:

\[W = \text{Constant}\]

Let us now apply the Hamiltonian principle to all the “varied” trajectories that lead from the given initial state A to the given final state B and that correspond to a determined value of the energy W. We can write, since W, t₁, and t₂ are constants:

\[\delta \int_{t_1}^{t_2} \mathscr{L} dt = \delta \int_{t_1}^{t_2} (\mathscr{L} + W) dt = 0\]

or else:

\[\delta \int_{t_1}^{t_2} \sum_i \frac{\partial \mathscr{L}}{\partial \dot{q}_i} \dot{q}_i \, dt = \delta \int_A^B \sum_i \frac{\partial \mathscr{L}}{\partial \dot{q}_i} dq_i = 0\]

the last integral being extended to all values of the q_i between those that define states A and B, so that time is eliminated; there is therefore no longer any need in the new form obtained to impose any restriction relative to the limits of time. On the other hand, the varied trajectories must all correspond to a single value W of the energy.

Let us set, following the classical notation of canonical equations: p_i = ∂ℒ/∂q̇_i. The p_i are the momenta conjugate to the variables q_i. The Maupertuisian principle is written: ¹⁹

¹⁹ Maupertuis’s principle: While Hamilton’s principle varies paths with fixed time endpoints, Maupertuis’s principle varies paths with fixed energy, eliminating time from the equations entirely.

\[\delta \int_A^B \sum_i p_i dq_i = 0\]

In classical dynamics where ℒ = E_kin − E_pot, E_pot is independent of the q̇_i and E_kin is a homogeneous quadratic function of them. By virtue of Euler’s theorem:

\[\sum_i p_i dq_i = \sum_i p_i \dot{q}_i dt = 2E_{\text{kin}} dt\]

For the material point, E_kin = (1/2)mv² and the principle of least action takes its oldest known form:

\[\delta \int_A^B mv \, dl = 0\]

dl being an element of trajectory.

III. The Two Principles of Least Action in Electron Dynamics

We shall now take up the question again for electron dynamics from the relativistic point of view. Here the word “electron” must be taken in the general sense of a material point carrying an electric charge. We shall suppose that the electron placed outside any field possesses a proper mass m₀; its electric charge is denoted by e.

We shall again consider space-time; the space coordinates will be called x¹, x², and x³, the time coordinate will be x⁴ = ct. The fundamental invariant “length element” is defined by:

\[ds = \sqrt{(dx^4)^2 - (dx^1)^2 - (dx^2)^2 - (dx^3)^2}\]

In this paragraph and in the following, we shall constantly use certain notations of tensor calculus.

A world line has at each point a tangent defined in direction by the vector “universe velocity” of unit length whose contravariant components are given by the relation:

\[u^i = \frac{dx^i}{ds} \qquad (i = 1, 2, 3, 4)\]

One verifies at once that uⁱu_i = 1.

Let a moving body describe the world line; when it passes through the point considered, it has a velocity v = βc with components v_x, v_y, v_z. The components of the universe velocity are:

\[u_1 = -u^1 = -\frac{v_x}{c\sqrt{1-\beta^2}} \qquad u_2 = -u^2 = -\frac{v_y}{c\sqrt{1-\beta^2}}\]

\[u_3 = -u^3 = -\frac{v_z}{c\sqrt{1-\beta^2}} \qquad u_4 = u^4 = \frac{1}{\sqrt{1-\beta^2}}\]

To define an electromagnetic field, we must introduce a second universe vector whose components are expressed in terms of the vector potential a⃗ and the scalar potential Ψ by the relations:

\[\varphi_1 = -\varphi^1 = -a_x; \quad \varphi_2 = -\varphi^2 = -a_y; \quad \varphi_3 = -\varphi^3 = -a_z\]

\[\varphi_4 = \varphi^4 = \frac{1}{c}\Psi\]

Let us now consider two points P and Q of space-time corresponding to given values of the space coordinates and time. We can consider a curvilinear integral taken along a world line going from P to Q; naturally the function to be integrated must be invariant. Let:

\[\int_P^Q (-m_0 c - e\varphi_i u^i) ds = \int_P^Q (-m_0 c u^i - e\varphi_i) u^i ds\]

be this integral. The Hamilton principle affirms that if the world line of a moving body passes through P and Q, it has a form such that the integral defined above has a stationary value.

Let us define a third universe vector by the relation:

\[J_i = m_0 c u_i + e\varphi_i \qquad (i = 1, 2, 3, 4)\]

The statement of least action becomes:

\[\delta \int_P^Q (J_1 dx^1 + J_2 dx^2 + J_3 dx^3 + J_4 dx^4) = \delta \int_P^Q J_i dx^i = 0\]

We shall give a physical meaning to the universe vector J a little further on.

For the moment, let us return to the usual form of the dynamical equations by replacing in the first form of the action integral ds by cdt√(1 − β²). We thus obtain:

\[\delta \int_{t_1}^{t_2} [-m_0 c^2 \sqrt{1-\beta^2} - ec\varphi_4 - e(\varphi_1 v_x + \varphi_2 v_y + \varphi_3 v_z)] dt = 0\]

t₁ and t₂ corresponding to points P and Q of space-time.

If there exists a purely electrostatic field, the quantities φ₁φ₂φ₃ are zero and the Lagrange function takes the often-used form:

\[\mathscr{L} = -m_0 c^2 \sqrt{1-\beta^2} - e\Psi\]

In all cases, the Hamilton principle always having the form δ∫_t1^t₂ ℒdt = 0, one is always led to the Lagrange equations:

\[\frac{d}{dt}\left(\frac{\partial \mathscr{L}}{\partial \dot{q}_i}\right) = \frac{\partial \mathscr{L}}{\partial q_i} \qquad (i = 1, 2, 3)\]

In all cases where the potentials do not depend on time, we recover the conservation of energy:

\[W = -\mathscr{L} + \sum_i p_i q_i = \text{Const.} \qquad p_i = \frac{\partial \mathscr{L}}{\partial \dot{q}_i} \quad (i = 1, 2, 3)\]

Following exactly the same procedure as above, we obtain Maupertuis’s principle:

\[\delta \int_A^B \sum_i p_i dq_i = 0\]

A and B being the two points of space that correspond for the reference system used to points P and Q of space-time.

The quantities p₁p₂p₃, equal to the partial derivatives of the function ℒ with respect to the corresponding velocities, can serve to define a vector p⃗ that we shall call the “momentum vector.” If there is no magnetic field (whether or not there is an electric field), the rectangular components of this vector are:

\[p_x = \frac{m_0 v_x}{\sqrt{1-\beta^2}} \qquad p_y = \frac{m_0 v_y}{\sqrt{1-\beta^2}} \qquad p_z = \frac{m_0 v_z}{\sqrt{1-\beta^2}}\]

It is therefore identical to the quantity of motion (momentum) and the Maupertuisian action integral has the simple form proposed by Maupertuis himself with the sole difference that the mass now varies with velocity according to Lorentz’s law.

If there is a magnetic field, we find for the components of the momentum vector the expressions:

\[p_x = \frac{m_0 v_x}{\sqrt{1-\beta^2}} + ea_x \qquad p_y = \frac{m_0 v_y}{\sqrt{1-\beta^2}} + ea_y \qquad p_z = \frac{m_0 v_z}{\sqrt{1-\beta^2}} + ea_z\]

There is no longer identity between the vector p⃗ and the momentum; consequently, the expression of the action integral becomes more complicated.

Let us consider a moving body placed in a field and whose total energy is given; at every point of the field that the moving body can reach, its speed is given by the energy equation, but a priori its direction can be arbitrary. The expression of p_x, p_y, and p_z shows that the momentum vector has the same magnitude at a point of an electrostatic field whatever the direction considered. This is no longer the case if there is a magnetic field: the magnitude of the vector p⃗ then depends on the angle between the chosen direction and the vector potential, as one sees by forming the expression p_x² + p_y² + p_z². This remark will be useful to us later.

To conclude this paragraph, we shall return to the physical meaning of the universe vector J on which the Hamiltonian integral depends. We defined it by the expression:

\[J_i = m_0 c u_i + e\varphi_i \qquad (i = 1, 2, 3, 4)\]

With the help of the values u_i and φ_i we find:

\[J_1 = -p_x \qquad J_2 = -p_y \qquad J_3 = -p_z \qquad J_4 = \frac{W}{c}\]

The contravariant components will be:

\[J^1 = p_x \qquad J^2 = p_y \qquad J^3 = p_z \qquad J^4 = \frac{W}{c}\]

We are therefore dealing with the famous “world momentum” vector which synthesizes energy and momentum.

From:

\[\delta \int_P^Q J_i dx^i = 0 \qquad (i = 1, 2, 3, 4)\]

we can immediately derive, if J₄ is constant:

\[\delta \int_A^B J_i dx^i = 0 \qquad (i = 1, 2, 3)\]

This is the most condensed way of passing from one of the statements of stationary action to the other.

IV. Propagation of Waves; Fermat’s Principle

We shall study the propagation of the phase of a sinusoidal phenomenon by a method parallel to that of the two preceding paragraphs. For this, we shall place ourselves at a very general point of view and again we shall have to consider space-time.

Consider the function sin φ in which the differential of φ is assumed to depend on the space and time variables xⁱ. There exists in space-time an infinity of world lines along which the function φ is constant.

The theory of undulations as it results notably from the work of Huygens and Fresnel teaches us to distinguish among these lines certain of them whose projections onto the space of an observer are for him the “rays” in the usual sense of optics.

Let P and Q be, as before, two points of space-time. If a universe ray passes through these two points, what will be the law that determines its form?

We shall consider the curvilinear integral ∫dφ and we shall take as the principle determining the universe ray the statement of Hamiltonian form: ²⁰

²⁰ Fermat’s principle: You may have seen Fermat’s principle in optics: light travels between two points along the path that takes the least time. This explains refraction (bending of light at interfaces) and reflection. De Broglie demonstrates something profound: if you apply Fermat’s principle to his phase waves and the principle of least action to particles, you get exactly the same trajectories. The ray paths of the wave are identical to the particle paths. This mathematical identity suggests that particles and waves are not opposites but deeply related descriptions of the same physical reality.

\[\delta \int_P^Q d\varphi = 0\]

The integral must indeed be stationary; otherwise, perturbations having left a certain point of space in phase concordance and crossing at another point after having followed slightly different paths would present different phases there.

The phase φ is an invariant; if therefore we set:

\[d\varphi = 2\pi(O_1 dx^1 + O_2 dx^2 + O_3 dx^3 + O_4 dx^4) = 2\pi O_i dx^i\]

the quantities O_i, generally functions of the xⁱ, will be the covariant components of a universe vector, the universe wave vector. If l is the direction of the ray in the ordinary sense, one is usually led to consider for dφ the form:

\[d\varphi = 2\pi \left(\nu dt - \frac{\nu}{V} dl\right)\]

ν is called frequency and V propagation velocity. One can then set:

\[O_1 = -\frac{\nu}{V}\cos(x,l), \quad O_2 = -\frac{\nu}{V}\cos(y,l),\]

\[O_3 = -\frac{\nu}{V}\cos(z,l), \quad O_4 = \frac{\nu}{c}\]

The universe wave vector thus decomposes into a time component proportional to the frequency and a space vector n⃗ carried in the direction of propagation and having length ν/V. We shall call it the “wave number” vector because it is equal to the inverse of the wavelength. If the frequency ν is constant, we are led to pass from the Hamiltonian form:

\[\delta \int_P^Q O_i dx^i = 0\]

to the Maupertuisian form:

\[\delta \int_A^B O_1 dx^1 + O_2 dx^2 + O_3 dx^3 = 0\]

where A and B are the points of space corresponding to P and Q.

Replacing O₁, O₂, and O₃ by their values, we get:

\[\delta \int_A^B \frac{\nu \, dl}{V} = 0\]

This Maupertuisian statement constitutes Fermat’s principle.

Just as in the preceding paragraph it sufficed to find the trajectory of a moving body of given total energy passing through two given points to know the distribution in the field of the vector p⃗, so here to find the ray of a wave of known frequency passing through two given points, it suffices to know the distribution in space of the wave number vector which determines at each point and for each direction the propagation velocity.

V. Extension of the Quantum Relation

We have reached the culminating point of this chapter. We had posed at its beginning the following question: “When a moving body moves in a force field with non-uniform motion, how does its phase wave propagate?” Instead of seeking by trial and error, as I had first done, to determine the propagation velocity at each point and for each direction, I am going to make an extension of the quantum relation that is perhaps somewhat hypothetical but whose profound agreement with the spirit of the theory of relativity is indisputable.

We have been constantly led to set hν = w, w being the total energy of the moving body and ν the frequency of its phase wave. On the other hand, the preceding paragraphs have taught us to define two universe vectors J and O which play perfectly symmetric roles in the study of the motion of a moving body and in that of the propagation of a wave.

By bringing in these vectors, the relation hν = w is written:

\[O_4 = \frac{1}{h} J_4\]

The fact that two vectors have one component equal does not prove that this is so for the others. However, by a quite natural generalization, we shall set:

²¹

²¹ De Broglie’s extension of the quantum relation: This is where λ = h/p comes from! De Broglie already established that the time component gives E = hν. Now he proposes that all components of the wave four-vector are proportional to the momentum four-vector. The space components give p = h/λ, or equivalently λ = h/p. This is the de Broglie wavelength you memorize in Gen Chem. For an electron (mass 9.11 × 10⁻³¹ kg) moving at 1% of light speed (v ≈ 3 × 10⁶ m/s), we get λ = (6.63 × 10⁻³⁴)/[(9.11 × 10⁻³¹)(3 × 10⁶)] ≈ 0.24 nm, comparable to atomic dimensions. This is why electron waves matter for atoms!

\[O_i = \frac{1}{h} J_i \qquad (i = 1, 2, 3, 4)\]

The variation dφ relative to an infinitely small portion of the phase wave has the value:

\[d\varphi = 2\pi O_i dx^i = \frac{2\pi}{h} J_i dx^i\]

Fermat’s principle therefore becomes:

\[\delta \int_A^B \sum_1^3 J_i dx^i = \delta \int_A^B \sum_1^3 p_i dx^i = 0\]

We thus arrive at the following statement:

“Fermat’s principle applied to the phase wave is identical to Maupertuis’s principle applied to the moving body; the dynamically possible trajectories of the moving body are identical to the possible rays of the wave.”

We think that this idea of a profound relation between the two great principles of geometrical optics and dynamics could be a precious guide for achieving the synthesis of waves and quanta.

The hypothesis of the proportionality of the vectors J and O is a sort of extension of the quantum relation whose current statement is manifestly insufficient since it involves energy without speaking of its inseparable companion, momentum. The new statement is much more satisfactory because it is expressed by the equality of two universe vectors.

