Louis de Broglie

How to use this course

Louis de Broglie is remembered for a single formula, \(\lambda=h/p\). Behind it lies a short burst of work in 1923 and 1924 that joined Einstein’s light quanta, special relativity and the old analogy between optics and mechanics. Erwin Schrödinger built his wave equation on it within two years, and electron diffraction confirmed it within three. De Broglie then spent sixty years arguing that the wave and the particle are both real, a view that was rejected in 1927 and revived by David Bohm in 1952.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations in modern notation and exercises with folded solutions. The lessons follow de Broglie’s own route: from the light quantum (1922) to the phase wave (1923), the thesis (1924), the experiments (1927), the pilot wave (1927 and 1952) and his theory of light (1934 onward). Readers who want only the story can read Parts I and III and the opening paragraph of each lesson in Part II.

Timeline

A ducal family, 1892–1911

Louis Victor Pierre Raymond de Broglie was born in Dieppe on 15 August 1892, the youngest of five children of Victor, 5th duc de Broglie, and Pauline d’Armaillé. The family was one of the most distinguished in France. It had produced marshals, ministers and diplomats since the seventeenth century; his great-grandfather Victor had been prime minister under Louis-Philippe, and his grandfather Albert prime minister under the Third Republic. The family name is pronounced roughly “de Broy”.

He was educated at the Lycée Janson-de-Sailly in Paris, which he completed in 1909. His father died in 1906, and his brother Maurice, seventeen years older, became head of the household. Maurice had left a naval career to do physics, against family expectations, and had set up a private laboratory in the family house in Paris. He took a close interest in his young brother’s education.

History first

The family tradition pointed to diplomacy, and Louis began by studying history at the Sorbonne. He took a licence in history in 1910, at eighteen, having worked on medieval history. A research topic on French political history did not hold him. He was drawn instead to the general ideas of science, first through the philosophical books of Henri Poincaré.

The turning point came in 1911. Maurice served as one of the scientific secretaries of the first Solvay Conference in Brussels, where Planck, Einstein, Lorentz, Marie Curie and others discussed radiation and quanta. Louis read the proceedings and decided to devote himself to the problem of the quantum. He switched to science and took his licence ès sciences in 1913.

Physics and the Eiffel Tower, 1913–1919

In 1913 de Broglie began his compulsory military service in the engineers, at Fort du Mont-Valérien outside Paris. Through Maurice’s influence he was transferred to the army’s radiotelegraphy service. When war broke out in 1914 his service simply continued, and he spent the whole war at the radio station in the Eiffel Tower, then the most important military transmitter in France.

The work was practical electronics: valve receivers, transmitters and, with Maurice and Léon Brillouin, communication with submarines. He was demobilized in August 1919 with the rank of adjudant, aged twenty-seven, having lost six years of study.

Waves and particles in the laboratory

The war years left de Broglie with a familiarity with electromagnetic waves, oscillators and resonance that shows in his later reasoning. His theory of matter waves rests on the language of frequency, phase and resonance that a radio engineer of 1915 used every day.

Maurice’s laboratory and the thesis, 1919–1924

After the war de Broglie worked in Maurice’s laboratory in the rue Lord-Byron, which specialized in X-ray spectroscopy and the photoelectric effect of X-rays. The brothers published together on X-ray absorption spectra and on the energies of photoelectrons. Maurice’s experiments showed X-rays behaving at once as waves, diffracted by crystals, and as particles, ejecting electrons with definite energies. Louis later said that conversations with his brother about this double behaviour had shaped his ideas.

In 1922 he published two notes treating black-body radiation as a gas of “light atoms”, particles with an extremely small mass. In September and October 1923 he published three notes in the Comptes rendus of the Académie des sciences: Ondes et quanta (10 September), Quanta de lumière, diffraction et interférences (24 September) and Les quanta, la théorie cinétique des gaz et le principe de Fermat (8 October). Together they proposed that every particle is accompanied by a wave, and the second note suggested that a beam of electrons passing through a small aperture should show diffraction.

The thesis

He expanded the notes into a doctoral thesis, Recherches sur la théorie des quanta. His formal advisor was Paul Langevin, who was impressed but uncertain. Langevin sent a copy to Einstein, who told him that it deserved serious attention and recommended it to Max Born. The defence took place at the Sorbonne on 25 November 1924, before a jury of Jean Perrin, the mathematician Élie Cartan, the crystallographer Charles Mauguin and Langevin. The thesis was published in the Annales de Physique in 1925.

