Einstein's Masterwork

by John Gribbin

  When he died, Kirchoff was the professor of Physics at the University of Berlin. His successor, who wasn’t appointed until 1889, was a 31-year-old physicist of the old school, Max Planck. Planck was so conservative, in scientific terms, that he might almost be described as a reactionary. He hated the way ideas involving probability and statistics, rather than the certainty of conventional mathematical equations, were being introduced into physics by people like Boltzmann, and had yet to be convinced of the reality of atoms. In 1882 he had stated dogmatically that: ‘despite the great success that the atomic theory has so far enjoyed, ultimately it will have to be abandoned in favour of the assumption of continuous matter.’6 What stuck in Planck’s throat was the idea that matter could come in discrete lumps, with gaps in between. He much preferred the image of electromagnetic radiation as smooth and continuous waves, and expected that matter would also be found to be smooth and continuous, regardless of how successful the kinetic theory and Boltzmann’s ideas on thermodynamics might seem to be.

  In 1897, though, J.J. Thomson, working at the Cavendish Laboratory in England, showed that the streams of radiation known as cathode rays were actually made up of tiny, electrically-charged particles, which soon came to be known as ‘electrons’. Whatever the reality of atoms, there could be no doubt that matter contained these little particles, and since they carry electric charge it seemed clear that there must be some connection between the behaviour of electrons in matter and the way matter radiated light. In particular, Maxwell’s equations told physicists that a charged particle vibrating to and fro (an electric oscillator) must radiate electromagnetic waves.

  By the end of the 19th century, though, the problem of how to describe black-body radiation mathematically had run in to a cul-de-sac – or rather, two cul-de-sacs. In the early 1890s, Wilhelm Wien, a lecturer at the University of Berlin, had come up with a mathematical description of black-body radiation that produced a graph which exactly matched one side of the spectrum, the short wavelength side of the hill, but was hopelessly wrong at describing the shape of the curve for longer wavelengths. In 1900, the English physicist Lord Rayleigh found another equation, based on different physical assumptions, that predicted a curve which exactly matched the black-body curve on the long wavelength side of the hill, but was hopeless when applied to the shorter wavelength side.g This discrepancy clearly showed that there was some fundamental misunderstanding of the nature of black-body radiation, and caused consternation among physicists. But in the same year, 1900, Planck, who had been studying the problem of black-body radiation intensively since 1897, came up with something that looked like it might be the answer to the puzzle.

  In October 1900, Planck came up with a formula – an equation – that described the entire black-body curve accurately, smoothing over the join between Wien’s law and Rayleigh’s law. This was essentially an empirical result, one worked out by trial and error, and it included two constants, one of which became known as Boltzmann’s constant and the other of which, given the label h, became known as Planck’s constant. Boltzmann’s constant usually turns up in connection with the properties of gases, but, unlike the case of the constant c in Maxwell’s equations, there was no obvious physical interpretation of the new constant h. Planck presented his results to a meeting of the Berlin Physical Society on 14 October, even though he was well aware that he still lacked any physical basis for the equation he had come up with. But he kept beavering away at the problem, and before the end of the year he had found a physical basis for the equation. It wasn’t particularly palatable to him, but it was the best he could do.

  Planck was working on the assumption that electromagnetic radiation was emitted or absorbed by matter because of the presence of the relatively newly discovered electrons jiggling about inside the matter. In trying to explain the nature of the cavity radiation studied by Kirchoff and others, he had to think of the walls of the cavity containing a large number of ‘harmonic oscillators’, each corresponding to a jiggling electron. Radiation would be radiated and absorbed, re-radiated and re-absorbed, re-re-radiated and re-re-absorbed, in a repeating process mixing up all the radiation to achieve a state of dynamic equilibrium, with the maximum amount of disorder, before it could escape from the pinhole.

  Unfortunately for Planck, this kind of equilibrium was described by the rules of thermodynamics – indeed, it is known as thermodynamic equilibrium – and describing it mathematically involved some of the statistical techniques developed by Boltzmann (which is where his constant comes in). Even more unfortunately, there was no getting away from the fact that electrons were individual particles, not a continuum, no matter how much Planck might like the idea that matter was continuous. The only way to take account of their collective behaviour was, once again, to use the statistical techniques developed by Boltzmann and Maxwell in connection with the study of the behaviour of large numbers of atoms and molecules. In particular, Planck was forced to use these statistical methods to calculate the property of the array of harmonic oscillators known as ‘entropy’. It’s worth elaborating a little on this, since it would also be central to Einstein’s work.

  Entropy is a very important concept in thermodynamics, but it can be understood very simply as a measure of the amount of disorder in a system. The entropy of an isolated system (which means anything left to its own devices, with no constructive input of energy from outside) always increases, which is a scientific way of saying that things wear out. If you build a house and leave it untended for a few hundred years, it will crumble away; but if you put a pile of bricks in a heap on the ground and leave them alone they will never spontaneously arrange themselves into a house. People, and other living things, can hold the increase of entropy at bay for a time by making use of the energy in the food we eat (which ultimately comes from the Sun and is stored in plants), but in the very long term everything wears out.

