## Recherches sur la théorie des quanta.

Paris: Masson et Cie. 1924.

First edition, rare, of de Broglie’s revolutionary doctoral thesis on the quantum theory, which, Einstein said, “lifted a corner of the great veil” (Isaacson, *Einstein: His Life and Universe*, p. 327). In this work he developed the startling and revolutionary idea that material particles such as electrons have a wave as well as a corpuscular nature, analogous to the dual behavior of light demonstrated by Einstein and others in the first two decades of the century. In his 1905 paper ‘Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt’ (‘On a Heuristic Viewpoint Concerning the Production and Transformation of Light’), “Einstein postulated that light is composed of individual quanta (later called photons) that, in addition to wavelike behavior, demonstrate certain properties unique to particles. In a single stroke he thus revolutionized the theory of light and provided an explanation for, among other phenomena, the emission of electrons from some solids when struck by light, called the photoelectric effect” (*Britannica*). “The central idea of de Broglie’s work, contained in ‘Ondes et quanta’ [*Comptes Rendus*, 10 September, 1923] was that the formula *E = hν* by which Einstein had related the frequency *ν *of light to the energy *E* of light quanta should not only apply to light but also to material particles. For a particle at rest with mass *m* he concluded, since its energy is *E = mc ^{2}*, that it performs an internal oscillation with frequency

*ν = mc*He considered the motion of a particle, carefully taking into account the effects of the special theory of relativity, and was able to construct a wave which was always

^{2}/h.*in phase*with the internal oscillation of the particle … At the end of [the present paper] he gave an application of his theory by showing that he could naturally explain the discrete electron orbits in Bohr’s model of the hydrogen atom. Each stable orbit should be closed in the sense that the same phase should be assumed by the matter wave after completion of an orbit” (Brandt,

*The Harvest of a Century*, Chapter 32, p. 133). De Broglie published two further notes on matter waves in the same volume of

*Comptes Rendus,*which gave applications of his theory of matter waves, but his ideas became widely known only with the publication of his doctoral thesis

*Recherches sur la théorie des quanta*in the summer of 1924, which is an elaboration of the content of the three notes. “Louis de Broglie achieved a worldwide reputation for his discovery of the wave theory of matter, for which he received the Nobel Prize for physics in 1929. His work was extended into a full-fledged wave mechanics by Erwin Schrödinger and thus contributed to the creation of quantum mechanics” (DSB). In his first paper on wave mechanics (January 1926), Schrödinger acknowledged his debt to de Broglie, writing: “I have recently shown that the Einstein gas theory can be founded on the consideration of standing waves which obey the dispersion law of de Broglie … The above considerations about the atom could have been presented as a generalization of these considerations” (quoted in Pais,

*loc cit*., p. 439). De Broglie’s theory predicted that streams of electrons passing through a small aperture should exhibit the phenomenon of diffraction. Following the experimental observation of this phenomenon by C. J. Davisson and L. H. Germer in 1927, de Broglie was awarded the 1929 Nobel Prize in physics “for his discovery of the wave nature of electrons.” To de Broglie's theory one can trace the development of quantum mechanics and modern physics, as well as the innumerable practical applications of electron diffraction in the analysis of microscopic structures, from the fields of medicine and biology to electronics and computer science. De Broglie’s book

*Ondes et mouvements*(1926), selected by Carter and Muir for the

*Printing and the Mind of Man*exhibition and catalogue (1967), was an expansion of ideas first published in his thesis. Unlike his book, de Broglie’s thesis was issued in a very small edition. De Broglie's thesis was reprinted within the year in the

*Annales de Physique*(vol. 3), with the type largely reset; offprints of this journal appearance are known and are easily distinguished from this first edition. Both are rare: ABPC/PBH list 5 copies of the thesis.