The jury approved it with some reserve: Perrin is said to have remarked to Maurice only that his brother was very intelligent. Einstein’s response was different. In his second paper on the quantum ideal gas, early in 1925, he used de Broglie’s waves to interpret a wave-like term in the density fluctuations of a gas. Through Einstein the idea reached Zürich, where Schrödinger turned it into an equation within the year.

Light quanta and the internal clock

For the general reader. Einstein had shown that light, a wave, also comes in particles. De Broglie asked the reverse question: if every particle has energy, and energy is tied to a frequency by Planck’s constant, then every particle has a frequency. Something in it must oscillate. He then found that relativity makes this “internal clock” behave in a puzzling way, and that the puzzle disappears only if the particle is accompanied by a wave.

The rest frequency

De Broglie started from two equations. Planck and Einstein had \(E=h\nu\) for light; relativity gives \(E=m_0c^2\) for a particle at rest. He postulated that a particle of rest mass \(m_0\) carries a periodic phenomenon of frequency

in its rest frame. In 1922 he had already treated light quanta as particles of very small but non-zero mass, so the postulate covered light and matter alike.

The paradox

Seen from a frame in which the particle moves with speed \(v\), and with \(\gamma=1/\sqrt{1-v^2/c^2}\), two different frequencies appear:

• The energy is \(\gamma m_0c^2\), so the quantum relation suggests a frequency \(\nu=\gamma\nu_0\), larger than \(\nu_0\).
• The internal clock is a moving clock, so time dilation makes it tick at \(\nu_1=\nu_0/\gamma\), smaller than \(\nu_0\).

De Broglie resolved the conflict by assigning the frequency \(\nu\) to a wave spread through space and \(\nu_1\) to the clock carried by the particle, and showing that the two stay in step.

Proof. At the particle, the wave phase is \(2\pi\gamma\nu_0\,t\,(1-v/V)\) and the clock phase is \(2\pi\nu_0t/\gamma\). Equality for all \(t\) means \(\gamma(1-v/V)=1/\gamma\), that is \(1-v/V=1-v^2/c^2\), so \(V=c^2/v\). \(\square\)

Since \(v<c\), the wave travels faster than light. De Broglie stressed that it carries no energy; he called it a phase wave (onde de phase), a wave of phase alone. In his old age he still called this theorem the central idea of his thesis.

The phase wave as a Lorentz transform

There is a cleaner way to see the result. In the rest frame, suppose the oscillation is in phase everywhere: \(\psi_0=\sin(2\pi\nu_0t_0)\), independent of \(x_0\). Transform to the frame where the particle moves, using \(t_0=\gamma(t-vx/c^2)\):

This is a travelling wave with frequency \(\gamma\nu_0\) and phase velocity \(c^2/v\). The “surfaces of simultaneity” of the rest frame are the wavefronts. The harmony of phases is automatic because the phase \(2\pi\nu_0t_0\) is a Lorentz scalar.

Phase waves, group velocity, λ = h/p

For the general reader. A wave faster than light sounds impossible. But no signal travels at the speed of the crests. Signals and energy travel with a group of waves, and de Broglie showed that a group of his phase waves moves exactly as fast as the particle. The wavelength of these waves, which turned out to be measurable, is Planck’s constant divided by the momentum.

From Lesson 04, the wave has frequency \(\nu=\gamma\nu_0\) and phase velocity \(V=c^2/v\), so its wavelength is

De Broglie wrote it in this form. For light, where \(p=E/c\), it reduces to the ordinary relation \(\lambda=c/\nu\).

The middle step uses \(E^2=p^2c^2+m_0^2c^4\), so \(E\,dE=c^2p\,dp\). The identity \(dE/dp=v\) is also one of Hamilton’s equations, \(\dot x=\partial H/\partial p\). De Broglie saw this as the key link: Lord Rayleigh’s group velocity of wave theory and Hamilton’s velocity of mechanics are the same quantity.

Wave packets

A localized particle corresponds to a superposition \(\psi(x,t)=\int A(k)\,e^{i(kx-\omega(k)t)}dk\) with \(A\) peaked at \(k_0\). Expanding \(\omega(k)\approx\omega(k_0)+v_g(k-k_0)\), the envelope moves rigidly at \(v_g\) to first order. The second-order term \(\tfrac12\omega''(k_0)(k-k_0)^2\) makes the packet spread, a fact that later counted against the idea that particles simply are wave packets.