  Disordered systems have more entropy than ordered systems. That’s why an ice cube placed in a glass of water melts, evening out the difference between the water and the ice, and why water diffuses the ‘wrong way’ through a semi-permeable membrane. It’s why the radiation inside one of Kirchoff ’s cavities gets scrambled up into a complete mess. A chess board painted in black and white squares is an ordered system with relatively low entropy, but the same piece of board painted with the same amount of black and white paint but mixed to a uniform grey colour has less order, and more entropy. If you pour a can of black paint and a can of white paint into a bucket, you don’t end up with black on one side of the bucket and white on the other side, but a grey mixture. This mixture has higher entropy. Similarly, the completely mixed up light that emerges from the pinhole in the form of cavity radiation has very high entropy. One of the most important features of the statistical approach to thermodynamics is that systems are more likely to be found in high entropy states than in low entropy states.

  You have to give Planck credit for biting the bullet and using the ideas and techniques he abhorred to work out a physical basis for the equation for black-body radiation. But what he discovered was startling. According to this interpretation of events, light was not being emitted or absorbed in a continuous fashion, but in the form of little lumps, which he called ‘quanta’. Speaking to the Berlin Physical Society again on 14 December 1900, Planck described his new work, and said:

  We therefore regard – and this is the most essential point of the entire calculation – energy to be composed of a very definite number of equal finite packages, making use for that purpose of a natural constant h = 6.55 × 10–27 erg sec.

  Don’t worry about the units for h; just notice how tiny it is – a decimal point followed by 26 zeroes before you get to the 6. And the ‘finite packages’ are only equal for each colour, or wavelength, of light. The size of each packet of energy E is given by the equation E = h, where is the frequency (proportional to 1 divided by the wavelength) of that particular colour of light. Instead of establishing that matter is continuous, on the
face of things Planck had found that electromagnetic radiation was not continuous! But he didn’t interpret his results that way.

  Looking back from 1931, Planck described his breakthrough as ‘an act of desperation’ and said the idea of the quantum of energy was:

  A purely formal assumption and I didn’t give it much thought, except only that, under all circumstances and at whatever cost, I had to produce a positive result.7

  He still thought of light as a continuous wave (after all, Young’s experiment still worked!), and regarded the quanta as some kind of mathematical tool, only useful in the statistical process of adding up the contributions of all the harmonic oscillators. Nobody else knew what else the quanta could actually be either; but Einstein, who had studied Kirchoff ’s work while he was at the ETH, heard news of Planck’s work while he was teaching at Winterthur in 1901, and was deeply puzzled. He kept the problem of what Planck’s quanta really were in his mind all the time he was developing his own statistical skills with the work described in his first scientific papers. Since nobody else had made any progress with interpreting Planck’s equation (indeed, physicists were largely too puzzled even to try to explain it), when he had learned enough to be able to tackle the puzzle properly at the beginning of 1905 Einstein was able to pick up the thread exactly where Planck had left off.

  The first great paper Einstein wrote in 1905 (he finished it on 17 March) is often referred to as the ‘photo-electric paper’, not least because when Einstein eventually got his Nobel Prize, the citation focused on that aspect of his work. But the section on the photoelectric effect was only a relatively small part of the paper: one of several examples that Einstein used to illustrate the importance of the concept of light quanta. Nevertheless, it was a very important part of the paper, and these ideas had also been turning over in his mind since 1901.

  It all started, as I hinted earlier, with the work of Philip Lenard, a German physicist who carried out a series of experiments, starting in 1899, which investigated the way ultraviolet light shining onto the surface of a metal in a vacuum could cause it to emit electrons (then still known as cathode rays; remember that electrons had only been identified as particles in 1897!). The immediate impact Lenard’s work (of which more shortly) had on Einstein when he learned about it in 1901 can be seen from the beginning of a letter he wrote to Mileva late in May that year:

  I just read a wonderful paper by Lenard on the generation of cathode rays by ultraviolet light. Under the influence of this beautiful piece I am filled with such happiness and joy that I must absolutely share some of it with you.8

  What’s especially interesting about that letter is that it is the first one he wrote to Mileva after she informed him that she was pregnant; he gets around to offering his response to that news only later on in the letter. It is quite clear where Einstein’s priorities lay. But the important point in terms of the genesis of the ‘very revolutionary’ paper on the light quantum is that this was not something that sprang suddenly into his mind fully formed at the beginning of 1905, but was a project that he had been working on, from time to time, for four years. Einstein was a genius, but he was a methodical and hard-working genius.

  In some ways, he was also cautious, as we have seen with his choice of title for the Brownian motion paper. Although he knew his work on the light quantum was revolutionary, he was careful to choose a title for the photoelectric paper that would not be an immediate turnoff to readers of the Annalen der Physik. He settled on ‘On a Heuristic Point of View Concerning the Production and Transformation of Light’, thereby giving the impression that all he was offering was a convenient mathematical device for carrying out calculations, not a suggestion that light might really be made up of a stream of particles. But he immediately pulled the rug from under that cosy assumption in the opening paragraphs of the paper.

