Einstein and the light quanta

In two previous posts (I and II) we learned how the attempt to explain the radiation emitted by hot bodies led the German physicist Max Planck in the late 1900s to propose a novel idea: energy can only be absorbed or emitted discontinuously, or as they say: in quanta of energy. This was the birth of quantum physics, as the German physicist Arnold Sommerfeld christened it years later. Thus, apparently, one of the problems that physicists had been struggling with for several years, one of the two famous little clouds of Lord Kelvin that we have already talked about in these entries, was closed. It is worth remembering, however, before moving on to the story we will deal with in this entry, that Planck’s quantum hypothesis only concerned the way in which bodies absorbed and emitted energy, and never the very nature of the radiation itself, which was made up of electromagnetic waves and which, as Maxwell’s laws of electromagnetism established, were continuous in space and time. If everything had remained like that, as a kind of mathematical artifice, perhaps we would still be living as we did at the beginning of the 20th century and we would not know… but let’s leave speculation aside and begin our story. Our protagonist this time is going to be probably the most famous (and certainly the most mediatic) scientist in history: Albert Einstein. Let’s start with a brief, but necessary, biographical sketch.

Einstein was born in the city of Ulm (Germany) on 14 March 1879 into a (non-practising) Jewish family. In 1880 the family moved to Munich where the young Einstein received his first lessons first at a Catholic primary schools (primary education) and then at the Luidpold Gymnasium (secondary education). In 1894 his family moved to Milan where Einstein followed them several months later, leaving the Gymnasium. His pre-university education was completed in Switzerland, where his family sent him to finish high school, after which, in 1896, he enrolled at the Swiss Federal Institute of Technology in Zurich (ETH) to study physics and mathematics. He graduated in 1900 but, although he intended to pursue an academic career, he was not accepted at any university and he really tried. What would have happened if Einstein had become an assistant to one of his professors at the ETH or at one of the other German universities he wrote to? We will never know, but it is very likely that the history of science would have been very different. For two reasons. The first is told by Einstein himself in his autobiography where, referring to his stay at the ETH, he wrote: “I had excellent teachers there, so that I could have acquired a profound mathematical education. I, however, spent most of my time working in the physics laboratory, fascinated by direct contact with experience”. The second was that after his unsuccessful attempts to work at the University, he got a job at the Patent Office in Bern, where he had plenty of free time to think about the theoretical physics problems he had been interested in. In other words, having no laboratory but plenty of time to think about the problems of theoretical physics, Einstein managed in just three years to solve three of the most elusive problems in physics at the time: two of them directly related to Lord Kelvin’s clouds and the third on the existence and dimensions of atoms, any one of which would have made him famous in the world of science. Einstein was finally awarded a professorship in 1909 at the University of Zurich, in 1914 a place at the Prussian Academy of Sciences in Berlin, and in 1921 the Nobel Prize in Physics for, in the words of the Nobel Committee.

“…for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.”

When the Nazis seized power in Germany in 1933 (with the appointment of Hitler as Chancellor), Einstein, who was visiting the United States, decided not to return to Germany, living and working at the Institute for Advanced Study in Princeton, New Jersey, until his death in 1955.

Our intention in this entry is to briefly discuss the paper that awarded Einstein his Nobel Prize: «Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt» (Concerning an heuristic point of view toward the emission and transformation of light), a manuscript that Einstein submitted to the Annalen der Physik (received on 18 March 1905 and published on 9 June 1905).

So what was this article, which in Einstein’s own words was very revolutionary, about?

In order to answer this question, we must briefly recall the state of the problem of black body radiation in the early years of the twentieth century. As we saw in the two previous entries, in the early 1900s there were two theoretical formulae, Wien’s law for high frequencies and Rayleigh’s law for low frequencies, both incompatible with each other. As we also saw there, Planck found a formula in October 1900 that made it possible to unify them, but in an attempt to give it a physical basis, Planck had to introduce the energy quanta, something rather artificial and with little foundation, and which in general even Planck himself did not like. This was apparently a good match between theory and experiment, but there were still a few loose ends to be ironed out. For example, what was the physical significance of the constant \(h\) that Planck had introduced?

Wien himself, in an article published on 11 November 1900 (shortly before Planck made public the deduction of his formula using the energy quanta) argued that his law (unlike Rayleigh’s) could not be obtained from electrodynamics. Einstein, influenced by Wien’s work, decided to assume Wein’s hypothesis, i.e., that black body radiation was divided into two types: one valid for low frequencies (Rayleigh’s) which could be described by the laws of classical electrodynamics, and another for high frequencies whose explanation was based on undiscovered laws. Einstein wanted to find out what these other laws were.

Einstein’s starting point was similar to Planck’s: to write an expression for the entropy S (see the previous post) of radiation in a volume V with a frequency \(\omega\) and an energy \(E_{\omega}\). Although the expression obtained by Einstein was similar to Planck’s, unlike him, Einstein took the dependence he had found between the entropy S and the volume V occupied by the radiation in a cavity (as the black body used to be called) to its ultimate consequences. Although we will not go into technical details (the interested reader can find an English version of Einstein’s article here which is very easy to follow), we will say that Einstein proved that if the monochromatic radiation of frequency \(\omega\) and energy \(E_\omega\) was enclosed inside the cavity of volume \(V_0\), the probability P that in an arbitrary time the total energy of radiation would be in a part V of the initial volume \(V_0\) must be

$$ P=\left(\frac{V}{V_0}\right)^{\frac{E_\omega}{h\omega}}$$

(Einstein did not use the constant h, but a combination of other constants of the kinetic theory of gases and the constant \(\beta\) which appeared in Wien’s formula, but for the sake of unification of notation and for the sake of better understanding we will use Planck’s h here).

