Einstein and the Quantum

by Stone, A. Douglas

  Fine, but do we really care that much about what happens when you swap atoms? Well, we should. Because it is very hard to think of atoms as particles in our usual everyday sense when they lack this individuality. After all, just as we could imagine painting one die red and the other blue (i.e., labeling them), can’t we somehow label atom one and atom two, and distinguish them? No, we can’t (according to Einstein). Atoms are fundamentally indistinguishable and impossible to label. Nature is such that they are not separate entities, with their own independent trajectories through space and time. They exist in an eerie, fuzzy state of oneness when aggregated. So the Bose-Einstein statistical worldview, coming from a different direction, reinforces the concept of wave-particle duality, in this case applied to both light and matter, and heralds the emerging discovery that the microscopic world exists in a bizarre mixture of potentiality and actuality.

  Einstein lays out this revolutionary idea in his second paper, read to the Prussian Academy on January 8, 1925, where he also predicts a totally unexpected condensation phenomenon that would have a profound influence on quantum physics up to the present. He introduces the new paper as follows: “When the Bose derivation of Planck’s radiation formula is taken seriously, then one is not permitted to ignore it as a theory of the ideal gas; when it is correctly applied, the radiation is recognized as a gas of quanta, so the analogy between the gas of quanta and the gas of molecules must be a complete one. In the following, the earlier development will be supplemented by something new, which seems to me to increase the interest of the subject.”

  The interesting “something new” is first presented as a mathematical paradox. In his first paper he derived an equation relating the density of the quantum ideal gas to the temperature of the gas. Upon close inspection one notices an odd feature of this equation. On the left-hand side of the equation sits the density of the gas in a container, a quantity that can be increased indefinitely simply by compressing the volume of the container, which is kept at a fixed temperature.4 But on the right-hand side is a mathematical expression that varies with temperature but cannot get larger than a certain maximum value if the temperature is fixed. This leads to an apparent contradiction, as Einstein points out: this equation violates the “self-evident requirement that the volume and temperature of an amount of gas can be given arbitrarily.” What happens, he asks, when at fixed temperature one lets the density increase by compressing it into a smaller volume until the density becomes greater than the maximum allowed?

  Having posed the question, he brilliantly resolves it with a bold hypothesis: “I maintain that in this case … an increasing number of the molecules go into the quantum state numbered 1 [the ground state], the state without kinetic energy…, a separation occurs; a part [of the gas] ‘condenses,’ the rest remains a ‘saturated ideal gas.’ ” Here he is making an analogy to an ordinary gas, like water vapor, which when cooled reaches a temperature at which it begins to condense partly into a liquid while still retaining a particular ratio of liquid to vapor.5 The reason that his hypothesis resolves the paradox is that in deriving the relation of density to pressure in his original paper he made an innocent mathematical transformation, which amounted to neglecting the single quantum state of the gas where each molecule has zero energy.6 Never before in the history of statistical physics had the neglect of a single state made any difference to the value of a thermodynamic property of a gas, such as its density. On the contrary, the number of states involved, as we saw earlier, is normally unimaginably large, and physicists routinely make approximations that neglect billions of states without giving it a second thought. But Einstein unerringly recognized that in this new world of Bose-Einstein statistics, this single zero-energy state would gobble up a macroscopic fraction of all the molecules, creating a novel quantum “liquid,” now known as a Bose-Einstein condensate.

  The generosity of the “Bose-Einstein” designation is not widely appreciated, as few physicists realize that Bose said not a word about the quantum ideal gas in his seminal paper. The paper that does predict quantum condensation belongs to Einstein alone, and it is a masterwork. The boldness of the young rebel combines with the technical virtuosity of the mature creator of general relativity to reach breathtaking conclusions with complete self-assurance. A lesser physicist would either not have noticed the subtle mathematical error introduced by the neglect of a single state or, even if noticed, would likely have dismissed its logical implications as so bizarre as to indicate some fundamental error. The reason that this condensation phenomenon seems so strange, even today, is that condensation of an ordinary gas is caused by the weak attraction between the gas molecules, which becomes important only when the gas is relatively dense. But Einstein is considering the theory of the ideal gas, in which such molecular interactions are assumed to be completely absent. His condensation phenomenon is driven purely by the newly discovered quantum “oneness” of identical particles, not by a force like electromagnetism, but by this strange statistical “pseudoforce” that Einstein was the first to recognize. Proposing it as a real physical phenomenon was an act of great courage.