  First, he spelled out the dilemma confronting physics: ‘While we consider the state of a body to be completely determined by the positions and velocities of an indeed very large yet finite number of atoms and electrons, we make use of continuous spatial functions to determine the electromagnetic state of a volume of space.’ Which means that this ‘electromagnetic state’ can never be described by a finite number of quantities, no matter how big that number is. Einstein spelled out that the energy of a material body cannot be broken down into an arbitrarily large number of arbitrarily small parts, but that in contrast – according to Maxwell’s equations – the energy of light spreading out from a source gets weaker and weaker indefinitely as it spreads. These two concepts of energy had to come into conflict where matter and light interact with one another, as in the interior of a ‘black-body’ cavity, or in the photoelectric effect.

  Then Einstein made a key point. Of course, he acknowledged, every traditional optical experiment (such as Young’s experiment) produced results that were consistent with the wave model of light. But: ‘One should keep in mind, however, that optical observations refer to time averages rather than instantaneous values.’ In other words, light could indeed be composed of tiny particles, provided those particles were so small that their combined effects averaged out on the scale of the usual optical experiments to give the appearance of a smooth continuum. Einstein then gave a list of several recent experiments, including studies of black-body radiation and Lenard’s work on the photoelectric effect, which conflicted with the classical view of light as a wave, and warned his readers what was coming:

  According to the assumption considered here, in the propagation of a light ray from a point source, the energy is not distributed continuously over ever-increasing volumes of space, but consists of a finite number of energy quanta localized at points of space that move without dividing, and can be absorbed or generated only as complete units.

  In other words, if you had infinitely sensitive eyes and looked at a source of light from very far away, you would not see a continuous faint glow, but individual flashes of light, with total darkness in between, as individual light quanta arrived at your eyes.h

  If that didn’t whet the appetite of his potential readers, nothing would. Without more ado, Einstein plunged into the mathematical part of his paper. He began by taking a fresh look at Planck’s calculations, correcting some errors in Planck’s own work and coming up with a new derivation of the key equation, describing the black-body curve, in which Planck’s constant appears. Then, in a tour de force argument at the heart of his paper (the bit for which he should have got the Nobel Prize), Einstein compares the entropy of a certain volume of monochromatic radiation (a box full of light of a single colour) with the entropy of a certain volume of gas. He calculates the entropy of the radiation for short-wavelength radiation (in the region where Wien’s law applies) and the entropy of an equivalent box of gas from, what were by then, the standard techniques developed by Boltzmann – and he gets the same answer. His conclusion is that:

  Monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves thermodynamically as if it consisted of mutually independent energy quanta.

  This is arguably the most revolutionary sentence written in science in the 20th century, given the success of Maxwell’s equations. What Einstein is saying is that, as far as thermodynamic properties such as entropy are concerned, a gas behaves as if it is made up of very many tiny particles (atoms and molecules), and electromagnetic radiation behaves as if it is made up of very many tiny particles (‘atoms of light,’ or photons). At this basic level, there is no fundamental difference between matter and light after all. And there is another point – the most important point of all – which might easily be missed by the casual reader. Einstein has reached this conclusion without having to assume anything at all about the way light interacts with the ‘harmonic oscillators’ that are at the heart of Planck’s treatment of the problem. He is explicitly saying that this graininess of light is an intrinsic property of the light itself, not something to do with the way light interacts with matter. He hasn’t ju
st reproduced Planck’s result, but has done something much more fundamental than Planck ever did.

  It is only after dropping this bombshell that Einstein goes on to consider the implications of his discovery for other areas of physics, most notably the photoelectric effect. He did not, as many people imagine, come up with the light quantum idea from the photoelectric effect, but used the photoelectric effect to demonstrate the power of this idea. And it still perfectly demonstrates the power of this ‘heuristic point of view’ concerning the nature of light.

  The curious thing that Lenard had discovered, which had made Einstein so excited in 1901, was that the energy of an electron produced by the photoelectric effect does not depend on the intensity of the light (how bright it is), but it does depend on the wavelength of the light. The effect only happens at all for ultraviolet light, which covers a range of wavelengths even shorter than the wavelength of blue light and cannot be seen by the human eye; so ‘colour’ is not really the right word to use, but in a sense the energy of the electrons produced by the photoelectric effect depends on the colour of the light shining on the metal surface. Since the energy of the electrons determines their speed, you can also say that the speed with which the electrons are ejected depends only on the colour of the light shining on the surface.

  Lenard’s discovery runs counter to common sense, because a bright light carries more energy than a dim light. You would expect a bright light to knock electrons out of the metal surface more energetically, so they would move away faster. But Lenard found no such effect. If he used ultraviolet light with a particular wavelength, the ejected electrons always escaped with the same speed. If he turned the brightness of the light up, more electrons were ejected, but still with the same speed; if he turned the brightness of the light down, fewer electrons were ejected, but still with the same speed.