From this he concluded that:

“Monochromatic radiation of low densities (within the range of validity of the Wien radiation formula) behaves with respect to the kinetic theory of heat, as if it consisted of mutually independent quanta of energy with magnitude \(h\omega\). […]”

That is, according to Einstein, light (radiation) of high frequencies \(\omega\) behaved as particles with an energy equal to \(h\omega\), something that was in contradiction with Maxwell’s laws of electromagnetism, since according to these laws electromagnetic radiation had to be described by a continuous function in space and time. Thus Einstein recovered the corpuscular property for light that Newton had introduced in his famous work on optics Opticks (published in 1704) and that was discarded in favour of the wave theory following the work of Young and Fresnel in the 19th century, which made it possible to explain a greater number of optical phenomena.

But Einstein was not satisfied with this. As he himself wrote at the end of section 6 of his article

“If now monochromatic radiation (with a sufficiently low density), as far as the volume entropy dependence is concerned, behaves as a discontinuous medium, which consists of energy quanta of size \(h\omega\), it is reasonable to investigate whether also the laws of light production and transformation are elaborated as if light consisted of such energy quanta.”

Indeed, in the rest of the article Einstein shows how his proposal to consider radiation as quanta of light of energy \(E_\omega= h\omega\) can explain certain phenomena that the wave theory was unable to do, just those where radiation was created or transformed. For this he chose three phenomena: 1) Stokes’ law of photoluminescence, 2) the photoelectric effect and 3) the ionisation of gases by ultraviolet light.

We will restrict ourselves to the second of these phenomena: the photoelectric effect. This phenomenon consists of the emission of electrons by a material when electromagnetic radiation is incident on it. It was discovered by Heinrich Hertz in 1887 and studied by several physicists throughout the 19th century, but it was the work of another German physicist, Philipp Lenard, who in 1902 published in a long article the results of his many experiments, among which were two very puzzling properties of the phenomenon: 1) that current (electrons) could only be obtained from certain frequencies of light, and 2) that the speed of the electrons increased with the frequency of the incident light and independently of the intensity of the incident light (as should be the case according to Maxwell’s electromagnetic theory).

To explain Leonard’s results, Einstein reasoned as follows: Suppose we irradiate the metal plate with a monochromatic light of frequency \(\omega\) composed of quanta of light energy \(E_\omega = h\omega\). If we denote by \(W\) the energy needed to extract an electron from the metal, then the kinetic energy of the electrons, \(E_c\), will be expressed by the formula

$$ E_c=\frac12 mv^2=h\omega -W, $$

where \(m\) is the mass of the electron and \(v\) its velocity.

This simple formula perfectly explained the two observations described above: the need for a minimum frequency (to overcome the amount of energy \(W\) needed to remove an electron from the metal) and that the energy of the electrons depended solely and exclusively on the frequency of the incident light (in a linear way, moreover). It must be said that Leonard’s results were rather qualitative due to the low precision of the measurements. It was not until 10 years later that the renowned American experimental physicist Robert Milikan carried out the experiments that confirmed that the Einsteinian theory was correct. Milikan himself wrote years later in 1949:

“I spent ten years of my life testing Einstein’s 1905 equation, and contrary to all my expectations I was driven, in 1915, to proclaim the undoubted experimental verification, however unreasonable it was, since it seemed to violate everything we knew about the interference of light.”

If Planck quantized the way light was absorbed and emitted, Einstein quantized light itself. This was the straw that broke the camel’s back, for it seemed that light, which was unanimously accepted as a continuous phenomenon, had a certain corpuscular nature. Einstein spent several more years studying other phenomena that could also be perfectly explained using his theory of light quanta, and he became increasingly convinced that light had a dual behaviour: on the one hand undulatory, as described by the laws of electrodynamics, and on the other hand corpuscular, as described by the laws of the kinetic theory of gases. He also discovered that both his quantum theory of light and Planck’s quantum theory rested on the same principles and should therefore somehow be unified in a new theory. That theory is what we know today as “quantum theory” or “quantum physics” and it had to have (according to Einstein) very different foundations from those of classical physics. But that is another story, which we will deal with another time.

Learn more:

Chapter 1 of J. Mehra y H. Rechenberg, The Historical Development of Quantum Theory, Vol 1. The Quantum Theory of Planck, Einstein, Bohr and Sommerfeld: Its Foundation and the Rise of Its Difficulties 1900-1925, 1982 Springer-Verlag New York Inc.

José Manuel Sánchez Ron, Historia de la física cuántica: I. El período fundacional (1860-1926), Drakontos, 2001.

Learn more about Einstein’s life:

Antonio J. Durán, El universo sobre nosotros. Un periplo fascinante desde el cielo de don Quijote al cosmos de Einstein, Crítica, 2015.