“De Broglie begins his thesis with a historical introduction. Newtonian mechanics, he notes, was eventually formulated in terms of the principle of least action, which was first given by Maupertuis and then later in another form by Hamilton. As for the science of light and optics, the laws of geometrical optics were eventually summarised by Fermat in terms of a principle whose form is reminiscent of the principle of least action. Newton tried to explain some of the phenomena of wave optics in terms of his corpuscular theory, but the work of Young and Fresnel led to the rise of the wave theory of light, in particular the successful wave explanation of the rectilinear propagation of light (which had been so clear in the corpuscular or ‘emission’ theory). On this, de Broglie comments (p. 25):

‘When two theories, based on ideas that seem entirely different, account for the same experimental fact with equal elegance, one can always wonder if the opposition between the two points of view is truly real and is not due solely to an inadequacy of our efforts at synthesis.’

“This remark is, of course, a hint that the aim of the thesis is to effect just such a synthesis. De Broglie then turns to the rise of electrodynamics, relativity and the theory of energy quanta. He notes that Einstein’s theory of the photoelectric effect amounts to a revival of Newton’s corpuscular theory. De Broglie then sketches Bohr’s 1913 theory of the atom, and goes on to point out that observations of the photoelectric effect for X- and γ -rays seem to confirm the corpuscular character of radiation. At the same time, the wave aspect continues to be confirmed by the observed interference and diffraction of X-rays. Finally, de Broglie notes the very recent corpuscular interpretation of Compton scattering. De Broglie concludes his historical introduction with a mention of his own recent work (p. 30):

‘… the moment seemed to have arrived to make an effort towards unifying the corpuscular and wave points of view and to go a bit more deeply into the true meaning of the quanta. That is what we have done recently …’

“De Broglie clearly regarded his own work as a synthesis of earlier theories of dynamics and optics, a synthesis increasingly forced upon us by accumulating experimental evidence.

“Chapter 1 of the thesis is entitled ‘The phase wave’. De Broglie begins by recalling the equivalence of mass and energy implied by the theory of relativity. Turning to the problem of quanta, he remarks (pp. 32–3):

‘It seems to us that the fundamental idea of the 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: energy = *h *× frequency, where h is Planck’s constant.’

“To make sense of the quantum relation, de Broglie proposes that (p. 33)

‘… to each energy fragment of proper mass *m*0 there is attached a periodic phenomenon of frequency ν_{0} such that one has: *h*ν_{0} = *m*_{0}*c*^{2},ν_{0} being measured, of course, in the system tied to the energy fragment.’

“De Broglie asks if the periodic phenomenon must be assumed to be localised inside the energy fragment. He asserts that this is not at all necessary, and that it will be seen to be ‘without doubt spread over an extensive region of space’ (p. 34) …

Chapter 2 is entitled ‘Maupertuis’ principle and Fermat’s principle’. The aim is to generalise the results of the first chapter to non-uniform, non-rectilinear motion. In the introduction to chapter 2 de Broglie writes (p. 45):

‘Guided by the idea of a deep unity between the principle of least action and that of Fermat, from the beginning of my investigations on this subject I was led to *assume *that, for a given value of the total energy of the moving body and therefore of the frequency of its phase wave, the dynamically possible trajectories of the one coincided with the possible rays of the other.’

“De Broglie discusses the principle of least action, in the different forms given by Hamilton and by Maupertuis, and also for relativistic particles in an external electromagnetic field … De Broglie then discusses wave propagation and Fermat’s principle from a spacetime perspective … [and] arrives at the following statement (p. 56):

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

“He adds that (p. 56):

‘We think that this idea of a deep relationship between the two great principles of Geometrical Optics and Dynamics could be a valuable guide in realising the synthesis of waves and quanta.’

“De Broglie then discusses some particular cases: the free particle, a particle in an electrostatic field, and a particle in a general electromagnetic field. He calculates the phase velocity, which depends on the electromagnetic potentials. He notes that the propagation of a phase wave in an external field depends on the charge and mass of the moving body. And he shows that the group velocity along a ray is still equal to the velocity of the moving body …

“Thus, de Broglie’s unification of the principles of Maupertuis and Fermat amounts to a new dynamical law, in which the phase of a guiding wave determines the particle velocity. This new law of motion is the essence of de Broglie’s new, first-order dynamics.