VI. Particular Cases; Discussions

The general conceptions of the preceding paragraph must now be applied to particular cases in order to clarify their meaning.

a) Let us first consider the rectilinear and uniform motion of a free moving body. The hypotheses made at the beginning of Chapter I allowed us, thanks to the restricted principle of relativity, to make a complete study of this case. Let us see if we can recover the predicted value for the propagation velocity of the phase wave:

\[V = \frac{c}{\beta}\]

Here we must set:

\[\nu = \frac{W}{h} = \frac{m_0 c^2}{h\sqrt{1-\beta^2}}\]

\[\frac{1}{h} \sum_1^3 p_i dq_i = \frac{1}{h} \frac{m_0 \beta^2 c^2}{\sqrt{1-\beta^2}} dt = \frac{1}{h} \frac{m_0 \beta c}{\sqrt{1-\beta^2}} dl = \frac{\nu \, dl}{V}\]

whence V = c/β. We have given an interpretation of this result from the point of view of space-time.

b) Let us consider an electron in an electrostatic field (Bohr atom). We must suppose the phase wave has a frequency ν equal to the quotient by h of the total energy of the moving body, that is:

\[W = \frac{m_0 c^2}{\sqrt{1-\beta^2}} + e\Psi = h\nu\]

The magnetic field being zero, we shall simply have:

\[p_x = \frac{m_0 v_x}{\sqrt{1-\beta^2}}, \text{ etc.}\]

\[\frac{1}{h} \sum_1^3 p_i dq_i = \frac{1}{h} \frac{m_0 \beta c}{\sqrt{1-\beta^2}} dl = \frac{\nu}{V} dl\]

whence

\[V = \frac{\frac{m_0 c^2}{\sqrt{1-\beta^2}} + e\Psi}{\frac{m_0 \beta c}{\sqrt{1-\beta^2}}} = \frac{c}{\beta}\left(1 + \frac{e\Psi \sqrt{1-\beta^2}}{m_0 c^2}\right)\]

\[= \frac{c}{\beta}\left(1 + \frac{e\Psi}{W - e\Psi}\right) = \frac{c}{\beta} \cdot \frac{W}{W - e\Psi}\]

This result calls for several remarks. From a physical point of view, it means that the phase wave of frequency ν = W/h propagates in the electrostatic field with a velocity that varies from one point to another according to the value of the potential. The velocity V depends on Ψ directly through the term (generally small compared to unity) eΨ/(W − eΨ) and indirectly through β which is calculated at each point as a function of W and Ψ.

Moreover, one will note that V is a function of the mass and charge of the moving body. This point may seem strange, but it is in reality less so than it seems. Let us consider an electron whose center C moves with velocity v; in the classical conception, at a point P whose coordinates in a system attached to the electron are known, there is a certain electromagnetic energy forming in some way part of the electron. Suppose that after having traversed a region R where a more or less complex electromagnetic field prevails, the electron is moving with the same velocity v but otherwise directed.

The point P of the system attached to the electron has come to P′ and one can say that the energy originally at P has been transported to P′. The displacement of this energy, even if one knows the fields prevailing in R, can be calculated only if the mass and charge of the electron are given. This indisputable conclusion could for a moment appear strange because we have the inveterate habit of considering mass and charge (as well as momentum and energy) as quantities linked to the center of the electron. Similarly, for the phase wave which, according to us, must be considered as a constituent part of the electron, the propagation in a field must depend on the charge and mass.

Let us now remember the results obtained in the preceding chapter in the case of uniform motion. We had then been led to consider the phase wave as due to the intersections by the present space of the stationary observer of the past, present, and future spaces of the moving observer. We might be tempted here again to recover the value of V given above by studying the successive “phases” of the moving body and by specifying the displacement for the stationary observer of the sections by his space of the equiphase states. Unfortunately, one encounters here very great difficulties. Relativity does not currently teach us how an observer carried along by a non-uniform motion cuts at each instant his space in space-time; it does not seem that there is much reason for this section to be plane as in uniform motion. But if this difficulty were resolved, we would still be embarrassed. Indeed, a moving body in uniform motion must be described in the same way by the observer attached to it, whatever the velocity of the uniform motion with respect to reference axes; this results from the principle that Galilean axes possessing uniform translational motions relative to one another are equivalent. If therefore our moving body in uniform motion is surrounded, for an attached observer, by a periodic phenomenon having everywhere the same phase, it must be the same for all velocities of uniform motion, and this is what justifies our method of Chapter I. But if the motion is not uniform, the description of the moving body made by the attached observer can no longer be the same, and we no longer know at all how he will define the periodic phenomenon and whether he will attribute the same phase to it at every point in space.

Perhaps one could reverse the problem, admit the results obtained in this chapter by entirely different considerations, and seek to deduce from them how the theory of relativity should envisage these questions of non-uniform motion in order to arrive at the same conclusions. We cannot tackle this difficult problem.

c) Let us take the general case of the electron in an electromagnetic field. We still have:

\[h\nu = W = \frac{m_0 c^2}{\sqrt{1 - \beta^2}} + e\Psi\]

Moreover, we have shown above that we must set:

\[p_x = \frac{m_0 v_x}{\sqrt{1 - \beta^2}} + ea_x, \text{ etc.}\]

a_x, a_y, and a_z being the components of the vector potential. Therefore:

\[\frac{1}{h} \sum_1^3 p_i dq^i = \frac{1}{h} \frac{m_0 \beta c}{\sqrt{1 - \beta^2}} dl + \frac{e}{h} a_l dl = \frac{\nu \, dl}{V}\]

One thus finds:

\[V = \frac{\frac{m_0 c^2}{\sqrt{1 - \beta^2}} + e\Psi}{\frac{m_0 \beta c}{\sqrt{1 - \beta^2}} + ea_l} = \frac{c}{\beta} \cdot \frac{W}{W - e\Psi} \cdot \frac{1}{1 + e\frac{a_l}{G}}\]

G being the momentum and a_l the projection of the vector potential on the direction l.

The medium at each point is no longer isotropic. The velocity V varies with the direction considered and the velocity of the moving body v does not have the same direction as the normal to the phase wave defined by the vector p⃗ = hn⃗. The ray no longer coincides with the normal to the wave, a classical conclusion of the optics of anisotropic media.

One may ask what becomes of the theorem on the equality of the velocity v = βc of the moving body and the group velocity of the phase waves.

Let us first note that the velocity V of the phase along the ray is defined by the relation:

\[\frac{1}{h} \sum_1^3 p_i dq^i = \frac{1}{h} \sum_1^3 p_i \frac{dq^i}{dl} dl = \frac{\nu}{V} dl\]

ν/V is not equal to p because here dl and p do not have the same direction.

We can, without loss of generality, take as the x-axis the direction of motion of the moving body at the point considered and call p_x the projection of the vector p⃗ on this direction. We then have the defining equation:

\[\frac{\nu}{V} = \frac{1}{h} p_x\]

The first of the canonical equations provides the equality:

\[\frac{dq}{dt} = v = \beta c = \frac{\partial W}{\partial p_x} = \frac{\partial(h\nu)}{\partial\left(h\frac{\nu}{V}\right)} = U\]

U being the group velocity along the ray.

The result of Chapter I, § 2, is therefore completely general and follows directly from the equations of Hamilton’s first group.

Chapter III: Quantum Stability Conditions of Trajectories ²²

I. The Bohr-Sommerfeld Stability Conditions

In his theory of the atom, Bohr was the first to put forward the idea that, among the closed trajectories that an electron can describe around a positive center, only certain ones are stable, the others being unrealizable in nature or at least so unstable that there is no need to take them into account. Limiting himself to circular trajectories involving a single degree of freedom, Bohr stated the following condition: ²³

²³ Bohr’s quantization condition: This is the original quantum condition Bohr proposed in 1913. It states that only certain “allowed” orbits exist where the angular momentum is quantized in units of h/2π.

“Only those circular trajectories are stable for which the angular momentum is an integer multiple of h/2π, h being Planck’s constant.” This is written:

\[m_0 \omega R^2 = n \frac{h}{2\pi} \qquad (n \text{ integer})\]

or else:

\[\int_0^{2\pi} p_\theta d\theta = nh\]

θ being the azimuth chosen as the Lagrange coordinate q, p_θ the corresponding momentum.

Sommerfeld and Wilson, to extend this statement to cases involving several degrees of freedom, showed that it is generally possible to choose coordinates q_i such that the quantization conditions for the orbits are: ²⁴

²⁴ Sommerfeld-Wilson quantization: While Bohr’s original condition (mvr = nℏ) works for circular orbits, Sommerfeld and Wilson generalized it for elliptical orbits and more complex motions. Their rule: ∮p_idq_i = n_ih (integrate momentum times position around a complete cycle, and the result must be an integer multiple of h). This gives rise to multiple quantum numbers. For hydrogen, this leads to the principal quantum number n (energy), azimuthal quantum number l (angular momentum shape), and magnetic quantum number m_l (orientation). These are the quantum numbers you use to label orbitals: 1s, 2p, 3d, etc.

\[\oint p_i dq_i = n_i h \qquad (n_i \text{ integers})\]

the sign ∮ indicating an integral extended over the entire domain of variation of the coordinate.

In 1917, Einstein gave the quantization condition an invariant form with respect to changes of coordinates.⁽³⁾ We shall state it for the case of closed trajectories; it is then as follows:

\[\oint \sum_1^3 p_i dq_i = nh \qquad (n \text{ integer})\]

the integral being extended over the entire length of the trajectory.

One recognizes the Maupertuisian action integral whose role thus becomes capital in quantum theory. This integral does not depend on the choice of space coordinates according to a known property which expresses the covariant character of the components p_i of the momentum vector. It is defined by Jacobi’s classical method as a complete integral of the partial differential equation:

\[H\left(\frac{\partial S}{\partial q_i}, q_i\right) = W \qquad i = 1, 2, \ldots f\]

a complete integral which contains f arbitrary constants of which one is the energy W. If there is a single degree of freedom, Einstein’s relation fixes the energy W; if there is more than one (and in the most important usual case, that of the motion of the electron in the intra-atomic field, there are a priori 3), one obtains only a relation between W and the integer n; this is what happens for Keplerian ellipses if one neglects the variation of mass with velocity. But if the motion is quasi-periodic, which moreover always occurs because of the said variation, it is possible to find coordinates which oscillate between limiting values (librations) and there exists an infinity of pseudo-periods approximately equal to integer multiples of the libration periods. At the end of each of these pseudo-periods, the moving body has returned to a state as close as desired to the initial state. Einstein’s equation applied to each of these pseudo-periods leads to an infinity of conditions which are compatible only if Sommerfeld’s multiple conditions are satisfied; these being equal in number to the degrees of freedom, all the constants are fixed and no indeterminacy remains.

For the calculation of Sommerfeld’s integrals, Jacobi’s equation and the residue theorem have been successfully used, as well as the conception of angle variables. These questions have been the subject of numerous works in recent years and are summarized in Sommerfeld’s beautiful book Atombau und Spectrallinien (French edition, translation by Bellenot, Blanchard publisher, 1923). We shall not dwell on them here and shall confine ourselves to noting that in the end, the quantization problem reduces entirely in principle to Einstein’s condition for closed orbits. If one succeeds in interpreting this condition, one will at the same time have clarified the entire question of stable trajectories.

II. Interpretation of Einstein’s Condition

The notion of the phase wave will allow us to provide an explanation of Einstein’s condition. It follows from the considerations of Chapter II that the trajectory of the moving body is one of the rays of its phase wave; the latter must run along the trajectory with a constant frequency (since the total energy is constant) and a variable velocity whose value we have learned to calculate. The propagation is therefore analogous to that of a liquid wave in a channel closed upon itself and of variable depth. It is physically evident that, to have a stable regime, the length of the channel must be in resonance with the wave; in other words, the portions of wave that follow each other at a distance equal to an integer multiple of the length l of the channel and which consequently find themselves at the same point of the channel must be in phase. The resonance condition is l = nλ if the wavelength is constant, and ∮ (ν/V)dl = n (integer) in the general case. ²⁵

²⁵ Standing wave condition: This is perhaps the most important insight in the thesis for understanding atomic structure. De Broglie shows that Bohr’s quantization condition (∮p dl = nh) is simply the requirement that the electron’s phase wave form a standing wave around the nucleus. The circumference of the orbit must contain a whole number of wavelengths: 2πr = nλ. If it doesn’t, the wave interferes destructively with itself after each orbit and cancels out. Only orbits where the wave “fits” are stable. This is exactly why electrons can only occupy certain energy levels! It’s like fitting standing waves on a guitar string: only certain wavelengths (frequencies, notes) work. The allowed orbits are those where the electron wave doesn’t destroy itself. This picture evolved directly into the orbital shapes (s, p, d, f) you learn in Gen Chem.

The integral that intervenes here is that of Fermat’s principle; now, we have shown that it should be considered as equal to the Maupertuisian action integral divided by h. The resonance condition is therefore identical to the stability condition required by quantum theory.

This beautiful result, whose demonstration is so immediate once one has admitted the ideas of the preceding chapter, is the best justification we can give for our manner of attacking the problem of quanta.

In the particular case of circular trajectories in the Bohr atom, one obtains m₀ ∮ v dl = 2πRm₀v = nh, or, since v = ωR if ω is the angular velocity,

\[m_0 \omega R^2 = n \frac{h}{2\pi}\]

This is indeed the simple form originally envisaged by Bohr.

We thus see clearly why certain orbits are stable, but we still do not know how the passage from one stable orbit to another takes place. The disturbed regime that accompanies this passage can only be studied with the help of an appropriately modified electromagnetic theory, and we do not yet possess it.

III. Sommerfeld Conditions for Quasi-Periodic Motions

I propose to demonstrate that, if the stability condition for a closed orbit is ∮ Σ₁³ p_idqⁱ = nh, the stability conditions for quasi-periodic motions are necessarily ∮ p_idqⁱ = n_ih (n_i integer, i = 1, 2, 3). The multiple conditions of Sommerfeld will thus also be reduced to the resonance of the phase wave.

We must first note that the electron having finite dimensions, if, as we admit, the stability conditions depend on the reactions exerted on it by its own phase wave, there must be phase agreement between all portions of the wave passing at a distance from the center of the electron less than a determined value, small but finite, of the order for example of its radius (10⁻¹³ cm). Not to admit this proposition would amount to saying: the electron is a geometric point without dimensions and the ray of its phase wave is a line of zero thickness. This is not physically admissible.

Let us now recall a known property of quasi-periodic trajectories. If M is the position of the center of the moving body at a given instant on the trajectory and if one draws from M as center a sphere of radius R arbitrarily chosen, small but finite, it is possible to find an infinity of time intervals such that at the end of each of them the moving body has returned within the sphere of radius R. Moreover, each of these time intervals or “approximate periods” τ can satisfy the relations:

\[\tau = n_1 T_1 + \varepsilon_1 = n_2 T_2 + \varepsilon_2 = n_3 T_3 + \varepsilon_3\]

where T₁, T₂, and T₃ are the periods of variation (libration) of the coordinates q¹, q², and q³. The quantities ε_i can always be made smaller than a certain fixed quantity η, small but finite. The smaller η is chosen, the longer will be the shortest of the periods τ.

Suppose that the radius R is chosen equal to the maximum distance of action of the phase wave on the electron, the distance defined above. Then one can apply to each approximate period τ the phase agreement condition in the form:

\[\int_0^\tau \sum_1^3 p_i dq^i = nh\]

which can also be written:

\[n_1 \int_0^{T_1} p_1 \dot{q}_1 dt + n_2 \int_0^{T_2} p_2 \dot{q}_2 dt + n_3 \int_0^{T_3} p_3 \dot{q}_3 dt + \varepsilon_1(p_1 \dot{q}_1)_\tau + \varepsilon_2(p_2 \dot{q}_2)_\tau + \varepsilon_3(p_3 \dot{q}_3)_\tau = nh\]

But a resonance condition is never rigorously satisfied. If the mathematician requires for resonance that a phase difference be exactly equal to n × 2π, the physicist must be content to write that it is equal to n.2π ± α, α being less than a quantity ε, small but finite, which measures, so to speak, the margin within which the resonance must be considered as physically realized.