Orders of magnitude

For a nonrelativistic particle of kinetic energy \(T\), \(\lambda=h/\sqrt{2mT}\). For electrons, \(\lambda\approx1.226/\sqrt{T/\text{eV}}\) nm: about 0.1 nm at 150 eV, the spacing of atoms in crystals. This is why crystals act as gratings for electrons (Lesson 08). For a 1 g object moving at 1 cm/s, \(\lambda\approx7\times10^{-29}\) m, which is why everyday objects show no wave behaviour.

Fermat, Maupertuis and Bohr’s orbits

For the general reader. Light takes the path of least time (Fermat, 1662). A thrown ball takes the path of least action (Maupertuis, 1744). De Broglie showed that, once a particle has a wave with \(\lambda=h/p\), these two principles are the same statement. He then explained Bohr’s mysterious rule for allowed orbits: an orbit is allowed when the electron’s wave closes on itself, like a standing wave on a ring.

Two principles

In a medium where the wave’s local wavelength is \(\lambda(\mathbf x)\), Fermat’s principle says that rays make the accumulated phase stationary:

Maupertuis’s principle, in the form given by Euler and Lagrange, says that a particle of fixed energy \(E\) in a static field moves so that the abbreviated action is stationary:

The proof is one line: \(1/\lambda=p/h\), and multiplying a functional by a constant does not change its stationary paths. De Broglie proved the relativistic version, including electromagnetic fields, where \(p\,ds\) is replaced by \((\mathbf p+q\mathbf A)\cdot d\mathbf s\) for a particle of charge \(q\). The wave guides the particle along the rays.

Stable orbits

On a closed orbit the wave returns to its starting point. De Broglie required it to be in resonance with itself: after one circuit the phase must change by a whole multiple of \(2\pi\).

Bohr had stated his condition as a postulate in 1913, and Arnold Sommerfeld had generalized it in 1916. De Broglie’s argument gave the first physical reason for it. In his note of 10 September 1923 he derived it from the resonance of the phase wave with the internal clock, and noted that for very fast electrons a separate analysis would be needed.

What the argument misses

The resonance picture assumes a well-defined trajectory and a wavelength that changes slowly along it. Schrödinger’s equation drops both assumptions. Its semiclassical limit (WKB) refines de Broglie’s condition to \(\oint p\,dq=(n+\tfrac12)h\) for oscillations between two smooth turning points, a correction from the phase loss at each turning point. For hydrogen, the exact Schrödinger levels coincide with Bohr’s, but the ground state has angular momentum zero, not \(\hbar\): the circular-orbit picture fails even where the energy is right.

Phase waves and quantum statistics

For the general reader. To count how many states a gas of particles can occupy, physicists divided phase space into cells of size \(h^3\), without knowing why \(h\) should appear. De Broglie explained the rule: the allowed states of a particle in a box are the standing waves that fit in the box, and counting standing waves gives exactly one state per cell. He also showed how treating light as a gas of particles could give either Wien’s law or Planck’s law, depending on how the particles share their waves. Einstein seized on this a few months later.

Counting states

For a particle in a cubical box of side \(L\) with periodic boundary conditions, the allowed waves are \(e^{i\mathbf k\cdot\mathbf x}\) with \(\mathbf k=\frac{2\pi}{L}(n_1,n_2,n_3)\), \(n_i\in\mathbb Z\). Each allowed \(\mathbf k\) occupies a volume \((2\pi/L)^3\) of \(k\)-space.

With \(\mathbf p=\hbar\mathbf k\), \(d^3k\,V/(2\pi)^3=V\,d^3p/(2\pi\hbar)^3=V\,d^3p/h^3\). The cell size \(h^3\), introduced by Planck and others as a rule, becomes the volume per standing wave. De Broglie stated this in the third 1923 note and in chapter 7 of the thesis.

Light as a gas

For light quanta \(p=h\nu/c\), so \(4\pi p^2dp=4\pi h^3\nu^2d\nu/c^3\), and the number of states per unit volume in \(d\nu\) is \(8\pi\nu^2d\nu/c^3\). Two assumptions about occupation give two laws.

De Broglie obtained Wien’s law for a gas of light atoms in 1922, and in the thesis recovered Planck’s law by allowing several quanta to be bound to the same phase wave. The second assumption is the essence of what Satyendra Nath Bose did in 1924 and Einstein extended to atoms: particles are not distributed independently among cells; what is counted is how many particles occupy each state.

