Einstein and the Quantum

by Stone, A. Douglas

  Consider further the previous example, where we have opened a partition and let the gas fill the entire box instead of just half the box. Now it is possible that if we wait long enough, all the collisions could work out just right, and all the gas molecules could reconvene on the left side. Is it worth waiting for this to occur? Not really. One can easily calculate that if all the states are equally likely and if we have only forty gas molecules in the box, it would take about the age of the universe for this to happen. With a trillion trillion molecules in the box? As they say in New York: fuggedaboutit.

  This was the subtle point that Einstein missed in trying to prove that the Second Law of thermodynamics, the increase of entropy, is an absolute law. It isn’t absolute; entropy is allowed to decrease. Just don’t bet on it.

  In fact, this overwhelmingly probable increase of entropy is how we determine the direction of time. Imagine that the gas in our box is colored and hence visible as it expands; if we saw a movie of the gas contracting back into half of the box, we would immediately assume that the movie was being run backward. Because the arrow of time is so fundamental, it was natural for physicists to assume that the increase of entropy was an absolute law of nature and not just a very, very, very, very, very … likely occurrence. When the young Einstein made this mistake, he was in good company; Boltzmann also got this wrong until his critics pointed it out to him. However, the canny Scot, Maxwell, was not fooled, and described the situation with colorful imagery: “the Second Law of Thermodynamics has the same degree of truth as the statement that if you throw a tumblerful of water into the sea, you cannot get the same tumblerful of water out again.”

  Maxwell invented an imaginary creature, dubbed “Maxwell’s demon,” to illustrate this point further. His demons were “lively beings incapable of doing work” (i.e., of adding energy to the gas). He imagined these miniature sprites hovering in the gas, which is uniformly distributed in the box, but now a partition is added to the box, in which the demon has cleverly fashioned a frictionless trapdoor. Whenever the demon sees a gas molecule coming at him from the right to left with high velocity, he lets it through, then closes the trapdoor before any molecules can escape from the left to right. In this way over time he groups the faster molecules on the left and the slower ones on the right. But for a gas the temperature is proportional to the average energy, so by doing this the demon has heated the left side and cooled the right side, without putting any energy in. In other words the demon has created a refrigerator on the right (and a heater on the left), neither of which require any fuel (Einstein definitely would have rejected this patent). And why did Maxwell create his demons? Not as a serious proposal for an invention. Instead his intention was “to show that the Second Law of Thermodynamics has only a statistical certainty.” Maxwell’s demons, spawned around 1870, did not fancy the trip across the Channel, and so the true meaning of the Second Law was not understood in Europe for several more decades.

  While Einstein did not recognize the “demonic” exception to the Second Law when he was reinventing Gibbs’s statistical mechanics in 1903–4, he did very much focus on what he called “Boltzmann’s principle,” the mathematical epitaph, S = k log W, mentioned above. In his third paper on statistical mechanics in 1904, he states, “I derive an expression for the entropy of a system, which is completely analogous to the expression found by Boltzmann for ideal gases [S = k log W] and assumed by Planck in his theory of radiation” (italics added). Later in that same paper he explicitly applies his results to Planck’s thermal radiation law, although in a manner that doesn’t yet refer to the quantum concept. This made Einstein the first physicist to extend the use of Planck’s law and to accept that statistical mechanics, which had previously been used only to describe gases, could also explain the properties of electromagnetic radiation. Radiation at this point was conceived to be a purely wave phenomenon, having nothing in common with the aggregate of particles (molecules) that make up a gas. Einstein now analyzed thermal radiation using his statistical methods, and he was beginning to see the problems with Planck’s “desperate” solution.