The quantities p_i and q_i remain finite during the motion and one can find six quantities P_i and Q̇_i such that one always has:

\[p_i < P_i \qquad \dot{q}_i < \dot{Q}_i \qquad (i = 1, 2, 3)\]

Let us choose the limit η such that η Σ₁³ P_iQ̇_i < εh/2π; we see that in writing the resonance condition for any of the approximate periods, it will be permissible to neglect the terms in ε_i and write:

\[n_1 \int_0^{T_1} p_1 \dot{q}_1 dt + n_2 \int_0^{T_2} p_2 \dot{q}_2 dt + n_3 \int_0^{T_3} p_3 \dot{q}_3 dt = nh\]

In the first member, n₁, n₂, n₃ are known integers; in the second member, n is any integer. We have an infinity of similar equations with different values of n₁, n₂, and n₃. To satisfy them, it is necessary and sufficient that each of the integrals

\[\int_0^{T_i} p_i \dot{q}_i dt = \oint p_i dq_i\]

be equal to an integer multiple of h.

These are indeed the Sommerfeld conditions.

The preceding demonstration appears rigorous. However, there is an objection to examine. The stability conditions can indeed only come into play after a time of the order of the shortest of the time intervals τ, which is already very large; if it were necessary to wait, for example, millions of years for them to intervene, one might as well say that they would never manifest themselves. This objection is not well founded because the periods τ are very large compared to the libration periods T_i, but can be very small compared to our usual scale of time measurement; in the atom, the periods T_i are, in fact, of the order of 10⁻¹⁵ to 10⁻²⁰ seconds.

One can get an idea of the order of magnitude of the approximate periods in the case of the L₂ trajectory of Sommerfeld for hydrogen. The rotation of the perihelion during one libration period of the radius vector is of the order of 10⁻⁵.2π. The shortest of the approximate periods would therefore be of the order of 10⁵ times the period of the radial variable (10⁻¹⁵ seconds), that is, of the order of 10⁻¹⁰ seconds. It therefore seems clear that the stability conditions will come into play in a time inaccessible to our experience and, consequently, that trajectories “without resonance” will appear to us as non-existent.

The principle of the demonstration developed above was borrowed from Léon Brillouin, who wrote in his thesis (p. 351): “For the Maupertuis integral taken over all the approximate periods τ to be an integer multiple of h, it is necessary that each of the integrals relative to each variable and taken over the corresponding period be equal to an integer number of quanta; this is indeed the way Sommerfeld writes his quantum conditions.”

²² Why this chapter matters for Gen Chem: This chapter explains why electrons can only exist in certain orbits (energy levels) in atoms. De Broglie shows that Bohr’s seemingly arbitrary quantization rule is actually a natural consequence of electrons having wave properties. The electron wave must form a standing wave around the nucleus, like a vibrating guitar string. Orbits where the wave doesn’t “fit” (destructively interferes with itself) are forbidden. This physical picture evolved directly into the quantum mechanical orbitals (1s, 2s, 2p, 3d, etc.) that you use to describe electron configurations. The number of nodes in these standing wave patterns gives rise to the quantum numbers n, l, and m_l.

²⁶ The real hydrogen atom: Chapter III treated the electron as orbiting a fixed nucleus. But in reality, both the nucleus (proton) and electron orbit their common center of mass. This chapter extends the analysis to this more realistic case. The correction is small (about 0.05%) because the proton is ~1836 times heavier than the electron, but it’s detectable in precision spectroscopy. This is why the Rydberg constant for hydrogen (R_H) differs slightly from the Rydberg constant for an infinitely heavy nucleus (R_∞). De Broglie shows that both the electron wave and the nucleus wave must satisfy standing wave conditions.

Chapter IV: Quantization of Simultaneous Motions of Two Electric Centers ²⁶

I. Difficulties Raised by This Problem

In the preceding chapters, we have constantly envisaged an “isolated piece of energy.” This expression is clear when dealing with an electric corpuscle (proton or electron) far from any other electrified body. But if electrified centers are in interaction, the concept of an isolated piece of energy becomes less clear. There is here a difficulty which is in no way peculiar to the theory contained in the present work and which is not elucidated in the current state of relativistic dynamics.

To understand this difficulty well, let us consider a proton (hydrogen nucleus) of proper mass M₀ and an electron of proper mass m₀. If these two entities are very far from each other such that their interaction is negligible, the principle of the inertia of energy applies without difficulties: the proton possesses the internal energy M₀c² and the electron m₀c². The total energy is therefore (M₀ + m₀)c². But if the two centers are close enough that their mutual potential energy −P (< 0) must be taken into account, how will the idea of inertia of energy be expressed? The total energy being obviously (M₀ + m₀)c² − P, can one admit that the proton always has a proper mass M₀ and the electron a proper mass m₀? Or should one instead share the potential energy between the two constituents of the system, attribute to the electron a proper mass m₀ − αP/c² and to the proton a proper mass M₀ − (1 − α)P/c²? In this case, what is the value of α and how does this quantity depend on M₀ and m₀?

In the theories of the Bohr and Sommerfeld atom, one admits that the electron always has the proper mass m₀ whatever its position in the electrostatic field of the nucleus. The potential energy always being much smaller than the internal energy m₀c², this hypothesis is approximately exact, but nothing says that it is rigorous. One can easily calculate the order of magnitude of the maximum correction (corresponding to n = 1) that would have to be made to the value of the Rydberg constant for the different terms of the Balmer series if one adopted the inverse hypothesis. One finds δR/R = 10⁻⁵. This correction would therefore be much smaller than the difference between the Rydberg constants for hydrogen and for helium (1/2000), a difference which Bohr has remarkably accounted for by consideration of the nuclear motion. However, given the extreme precision of spectroscopic measurements, it is perhaps permissible to think that the variation of the Rydberg constant due to the variation of the proper mass of the electron as a function of its potential energy could be detected if it exists.

II. Nuclear Motion in the Hydrogen Atom

A question closely related to the preceding one is that of the method to be used to apply the quantum conditions to an ensemble of electric centers in relative motion. The simplest case is that of the motion of the electron in the hydrogen atom when account is taken of the simultaneous displacements of the nucleus. Bohr was able to treat this problem by relying on the following theorem of rational mechanics: “If the motion of the electron is referred to axes of fixed directions attached to the nucleus, this motion is the same as if these axes were Galilean and if the electron possessed a mass μ₀ = m₀M₀/(m₀ + M₀).” ²⁷

²⁷ Reduced mass: When both the nucleus and electron move around their common center of mass, the problem simplifies if we use the “reduced mass” μ = mM/(m + M). This is a standard technique in classical mechanics for two-body problems.

In the system of axes attached to the nucleus, the electrostatic field acting on the electron can be considered as constant at every point of space and one is thus brought back to the problem without nuclear motion thanks to the substitution of the fictitious mass μ₀ for the real mass m₀. In Chapter II of the present work, we established a general parallelism between the fundamental quantities of dynamics and those of wave theory; the theorem stated above therefore determines what values must be attributed to the frequency of the electronic phase wave and to its velocity in the system attached to the nucleus, a system which is not Galilean. Thanks to this artifice, the quantum stability conditions can be considered in this case too as being interpretable by the resonance of the phase wave.

We shall make this precise by considering the case where nucleus and electron describe circular orbits around their common center of gravity. The plane of these orbits will be taken as the plane of coordinates with indices 1 and 2 in both systems. The space coordinates in the Galilean system attached to the center of gravity will be x¹, x², and x³; those of the system attached to the nucleus will be y¹, y², and y³. Finally, we shall have x⁴ = y⁴ = ct.

Let us call ω the angular velocity of rotation of the line NE around the point G.

Let us set by definition:

\[\eta = \frac{M_0}{m_0 + M_0}\]

The formulas allowing passage from one system of axes to the other are as follows:

\[y^1 = x^1 + R \cos \omega t \qquad y^2 = x^2 + R \sin \omega t \qquad y^3 = x^3 \qquad y^4 = x^4\]

From this one deduces:

\[ds = (dx^4)^2 - (dx^1)^2 - (dx^2)^2 - (dx^3)^2\]

\[= \left(1 - \frac{\omega^2 R^2}{c^2}\right)(dy^4)^2 - (dy^1)^2 - (dy^2)^2 - (dy^3)^2 - 2\frac{\omega R}{c} \sin \omega t \, dy^1 dy^4 + 2\frac{\omega R}{c} \cos \omega t \, dy^2 dy^4\]

The components of the momentum four-vector are defined by the relations:

\[u^i = \frac{dy^i}{ds} \qquad p_i = m_0 c u_i + e\varphi_i = m_0 c g_{ij} u^j + e\varphi_i\]

One finds easily:

\[p_1 = \frac{m_0}{\sqrt{1 - \eta^2 \beta^2}} \left[\frac{dy^1}{dt} + \omega R \sin \omega t\right]\]

\[p_2 = \frac{m_0}{\sqrt{1 - \eta^2 \beta^2}} \left[\frac{dy^2}{dt} - \omega R \cos \omega t\right] \qquad p_3 = 0\]

The resonance of the phase wave is expressed according to the general ideas of Chapter II by the relation:

\[\left| \oint \frac{1}{h} (p_1 dy^1 + p_2 dy^2) \right| = n \qquad (n \text{ integer})\]

the integral being extended to the circular trajectory of radius R + r described by the electron around the nucleus.

Since we have:

\[\frac{dy^1}{dt} = -\omega(R + r) \sin \omega t \qquad \frac{dy^2}{dt} = \omega(R + r) \cos \omega t\]

it comes:

\[\frac{1}{h} \oint (p_1 dy^1 + p_2 dy^2) = \frac{1}{h} \oint \frac{m_0}{\sqrt{1 - \eta^2 \beta^2}} (v \, dl - \omega R v \, dt)\]

denoting by v the velocity of the electron relative to the axes y and by dl the length element of its trajectory,

\[v = \omega(R + r) = \frac{dl}{dt}\]

Finally the resonance condition becomes:

\[\frac{m_0}{\sqrt{1 - \eta^2 \beta^2}} \omega(R + r) \left(1 - \frac{\omega R}{v}\right) \cdot 2\pi(R + r) = nh\]

or, assuming with classical mechanics that β² is negligible compared to unity,

\[2\pi m_0 \frac{M_0}{m_0 + M_0} \omega(R + r)^2 = nh\]

This is indeed Bohr’s formula which is deduced from the theorem stated above and which can therefore here too be regarded as a resonance condition for the electronic wave written in the system attached to the nucleus of the atom.

III. The Two Phase Waves of the Nucleus and the Electron

In what precedes, the introduction of axes attached to the nucleus has allowed us in a way to eliminate the motion of the latter and to consider the displacement of the electron in a constant electrostatic field; we have thus been brought back to the problem treated in Chapter II.

But if we pass to other axes attached for example to the center of gravity, the nucleus and the electron will both describe closed trajectories and the ideas that have guided us until now must necessarily lead us to conceive the existence of two phase waves: that of the electron and that of the nucleus; we must examine how the resonance conditions of these two waves should be expressed and why they are compatible.

Let us first consider the phase wave of the electron. In the system attached to the nucleus, the resonance condition for this wave is:

\[\oint p_1 dy^1 + p_2 dy^2 = 2\pi \frac{m_0 M_0}{m_0 + M_0} \omega(R + r)^2 = nh\]

the integral being taken at constant time along the circle of center N and radius R + r, the relative trajectory of the moving body and ray of its wave. If we pass to axes attached to point G, the relative trajectory becomes a circle of center G and radius r; the ray of the phase wave passing through E is at each instant the circle of center N and radius R + r, but this circle is mobile because its center rotates with uniform motion around the origin of coordinates. The resonance condition of the electronic wave at a given instant is not modified; it is always written:

\[2\pi \frac{m_0 M_0}{m_0 + M_0} \omega(R + r)^2 = nh\]

Let us pass to the wave of the nucleus. In all that precedes, nucleus and electron play a perfectly symmetric role and one must obtain the resonance condition by interchanging M₀ and m₀, and R and r. One therefore falls back on the same formula.

In summary, we see that Bohr’s condition can be interpreted as the expression of the resonance of each of the waves present. The stability conditions for the motions of the nucleus and the electron considered separately are compatible because they are identical.

It is instructive to draw in the system of axes attached to the center of gravity the rays at instant t of the two phase waves (solid line) and the trajectories described over time by the two moving bodies (dotted line). One then succeeds very well in representing how each moving body describes its trajectory with a velocity that is at every instant tangent to the ray of the phase wave.

Let us insist on one last point. The rays of the wave at instant t are the envelopes of the propagation velocity, but these rays are not the trajectories of the energy; they are only tangent to them at each point. This recalls known conclusions of hydrodynamics where the streamlines, envelopes of velocities, are the trajectories of the fluid particles only if their form is invariable, in other words if the motion is steady.

Chapter V: Light Quanta ²⁸

I. The Atom of Light

As we said in the introduction, the development of radiation physics has been proceeding for several years in the direction of a return, at least partial, to the corpuscular theory of light. An attempt made by us to obtain an atomic theory of black-body radiation, published by the Journal de Physique in November 1922 under the title “Light Quanta and Black-Body Radiation” and whose principal results will be given in Chapter VII, had confirmed us in the idea of the real existence of the atom of light. The ideas expounded in Chapter I, and whose deduction of the stability conditions in the Bohr atom in Chapter III seem to bring such an interesting confirmation, appear to make us take a small step toward the synthesis of the conceptions of Newton and Fresnel. ²⁹

²⁹ Light quanta (photons): De Broglie applies his ideas symmetrically: just as electrons (particles) have wave properties, light (waves) has particle properties. These “atoms of light” were named photons by Gilbert Lewis in 1926. A photon of frequency ν has energy E = hν and, remarkably, momentum p = h/λ = hν/c, even though it has zero rest mass. This momentum is real, it’s what makes solar sails work and it’s transferred in the Compton effect. De Broglie proposes that photons have an extremely small but nonzero rest mass, which would make them travel slightly slower than the limiting speed c. Modern experiments constrain the photon mass to be less than 10⁻⁵⁴ kg, effectively zero.

Without concealing from ourselves the difficulties raised by such boldness, we shall try to specify how one can currently represent the atom of light.

We conceive it in the following way: for an observer attached to it, it appears as a small region of space around which the energy is very strongly condensed and forms an indivisible whole. This agglomeration of energy having as total value ε₀ (measured by the attached observer), one must, according to the principle of the inertia of energy, attribute to it a proper mass:

\[m_0 = \frac{\varepsilon_0}{c^2}\]

This definition is entirely analogous to that which one can give for the electron. There remains, however, an essential difference of structure between the electron and the atom of light. While the electron must be considered until now as endowed with spherical symmetry, the atom of light must possess an axis of symmetry corresponding to polarization. We shall therefore represent the light quantum as possessing the same symmetry as a doublet of electromagnetic theory. This representation is entirely provisional and one will only be able, if necessary, to specify with some chance of accuracy the constitution of the luminous unit after having made profound modifications to electromagnetism, and this work is not accomplished.

In conformity with our general ideas, we shall suppose that there exists in the very constitution of the light quantum a periodic phenomenon whose proper frequency ν₀ is given by the relation:

\[\nu_0 = \frac{1}{h} m_0 c^2\]

The phase wave corresponding to the motion of this quantum with velocity βc will have frequency:

\[\nu = \frac{1}{h} \frac{m_0 c^2}{\sqrt{1 - \beta^2}}\]

and it is quite natural to suppose that this wave is identical to that of the wave theories or, more exactly, that the distribution conceived in the classical manner of waves in space is a sort of time average of the real distribution of phase waves accompanying the atoms of light.

It is an experimental fact that luminous energy moves with a velocity indiscernible from the limiting value c. The velocity c being a velocity that energy can never attain by reason of the very law of variation of mass with velocity, we are quite naturally led to suppose that radiations are formed of atoms of light moving with velocities very close to c, but slightly less.