Einstein’s fluctuations

For radiation obeying Planck’s law, Einstein had shown in 1909 that the energy fluctuations in a small volume contain two terms, one like a gas of particles and one like interfering waves. In his 1925 paper on the ideal quantum gas he found the same two terms for a gas of atoms obeying Bose’s statistics, and cited de Broglie’s thesis to explain the wave term. That citation drew Schrödinger’s attention to the thesis.

Electron diffraction and matter-wave optics

For the general reader. In 1927, two laboratories found that electrons bounce off crystals in the patterns that waves make, with exactly the wavelength de Broglie predicted. Since then the same has been shown for neutrons, atoms and molecules of hundreds of atoms. Electron microscopes, which see individual atoms, work because electron waves are far shorter than light waves.

Davisson and Germer

At Bell Telephone Laboratories, Clinton Davisson and Lester Germer scattered slow electrons from a nickel crystal. After an accident in 1925 that annealed their sample into large crystals, the scattered intensity showed sharp peaks. Davisson learned of de Broglie’s idea at a meeting in Oxford in 1926 and the experiment was redirected to test it. In 1927 they reported a strong peak for 54 eV electrons at 50° from the incident beam, in agreement with diffraction from rows of atoms of spacing \(d=0.215\) nm.

In the same year, George Paget Thomson and Alexander Reid in Aberdeen sent electrons of tens of keV through thin metal films and photographed rings, the electron analogue of X-ray powder patterns. Davisson and Thomson shared the 1937 Nobel Prize in Physics. Atoms followed: in 1930 Immanuel Estermann and Otto Stern diffracted helium atoms from a lithium fluoride crystal.

Relativistic wavelengths

An electron accelerated through a potential difference \(U\) has kinetic energy \(T=eU\). The relativistic relation \(E^2=p^2c^2+m^2c^4\) with \(E=T+mc^2\) gives

Ernst Ruska and Max Knoll built the first electron microscope in 1931. Modern transmission microscopes at 100–300 kV use wavelengths of a few picometres and resolve single atoms, limited by lens aberrations rather than by \(\lambda\).

Large molecules

In 1999 Markus Arndt, Anton Zeilinger and colleagues in Vienna observed interference of C\(_{60}\) fullerene molecules passing through a grating, with de Broglie wavelengths of a few picometres. Later experiments in the same group extended this to molecules of thousands of atomic mass units. Each molecule is a hot, complex object with internal vibrations; the interference shows that the centre-of-mass motion keeps its coherence as long as no which-path information leaks to the environment.

The pilot wave and the double solution

For the general reader. For de Broglie the wave did not replace the particle; it guided it. In 1927 he proposed that every particle has a definite position and is steered by its wave, like a cork carried by a current. The leading physicists of the day rejected the idea at the Solvay Conference that October, and he abandoned it. Twenty-five years later David Bohm rediscovered and completed it, and de Broglie spent the rest of his life defending a realist account of quantum theory.

The guidance equation

Write the wave function in polar form \(\psi=R\,e^{iS/\hbar}\) with real \(R\ge0\) and \(S\). Substituting into the Schrödinger equation \(i\hbar\partial_t\psi=-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi\) and separating real and imaginary parts gives two equations:

The first is a Hamilton–Jacobi equation with an extra “quantum potential” \(Q\); the second is a continuity equation for \(\rho=R^2=|\psi|^2\) with velocity field \(\nabla S/m\).

Equivariance follows from the continuity equation: \(|\psi|^2\) and the density of an ensemble of trajectories moving with velocity \(\nabla S/m\) satisfy the same first-order equation, so they agree for all time if they agree once. Note that the velocity of particle \(k\) depends on the positions of all the others: the theory is explicitly nonlocal, as Bell later stressed.

Solvay 1927

De Broglie presented a version of this theory, which he called the pilot wave (onde pilote), at the Fifth Solvay Conference in Brussels in October 1927. He had arrived at it as a simplified fallback from a more ambitious “double solution” theory he had outlined earlier that year in the Journal de Physique. In the discussion Wolfgang Pauli objected that the theory could not handle inelastic collisions, using an example of scattering by a rotator. De Broglie could not answer convincingly at the time. The objection is now known to be mistaken: a full treatment of the measuring apparatus within the theory reproduces the quantum predictions. Discouraged and without support, de Broglie soon gave up the pilot wave and taught the standard interpretation for the next twenty-five years.