  So what had Planck actually assumed about radiation, and how had he used Boltzmann’s principle to justify the formula that he had initially guessed by fitting the data? Planck had not been as bold as Einstein; he did not apply statistical mechanics to radiation but rather to the matter that exchanged energy with radiation. The Planck radiation law is, strictly speaking, only completely correct for what physicists call a “blackbody.” We all learn in school that the color white is a mixture of all colors and that black is the absence of color. A perfectly black body absorbs all light that falls upon it; hence no light of any color is directly reflected from it, and it appears black. In contrast, a surface that looks blue to us absorbs most of the red and yellow light incident on it and reflects the blue to our eye. But does the black object actually emit no light? Well, yes and no. It doesn’t emit any visible light, but it does send out a lot of electromagnetic radiation; however, as we learned earlier, if the object is at room temperature, the radiation is mainly at infrared wavelengths, which we can’t see. As already mentioned, the radiation law is precisely the rule for how much EM radiation of a given wavelength a blackbody emits at a given temperature.

  To test this ideal behavior, physicists had to find a perfectly black body, not just for visible radiation but for all possible wavelengths. Unfortunately all real materials reflect EM radiation at some wavelengths, so soot, oil, burnt toast, and the other obvious candidates don’t actually do the job. So the experimenters came up with a clever idea: instead of using the surface of a material, they would use the inside of a kind of furnace with a small hole. Any radiation that went in through the hole would bounce around, being reflected many times, but eventually it would be absorbed before escaping. Thus any light coming out of the hole must have been emitted from the walls and would be representative of a perfect blackbody.

  It was this kind of ideal black box or “radiation cavity” that Planck analyzed between 1895 and 1900. And one of his first ideas was to transfer his ignorance of the blackbody law from radiation to matter. He assumed that the walls of the cavity were made of molecules that would vibrate at a certain frequency in response to the EM radiation that fell upon them. Then by a clever argument he related the density of the energy of EM radiation at a given frequency (his goal), to the average energy of the vibrating molecules at the same frequency.3 He thus no longer had to deal with Maxwell’s equations describing the electromagnetic waves; he could assume that Newton’s laws held for the molecular vibrations, and he could use statistical mechanics. However, instead of doing the obvious thing and calculating the average energy of a molecule from statistical mechanics à la Boltzmann, he chose to find the entropy of the molecules.

  He did this for a strange historical reason. When he began studying blackbody radiation in 1895, he hoped to find the missing principle that would restore the perfection of the Second Law and make the increase of entropy and hence the arrow of time an absolute principle of nature. While he had been convinced that the equations of matter allowed for entropy to decrease (although very rarely!), he hoped that those of Maxwell would prevent this from ever happening. This turned out to be a vain hope, as Boltzmann himself was able to demonstrate to Planck. However, having committed himself to the study of the entropy of radiation, and since the actual radiation law was still not definitively known, Planck continued his investigations. He knew that if he could find the average entropy of thermal radiation, it was related by straightforward mathematical steps to the average energy density, and hence to the correct radiation law.

  At first Planck made an incorrect argument, which did not rely on Boltzmann’s principle. This led him to what was then called the Planck-Wien4 radiation law, and the embarrassing retraction in October 1900 after that law was ruled out by the experiments of his friends Rubens and Kurlbaum. At that point, guided by their experimental results and his mathematical intuition, as we saw ear
lier, he guessed the right form of the radiation law. Now, working backward from his apparently correct guess for the energy density of the radiation, he could figure out what the corresponding mathematical expression for the entropy of the radiation had to be. So this distinguished physicist was in a position oddly familiar to novice physics students, who might find the correct answer to a problem listed in the solutions at the back of their textbook but can’t quite figure out how to get that answer based on the principles they are supposed to have learned.

  Faced with this quandary, for the first time in his career Planck resorted to Boltzmann’s principle. By accepting and using that principle (the formula S = k log W), he now had an approach to justify his empirical guess from the fundamental laws of statistical physics. What he needed to do was to count the possible states of molecular vibration, W, and show that when plugged into Boltzmann’s formula, it gave the answer that he “knew” was correct.