If a body has an extraordinarily small proper mass, to give it an appreciable kinetic energy, it will be necessary to give it a velocity very close to c; this results from the expression for kinetic energy:

\[E = m_0 c^2 \left(\frac{1}{\sqrt{1 - \beta^2}} - 1\right)\]

Moreover, to velocities comprised in a very small interval c − ε, c, correspond energies having all values from 0 to +∞. We therefore conceive that by supposing m₀ extraordinarily small (we shall specify later), the atoms of light possessing an appreciable energy will all have a velocity very close to c and, despite the near equality of their velocities, will have very different energies.

Since we make the phase wave correspond to the classical light wave, the frequency ν of the radiation will be defined by the relation:

\[\nu = \frac{1}{h} \frac{m_0 c^2}{\sqrt{1 - \beta^2}}\]

Let us note, a fact that one must remember each time it concerns atoms of light, the extreme smallness of m₀c² compared to m₀c²/√(1 − β²); the kinetic energy can therefore here be written simply:

\[\frac{m_0 c^2}{\sqrt{1 - \beta^2}}\]

The light wave of frequency ν would therefore correspond to the displacement of an atom of light with velocity v = βc related to ν by the relation:

\[v = \beta c = c\sqrt{1 - \frac{m_0^2 c^4}{h^2 \nu^2}}\]

Except for extremely slow vibrations, m₀c²/hν and a fortiori its square will be very small and one can set:

\[v = c\left(1 - \frac{m_0^2 c^4}{2h^2 \nu^2}\right)\]

We can try to fix an upper limit for the value of m₀. Indeed, radio experiments have shown that radiations of a few kilometers wavelength still propagate sensibly with velocity c. Let us admit that waves for which 1/ν = 10⁻⁴ seconds have a velocity differing from c by less than one hundredth. The upper limit of m₀ will be:

\[(m_0)_{\max} = \frac{\sqrt{2}}{10} \frac{h\nu}{c^2}\]

that is, approximately 10⁻⁴⁴ grams. It is even probable that m₀ should be chosen still smaller; perhaps one can hope that one day by measuring the velocity in vacuum of waves of very low frequency, one will find numbers appreciably less than c.

One must not forget that the propagation velocity just discussed is not that of the phase wave, always greater than c, but that of the displacement of energy, the only one experimentally detectable.⁽⁵⁾

II. The Motion of the Atom of Light

Atoms of light for which β = 1 sensibly would therefore be accompanied by phase waves whose velocity c/β would also be sensibly equal to c; it is, we think, this coincidence that would establish between the atom of light and its phase wave a particularly close connection expressed by the dual corpuscular and wave aspect of radiations. The identity of Fermat’s principle and the principle of least action would explain why the rectilinear propagation of light is compatible with both points of view at once.

The trajectory of the luminous corpuscle would be one of the rays of its phase wave. There are reasons to believe, as we shall see later, that several corpuscles could have the same phase wave; their trajectories would then be various rays of this wave. The old idea that the ray is the trajectory of energy would thus be confirmed and made more precise.

However, rectilinear propagation is not an absolutely general fact; a light wave falling on the edge of a screen diffracts and penetrates into the geometrical shadow; rays that pass at distances from the screen small compared to the wavelength are deviated and no longer follow Fermat’s law. From the wave point of view, the deviation of the rays is explained by the imbalance introduced between the actions of the various zones very close to the wave as a result of the presence of the screen. Placed at the opposite point of view, Newton supposed a force exerted by the edge of the screen on the corpuscle. It seems that we can arrive at a synthetic view: the ray of the wave would curve as the wave theory predicts, and the moving body, for which the principle of inertia would no longer be valid, would undergo the same deviation as the ray with whose motion it is bound; perhaps one could say that the wall exerts a force on it if one takes the curvature of the trajectory as the criterion for the existence of a force.

In what precedes, we have been guided by the idea that the corpuscle and its phase wave are not different physical realities. If one reflects, one will see that the following conclusion seems to result: “Our dynamics (including its Einsteinian form) has remained behind optics: it is still at the stage of geometrical optics.” If it seems to us today fairly probable that every wave involves concentrations of energy, on the other hand the dynamics of the material point doubtless conceals a wave propagation, and the true meaning of the principle of least action is to express a phase concordance.

It would be very interesting to seek the interpretation of diffraction in space-time, but here one encounters the difficulties mentioned in Chapter II regarding non-uniform motion, and we have not been able to specify the question in a satisfactory way.

III. Some Concordances Between the Opposing Theories of Radiation

We shall show by some examples with what facility the corpuscular theory of radiations accounts for a certain number of results known from wave theories.

a) Doppler effect by motion of the source:

Consider a light source in motion with velocity v = βc in the direction of an observer considered stationary. This source is supposed to emit atoms of light; the frequency of the phase waves is ν and the velocity c(1 − ε) where ε = (1/2)(m₀²c⁴)/(h²ν²). For the stationary observer, these quantities have the values ν′ and c(1 − ε′). The theorem of velocity addition gives:

\[c(1 - \varepsilon') = \frac{c(1 - \varepsilon) + v}{1 + \frac{c(1 - \varepsilon) \cdot v}{c^2}}\]

or

\[1 - \varepsilon' = \frac{1 - \varepsilon + \beta}{1 + (1 - \varepsilon)\beta}\]

or else, neglecting εε′:

\[\frac{\varepsilon}{\varepsilon'} = \frac{\nu'^2}{\nu^2} = \frac{1 + \beta}{1 - \beta}, \qquad \frac{\nu'}{\nu} = \sqrt{\frac{1 + \beta}{1 - \beta}}\]

If β is small, one recovers the formulas of classical optics:

\[\frac{\nu'}{\nu} = 1 + \beta, \qquad \frac{T'}{T} = 1 - \beta = 1 - \frac{v}{c}\]

It is also easy to find the ratio of the intensities emitted for the two observers. During unit time, the moving observer sees the source emit n atoms of light per unit surface. The energy density of the beam evaluated by this observer is therefore nhν/c, and the intensity is I = nhν. For the stationary observer, the n atoms are emitted in a time equal to 1/√(1 − β²) and they fill a volume c(1 − β) × 1/√(1 − β²) = c√[(1 − β)/(1 + β)]. The energy density of the beam appears to him to be:

\[\frac{nh\nu'}{c} \sqrt{\frac{1 + \beta}{1 - \beta}}\]

and the intensity:

\[I' = nh\nu' \sqrt{\frac{1 + \beta}{1 - \beta}} = nh\nu' \cdot \frac{\nu'}{\nu}\]

whence

\[\frac{I'}{I} = \left(\frac{\nu'}{\nu}\right)^2\]

All these formulas are demonstrated from the wave point of view in Laue’s book, Die Relativitätstheorie, vol. I, 3rd ed., p. 119.

b) Reflection on a moving mirror:

Consider the reflection of light corpuscles falling normally on a perfectly reflecting plane mirror which moves with velocity βc in the direction perpendicular to its surface.

Let ν′₁ be the frequency of the phase waves accompanying the incident corpuscles for the stationary observer, and c(1 − ε′₁) their velocity. The same quantities for the attached observer will be ν₁ and c(1 − ε₁).

If we consider the reflected corpuscles, the corresponding values will be called ν₂, c(1 − ε₂), ν′₂, and c(1 − ε′₂).

The composition of velocities gives:

\[c(1 - \varepsilon_1) = \frac{c(1 - \varepsilon'_1) + \beta c}{1 + \beta(1 - \varepsilon'_1)}, \qquad c(1 - \varepsilon_2) = \frac{c(1 - \varepsilon'_2) - \beta c}{1 - \beta(1 - \varepsilon'_2)}\]

For the attached observer, there is reflection on a stationary mirror without change of frequency since energy is conserved. Whence:

\[\nu_1 = \nu_2, \quad \varepsilon_1 = \varepsilon_2, \qquad \frac{1 - \varepsilon'_1 + \beta}{1 + \beta(1 - \varepsilon'_1)} = \frac{1 - \varepsilon'_2 - \beta}{1 - \beta(1 - \varepsilon'_2)}\]

Neglecting ε′₁ε′₂, we get:

\[\frac{\varepsilon'_1}{\varepsilon'_2} = \left(\frac{\nu'_2}{\nu'_1}\right)^2 = \left(\frac{1 + \beta}{1 - \beta}\right)^2 \quad \text{or} \quad \frac{\nu'_2}{\nu'_1} = \frac{1 + \beta}{1 - \beta}\]

If β is small, one falls back on the classical formula:

\[\frac{T_2}{T_1} = 1 - 2\frac{v}{c}\]

It would be easy to treat the problem by assuming oblique incidence.

Let us denote by n the number of corpuscles reflected by the mirror during a given time. The total energy of the n corpuscles after reflection E′₂ is to their total energy before reflection E′₁ in the ratio:

\[\frac{nh\nu'_2}{nh\nu'_1} = \frac{\nu'_2}{\nu'_1}\]

Electromagnetic theory also gives this relation, but here it is entirely evident.

If the n corpuscles occupied a volume V′₁ before reflection, they will occupy after reflection a volume V′₂ = V′₁(1 − β)/(1 + β), as a very simple geometric reasoning shows. The intensities I′₁ and I′₂ before and after reflection are therefore in the ratio:

\[\frac{I'_2}{I'_1} = \frac{nh\nu'_2}{nh\nu'_1} \cdot \frac{1 + \beta}{1 - \beta} = \left(\frac{\nu'_2}{\nu'_1}\right)^2\]

All these results are demonstrated from the wave point of view by Laue, page 124.

c) Radiation pressure of black-body radiation:

Let there be an enclosure filled with black-body radiation at temperature T. What is the pressure supported by the walls of the enclosure? For us, black-body radiation will be a gas of atoms of light, and we shall suppose the distribution of velocities to be isotropic. Let u be the total energy (or, what here amounts to the same thing, the total kinetic energy) of the atoms contained in unit volume. Let ds be a surface element of the wall, dv a volume element, r their distance, θ the angle of the line joining them with the normal to the surface element.

The solid angle under which the element ds is seen from the point O, center of dv, is:

\[d\Omega = \frac{ds \cos \theta}{r^2}\]

Consider only those atoms of the volume dv whose energy is comprised between w and w + dw, numbering n_wdw dv; the number of those among them whose velocity is directed toward ds is, by reason of isotropy:

\[\frac{d\Omega}{4\pi} \times n_w dw \, dv = n_w dw \frac{ds \cos \theta}{4\pi r^2} dv\]

Taking a system of spherical coordinates with the normal to ds as polar axis, one finds:

\[dv = r^2 \sin \theta \, d\theta \, d\psi \, dr\]

Moreover, the kinetic energy of an atom of light being m₀c²/√(1 − β²) and its momentum G = m₀v/√(1 − β²) with v = c very approximately, one has:

\[\frac{W}{c} = G\]

Therefore, the reflection at angle θ of an atom of energy W imparts to ds an impulse 2G cos θ = 2(W/c) cos θ. The atoms of volume dv having this energy will therefore impart to it by reflection an impulse equal to:

\[2 \frac{W}{c} \cos \theta \, n_w dw \, r^2 \sin \theta \, d\theta \, d\psi \, dr \frac{ds \cos \theta}{4\pi r^2}\]

Integrating with respect to W from 0 to ∞, noting that ∫₀^+∞ wn_wdw = u, with respect to the angles ψ and θ respectively from 0 to 2π and from 0 to π/2, finally with respect to r from 0 to c. We thus obtain the total impulse received in one second by the element ds, and, dividing by ds, the radiation pressure:

\[p = u \cdot \int_0^{\frac{\pi}{2}} \cos^2 \theta \sin \theta \, d\theta = \frac{u}{3}\]

The radiation pressure is equal to one third of the energy contained in unit volume, a result known from classical theories.

The ease with which we have just recovered in this paragraph certain results equally furnished by wave conceptions of radiation reveals to us the existence between the two apparently opposed points of view of a secret harmony whose nature the notion of phase wave lets us foresee.

IV. Wave Optics and Light Quanta⁽⁶⁾

The stumbling block of the theory of light quanta is the explanation of the phenomena which constitute wave optics. The essential reason for this is that this explanation requires the intervention of the phase of periodic phenomena; it may therefore seem that we have made a very great step forward on the question by managing to conceive a close connection between the movement of a light corpuscle and the propagation of a certain wave. It is very probable, indeed, that if the theory of light quanta one day manages to explain the phenomena of wave optics, it is by conceptions of this kind that it will succeed. Unfortunately, it is still impossible to arrive at satisfactory results in this order of ideas, and only the future will be able to tell us whether Einstein’s bold conception, judiciously softened and completed, will be able to accommodate within its framework the numerous phenomena whose study of wonderful precision had led 19th-century physicists to consider the wave hypothesis as definitively established.

Let us content ourselves with circling around this difficult problem without seeking to attack it head-on. To progress along the path followed until now, one would have to establish, as we have said, a certain connection, probably statistical in nature, between the wave conceived in the classical manner and the superposition of phase waves; this would certainly lead to attributing to the phase wave, and consequently also to the periodic phenomenon defined in Chapter I, an electromagnetic nature.

It can be considered as proven with near certainty that the emission and absorption of radiation take place discontinuously. Electromagnetism, or more precisely electron theory, therefore gives us an inexact view of the mechanism of these phenomena. However, Bohr, through his correspondence principle, has taught us that if one considers the predictions of this theory for the radiation emitted by an ensemble of electrons, they probably possess a sort of global accuracy. Perhaps all of electromagnetic theory would have only statistical value; Maxwell’s laws would then appear as a continuous approximation of a discontinuous reality, somewhat in the same way (but only somewhat) that the laws of hydrodynamics give a continuous approximation of the very complex and very rapidly variable movements of fluid molecules. This idea of correspondence, which still appears rather imprecise and rather elastic, will have to serve as a guide to the bold researchers who will want to constitute a new electromagnetic theory more in accord than the present one with quantum phenomena.

We shall reproduce in the following paragraph considerations that we have put forward on interference; to speak frankly, they must be considered as vague suggestions rather than as true explanations.

V. Interference and Coherence

We shall first ask how the presence of light at a point in space is detected. One can place there a body on which the radiation can exert a photoelectric, chemical, calorific effect, etc.; it is moreover possible that in the final analysis all effects of this kind are photoelectric. One can also observe the scattering of waves produced by matter at the point of space considered. We can therefore say that where radiation cannot react on matter, it is experimentally undetectable. Electromagnetic theory admits that photographic actions (Wiener’s experiments) and scattering are linked to the intensity of the resulting electric field; where the electric field is zero, if there is magnetic energy, it is undetectable.

The ideas developed here lead to assimilating phase waves to electromagnetic waves, at least as regards the distribution of phases in space, the question of intensities being reserved. This idea, joined to that of correspondence, leads us to think that the probability of reactions between atoms of matter and atoms of light is at each point linked to the resultant (or rather to the mean value of the latter) of one of the vectors characterizing the phase wave; where this resultant is zero, light is undetectable; there is interference. One therefore conceives that an atom of light traversing a region where the phase waves interfere could be absorbed by matter at certain points and at others could not be. There is here the principle, still very qualitative, of an explanation of interference compatible with the discontinuity of radiant energy. Norman Campbell in his book Modern Electrical Theory (1913) seems to have glimpsed a solution of the same kind when he wrote:

“The corpuscular theory alone can explain how the energy of radiation is transferred from one place to another, while the wave theory alone can explain why the transfer along one trajectory depends on that which takes place on another. It almost seems that energy itself is transported by corpuscles while the power to absorb it and make it perceptible to experiment is transported by spherical waves.”