Bohm and the return

In 1952 David Bohm, unaware of de Broglie’s 1927 work, published the same theory in two papers in the Physical Review, together with a theory of measurement that answered Pauli. De Broglie returned to his old ideas, working with Jean-Pierre Vigier at the Institut Henri Poincaré. In 1956 he published Une tentative d’interprétation causale et non linéaire de la mécanique ondulatoire: la théorie de la double solution. In the double solution the particle is a localized singular or highly concentrated solution \(u\) of a nonlinear wave equation, moving in step with the regular statistical wave \(\psi\) through the “guidance formula”. This programme was never completed. His last major idea, the “hidden thermodynamics” of the isolated particle (1964), proposed links between the action of a particle and the entropy of a hidden thermostat.

A massive photon

For the general reader. De Broglie’s physics began with light quanta that have a tiny mass, and he never gave the idea up. From 1934 he tried to build the photon from a pair of more elementary particles. His student Alexandru Proca wrote in 1936 the field equations for a particle of spin 1 and non-zero mass, which now describe the W and Z bosons of the weak force. Experiments have pushed any photon mass far below de Broglie’s 1923 bound, but the question remains a standard test of electromagnetism.

The Proca equations

With signature \((+,-,-,-)\), \(F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu\), and \(\mu=m_\gamma c/\hbar\) the inverse reduced Compton wavelength, the Proca Lagrangian density in Heaviside–Lorentz units with \(c=1\) is

Proof. Apply \(\partial_\nu\): the term \(\partial_\nu\partial_\mu F^{\mu\nu}\) vanishes by antisymmetry, leaving \(\mu^2\partial_\nu A^\nu=\partial_\nu J^\nu=0\). Then \(\partial_\mu F^{\mu\nu}=\Box A^\nu-\partial^\nu(\partial_\mu A^\mu)=\Box A^\nu\). The Lorenz condition removes one of the four components, leaving three, against two for the massless photon. \(\square\)

The mass term breaks gauge invariance: \(A_\mu\) itself, and not only \(F_{\mu\nu}\), becomes physical. In the Standard Model the W and Z masses arise instead through the Higgs mechanism, which keeps the gauge symmetry of the Lagrangian.

Consequences

For a static point charge, the time component obeys \((\nabla^2-\mu^2)\phi=-\rho\), whose solution is the Yukawa potential

so Coulomb’s law is cut off beyond the range \(1/\mu=\hbar/m_\gamma c\). Plane waves obey \(\omega^2=c^2k^2+\mu^2c^2\): light would be dispersive, with low frequencies travelling more slowly, exactly as de Broglie supposed in 1923. Current limits come from planetary and solar-wind magnetic fields; the Particle Data Group quotes \(m_\gamma<10^{-18}\) eV/\(c^2\).

Nobel Prize and the Paris school, 1925–1939

After the thesis, de Broglie gave free courses at the Sorbonne for two years. In 1926 he published Ondes et mouvements, and his report to the Fifth Solvay Conference in October 1927 set out the pilot wave. The same year electron diffraction confirmed his central prediction. In 1928 he became professor of theoretical physics at the new Institut Henri Poincaré, built with Rockefeller and Rothschild funds as a centre for mathematics and theoretical physics in Paris.

In 1929 he received the Nobel Prize in Physics “for his discovery of the wave nature of electrons”, five years after his thesis, as well as the first Henri Poincaré Medal of the Académie des sciences. In 1932 he succeeded to the chair of theoretical physics at the Faculty of Sciences of the University of Paris, which he held for thirty years, and in 1933 he was elected to the Académie des sciences, its youngest member.

Teaching and students

His seminar at the Institut Henri Poincaré became the centre of French theoretical physics between the wars. His books, written in a clear and elegant French, explained wave mechanics, Dirac’s electron theory and nuclear physics to a generation of French students. His doctoral students included Alexandru Proca, Marie-Antoinette Tonnelat, Olivier Costa de Beauregard, Bernard d’Espagnat, Cécile DeWitt-Morette, Satosi Watanabe and Nicolás Cabrera.

From 1934 he worked on a “neutrino theory of light”, in which the photon is a bound pair of spin-½ particles. The programme did not succeed, but it produced Proca’s equations (Lesson 10). He received the Max Planck Medal in 1938.