  The mathematical problem he faced can be posed as follows. Planck assumed that all the molecules in the walls of the blackbody cavity had a fixed total amount of energy, which we can think of as a quantity of liquid, such as ten gallons of milk. For simplicity, imagine that there are one hundred molecules in the walls and that each molecule corresponds to a container that can hold up to the entire ten gallons. The question is how many ways can the ten gallons be shared among the hundred containers? If milk (and energy) are assumed to be continuous, infinitely divisible quantities, then the obvious answer is an infinite number of ways. But this didn’t deter Planck. The number of places you can put a gas molecule in a box is also infinite, but Boltzmann had found that his answer for the entropy of a system didn’t depend in any important way on how he divided the box into smaller boxes. So Planck essentially put little tick marks on the molecular energy containers, saying, for our imaginary example, that milk could only be distributed one fluid ounce at a time. Now he could go ahead and calculate the finite number of ways the milk could be shared and how that number depended on both the total amount of milk (the energy), the number of containers (molecules), and the size of the tick marks (the minimum “quantum” of energy). He was expecting that, as for Boltzmann’s gas calculation, nothing crucial would depend on the size of the tick marks. He was mistaken.

  Try as he might, if he let the spacing of the tick marks get smaller and smaller, the calculation yielded the wrong entropy and the wrong radiation law. Finally he was forced to the conclusion that there must be some smallest spacing of the tick marks, that is, that energy could only be distributed among the molecules in some smallest “quantized” unit. Since there was absolutely zero justification for this final hypothesis, it is clear why Planck called it “an act of desperation.” To his credit, however, Planck did not shy away from stating clearly his unprecedented conclusion in his famous lecture of December 14, 1900, on the blackbody law:

  We consider, however—this is the most essential point of the whole calculation—[the energy] E to be composed of a very definite number of equal parts and use thereto the constant of nature h = 6.55 × 10−27 erg-sec. This constant, multiplied by the frequency ν … gives us the energy element, ε.

  Now we can understand fully this cryptic statement. The “definite number of equal parts” were the “tick marks,” that is, the minimum quantum of molecular vibrational energy, ε. Moreover it was clear from other considerations that in order to get the right radiation law, this minimum energy must be proportional to the frequency at which the molecules vibrated; thus he was forced to the conclusion that ε = hν. Here the Greek letter ν stands for the vibration frequency, and h (as Planck says) is a new constant of nature, undreamt of in our previous natural philosophies. Finally, because the radiation law was measured experimentally, he could go to the data and quickly figure out the actual value of the constant h (quoted above), which is now known as Planck’s constant and is the signature of all things quantum.

  Planck later said that the radiation law had to be justified “no matter how high the cost.” Although he didn’t emphasize it at the time, the cost was very high. Planck’s little, technical fudge, if taken seriously, said something very, very strange about forces and motion at the atomic scale. It said that the Newtonian picture could not be right. For all intents and purposes, Planck had described molecules as little balls on springs, which stored energy by being compressed, and when the springs vibrated the energy was transferred back and forth between this stored (potential) energy and the kinetic energy of motion of the molecules, but in such a way that the sum of these energies, the total energy, was conserved. This much is standard Newtonian physics.

  But in Newtonian physics the initial amount of total energy can vary continuously; all you need to do is compress the spring a little more, and it will have a little more energy. The fact that it can have any amount of energy (between some limits) appears intuitively to be related to the very fact that space is continuous. Nothing in Newtonian physics could explain quantized amounts of energy, the idea that the spring could only be compressed, say, precisely 1 or 2 or 3 or … inches but nothing in between. This was like imagining a car that can only go 0, 10, 20, … miles per hour and nothing in between. The obvious question is: how does it get from 0 to 10 miles per hour without passing through the intermediate values as it accelerates?

