Chapter 18 The Photon

Whereas the main Business of natural Philosophy, is to argue from Phaenomena without feigning Hypotheses, and to deduce cause from effect until we come to the very first Cause, which certainly is not mechanical; and not only to unfold the Mechanism for the World, but chiefly to resolve these and such like Questions.[1]

Isaac Newton demonstrated with his famous seventeenth century prism experiment that white light is composed of the colors of the rainbow. Based on this, and a whole host of other experiments, he put forth a particle theory of light in his works Principia (1687) and Opticks (1704) that he used to explain rectilinear propagation, reflection, refraction, and optical dispersion. In his corpuscular theory, Newton rejected the wave model of Huygens, largely because the medium required for such a wave was incompatible with the motion of the planets. Instead he proposed that light consists of very small particles endowed with a tiny mass, and that these corpuscles can therefore experience force, impulse, and convey momentum. And, most importantly, Newton’s corpuscle model clearly explained cause and effect—even across the vast distances of space.

James Bradley’s 1727 discovery of stellar aberration supported Newton’s corpuscular model, as it could easily be explained from vector addition of particle velocities, so Newton’s theory of light dominated through the rest of the century—especially in the Anglosphere.

As the nineteenth century dawned, Newton’s corpuscle theory set. This commenced in 1804 when Thomas Young presented his famous experiment in which light falls on an opaque screen after pasting through a system of two slits.[2] Young also suggested, in place of the idea that different colors were associated with different velocities or corpuscular masses, that each color has a specific wavelength.

After that, things just got worse and worse for the corpuscular theory of light. In 1862, Foucault measured the speed of light accurately in both air and water.[3] The result was decisive! Light moves slower in a denser medium, exactly as predicted by wave theory. Finally, Maxwell’s 1864 paper, “Dynamical Theory of the Electromagnetic Field,” provided a clear wave mechanism, and the question of the nature of light was totally solved.

In the last decade of the century, there were plenty of other problems to solve. Most of these had to do with the detailed nature of matter, including the medium that pervades free space—the aether. Not only did this decide the nature of light, but also it solved the action at a distance problem once and for all, a point Lord Kelvin made clear in his preface to the 1893 edition of the English translation of Hertz’s treatise that we have reproduced in Appendix B. Things would soon change, and quickly too. In 1900, the elderly Kelvin changed his tune, and now spoke of two ominous dark clouds looming over physics. The first cloud shed doubt on the existence of the aether, and the second cloud shed doubt on “the Maxwell-Boltzmann doctrine regarding the partition of energy.”

In hindsight, Max Planck became the first heretic when he used energy quanta to derive the correct formula for the spectrum of blackbody radiation. Perhaps because Planck did not dwell on this radical assumption in his original papers of 1900 and 1901, few scientists recognized the complete break from what we now call classical physics. In fact, during the five or six years immediately following the publication of Planck’s papers, only a handful of scientists publicly questioned Planck’s result or the route by which he had obtained it. Among these scientists were Sir James Jeans, Lord Rayleigh, and Albert Einstein, all of whom pointed out that any consistent derivation that recognized energy equipartition should, instead, lead to the following result for the spectral energy density of the cavity oscillators:
u  \left( \omega ,\,T \right) = \frac{{{\omega }^{2}}}{{{\pi }^{2}}{{c}^{3}}}{{k}_{B}}T.

Rayleigh, who had been critical of the equipartition theorem, showed that this formula failed at short wavelengths and suggested it might be due to a failure of the equipartition theorem at high frequencies. To remedy what Paul Ehrenfest would later refer to as “the ultraviolet catastrophe,” Rayleigh proposed an exponential cutoff that modified the temperature and frequency dependence of Planck’s expression but still managed to agree with most experimental data. Despite including a plausible, but ad hoc, exponential factor that mimicked the asymptotic behavior observed in blackbody curves at high frequencies, Rayleigh’s modified formula failed to reproduce Wien’s displacement law that {{\lambda }_{\max }}T={\rm constant}. This was a severe drawback since Wien’s displacement law was known to be in excellent accord with the experimental facts. For this reason, Rayleigh’s more general formula is largely forgotten. The name Rayleigh-Jean’s law is attached to the above expression after Rayleigh, and Jeans who corrected a factor in Rayleigh’s original equation.

The so-called Rayleigh-Jeans law might be called the Rayleigh-Jeans-Einstein law instead, since Einstein obtained the above expression for the thermal radiation energy density \left( u \left(\omega ,\,T \right) \right) independently of Rayleigh and Jeans, and (unlike Rayleigh and Jeans) demonstrated explicitly that it diverges for high frequencies. He also argued that Planck’s formula, though in excellent agreement with data at all wavelengths, could only be derived by assuming the discontinuous transfer of energy between resonators and radiation within the cavity. In fact, Einstein demonstrated, had Planck maintained a consistently classical approach in his derivation, he would have arrived at the Rayleigh-Jeans expression for \left( u \left(\omega ,\,T \right) \right) rather than his experimentally correct formula.