Academies, return to realism, legacy, 1940–1987

During the German occupation de Broglie remained in Paris. In 1941 he was named to the National Council of the Vichy regime, a consultative body, and in 1942 he became permanent secretary of the Académie des sciences for the mathematical sciences, which made him one of the most influential figures in French science. In 1944 he joined the Bureau des Longitudes. On 12 October 1944, weeks after the liberation of Paris, the Académie française elected him to the seat of Émile Picard, and on 31 May 1945 he was received by his own brother Maurice, already a member. In 1945 he became an adviser to the newly founded French Atomic Energy Commission.

He was an untiring popularizer, and in 1952 he received the first Kalinga Prize of UNESCO for the popularization of science. He became a Foreign Member of the Royal Society in 1953, received the gold medal of the CNRS in 1956, and was made Grand Cross of the Légion d’honneur in 1961. He wrote more than twenty-five books, among them Matière et lumière (1937) and Certitudes et incertitudes de la science (1966).

After Bohm’s 1952 papers, de Broglie returned to the causal view of his youth, convinced that statistical quantum mechanics hides a deterministic reality. He worked on the double solution and the hidden thermodynamics with a small group of collaborators. On Maurice’s death in 1960 he became the 7th duc de Broglie. He never married. He retired from his chair in 1962. The Fondation Louis de Broglie was set up in 1973 to continue his programme. He died at Louveciennes, near Paris, on 19 March 1987, aged 94. The ducal title passed to Victor-François de Broglie.

Legacy

• Matter waves: \(\lambda=h/p\) is the starting point of Schrödinger’s wave mechanics, and of electron microscopy, neutron scattering and atom interferometry.
• Quantum statistics: his counting of phase waves in a box gave the physical reason for the \(h^3\) cell and influenced Einstein’s Bose gas papers.
• Foundations: the pilot-wave theory, revived by Bohm and studied by John Bell, is a complete and consistent alternative formulation of nonrelativistic quantum mechanics. Bell’s work on nonlocality began from it. Walking-droplet experiments since 2005 have given hydrodynamic analogues of the pilot-wave idea.
• Honours: Nobel Prize (1929), Henri Poincaré Medal (1929), Albert I of Monaco Prize (1932), Max Planck Medal (1938), Kalinga Prize (1952), Foreign Member of the Royal Society (1953), and international membership of the US National Academy of Sciences (1948).

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Louis de Broglie, MacTutor History of Mathematics.
2. Louis de Broglie, Wikipedia (with extensive references).
3. Louis de Broglie, biographical, NobelPrize.org.
4. Académie française, Louis de Broglie (fauteuil 1).
5. G. Lochak, Louis de Broglie: un prince de la science, Flammarion, 1992.
6. A. Abragam, Louis Victor Pierre Raymond de Broglie, Biographical Memoirs of Fellows of the Royal Society 34 (1988).
7. Louis de Broglie und die Quantenmechanik, arXiv:physics/9807012.

Physics

1. L. de Broglie, Ondes et quanta, Comptes rendus 177 (1923) 507–510; Quanta de lumière, diffraction et interférences, 548–550; Les quanta, la théorie cinétique des gaz et le principe de Fermat, 630–632.
2. L. de Broglie, Recherches sur la théorie des quanta, thesis, Paris, 1924; Annales de Physique (10) 3 (1925) 22–128.
3. L. de Broglie, La mécanique ondulatoire et la structure atomique de la matière et du rayonnement, Journal de Physique et le Radium 8 (1927) 225–241.
4. L. de Broglie, The wave nature of the electron, Nobel Lecture, 1929.
5. C. Davisson and L. H. Germer, Diffraction of electrons by a crystal of nickel, Physical Review 30 (1927) 705–740.
6. G. P. Thomson and A. Reid, Diffraction of cathode rays by a thin film, Nature 119 (1927) 890.
7. A. Proca, Sur la théorie ondulatoire des électrons positifs et négatifs, Journal de Physique et le Radium 7 (1936) 347–353.
8. D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variables I and II, Physical Review 85 (1952) 166–193.
9. G. Bacciagaluppi and A. Valentini, Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, Cambridge University Press, 2009; arXiv:quant-ph/0609184.
10. M. Arndt et al., Wave–particle duality of C\(_{60}\) molecules, Nature 401 (1999) 680–682.
11. L. de Broglie, Une tentative d’interprétation causale et non linéaire de la mécanique ondulatoire, Gauthier-Villars, 1956.
12. Particle Data Group, Review of Particle Physics, gauge boson listings (photon mass).