  There was nothing innocent about Planck’s explanation of the radiation law. If it were the real explanation, it was a time bomb hidden in a thicket of algebra, which would explode with earth-shattering implications. Atoms and molecules were not little Newtonian billiard balls; they obeyed completely different and counterintuitive laws.

  But Planck did not insist that his quantum hypothesis was a statement about the real mechanics of actual molecules. In fact he dropped a small hint in his lecture that perhaps energy is not really quantized. He denoted the total energy of his molecules as E and stated, “dividing E by ε we get the number P of energy elements which must be divided over the N resonators [molecules]. If this ratio is not an integer, we take for P an integer in the neighborhood” (italics added). But if molecular vibrations were really quantized, then E/ε would have to be a whole number! Planck was hedging his bets, signaling that one didn’t have to take this crazy energy element too seriously. Planck thought the constant of nature he had discovered, h, was very important, but there is no evidence that he believed his derivation invalidated Newtonian mechanics on the atomic scale.

  Why not? Theoretical physics is a tricky business; sometimes one can get the right answer with assumptions that are wrong, or at least with stronger assumptions than one really needs. Perhaps another line of argument would occur to Planck, one that would preserve the welcome constant h but dispense with the uncomfortable assumption of the energy quantum, ε. Perhaps this weird, apparent quantization of energy only involved the interaction of radiation with matter but not mechanics per se. After all, there had been no obvious evidence of Planck’s constant in other areas of physics. It could be a new embarrassment if he trumpeted this energy quantum as a breakthrough in atomic physics and it turned out not to be so. No, best to play it safe, thought Planck; no need to cry wolf.

  So, remarkably, Planck said nothing more in print for five full years about his great discovery, and the strange assumption buried in his derivation remained almost unnoticed. Except in Bern. There the unknown patent clerk’s searching investigations into the foundations of statistical mechanics were placing Planck’s Rube Goldberg mechanism on the witness stand and returning a verdict: not innocent.

  1 Experts will know that in this equation the base of the logarithm is not the usual base 10 version, but is what is called the natural logarithm. The difference is not essential for understanding the meaning of entropy.

  2 In chapters 24–25 we will learn that under certain circumstances the method for counting the states of quantum gases can differ from this classical reasoning. However, Boltzmann’s equation for entropy still holds, just with a different counting method for W.

 
3 Planck did not call his vibrating entities molecules but used the term “resonators” instead to emphasize that they were idealized microscopic oscillators and that he was not committing himself to any atomic theory. At this point the composition of the atom, with a compact nucleus and electrons bound to it, was not known, although, as we saw from Maxwell, the concept of atoms and molecules was widely accepted by the leading statistical physicists of the time.

  4 Recall that after Planck came up with his new radiation law, which agreed with experiment, his name became attached to the new correct law and was dropped from the older law, now referred to as simply Wien’s law.

  CHAPTER 8

  THOSE FABULOUS MOLECULES

  One of the great open questions in the history of science is how Einstein came to the core idea of his paradigm-shifting paper of 1905. No, not his paper on special relativity or his paper proposing the famous equation E = mc2. Einstein was asked over and over again how he had developed the key insights leading to the special and general theories of relativity, and he answered with various charming anecdotes that have become part of his legend. As far as we know, he never went on record as to how he came up with the basic conception for his first paper of the annus mirabilis, a radical alternative to Maxwell’s theory of electromagnetic waves, which is the only one of his discoveries that he himself labeled as “revolutionary.” He says nothing directly about how he arrived at his first work on quantum theory in either his contemporary correspondence or in the papers preceding it. However, there are a few clues in the historical record, and these suggest that the key insight was his realization that the Planck radiation law was absolutely incompatible with statistical mechanics, at least in the form developed by Maxwell, Boltzmann, and Gibbs. This understanding likely matured during the year 1904 and early in 1905, when he was living a comfortable married life with Mileva and, as he was unknown to the wider physics community, his scientific correspondence was quite thin, leaving few traces of his profound ruminations.