Einstein publicly revealed Planck’s tentative heresy by showing that Planck’s distribution function implied light should be conceived of as a collection of particles, or quanta—despite the previous century’s worth of overwhelming evidence to the contrary. As Einstein puts it in his famous 1905 paper:

It seems to me that the observations associated with blackbody radiation, fluorescence, the production of cathode rays by ultraviolet light, and other related phenomena connected with the emission or transformation of light are more readily understood if one assumes that the energy of light is discontinuously distributed in space. In accordance with the assumption to be considered here, the energy of a light ray spreading out from a point source is not continuously distributed over an increasing space but consists of a finite number of energy quanta which are localized at points in space, which move without dividing, and which can only be produced and absorbed as complete units.

In the following I wish to present the line of thought and the facts which have led me to this point of view, hoping that this approach may be useful to some investigators in their research.[4]

Einstein’s proposal of light quanta faced strong opposition, even from those scientists who enthusiastically embraced relativity, and it is not difficult to see why this was the case. Although the particle hypothesis explains the photoelectric effect and blackbody radiation, it is inconsistent with the overwhelming number of experiments that appear to confirm the wave nature of light.

Einstein, of course, was well aware of the success of the wave theory of light, about which he wrote:

The wave theory of light, which operates with continuous spatial functions, has worked well in the representation of purely optical phenomena and will probably never be replaced by another theory. It should be kept in mind, however, that the optical observations refer to time averages rather than instantaneous values. In spite of the complete experimental confirmation of the theory as applied to diffraction, reflection, refraction, dispersion, etc., it is still conceivable that the theory of light which operates with continuous spatial functions may lead to contradictions with experience when it is applied to the phenomena of emission and transformation of light.[5]

Einstein’s hypothesis was actually well reasoned, and in retrospect his argument is quite convincing. After confirming that the Rayleigh-Jeans law agrees with Planck’s distribution at low frequencies, Einstein considered the high frequency limit of Planck’s law. In the region of high frequencies, for which \hbar \omega \gg {{k}_{B}}T\ ,  Planck’s radiation law takes the following form:
u \left( \omega ,\,T \right)=\frac{\hbar {{\omega }^{3}}}{{{\pi }^{3}}{{c}^{3}}}{{e}^{-\hbar \omega /{{k}_{B}}T}}

Based on experimental data taken at high frequency, Wilhelm Wien had actually proposed this function (without explicit reference to the constant \hbar ) as the correct expression for the spectral energy density of blackbody radiation at all frequencies. In the limit where this expression applies, Einstein demonstrated that the entropy of radiation does not behave like that of waves, but rather like particles of energy \mathbb{E}=\hbar \omega !

Where Planck had assumed only that the energy absorbed, or emitted, by the material oscillators of the cavity walls was restricted to integer multiples of , Einstein now proposed that it was the radiation in the cavity whose energy was actually quantized. Not only that, but he broke entirely with the received physics of his day by suggesting a new heuristic viewpoint that would treat light as if it were comprised particles whose energy is proportional to the frequency of that light.

Einstein adduced such phenomena as fluorescence and the photoionization of gases to argue for the plausibility of his heuristic light-quanta, which were later dubbed photons. Most significantly, he explained the ejection of electrons from the surface of an irradiated metal, which is called the photoelectric effect, which had been studied by the Phillip Lenard whose figures from his 1905 Nobel Prize lecture are shown above.[6]

In the end, it was the skeptical Robert Millikan who convinced the physics community to take the photon concept seriously by thoroughly testing the photoelectric effect. He eloquently describes his understanding of the relevant physics in his 1916 paper:

Quantum theory was not originally developed for the sake of interpreting photoelectric phenomena. It was solely a theory as to the mechanism of absorption and emission of electromagnetic waves by resonators of atomic or subatomic dimensions. It had nothing whatever to say about the energy of an escaping electron or about the conditions under which such an electron could make its escape, and up to this day the form of the theory developed by its author has not been able to account satisfactorily for the photoelectric facts presented here-with. We are confronted, however, by the astonishing situation that these facts were correctly and exactly predicted nine years ago by a form of quantum theory which has now been pretty generally abandoned.