For interference to be able to occur regularly, it seems necessary to establish a sort of dependence between the emissions of the various atoms of the same source. We have proposed to express this dependence by the following postulate: “The phase wave linked to the motion of an atom of light can, by passing over excited material atoms, trigger the emission of other atoms of light whose phase will be in agreement with that of the wave.” A wave could thus transport many small centers of energy condensation which would, moreover, slide slightly over its surface while always remaining in phase with it. If the number of transported atoms were extremely large, the structure of the wave would approach classical conceptions as a sort of limit.

VI. Bohr’s Frequency Law. Conclusions

From whatever point of view one places oneself, the detail of the internal transformations undergone by the atom when it absorbs or emits cannot yet be imagined in any way. Let us always admit the granular hypothesis: we do not know whether the quantum absorbed by the atom merges in some way with it or whether it subsists in its interior in the state of an isolated unit; neither do we know whether emission is the expulsion of a quantum pre-existing in the atom or the creation of a new unit at the expense of the internal energy of the latter. Be that as it may, it appears certain that emission involves only a single quantum; henceforth, the total energy of the corpuscle, equal to h times the frequency of the phase wave that accompanies it, should, to safeguard the conservation of energy, be equal to the decrease in the total energy content of the atom, and this gives us Bohr’s frequency law: ³⁰

³⁰ Bohr’s frequency condition: When an atom transitions between energy levels, it emits or absorbs a photon whose energy exactly matches the energy difference: hν = |E₁ − E₂|. This explains atomic emission and absorption spectra. When you excite hydrogen gas (in a discharge tube) and see those specific colored lines (red at 656 nm, blue-green at 486 nm, etc.), each line corresponds to electrons jumping between specific energy levels. The Rydberg formula you learn in Gen Chem, 1/λ = R_H(1/n₁² − 1/n₂²), is a direct consequence of Bohr’s condition combined with quantized energy levels.

\[h\nu = W_1 - W_2\]

We see therefore that our conceptions, after having led us to a simple explanation of the stability conditions, also allow us to obtain the frequency law, provided however that we admit that emission always involves a single corpuscle.

Let us note that the image of emission furnished by the theory of light quanta seems confirmed by the conclusions of Einstein and Léon Brillouin⁽⁷⁾ who have shown the necessity of introducing into the analysis of reactions between black-body radiation and a free particle the idea of a strictly directed emission.

What should we conclude from this entire chapter? Certainly, such a phenomenon as dispersion, which seemed incompatible with the notion of light quanta in its simplistic form, now seems to us less impossible to reconcile with it thanks to the introduction of a phase. The recent theory of X-ray and γ-ray scattering given by A. H. Compton, which we shall expound later, seems to rest on serious experimental evidence and to make tangible the existence of luminous corpuscles in a domain where wave schemes reigned supreme. It is nonetheless incontestable that the conception of grains of luminous energy still does not at all succeed in solving the problems of wave optics and that it encounters very serious difficulties there; it would seem premature to us to pronounce on the question of whether it will or will not succeed in overcoming them.

²⁸ Extending duality to light: Having shown that matter (electrons) has wave properties, de Broglie now examines the reverse: treating light as particles. He explores what properties “atoms of light” (photons) must have. This chapter develops the photon concept further than Einstein’s 1905 treatment and considers how wave optics (interference, diffraction) can be reconciled with a particle picture of light. The difficulties de Broglie identifies here were eventually resolved by quantum electrodynamics in the 1940s.⁽⁴⁾

Chapter VI: The Scattering of X-Rays and γ-Rays

I. J. J. Thomson’s Theory⁽⁸⁾

In this chapter, we wish to study the scattering of X-rays and γ-rays and show by this particularly suggestive example the respective current positions of electromagnetic theory and that of light quanta.

Let us begin by defining the very phenomenon of scattering: when a beam of rays is sent onto a piece of matter, part of the energy is, in general, scattered in all directions. It is said that there is scattering and weakening by scattering of the beam during its passage through the substance.

Electronic theory interprets this phenomenon very simply. It supposes (which moreover seems in direct opposition to Bohr’s atomic model) that the electrons contained in an atom are subjected to quasi-elastic forces and possess a well-determined vibration period. Henceforth, the passage of an electromagnetic wave over these electrons will impart to them an oscillatory motion whose amplitude will depend in general both on the frequency of the incident wave and on the proper frequency of the electronic resonators. In conformity with the theory of the acceleration wave, the motion of the electron will be constantly damped by the emission of a wave with cylindrical symmetry. An equilibrium regime will be established in which the resonator will draw from the incident radiation the energy necessary to compensate this damping. The final result will therefore be a scattering of a fraction of the incident energy in all directions of space.

To calculate the magnitude of the scattering phenomenon, one must first determine the motion of the vibrating electron. For this, one must express the equilibrium between the resultant of the inertial force and the quasi-elastic force on one hand, and the electric force exerted by the incident radiation on the electron on the other hand. In the visible domain, examination of numerical values shows that one can neglect the inertial term compared to the quasi-elastic term, and one is thus led to attribute to the amplitude of the vibratory motion a value proportional to the amplitude of the exciting light, but independent of its frequency. The theory of dipole radiation then teaches that the total secondary radiation varies inversely as the 4th power of the wavelength; radiations are therefore the more scattered the higher their frequency. It is on this conclusion that Lord Rayleigh based his beautiful theory of the blue color of the sky.⁽⁹⁾

In the domain of very high frequencies (X-rays and γ-rays), it is on the contrary the quasi-elastic term that is negligible compared to the inertial term. Everything happens as if the electron were free and the amplitude of its vibratory motion is proportional not only to the incident amplitude but also to the 2nd power of the wavelength. It follows that the total scattered intensity is this time independent of wavelength. It was J. J. Thomson who first drew attention to this fact and constituted the first theory of X-ray scattering. The two principal conclusions were the following: ³¹

³¹ Thomson scattering: In classical physics, when an electromagnetic wave hits an electron, the electron oscillates and re-radiates in all directions. This is called Thomson scattering. The key prediction: the scattered light has the same wavelength as the incident light, regardless of angle. This works well for visible light and soft X-rays. But for hard X-rays and gamma rays, experiments showed the scattered wavelength increases (shifts to lower energy), which classical theory couldn’t explain. This failure led to Compton’s quantum explanation.

1° If one denotes by θ the angle of the extension of the direction of incidence with the direction of scattering, the scattered energy varies as a function of θ like (1 + cos²θ)/2.

2° The total energy scattered by an electron in one second is to the incident intensity in the ratio:

\[\frac{I_\sigma}{I} = \frac{8\pi}{3} \frac{e^4}{m_0^2 c^4}\]

e and m₀ being the constants of the electron, c the speed of light.

An atom certainly contains several electrons; today there are good reasons to believe their number p equal to the atomic number of the element. Thomson supposed “incoherent” the waves emitted by the p electrons of the same atom and, consequently, considered the energy scattered by an atom as equal to p times that which a single electron would scatter. From the experimental point of view, scattering is manifested by a gradual weakening of the beam intensity, and this weakening obeys an exponential law:

\[I_x = I_0 e^{-sx}\]

s is the coefficient of weakening by scattering or more briefly the scattering coefficient. The quotient S of this number by the density of the scattering body is the mass scattering coefficient. If one calls atomic scattering coefficient σ the ratio between the energy scattered in a single atom and the intensity of the incident radiation, one easily sees that it is related to s by the equation:

\[\sigma = \frac{s}{\rho} A m_H\]

A is here the atomic weight of the scatterer, m_H the mass of the hydrogen atom. Substituting the numerical values in the factor (8π/3)(e⁴/m₀²c⁴), one finds:

\[\sigma = 0.54 \times 10^{-24} p\]

Now, experiment has shown that the ratio s/ρ is very close to 0.2, so that one should have:

\[\frac{A}{p} = \frac{0.54 \times 10^{-24}}{0.2 \times 1.46 \times 10^{-24}} = \frac{0.54}{0.29}\]

This figure is close to 2, which is entirely in accord with our current conception of the relation between the number of intra-atomic electrons and the atomic weight. Thomson’s theory has therefore led to interesting coincidences, and the work of various experimenters, notably that of Barkla, demonstrated long ago that it was verified to a large extent.⁽¹⁰⁾

II. Debye’s Theory⁽¹¹⁾

Difficulties remained. In particular, W. H. Bragg had found in certain cases a scattering much stronger than that accounted for by the preceding theory, and he had concluded that the scattered energy was proportional not to the number of atomic electrons but to the square of this number. Debye presented a more complete theory compatible at once with the results of Bragg and Barkla.

Debye considers intra-atomic electrons as distributed regularly in a volume whose dimensions are of the order of 10⁻⁸ cm; to facilitate calculations, he even supposes them all distributed on a single circle. If the wavelength is large compared to the mean distances of the electrons, their motions must be almost in phase, and in the total wave, the amplitudes radiated by each of them will add up. The scattered energy will then be proportional to p² and no longer to p, so that the coefficient σ will be written:

\[\sigma = \frac{8\pi}{3} \frac{e^4}{m_0^2 c^4} p^2\]

As for the distribution in space, it will be identical to that which Thomson had predicted.

For waves of progressively decreasing wavelength, the distribution in space will become asymmetric, the energy scattered in the direction from which the radiation comes being much weaker than in the opposite direction. Here is the reason: one can no longer regard the vibrations of the various electrons as in phase when the wavelength becomes comparable to the mutual distances. The amplitudes radiated in the various directions will no longer add up because they are out of phase, and the scattered energy will be less. However, in a cone of small opening surrounding the extension of the direction of incidence, there will always be phase agreement and the amplitudes will add up; therefore, for directions contained in this cone, the scattering will be much greater than for others. Debye moreover predicted a curious phenomenon: when one moves progressively away from the axis of the cone defined above, the scattered intensity does not immediately decrease regularly but first undergoes periodic variations; one should therefore observe on a screen placed perpendicular to the transmitted beam bright and dark rings centered on the direction of the beam. Although Debye at first thought he recognized this phenomenon in certain experimental results of Friedrich, it does not seem to have been clearly observed until now.

For short wavelengths, the phenomena should simplify. The cone of strong scattering narrows more and more, the distribution becomes symmetric again and must now satisfy Thomson’s formulas because the phases of the various electrons become entirely incoherent; it is therefore the energies and no longer the amplitudes that add up.

The great interest of Debye’s theory is to have explained the strong scattering of soft X-rays and to have shown how the passage from this phenomenon to that of Thomson must occur when the frequency rises. But it is essential to note that according to Debye’s ideas, the higher the frequency, the better the symmetry of the scattered radiation and the value 0.2 of the coefficient s/ρ should be realized. Now, we shall see in the following paragraph that this is not at all the case.

III. Recent Theory of Debye and Compton⁽¹²⁾

Experiments in the domain of hard X-rays and γ-rays have revealed facts very different from those that the preceding theories can predict. First, the higher the frequency rises, the more the asymmetry of the scattered radiation is accentuated; on the other hand, the total scattered energy decreases, the value of the mass coefficient s/ρ tends to decrease rapidly as soon as the wavelength falls below 0.3 or 0.2 Å and becomes very small for γ-rays. Thus, where Thomson’s theory should apply better and better, it applies less and less.

Two other phenomena have been brought to light by recent experimental research, foremost among which must be placed that of A. H. Compton. This has in fact shown that scattering appears to be accompanied by a lowering of the frequency variable moreover with the direction of observation, and on the other hand, that it seems to cause electrons to be set in motion. Almost simultaneously and independently of each other, Debye and Compton succeeded in giving an interpretation of these departures from the classical laws based on the notion of light quanta. ³²

³² The Compton effect: When an X-ray photon collides with an electron, it bounces off like a billiard ball, losing some energy to the electron. Since E = hν = hc/λ, losing energy means the wavelength increases (shifts toward red). The wavelength shift depends on scattering angle: Δλ = (h/m_ec)(1 − cosθ). At 90°, the shift is about 0.00243 nm (the “Compton wavelength” of the electron). This was conclusive proof that photons carry momentum and behave like particles in collisions. Compton received the Nobel Prize in 1927 for this work. The effect is used today in medical imaging (Compton cameras) and astrophysics (inverse Compton scattering from cosmic rays).

Here is the principle: if a light quantum is deflected from its rectilinear path by passing near an electron, we must suppose that during the time when the two energy centers are sufficiently close, they exert a certain action on each other. When this action ends, the electron, initially at rest, will have borrowed a certain energy from the luminous corpuscle; according to the quantum relation, the scattered frequency will therefore be less than the incident frequency. The conservation of momentum completes the determination of the problem. Suppose that the scattered quantum moves in a direction making angle θ with the extension of the direction of incidence. The frequencies before and after scattering being ν₀ and ν_θ, and the proper mass of the electron being m₀, we shall have:

\[h\nu_\theta = h\nu_0 - m_0 c^2 \left(\frac{1}{\sqrt{1 - \beta^2}} - 1\right)\]

\[\left(\frac{m_0 \beta c}{\sqrt{1 - \beta^2}}\right)^2 = \left(\frac{h\nu_0}{c}\right)^2 + \left(\frac{h\nu_\theta}{c}\right)^2 - 2\frac{h\nu_0}{c} \cdot \frac{h\nu_\theta}{c} \cos \theta\]

This second relation is read at once from the adjoining figure. The velocity v = βc is that which the electron acquires by this process.

Let us denote by α the ratio hν₀/(m₀c²) equal to the quotient of ν₀ by what we call the proper frequency of the electron. We get:

\[\nu_\theta = \frac{\nu_0}{1 + 2\alpha \sin^2 \frac{\theta}{2}}\]

or

\[\lambda_\theta = \lambda_0 \left(1 + 2\alpha \sin^2 \frac{\theta}{2}\right)\]

One can also, with the help of these formulas, study the velocity of projection and direction of the “recoiling” electron. One finds that to scattering directions varying from 0 to π correspond for the electron recoil angles varying from π/2 to 0, the velocity varying simultaneously from 0 to a certain maximum.

Compton, invoking hypotheses inspired by the correspondence principle, thought he could calculate the value of the total scattered energy and thus explain the rapid decrease of the coefficient s/ρ. Debye applies the idea of correspondence in a somewhat different form but also succeeds in interpreting this same phenomenon.

In an article in the Physical Review of May 1923 and in a more recent article in the Philosophical Magazine (November 1923), Compton showed that the new ideas expounded above accounted for many experimental facts and that in particular for hard rays and light bodies, the predicted wavelength variation was quantitatively verified. For heavier bodies and softer radiations, there seems to be coexistence of a line scattered without change of frequency and another line scattered according to the Compton-Debye law. For low frequencies, the former becomes predominant and often even seems to be the only one to exist. Experiments by Ross on the scattering of the MoK_α line and green light by paraffin confirm this way of seeing things. The K_α line gives a strong line scattered according to Compton’s law and a weak line at unmodified frequency; the latter appears to be the only one to exist for green light.

The existence of an unshifted line seems to explain why crystal reflection (Laue phenomenon) is not accompanied by a wavelength variation. Jauncey and Wolfers have, in fact, recently shown that if the lines scattered by the crystals usually employed as reflectors underwent the Compton-Debye effect to an appreciable extent, precision measurements of X-ray wavelengths would already have revealed the phenomenon. It must therefore be supposed that in this case, scattering takes place without degradation of the quantum.