Chapter 3 of de Broglie’s thesis is entitled ‘The quantum conditions for the stability of orbits’. De Broglie reviews Bohr’s condition for circular orbits, according to which the angular momentum of the electron must be a multiple of h/2π … He also reviews Sommerfeld’s generalisation, and Einstein’s invariant formulation. De Broglie then provides an explanation for Einstein’s condition. The trajectory of the moving body coincides with one of the rays of its phase wave, and the phase wave moves along the trajectory with a constant frequency (because the total energy is constant) and with a variable speed whose value has been calculated. To have a stable orbit, claims de Broglie, the length *l *of the orbit must be in ‘resonance’ with the wave … (Note that the simple argument commonly found in textbooks, about the fitting of whole numbers of wavelengths along a Bohr orbit, originates in this work of de Broglie’s.)

“De Broglie thought that his explanation of the stability or quantisation conditions constituted important evidence for his ideas. As he puts it (p. 65):

‘This beautiful result, whose demonstration is so immediate when one has accepted the ideas of the preceding chapter, is the best justification we can give for our way of attacking the problem of quanta.’

“Certainly, de Broglie had achieved a concrete realisation of his initial intuition that quantisation conditions for atomic energy levels could arise from the properties of waves.

“In his chapter 4, de Broglie considers the two-body problem, in particular the hydrogen atom. He expresses concern over how to define the proper masses, taking into account the interaction energy. He discusses the quantisation conditions for hydrogen from a two-body point of view: he has two phase waves, one for the electron and one for the nucleus.

“The subject of chapter 5 is light quanta. De Broglie suggests that the classical (electromagnetic) wave distribution in space is some sort of time average over the true distribution of phase waves. His light quantum is assigned a very small proper mass: the velocity v of the quantum and the phase velocity *c*^{2}/v of the accompanying phase wave are then both very close to *c*.

“De Broglie points out that radiation is sometimes observed to violate rectilinear propagation: a light wave striking the edge of a screen diffracts into the geometrical shadow, and rays passing close to the screen deviate from a straight line. De Broglie notes the two historical explanations for this phenomenon – on the one hand the explanation for diffraction given by the wave theory, and on the other the explanation given by Newton in his emission theory: ‘Newton assumed [the existence of] a force exerted by the edge of the screen on the corpuscle’ (p. 80). De Broglie asserts that he can now give a unified explanation for diffraction, by abandoning Newton’s first law of motion (p. 80):

“… the ray of the wave would curve as predicted by the theory of waves, and the moving body, for which the principle of inertia would no longer be valid, would suffer the same deviation as the ray with which its motion is bound up [solidaire] …’

“After considering the Doppler effect, reflection by a moving mirror, and radiation pressure, all from a photon viewpoint, de Broglie turns to the phenomena of wave optics, noting that (p. 86):

‘The stumbling block of the theory of light quanta is the explanation of the phenomena that constitute wave optics.’

“Here it becomes apparent that, despite his understanding of how non-rectilinear particle trajectories arise during diffraction and interference, de Broglie is not sure of the details of how to explain the observed bright and dark fringes in diffraction and interference experiments with light. In particular, de Broglie did not have a precise theory of the assumed statistical relationship between his phase waves and the electromagnetic field. Even so, he went on to make what he called ‘vague suggestions’ (p. 87) towards a detailed theory of optical interference. De Broglie’s idea was that the phase waves would determine the probability for the light quanta to interact with the atoms constituting the equipment used to observe the radiation, in such a way as to account for the observed fringes (p. 88):

“… the probability of reactions between atoms of matter and atoms of light is at each point tied to the resultant (or rather to the mean value of this) of one of the vectors characterising the phase wave; where this resultant vanishes the light is undetectable; there is interference. One then conceives that an atom of light traversing a region where the phase waves interfere will be able to be absorbed by matter at certain points and not at others. This is the still very qualitative principle of an explanation of interference …’

“De Broglie’s chapter 6 discusses the scattering of X- and γ -rays.