It was in 1905 that Einstein made the first coupling of photo effects and with any form of quantum theory by bringing forward the bold, not to say the reckless, hypothesis of an electro-magnetic light corpuscle of energy which energy was transferred upon absorption to an electron. This hypothesis may well be called reckless first because an electromagnetic disturbance which remains localized in space seems a violation of the very conception of an electromagnetic disturbance, and second because it flies in the face of the thoroughly established facts of interference. The hypothesis was apparently made solely because it furnished a ready explanation of one of the most remarkable facts brought to light by recent investigations, viz., that the energy with which an electron is thrown out of a metal by ultra-violet light or X-rays is independent of the intensity of the light while it depends on its frequency. This fact alone seems to demand some modification of classical theory or, at any rate, it has not yet been interpreted satisfactorily in terms of classical theory.[7]

Despite recklessly flying in the face of the thoroughly established facts of interference, Millikan had to conclude that Einstein’s particle model correctly predicted the key facts of photoelectric emission. Or, as Millikan put it in his conclusions:

1. Einstein’s photoelectric equation has been subjected to very searching tests and it appears in every case to predict exactly the observed results.

2. Planck’s h has been photoelectrically determined with a precision of about .5 per cent. and is found to have the value h=6.57\times {{10}^{-27}}\ \text{erg}\cdot \text{s}\,\text{.}

Now that Einstein’s model had to be taken seriously, the interference problem became more acute.[8] In 1924, the French graduate student, Louis deBroglie, proposed his own bold and reckless hypothesis. If waves act like particles, perhaps particles act like waves?

Meanwhile two Americans, Clinton Davisson and Lester Germer, were working on the problem of using electrons as a probe to understand the atomic structure of nickel. Against expectations, they observed electron diffraction patterns, which confirmed deBroglie’s hypothesis.

At nearly the same moment that physicists were coming to grips with the idea of light quanta, the analysis of Louis de Broglie and the electron diffraction experiments of Davisson and Germer forced them to consider the wave properties of electrons. The new quantum physics that eventually emerged from this crucible compels us to accept a fusion of the concepts of wave and particle when explaining light, atoms, and subatomic particles. Electrons are clearly detected as whole particles: even in the double slit experiment, a fraction of an electron is never observed. Yet, even when only one electron at a time is allowed to pass through the diffracting system, an interference pattern typical of classical waves appears after a sufficiently large number of electrons have passed through. Real particles in nature, such as electrons, do not behave like classical particles! In hindsight, it is clear that classically intuitive notions of particles and waves are abstracted from our everyday experiences with macroscopic systems such as rocks and ponds, and we shouldn’t be surprised when these concepts do not literally describe the subatomic world of which we have no direct experience.

Oddly enough, Newton’s theory of light was, at least in principle, right after all: light is composed of particles. Newton’s own laws of motion, however, do not govern these photons. Rather, both matter and light must be governed by a whole new theory of quantum mechanics and quantum electrodynamics.

[1] Isaac Newton, Opticks or A Treatise on the Reflections, Refractions, Inflections and Colors of Light, Book III, Part I, Query 28

[2] Young, Thomas, “The Bakerian Lecture: Experiments and calculations relative to physical optics”, Philosophical Transactions of the Royal Society of London (Royal Society of London) 94 (1804), 1–16.

[3] Foucalt, Comptes Rendus, “Détermination expérimentale de la vitesse e la lumière; parallaxe du Soleil”(1862), 55, 501-3, 792-96. English translation is available in W.F. Magie, Source Book in Physics, (Cambridge: Harvard University Press, 1935) 343-44.

[4] Albert Einstein, “Concerning a Heuristic Point of View Toward the Emission and Transformation of Light,” Annalen der Physik 17 (1905), 132-148.

[5] Albert Einstein, “Concerning a Heuristic Point of View Toward the Emission and Transformation of Light,” Annalen der Physik 17 (1905), 132-148.

[6]P. Lenard, Über Kathodenstrahlen Nobel-Vorlesung (Verlag von Johann Ambrosius Barth, 19) He won the 1905 Nobel prize for his work that took place around 1890.

[7]R.A. Millikan, “A Direct Photoelectric Determination of Planck’s ‘h’,” Physical Review 7 (1916), 355-388.

[8]The results of the double slit experiment turn out to be the same whether several photons or only one photon passes through the double slit apparatus. The British physicist G.I. Taylor who conducted painstaking observations using an extremely dim light source first obtained evidence for this startling result. The resulting diffraction patterns that Taylor obtained were one reason Dirac famously wrote, “Each photon then interferes only with itself.” See G.I. Taylor, “Interference fringes with feeble light,” Proceedings of the Cambridge Philosophical Society 15 (1909), 114-115. For a thorough discussion of the quantum mechanical double slit experiment, see Richard Feynman, Robert Leighton and Matthew Sands, The Feynman Lectures on Physics (Addison-Wesley), Vol. III, pp. I-1-11. Feynman discusses the double slit experiment using matter (electrons) instead of light, but the pertinent physics is the same.