At first sight, one is tempted to explain the existence of the two kinds of scattering in the following way: the Compton effect would occur each time the scattering electron is free, or at least its binding to an atom corresponds to an energy small compared to that of the incident quantum; in the opposite case, there would be scattering without change of wavelength because then the entire atom would take part in the process without acquiring appreciable velocity by reason of its large mass. Compton finds difficulties in admitting this idea and prefers to explain the unmodified line by the intervention of several electrons in the deflection of a single quantum; it would then be the high value of the sum of their masses that would prevent the passage of notable energy from the radiation to matter. Be that as it may, one conceives well why heavy elements and hard rays behave differently from light elements and soft rays.

As for the way of rendering compatible the conception of scattering as being the deflection of a luminous particle and the conservation of phase necessary for the explanation of Laue figures, it raises the considerable and as yet unresolved difficulties that we pointed out in the previous chapter regarding wave optics.

When dealing with hard X-rays and light elements as occurs in practice in radiotherapy, the phenomena must be completely modified by the Compton effect, and this is indeed what seems to occur. We shall give an example of this. It is known that in addition to weakening by scattering, an X-ray beam passing through matter undergoes weakening by absorption, a phenomenon that is accompanied by an emission of photoelectrons. An empirical law due to Messrs. Bragg and Pierce tells us that this absorption varies as the cube of the wavelength and undergoes abrupt discontinuities for all the characteristic frequencies of the intra-atomic levels of the substance considered; moreover, for the same wavelength and various elements, the atomic absorption coefficient varies as the fourth power of the atomic number.

This law is well verified in the middle domain of Röntgen frequencies and it seems quite probable that it should apply to hard rays. Since, according to ideas accepted before the Compton-Debye theory, scattering was only a dispersal of radiation, only the energy absorbed according to Bragg’s law could produce ionization in a gas, the photoelectric electrons animated with high velocities ionizing by collision the atoms encountered. The Bragg-Pierce law would therefore allow us to calculate the ratio of ionizations produced by the same hard radiation in two bulbs containing one a heavy gas (for example CH₃I) and the other a light gas (for example air). Even taking into account the numerous accessory corrections, this ratio was found experimentally to be much smaller than predicted. M. Dauvillier had observed this phenomenon for X-rays and its interpretation puzzled us for a long time.

The new theory of scattering appears to explain this anomaly well. If, indeed, at least in the case of hard rays, part of the energy of the light quanta is transferred to the scattering electron, there will be not only dispersal of the radiation, but also “absorption by scattering.” The ionization of the gas will be due both to electrons expelled from the atom by the mechanism of absorption proper and to electrons set in recoil motion by scattering. In a heavy gas (CH₃I), Bragg absorption is intense and that of Compton is almost negligible in comparison. For a light gas (air), it is no longer at all the same; the first absorption because of its variation as N⁴ is very weak and the second which is independent of N becomes the most important. The ratio of total absorptions and consequently of ionizations in the two gases must therefore be much smaller than previously predicted. It is even possible to account quantitatively in this way for the ratio of ionizations. One sees therefore from this example the very great practical interest of the new ideas of Messrs. Compton and Debye. The recoil of scattering electrons seems moreover to provide the key to many other unexplained phenomena.

IV. Scattering by Moving Electrons

One can generalize the Compton-Debye theory by considering the scattering of a quantum of radiation by an electron in motion. Let us take as the x-axis the direction of primitive propagation of a quantum of initial frequency ν₁, the y and z axes being chosen arbitrarily at right angles to each other in a plane normal to ox and passing through the point where the scattering occurs. The direction of the velocity β₁c of the electron before the collision being defined by the direction cosines a₁b₁c₁, we shall call θ₁ the angle it makes with ox, so that a₁ = cos θ₁; after the collision, the scattered radiation quantum of frequency ν₂ propagates in a direction with direction cosines pqr making angle φ with the direction of the initial velocity of the electron (cos φ = a₁p + b₁q + c₁r) and angle θ with the axis ox (p = cos θ). Finally the electron will possess a final velocity β₂c whose direction cosines will be a₂b₂c₂.

The conservation of energy and of momentum during the collision allow us to write the equations:

\[h\nu_1 + \frac{m_0 c^2}{\sqrt{1 - \beta_1^2}} = h\nu_2 + \frac{m_0 c^2}{\sqrt{1 - \beta_2^2}}\]

\[\frac{h\nu_1}{c} + \frac{m_0 \beta_1 c}{\sqrt{1 - \beta_1^2}} a_1 = \frac{h\nu_2}{c} p + \frac{m_0 \beta_2 c}{\sqrt{1 - \beta_2^2}} a_2\]

\[\frac{m_0 \beta_1 c}{\sqrt{1 - \beta_1^2}} b_1 = \frac{h\nu_2}{c} q + \frac{m_0 \beta_2 c}{\sqrt{1 - \beta_2^2}} b_2\]

\[\frac{m_0 \beta_1 c}{\sqrt{1 - \beta_1^2}} c_1 = \frac{h\nu_2}{c} r + \frac{m_0 \beta_2 c}{\sqrt{1 - \beta_2^2}} c_2\]

Let us eliminate a₂b₂c₂ by means of the relation a₂² + b₂² + c₂² = 1; then, between the relation thus obtained and that which expresses the conservation of energy, let us eliminate β₂. Setting with Compton α₁ = hν₁/(m₀c²), we get:

\[\nu_2 = \nu_1 \frac{1 - \beta_1 \cos \theta_1}{1 - \beta_1 \cos \varphi + 2\alpha_1 \sqrt{1 - \beta_1^2} \sin^2 \frac{\theta}{2}}\]

If the initial velocity of the electron is zero or negligible, we find Compton’s formula:

\[\nu_2 = \nu_1 \frac{1}{1 + 2\alpha \sin^2 \frac{\theta}{2}}\]

In the general case, the Compton effect represented by the term in α persists but diminished; moreover, a Doppler effect is added to it. If the Compton effect is negligible, one finds:

\[\nu_2 = \nu_1 \frac{1 - \beta_1 \cos \theta_1}{1 - \beta_1 \cos \varphi}\]

Since, in this case, the scattering of the quantum does not disturb the motion of the electron, one can expect to find a result identical to that of electromagnetic theory. This is indeed what occurs. Let us calculate the scattered frequency according to electromagnetic theory (taking into account Relativity). The incident radiation possesses for the electron the frequency:

\[\nu' = \nu_1 \frac{1 - \beta_1 \cos \theta_1}{\sqrt{1 - \beta_1^2}}\]

If the electron, while keeping the translational velocity β₁c, begins to vibrate at frequency ν′, the observer who receives the scattered radiation in a direction making angle φ with the velocity of the source, attributes to it the frequency:

\[\nu_2 = \nu' \frac{\sqrt{1 - \beta_1^2}}{1 - \beta_1 \cos \varphi}\]

and we do indeed have:

\[\nu_2 = \nu_1 \frac{1 - \beta_1 \cos \theta_1}{1 - \beta_1 \cos \varphi}\]

The Compton effect remains in general rather small; on the contrary, the Doppler effect can attain for electrons accelerated by potential drops of several hundred kilovolts very large values (increase of one third of the frequency for 200 kilovolts).

We have here an elevation of the quantum because the scattering body being animated with a high velocity, can yield energy to the atom of radiation. The conditions of application of Stokes’ rule are not realized. It is not impossible that certain of the conclusions stated above could be submitted to experimental verification at least as regards X-rays.

Chapter VII: Statistical Mechanics and Quanta

I. Review of Some Results of Statistical Thermodynamics

The interpretation of the laws of thermodynamics with the help of statistical considerations is one of the most beautiful successes of scientific thought, but it is not without some difficulties and some objections. It does not fall within the scope of the present work to make a critique of these methods; we shall content ourselves here, after having recalled in their form most employed today certain fundamental results, with examining how our new ideas could be introduced into the theory of gases and into that of black-body radiation.

Boltzmann showed, first, that the entropy of a gas in a determined state is, to within an additive constant, the product of the logarithm of the probability of that state by the constant k called “Boltzmann’s constant” which depends on the choice of temperature scale; he had first arrived at this conclusion by analyzing the collisions between atoms under the hypothesis of a completely disordered agitation of these. Today, following the work of Messrs. Planck and Einstein, one considers rather the relation S = k log P as the very definition of the entropy S of a system. In this definition, P is not the mathematical probability equal to the quotient of the number of microscopic configurations giving the same total macroscopic configuration by the total number of possible configurations; it is the “thermodynamic probability” simply equal to the numerator of this fraction. This choice of the meaning of P amounts to fixing in a certain way (ultimately arbitrary) the constant of the entropy. This postulate admitted, we shall recall a well-known demonstration of the analytical expression of thermodynamic quantities, a demonstration which has the advantage of being valid as well when the series of possible states is discontinuous as in the opposite case. ³³

³³ Statistical mechanics and entropy: De Broglie is reviewing the Boltzmann-Gibbs framework where entropy S = k ln P relates macroscopic thermodynamic properties to microscopic probability distributions. This foundation underlies modern physical chemistry.

Consider for this purpose N objects that can be distributed arbitrarily among m “states” or “cells” considered a priori as equally probable. A certain configuration of the system will be realized by placing n₁ objects in cell 1, n₂ in cell 2, etc. The thermodynamic probability of this configuration will be:

\[P = \frac{N!}{n_1! \, n_2! \ldots n_m!}\]

If N and all the n_i are large numbers, Stirling’s formula gives for the entropy of the system:

\[S = k \log P = kN \log N - k \sum_1^m n_i \log n_i\]

Suppose that to each cell corresponds a given value of a certain function ε that we shall name “the energy of an object placed in this cell.” Let us consider a modification of the distribution of objects among cells subject to the condition of leaving invariable the sum of the energies. The entropy S will vary by:

\[\delta S = -k \delta \left[ \sum_1^m n_i \log n_i \right] = -k \sum_1^m \delta n_i - k \sum_1^m \log n_i \delta n_i\]

with the adjoined conditions: Σ₁^m *δ**n_i* = 0 and Σ₁^m ε_i*δ**n_i* = 0.

The maximum entropy is determined by the relation: δS = 0. The method of indeterminate coefficients teaches us that, to realize this condition, one must satisfy the equation:

\[\sum_1^m \left[ \log n_i + \eta + \beta \varepsilon_i \right] \delta n_i = 0\]

where η and β are constants, and this whatever the *δ**n_i* may be.

One concludes from this that the most probable distribution, the only one realized in practice, is governed by the law:

\[n_i = \alpha e^{-\beta \varepsilon_i} \qquad (\alpha = e^{-\eta})\]

This is the so-called “canonical” distribution. The thermodynamic entropy of the system corresponding to this most probable distribution is given by:

\[S = kN \log N - \sum_1^m \left[ k\alpha e^{-\beta \varepsilon_i} (\log \alpha - \beta \varepsilon_i) \right]\]

or since Σ₁^m n_i = N and Σ₁^m ε_in_i = total energy E:

\[S = kN \log \frac{N}{\alpha} + k\beta E = kN \log \sum_1^m e^{-\beta \varepsilon_i} + k\beta E\]

To determine β we shall use the thermodynamic relation:

\[\frac{1}{T} = \frac{dS}{dE} = \frac{\partial S}{\partial \beta} \cdot \frac{\partial \beta}{\partial E} + \frac{\partial S}{\partial E}\]

and, because

\[N \frac{\sum_1^m \varepsilon_i e^{-\beta \varepsilon_i}}{\sum_1^m e^{-\beta \varepsilon_i}} = N\bar{\varepsilon} = E\]

\[\frac{1}{T} = k\beta, \qquad \beta = \frac{1}{kT}\]

The free energy is calculated by the relation:

\[F = E - TS = E - kNT \log \left[ \sum_1^m e^{-\beta \varepsilon_i} \right] - \beta kTE\]

\[= -kNT \log \left[ \sum_1^m e^{-\beta \varepsilon_i} \right]\]

The mean value of the free energy per object is therefore:

\[\bar{F} = -kT \log \left[ \sum_1^m e^{-\beta \varepsilon_i} \right]\]

Let us apply these general considerations to a gas formed of identical molecules of mass m₀. Liouville’s theorem (equally valid in relativistic dynamics) teaches us that the element of phase space extension of a molecule equal to dxdydzdpdqdr (where x, y, and z are the coordinates, p, q, r the corresponding momenta) is an invariant of the equations of motion whose value is independent of the choice of coordinates. One has consequently been led to admit that the number of states of equal probability represented by an element of this phase space extension was proportional to its size. This leads immediately to Maxwell’s distribution law giving the number of atoms whose representative point falls in the element dxdydzdpdqdr:

\[dn = \text{Cte} \cdot e^{-\frac{w}{kT}} dxdydzdpdqdr\]

w being the kinetic energy of these atoms.

Suppose the velocities are sufficiently small to justify the use of classical dynamics; we then find:

\[w = \frac{1}{2} m_0 v^2 \qquad dpdqdr = 4\pi G^2 dG\]

where G = m₀v = √(2m₀w) is the momentum. Finally, the number of atoms contained in the volume element whose energy is comprised between w and w + dw is given by the classical formula:

\[dn = \text{Cte} \cdot e^{-\frac{w}{kT}} 4\pi m_0^{\frac{3}{2}} \sqrt{2w} \, dw \, dxdydz\]

It remains to calculate the free energy and the entropy. For this, we shall take as the object of the general theory not an isolated molecule, but an entire gas formed of N identical molecules of mass m₀ whose state is consequently defined by 6N parameters. The free energy of the gas in the thermodynamic sense will be defined in the manner of Gibbs, as the mean value of the free energy of the N gases, namely:

\[\bar{F} = -kT \log \left[ \sum_1^m e^{-\beta \varepsilon_i} \right] \qquad \beta = \frac{1}{kT}\]

M. Planck has specified how this sum should be carried out; it can be expressed by an integral extended to all of phase space with 6N dimensions, an integral which itself is equivalent to the product of N sextuple integrals extended to the phase space of each molecule; but one must take care to divide the result by N! by reason of the identity of the molecules. The free energy being thus calculated, one deduces from it the entropy and the energy by the classical thermodynamic relations:

\[S = -\frac{\partial F}{\partial T} \qquad E = F + TS\]

To carry out the calculations, one must specify what is the constant whose product by the element of phase space gives the number of equally probable states represented by points of this element. This factor has the dimensions of the inverse of the cube of an action. M. Planck determines it by the following somewhat disconcerting hypothesis: “The phase space extension of a molecule is divided into cells of equal probability whose value is finite and equal to h³.” One can say either that in the interior of each cell, there is a single point whose probability is not zero, or that all points of the same cell correspond to states impossible to distinguish physically.

Planck’s hypothesis leads to writing for the free energy:

\[F = -kT \log \left[ \frac{1}{N!} \left( \int \int \int_{-\infty}^{+\infty} \int \int \int e^{-\frac{\varepsilon}{kT}} \frac{dxdydzdpdqdr}{h^3} \right)^N \right]\]

\[= -NkT \log \left[ \frac{e}{N} \int \int \int_{-\infty}^{+\infty} \int \int \int \frac{1}{h^3} e^{-\frac{\varepsilon}{kT}} dxdydzdpdqdr \right]\]

One finds by carrying out the integration:

\[F = Nm_0 c^2 - kNT \log \left[ \frac{eV}{Nh^3} (2\pi m_0 kT)^{\frac{3}{2}} \right] \qquad V = \text{total volume of the gas}\]

and, consequently,

\[S = kN \log \left[ \frac{e^{\frac{5}{2}} V}{Nh^3} (2\pi m_0 kT)^{\frac{3}{2}} \right]\]

\[E = Nm_0 c^2 + \frac{3}{2} kNT\]

At the end of his book “Wärmestrahlung” (4th ed.), Planck shows how one deduces from this the “chemical constant” intervening in the equilibrium of a gas with its condensed phase. Measurements of this chemical constant have given strong support to Planck’s method.