“In his chapter 7, de Broglie turns to statistical mechanics, and shows how the concept of statistical equilibrium is to be modified in the presence of phase waves. If each particle or atom in a gas is accompanied by a phase wave, then a box of gas will be ‘criss-crossed in all directions’ (p. 110) by the waves. De Broglie finds it natural to assume that the only stable phase waves in the box will be those that form stationary or standing waves, and that only these will be relevant to thermodynamic equilibrium … De Broglie then turns to the photon gas, for which he obtains Wien’s law. He claims that, in order to get Planck’s law, the following further hypothesis is required (p. 116):

‘If two or several atoms [of light] have phase waves that are exactly superposed, of which one can therefore say 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 separate units in calculating the probabilities’ …

“De Broglie’s thesis ends with a summary and conclusions (pp. 125–8). The seeds of the problem of quanta have been shown, he claims, to be contained in the historical ‘parallelism of the corpuscular and wave-like conceptions of radiation’. He has postulated a periodic phenomenon associated with each energy fragment, and shown how relativity requires us to associate a phase wave with every uniformly moving body. For the case of non-uniform motion, Maupertuis’ principle and Fermat’s principle ‘could well be two aspects of a single law’, and this new approach to dynamics led to an extension of the quantum relation, giving the speed of a phase wave in an electromagnetic field. The most important consequence is the interpretation of the quantum conditions for atomic orbits in terms of a resonance of the phase wave along the trajectories: ‘this is the first physically plausible explanation proposed for the Bohr–Sommerfeld stability conditions’. A ‘qualitative theory of interference’ has been suggested. The phase wave has been introduced into statistical mechanics, yielding a derivation of Planck’s phase volume element, and of the blackbody spectrum. De Broglie has, he claims, perhaps contributed to a unification of the opposing conceptions of waves and particles, in which the dynamics of the material point is understood in terms of wave propagation. He adds that the ideas need further development: first of all, a new electromagnetic theory is required that takes into account the discontinuous structure of radiation and the physical nature of phase waves, with Maxwell’s theory emerging as a statistical approximation. The final paragraph of de Broglie’s thesis emphasises the incompleteness of his theory at the time:

‘I have deliberately left rather vague the definition of the phase wave, and of the periodic phenomenon of which it must in some sense be the translation, as well as that of the light quantum. The present theory should therefore be considered as one whose physical content is not entirely specified, rather than as a consistent and definitively constituted doctrine’ …

“In his thesis de Broglie does not explicitly discuss diffraction or interference experiments with electrons, even though in his second communication of 1923 he had suggested electron diffraction as an experimental test. According to de Broglie’s later recollections, at his thesis defence on 25 November 1924:

‘Mr Jean Perrin, who chaired the committee, asked me if my ideas could lead to experimental confirmation. I replied that yes they could, and I mentioned the diffraction of electrons by crystals. Soon afterwards, I advised Mr Dauvillier … to try the experiment, but, absorbed by other research, he did not do it. I do not know if he believed, or if he said to himself that it was perhaps very uncertain, that he was going to go to a lot of trouble for nothing – it’s possible … But the following year it was discovered in America by Davisson and Germer” (Bacciagaluppi & Valentini, pp. 39-48).

En français dans le texte 353; Norman 347. Bacciagaluppi & Valentini, *Quantum Theory at the Crossroads*: *Reconsidering the 1927 Solvay Conference*, 2009. , pp. 39-48.

8vo (228 x 144) mm, pp. [iv], 111, [1]. Original printed wrappers. A fine copy.

Item #5444

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Price:
$32,500.00
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