Until now we have brought in neither Relativity nor our ideas on the connection of dynamics with wave theory. We shall seek how the preceding formulas are modified by the introduction of these two notions.

II. New Conception of the Statistical Equilibrium of a Gas

If the motion of gaseous atoms is accompanied by a propagation of waves, the vessel containing the gas is going to be traversed in all directions by these waves. We are naturally led to consider, as in the conception of black-body radiation developed by M. Jeans, the phase waves forming stationary systems (that is to say, resonating on the dimensions of the enclosure) as being the only stable ones; they alone would intervene in the study of thermodynamic equilibrium. This is something analogous to what we encountered regarding Bohr’s atom; there too, the stable trajectories were defined by a resonance condition and the others had to be considered as normally unrealizable in the atom. ³⁴

³⁴ Standing wave condition for matter waves: Just as only certain standing waves fit in a cavity (this is what quantizes light in black-body radiation), de Broglie proposes that only certain matter waves can exist in a container. In the limit of a small box, only specific wavelengths fit, leading to quantized energies. This is the conceptual foundation for the “particle in a box” model you encounter in quantum chemistry. The energy levels go as E_n = n²h²/(8mL²), where n = 1, 2, 3… This simple model explains why conjugated π systems (like in dyes) absorb specific colors: electrons are confined to “boxes” of specific lengths.

One might ask how there can exist in a gas stationary systems of phase waves since the motion of atoms is constantly disturbed by their mutual collisions. One can first reply that thanks to the lack of coordination of molecular motion, the number of atoms diverted from their primitive direction during time dt by the effect of collisions is exactly compensated by the number of those whose motion is brought back by said effect in the same direction; everything happens in sum as if the atoms described a rectilinear trajectory from one wall to the other since their identity of structure dispenses with taking account of their individuality. Moreover, during the duration of the free path, the phase wave can traverse several times the length of a vessel even of large dimension; if, for example, the mean velocity of the atoms of a gas is 10⁵ cm/sec and the mean free path 10⁻⁵ cm, the mean velocity of the phase waves will be c²/v = 9 × 10¹⁵ cm/sec and during the time 10⁻¹⁰ second necessary on average for the free path, it will progress 9 × 10⁵ cm or 9 kilometers. It therefore seems possible to imagine the existence of stationary phase waves in a gaseous mass in equilibrium.

To better understand the nature of the modifications that we are going to have to make to statistical mechanics, we shall first consider the simple case where molecules move along a line AB of length l reflecting at A and B. The initial distribution of positions and velocities is supposed regulated by chance. The probability that a molecule is found on an element dx of AB is therefore dx/l. In the classical conception, one must moreover take the probability of a velocity comprised between v and v + dv proportional to dv; therefore if one constitutes a phase space taking as variables x and v, all equal elements dxdv will be equally probable. It is quite otherwise when one introduces the stability conditions considered above. If the velocities are sufficiently small to permit neglecting the terms of Relativity, the wavelength linked to the motion of a molecule whose velocity is v, will be:

\[\lambda = \frac{c/\beta}{m_0 c^2 / h} = \frac{h}{m_0 v}\]

and the resonance condition will be written:

\[l = n\lambda = n \frac{h}{m_0 v} \qquad (n, \text{ integer})\]

Setting h/(m₀l) = v₀, we get:

\[v = nv_0\]

The velocity can therefore only take values equal to integer multiples of v₀.

The variation δn of the integer n corresponding to a variation δv of the velocity gives the number of states of a molecule compatible with the existence of stationary phase waves. One sees at once that:

\[\delta n = \frac{m_0 l}{h} \delta v\]

Everything will happen as if, to each element of the phase space, there corresponded (m₀/h) δxδv possible states, which is the classical expression for the element of phase space divided by h. Examination of numerical values shows that to a value of δv even extremely small on the scale of our experimental measurements, corresponds a large interval δn; any rectangle even very small of the phase space corresponds to an enormous number of “possible” values of v. One can therefore in general treat the quantity δxδv as a differential in calculations.

But, in principle, the distribution of representative points is no longer at all that which statistical mechanics imagines; it is discontinuous and supposes that, by the action of a mechanism still impossible to specify, the motions of atoms which would be linked to non-stationary systems of phase waves are automatically eliminated.

Let us now pass to the more realistic case of the three-dimensional gas. The distribution of phase waves in the enclosure will be entirely analogous to that which the old theory of black-body radiation gave for thermal waves. One can, just as M. Jeans did in this case, calculate the number of stationary waves contained in unit volume and whose frequencies are comprised between ν and ν + δν. One finds for this number, distinguishing the group velocity U from the phase velocity V, the following expression:

\[n_\nu \delta \nu = \gamma \cdot \frac{4\pi}{UV^2} \nu^2 \delta \nu\]

γ being equal to 1 for longitudinal waves and to 2 for transverse waves. The preceding expression should not delude us: all values of ν are not present in the system of waves and, if it is permissible to consider in calculations the expression above as a differential, it is because in general, in a very small frequency interval, there will be an enormous number of admissible values for ν.

The moment has come to make use of the theorem demonstrated in Chapter I, paragraph II. To an atom of velocity U = βc, corresponds a wave having for phase velocity V = c/β, for group velocity U = βc, and for frequency ν = (1/h) × m₀c²/√(1 − β²). If w denotes the kinetic energy, one finds by the formulas of Relativity:

\[h\nu = \frac{m_0 c^2}{\sqrt{1 - \beta^2}} = m_0 c^2 + w = m_0 c^2 (1 + \alpha) \qquad \left( \alpha = \frac{w}{m_0 c^2} \right)\]

Whence:

\[n_w dw = \gamma \cdot \frac{4\pi}{UV^2} \nu^2 d\nu = \gamma \cdot \frac{4\pi}{h^3} m_0^2 c(1 + \alpha) \sqrt{\alpha(\alpha + 2)} \, dw\]

If one applies to the ensemble of atoms the law of canonical distribution demonstrated above, one obtains for the number of those which are contained in the volume element dxdydz and whose kinetic energy is comprised between w and w + dw:

\[(1) \qquad \text{Cte} \cdot \gamma \cdot \frac{4\pi}{h^3} m_0^2 c(1 + \alpha) \sqrt{\alpha(\alpha + 2)} \, e^{-\frac{w}{kT}} dw \, dxdydz\]

For material atoms, the phase waves must by reason of symmetry be analogous to longitudinal waves; let us therefore set γ = 1. Moreover, for these atoms (except for some negligible in number at ordinary temperatures), the proper energy m₀c² is infinitely greater than the kinetic energy. We can therefore confuse 1 + α with unity and find for the number defined above:

\[\text{Cte} \cdot \frac{4\pi}{h^3} m_0^{\frac{3}{2}} \sqrt{2w} \, e^{-\frac{w}{kT}} dw \, dxdydz =\]

\[= \text{Cte} \cdot e^{-\frac{w}{kT}} \int_w^{w+dw} \frac{dxdydzdpdqdr}{h^3}\]

It is visible that our method leads us to take as the measure of the number of possible states of the molecule corresponding to an element of its phase space not the size of this element itself but this size divided by h³. We therefore justify M. Planck’s hypothesis and, consequently, the results obtained by this scientist and expounded above. One will note that it is the values found for the velocities Vand U of the phase wave which have permitted arriving at this result starting from Jeans’ formula.⁽¹³⁾

III. The Gas of Light Atoms

If light is divided into atoms, black-body radiation can be considered as a gas of such atoms in equilibrium with matter somewhat like a saturated vapor is in equilibrium with its condensed phase. We have already shown in Chapter III that this idea leads to an exact prediction of radiation pressure.

Let us try to apply to such a gas of light the general formula (1) of the preceding paragraph. Here one must set γ = 2 by reason of the symmetry of the luminous unit on which we insisted in Chapter IV. Moreover, α is very large compared to unity, if one excepts a few atoms negligible in number at ordinary temperatures, which permits confusing α + 1 and α + 2 with α. One would therefore obtain for the number of atoms per volume element, of energy comprised between hν and h(ν + dν):

\[\text{Cte} \frac{8\pi}{c^3} \nu^2 e^{-\frac{h\nu}{kT}} d\nu \, dxdydz\]

and for the energy density corresponding to the same frequencies:

\[u_\nu d\nu = \text{Cte} \frac{8\pi h}{c^3} \nu^3 e^{-\frac{h\nu}{kT}} d\nu\]

It would moreover be easy to show that the constant is equal to 1 by following a reasoning contained in my article “Light Quanta and Black-Body Radiation” published in the Journal de Physique in November 1922.

Unfortunately, the law thus obtained is Wien’s law which is only the first term of the series that constitutes the experimentally exact Planck law. ³⁵

³⁵ Wien’s law vs. Planck’s law: The black-body radiation spectrum has a characteristic shape: intensity rises with frequency, peaks, then falls. Wien’s law (1896) correctly describes the high-frequency side but fails at low frequencies. Rayleigh-Jeans law works at low frequencies but gives the ultraviolet catastrophe at high frequencies. Planck’s law (1900) gets the entire spectrum right: u(ν) = (8πhν³/c³)/(e^hν/kT − 1). De Broglie shows that treating photons as independent particles gives Wien’s law. To get Planck’s law, he introduces a “coherence hypothesis”: photons traveling together on the same wave are not independent. This foreshadows Bose-Einstein statistics, which correctly describes photons (and all bosons).

This should not surprise us because, in supposing the motions of the atoms of light completely independent, we must necessarily arrive at a law whose exponential factor is identical to that of Maxwell’s law.

We know moreover that a continuous distribution of radiant energy in space would lead to the Rayleigh law as Jeans’ reasoning shows. Now, Planck’s law admits the expressions proposed by Messrs. Wien and Lord Rayleigh as limiting forms valid respectively for very large and very small values of the quotient hν/(kT). To recover Planck’s result, it will therefore be necessary here to make a new hypothesis which without distancing us from the conception of light quanta, permits us to explain how the classical formulas can be valid in a certain domain. We state this hypothesis in the following way:

“If two or more atoms have phase waves that superimpose exactly, of which one can say consequently that they are transported by the same wave, their motions can no longer be considered as entirely independent and these atoms can no longer be treated as distinct units in probability calculations.” The motion of these atoms “in wave” would therefore present a sort of coherence as a result of interactions impossible to specify, but probably related to the mechanism which would render unstable the motion of atoms whose phase wave would not be stationary.

This coherence hypothesis obliges us to take up again entirely the demonstration of Maxwell’s law. Since we can no longer take each atom as “object” of the general theory, it is the elementary stationary phase waves that must play this role. What do we call an elementary stationary wave? A stationary wave can be regarded as due to the superposition of two waves of formulas:

\[\sin \left[ 2\pi \left( \nu t - \frac{x}{\lambda} + \varphi_0 \right) \right] \quad \text{and} \quad \sin \left[ 2\pi \left( \nu t + \frac{x}{\lambda} + \varphi_0 \right) \right]\]

where φ₀ can take all values from 0 to 1. By giving to ν one of the permitted values and to φ₀ an arbitrary value between 0 and 1, one defines an elementary stationary wave. Consider a determined value of φ₀ and all the permitted values of ν comprised in a small interval dν. Each elementary wave can transport 0, 1, 2… atoms and, since the law of canonical distribution must be applicable to the waves considered, we find for the number of atoms corresponding:

\[N_\nu d\nu = n_\nu d\nu \frac{\sum_1^\infty p e^{-p \frac{h\nu}{kT}}}{\sum_0^\infty e^{-p \frac{h\nu}{kT}}}\]

By giving to φ₀ other values, one will obtain other stable states and by superposing several of these stable states in such a way that a single stationary wave corresponds to several elementary waves, one will obtain yet another stable state. We conclude from this that the number of atoms whose total energy corresponds to frequencies comprised between ν and ν + dν is:

\[N_\nu d\nu = A\gamma \frac{4\pi}{h^3} m_0^2 c(1 + \alpha) \sqrt{\alpha(\alpha + 2)} \, dw \sum_1^\infty e^{-p \frac{m_0 c^2 + w}{kT}}\]

per unit volume. A can be a function of the temperature.

For a gas in the ordinary sense of the word, m₀ is so large that one can neglect all terms of the series except the first. One indeed recovers formula (1) of the preceding paragraph.

For the gas of light, one will now find:

\[N_\nu d\nu = A \frac{8\pi}{c^3} \nu^2 \sum_1^\infty e^{-p \frac{h\nu}{kT}} d\nu\]

and, consequently, for the energy density:

\[u_\nu d\nu = A \cdot \frac{8\pi h}{c^3} \nu^3 \sum_1^\infty e^{-p \frac{h\nu}{kT}} d\nu\]

This is indeed Planck’s form. But it must be shown that in this case A = 1. First of all, A is here certainly a constant and not a function of temperature. Indeed, the total energy of radiation per unit volume is:

\[u = \int_0^{+\infty} u_\nu d\nu = A \cdot \frac{48\pi h}{c^3} \left( \frac{kT}{h} \right)^4 \sum_1^\infty \frac{1}{p^4}\]

and the total entropy is given by:

\[dS = \frac{1}{T} \left[ d(uV) + P \, dV \right] = V \frac{du}{T} + (u + P) \frac{dV}{T} \qquad (V \text{ total volume})\]

\[= \frac{V}{T} \frac{du}{dT} dT + \frac{4}{3} u \frac{dV}{T}\]

because u = f(T) and P = (1/3)u; dS being an exact differential, the integrability condition is written:

\[\frac{1}{T} \frac{du}{dT} = \frac{4}{3} \frac{1}{T} \frac{du}{dT} - \frac{4}{3} \frac{u}{T^2} \quad \text{or} \quad 4 \frac{u}{T} = \frac{du}{dT} \qquad u = aT^4\]

This is the classical Stefan law which obliges us to set A = Constant. The preceding reasoning furnishes us the values of the entropy and the free energy:

\[S = A \cdot \frac{64\pi}{c^3 h^3} k^4 T^3 V \sum_1^\infty \frac{1}{p^4}\]

\[F = U - TS = -A \cdot \frac{16\pi}{c^3 h^3} k^4 T^4 V \sum_1^\infty \frac{1}{p^4}\]

It remains to determine the constant A. If we succeed in demonstrating that it is unity, we shall have recovered all the formulas of Planck’s theory.

As we said above, if one neglects the terms where p > 1, the matter is easy; the distribution of atoms obeying the simple canonical law:

\[A \cdot \frac{8\pi}{c^3} \nu^2 e^{-\frac{h\nu}{kT}} d\nu\]

one can carry out the calculation of free energy by Planck’s method as for an ordinary gas and, by identifying the result with the expression above, one finds A = 1.

In the general case, one must employ a more roundabout method. Consider the pth term of Planck’s series:

\[u_{\nu p} d\nu = A \cdot \frac{8\pi}{c^3} h\nu^3 e^{-p \frac{h\nu}{kT}} d\nu\]

One can also write it:

\[A \frac{8\pi}{c^3 p} \nu^2 e^{-p \frac{h\nu}{kT}} d\nu \cdot p \cdot h\nu\]

which permits saying:

“Black-body radiation can be considered as the mixture of an infinity of gases each characterized by an integer value p and enjoying the following property: the number of possible states of a gaseous unit situated in a volume element dxdydz and having an energy comprised between phν and ph(ν + dν) is equal to (8π/c³p) ν²dν dxdydz.” Consequently, one can calculate the free energy by the method of the first paragraph. One obtains:

\[F = \sum_1^\infty F_p = -kT \sum_1^\infty \log \left[ \frac{1}{n_p!} \left( V \int_0^\infty \frac{8\pi}{c^3 p} \nu^2 e^{-p \frac{h\nu}{kT}} d\nu \right)^{n_p} \right]\]

\[= -kT \sum_1^\infty n_p \log \left[ \frac{e}{n_p} V \int_0^{+\infty} \frac{8\pi}{c^3 p} \nu^2 e^{-p \frac{h\nu}{kT}} d\nu \right]\]

where

\[n_p = V \int_0^{+\infty} A \frac{8\pi}{pc^3} \nu^2 e^{-p \frac{h\nu}{kT}} d\nu = A \cdot \frac{16\pi}{c^3} \cdot \frac{k^3 T^3}{h^3} \cdot \frac{1}{p^4} \cdot V\]

Therefore:

\[F = -A \frac{16\pi}{c^3 h^3} k^4 T^4 \log \left( \frac{e}{A} \right) \sum_1^\infty \frac{1}{p^4} \cdot V\]

and, by identification with the expression previously found:

\[\log \left( \frac{e}{A} \right) = 1 \qquad A = 1\]

This is what we wanted to demonstrate.

The coherence hypothesis adopted above has therefore brought us to port safely by preventing us from running aground on the Rayleigh law or on that of Wien. The study of fluctuations in black-body radiation will provide us with a new proof of its importance.

IV. Energy Fluctuations in Black-Body Radiation⁽¹⁴⁾

If energy grains of value q are distributed in very large numbers in a certain space and if their positions vary ceaselessly according to the laws of chance, a volume element will normally contain n̄ grains, that is an energy Ē = n̄q. But the real value of n will constantly deviate from n̄ and we shall have (n − n̄)² = n̄ according to a known theorem of probability theory and, consequently, the mean quadratic fluctuation of the energy will be:

\[\overline{\varepsilon^2} = \overline{(n - \bar{n})^2} q^2 = \bar{n}q^2 = \bar{E}q\]

On the other hand, we know that the energy fluctuations in a volume V of black-body radiation are governed by the law of statistical thermodynamics:

\[\overline{\varepsilon^2} = kT^2 V \frac{d(u_\nu d\nu)}{dT}\]

in so far as they relate to the frequency interval ν, ν + dν. If one admits the Rayleigh law:

\[u_\nu = \frac{8\pi k}{c^3} \nu^2 T, \qquad \overline{\varepsilon^2} = \frac{c^3}{8\pi \nu^2 d\nu} \cdot \frac{(u_\nu d\nu V)^2}{V}\]

and this result, as one should expect, coincides with that which the calculation of interferences furnishes according to the rules of electromagnetic theory.

If, on the contrary, one adopts Wien’s law which corresponds to the hypothesis of a radiation formed of entirely independent atoms, one finds:

\[\overline{\varepsilon^2} = kT^2 V \frac{d}{dT} \left( \frac{8\pi h}{c^3} \nu^3 e^{-\frac{h\nu}{kT}} d\nu \right) = (u_\nu V d\nu) h\nu\]

a formula which is also deduced from ε̄² = Ēhν.

Finally, in the real case of Planck’s law, one arrives, as Einstein first remarked, at the expression:

\[\overline{\varepsilon^2} = (u_\nu V d\nu) h\nu + \frac{c^3}{8\pi \nu^2 d\nu} \cdot \frac{(u_\nu d\nu V)^2}{V}\]

ε̄² therefore appears as the sum of what it would be: 1° if the radiation were formed of independent quanta hν; 2° if the radiation were purely wave-like.

On the other hand, the conception of groupings of atoms “in waves” leads us to write Planck’s law:

\[u_\nu d\nu = \sum_1^\infty \frac{8\pi h}{c^3} \nu^3 e^{-p \frac{h\nu}{kT}} d\nu = \sum_1^\infty n_{p,\nu} d\nu \cdot ph\nu\]

and, applying to each kind of grouping the formula ε̄² = n̄q², one obtains:

\[\overline{\varepsilon^2} = \sum_1^\infty n_{p,\nu} d\nu (ph\nu)^2\]

Naturally this expression is fundamentally identical to that of Einstein; only the manner of writing differs. But its interest is to lead us to the following statement: “One can equally evaluate correctly the fluctuations of black-body radiation without making any appeal to the theory of interference, but by introducing the coherence of atoms linked to the same phase wave.”

It therefore seems almost certain that any attempt at reconciliation between the discontinuity of radiant energy and interference would have to involve the coherence hypothesis of the last paragraph.

Appendix to Chapter V: On Light Quanta

We have proposed to consider light atoms as small centers of energy characterized by a very small proper mass m₀ and animated with velocities generally very close to c, in such a way that there exists between the frequency ν, the proper mass m₀, and the velocity βc the relation:

\[h\nu = \frac{m_0 c^2}{\sqrt{1 - \beta^2}}\]

from which one deduces:

\[\beta = \sqrt{1 - \left( \frac{m_0 c^2}{h\nu} \right)^2}\]

This way of seeing has led us to remarkable concordances concerning the Doppler effect and radiation pressure.

Unfortunately, it raises a great difficulty: for frequencies ν that become smaller and smaller, the velocity βc of the radiant energy would become smaller and smaller, would vanish for hν = m₀c² and would then become imaginary (?). This is all the more difficult to admit since, in the domain of very low frequencies, one should expect to recover the conclusions of the old theories which assign to radiant energy the velocity c.

This objection is very interesting because it draws attention to the passage from the purely corpuscular form of light manifesting itself in the domain of high frequencies to the purely wave-like form of very low frequencies. We have shown in Chapter VII that the purely corpuscular conception leads to Wien’s law while, as is well known, the purely wave-like conception leads to Rayleigh’s law. The passage from one to the other of these laws must, it seems to me, be closely linked to the answers which can be given to the objection stated above.

I am going to develop, more by way of example than in the hope of providing a satisfactory solution, an idea suggested by the preceding reflections.

In Chapter VII, I showed that it was possible to interpret the passage from Wien’s law to Rayleigh’s law by conceiving the existence of ensembles of light atoms linked to the propagation of a single phase wave. I insisted on the resemblance that such a wave carrying numerous quanta will take with the classical wave when the number of quanta increases indefinitely. However, this resemblance would be limited in the conception expounded in the text by the fact that each grain of energy would retain the very small, but finite, proper mass m₀ while electromagnetic theory attributes to light a zero proper mass. The frequency of the wave with multiple energy centers is determined by the relation:

\[h\nu = \frac{\mu_0 c^2}{\sqrt{1 - \beta^2}}\]

where μ₀ is the proper mass of each of the centers: this seems necessary to account for the emission and absorption of energy in finite quantities hν. But we could perhaps suppose that the mass of the energy centers linked to a single wave differs from the proper mass m₀ of an isolated center and depends on the number of other centers with which they find themselves in interaction. One would then have:

\[\mu_0 = f(p) \qquad \text{with} \quad f(1) = m_0\]

designating by p the number of centers carried by the wave.

The necessity of falling back on the formulas of electromagnetism for very low frequencies would lead to supposing that f(p) is a decreasing function of p tending toward 0 when p tends toward infinity. The velocity of the ensemble of p centers forming a wave would then be:

\[\beta c = c\sqrt{1 - \left[ \frac{f(p)c^2}{h\nu} \right]^2}\]

For very high frequencies, p would almost always be equal to 1, the energy grains would be isolated, one would have Wien’s law for black-body radiation and the formula of the text β = √[1 − m₀²c⁴/(h²ν²)] for the velocity of radiant energy.

For very low frequencies, p would always be very large, the grains would be gathered in very numerous groups on a single wave. Black-body radiation would obey Rayleigh’s law and the velocity would tend toward c when ν tends toward 0.

The preceding hypothesis somewhat destroys the simplicity of the conception of the “light quantum,” but this simplicity certainly cannot be entirely conserved if one wants to be able to connect electromagnetic theory with the discontinuity revealed by photoelectric phenomena. This connection would be obtained, it seems to me, by the introduction of the function f(p) because, for a given energy, a wave will have to comprise a number p of grains increasingly large when ν and hν decrease; when the frequency becomes smaller and smaller, the number of grains must increase indefinitely, their proper mass tending toward 0 and their velocity toward c, so that the wave transporting them would become more and more analogous to the electromagnetic wave.

It must be admitted that the real structure of luminous energy still remains very mysterious.

Summary and Conclusions

In a brief history of the development of Physics since the 17th century and in particular of Dynamics and Optics, we have shown how the problem of quanta was in some way contained in germ in the parallelism of the corpuscular and wave-like conceptions of radiation; then, we recalled with what ever-increasing intensity the notion of quanta had imposed itself on the attention of 20th-century scientists.

In the first chapter, we admitted as a fundamental postulate the existence of a periodic phenomenon linked to every isolated piece of energy and depending on its proper mass by the Planck-Einstein relation. The theory of Relativity then showed us the necessity of associating with the uniform motion of any moving body the propagation at constant velocity of a certain “phase wave” and we were able to interpret this propagation by the consideration of Minkowski space-time.

Taking up again in Chapter II the same question in the more general case of an electrically charged body moving with varied motion in an electromagnetic field, we showed that, according to our ideas, the principle of least action in its Maupertuisian form and the principle of phase concordance due to Fermat could well be two aspects of a single law; this led us to conceive an extension of the quantum relation giving the velocity of the phase wave in the electromagnetic field. Certainly, this idea that the motion of a material point always conceals the propagation of a wave would need to be studied and completed, but if one succeeded in giving it an entirely satisfactory form, it would represent a synthesis of great rational beauty.

The most important consequence that one can draw from it is expounded in Chapter III. After having recalled the laws of stability of quantized trajectories as they result from numerous recent works, we showed that they can be interpreted as expressing the resonance of the phase wave over the length of closed or quasi-closed trajectories. We believe that this is the first physically plausible explanation proposed for these Bohr-Sommerfeld stability conditions.

The difficulties raised by the simultaneous displacements of two electric centers are studied in Chapter IV, in particular in the case of the circular motions of the nucleus and of the electron around their center of gravity in the hydrogen atom.

In Chapter V, guided by the results previously obtained, we seek to represent to ourselves the possibility of a concentration of radiant energy around certain singular points and we show what profound harmony seems to exist between the opposing points of view of Newton and of Fresnel and to be revealed by the identity of numerous predictions. Electromagnetic theory cannot be integrally conserved in its present form, but its reworking is a difficult task; we suggest in this connection a qualitative theory of interference.

In Chapter VI, we summarize the various successive theories of the scattering of X-rays and γ-rays by amorphous bodies, insisting particularly on the very recent theory of Messrs. P. Debye and A. H. Compton which renders, it seems, almost tangible the existence of light quanta.

Finally, in Chapter VII, we introduce the phase wave into Statistical Mechanics, we also recover the value of the element of phase space extension that Planck had proposed, and we obtain the law of black-body radiation as Maxwell’s law of a gas formed of light atoms, provided however that we admit a certain coherence between the motions of certain atoms, a coherence whose interest the study of energy fluctuations also seems to show.

In short, I have developed new ideas which can perhaps contribute to hastening the necessary synthesis which will once again unify the physics of radiation, today so strangely split into two domains where two opposing conceptions respectively reign: the corpuscular conception and that of waves. I have sensed that the principles of the Dynamics of the material point, if one knew how to analyze them correctly, would doubtless present themselves as expressing propagations and phase concordances, and I have sought, as best I could, to draw from this the explanation of a certain number of enigmas posed by the theory of Quanta. In attempting this effort, I have arrived at some interesting conclusions which perhaps permit hoping to arrive at more complete results by pursuing the same path. But it would first be necessary to constitute a new electromagnetic theory conforming naturally to the principle of Relativity, accounting for the discontinuous structure of radiant energy and for the physical nature of phase waves, finally leaving to Maxwell-Lorentz theory a character of statistical approximation which would explain the legitimacy of its use and the accuracy of its predictions in a very great number of cases.

I have intentionally left rather vague the definitions of the phase wave and of the periodic phenomenon of which it would be in some way the translation, as well as that of the light quantum. The present theory must therefore be considered rather as a form whose physical content is not entirely specified than as a homogeneous doctrine definitively constituted.

Notes

1. Let us cite here some works where questions relating to quanta are treated:

   J. Perrin, Les atomes, Alcan, 1913.

   H. Poincaré, Dernières pensées, Flammarion, 1913.

   E. Bauer, Recherches sur le rayonnement, Doctoral Thesis, 1912.

   La théorie du rayonnement et les quanta (1st Solvay Congress, 1911), published by P. Langevin and M. de Broglie.

   M. Planck, Theorie der Wärmestrahlung, J.-A. Barth, Leipzig, 1921 (4th ed.).

   L. Brillouin, La théorie des quanta et l’atome de Bohr (Conference reports), 1921.

   F. Reiche, Die Quantentheorie, J. Springer, Berlin, 1921.

   A. Sommerfeld, La constitution de l’atome et les raies spectrales. Trans. Bellenot, A. Blanchard, 1923.

   A. Landé, Vorschritte der Quantentheorie, F. Steinhopff, Dresden, 1922.

   Atomes et électrons (3rd Solvay Congress), Gauthier-Villars, 1923.

2. Regarding the difficulties that arise in the interaction of several electrified centers, see Chapter IV below.

3. A. Einstein, “Zum Quantensatz von Sommerfeld und Epstein” (Ber. deutschen. Phys. Ges., 1917, p. 82).

4. See A. Einstein, Ann. d. Phys., 17, 132 (1905); Phys. Zeitsch., 10, 185 (1909).

5. Regarding the objections raised by the ideas contained in this paragraph, see the appendix.

6. Regarding the difficulties raised for the theory of light quanta by the phenomena of wave optics, see H. A. Lorentz, Phys. Zeitschr., 11, 1234 (1910); A. Einstein, Ann. d. Phys., 20, 199 (1906) and Phys. Zeitschr., 10, 185 (1909); W. H. Bragg, Studies in Radioactivity, Macmillan (1912).

7. A. Einstein, Phys. Zeitschr., 18, 121 (1917); L. Brillouin, J. de Phys. (6), 2, 142 (1921).

8. J. J. Thomson, Conduction of Electricity through Gases (2nd ed.), Cambridge, 1906, p. 325.

9. Regarding the electromagnetic theory of scattering, see Lord Rayleigh, Phil. Mag. (5), 12, 81 (1881); J. J. Thomson, loc. cit.; H. A. Lorentz, Theory of Electrons, Leipzig, 1909.

10. C. G. Barkla, Phil. Mag., 7, 543 (1904); 21, 648 (1911).

11. P. Debye, Ann. d. Phys., 46, 809 (1915).

12. A. H. Compton, Phys. Rev., 21, 484 (1923) and 22, 409 (1923); Phil. Mag., 46, 897 (1923). P. Debye, Phys. Zeitschr., 24, 161 (1923). See also the reports of the 1923 Brussels Congress.

13. On the subject of this paragraph, see: O. Sackur, Ann. d. Phys., 36, 958 (1911) and 40, 67 (1913); H. Tetrode, Phys. Zeitschr., 14, 212; Ann. d. Phys., 38, 434 (1912); W. H. Keesom, Phys. Zeitschr., 15, 695 (1914); O. Stern, Phys. Zeitschr., 14, 629 (1913); E. Brody, Zeitschr. f. Phys., 16, 79 (1921).

14. La théorie du Rayonnement noir et les quanta, Solvay Reunion, report by M. Einstein, p. 419; Les théories statistiques en thermodynamique, Lectures by M. H. A. Lorentz at the Collège de France, Teubner, 1916, pp. 70 and 114.