  Bose-Einstein condensation is now one of the fundamental pillars of condensed-matter physics; it underlies the phenomena of superconductivity of solids and superfluidity of liquids such as helium at low temperatures,7 which have been the subject of five Nobel prizes. These substances have substantial interaction forces between atoms and electrons, unlike the ideal gas of Einstein’s theory, although it is clear theoretically that the “statistical attraction” of bosonic particles plays the key role in generating their unique properties. Nonetheless, it was big news and Nobel-worthy yet again when, in 1995, atomic physicists finally realized a holy grail of the field. They created an atomic gas with negligible interactions, cold enough8 to observe pure Bose-Einstein condensation—Einstein’s last great experimental prediction, coming to fruition a lifetime after its first statement.

  And again, as before, Einstein’s progress was a step too far for the physics world; it was ignored by the leading atomic physicists, such as Bohr, Sommerfeld, and Max Born, and by the soon-to-be-famous Werner Heisenberg and Wolfgang Pauli. Even Einstein’s admired colleague Planck and his great friend Ehrenfest, the former student of Boltzmann, thought it was unacceptable; they believed that the Bose-Einstein statistical method was just plain wrong. In the condensate paper Einstein responds explicitly to their criticism: “Ehrenfest and others have reported that in the Bose theory of radiation and in my [ideal gas theory], the quanta or molecules do not act in a manner statistically independent of each other … this is entirely correct…. The formula [for counting states according to Bose and Einstein] expresses indirectly an implicit hypothesis about the mutual influence of the molecules of a totally new and mysterious kind.” That’s it. Statistical independence (which here means the classical method of counting configurations) is out; statistical attraction is in. Quantum particles bunch; get used to it.

  Einstein then goes on to show that only Bose-Einstein statistics can save two things statistical physicists hold dear: Nernst’s Third Law of thermodynamics and the additivity of entropy (i.e., when one combines two systems their total entropy is the sum of their individual entropies), a less obvious but fundamental requirement for thermodynamics to make sense. Having carried physics to the very brink of the new quantum theory, he rests his case.

  While Einstein’s breakthrough did not have much impact on the mainstream of quantum research, a lesser-known but already well-respected physicist had been following carefully Einstein’s papers on the quantum ideal gas, the Austrian Erwin Schrödinger, who had recently been appointed professor of theoretical physics at the University of Zurich (Einstein’s first academic position). Schrödinger had previously worked on both general relativity theory and on the quantum ideal gas, and he had met Einstein recently in Innsbruck, shortly after his first paper on the ideal gas had been published. He wrote to Einstein in February of 1925, when he did not yet know of Einstein’s land
mark second paper on condensation, expressing skepticism about the validity of his application of Bose’s method to atoms in his first paper. Einstein replied a few weeks later with characteristic good humor: “your reproach is not unjustified, although I have not made a mistake in my paper…. In the Bose statistics, which I use, the quanta or molecules are not considered as being mutually independent objects.” He then drew a small diagram illustrating the case of just two particles in two states, pointing out that there is only one configuration in which the particles are in different states instead of two, as there would be in classical statistics. To Schrödinger, Einstein’s reply was a revelation: “only through your letter did the uniqueness and originality of your statistical method of calculation become clear to me,” he wrote in November of 1925. “I had not grasped it before at all despite the fact that Bose’s paper had already come out…. [Bose’s work] did not seem particularly interesting to me. Only your theory of gas degeneracy is really something fundamentally new.”

  FIGURE 25.1. First page of Einstein’s letter to Schrödinger of February 28, 1925 in which he explains how Bose statistics differs from classical statistics, using the diagram at the bottom of the page. The example is equivalent to two quantum coins: on the right Einstein lists the four states of two ordinary coins, whereas on the left he lists the three states of quantum coins, since heads-tails and tails-head are indistinguishable quantum mechanically. Austrian Central Library for Physics.

  Schrödinger wanted to follow up on the suggestion, which Einstein had made at the end of the paper, that the strange statistics could be understood using the concept of matter waves, recently introduced by the young French physicist Louis de Broglie. Within months this newly inspired research direction would fundamentally change the emerging modern form of quantum theory, and enshrine Herr Schrödinger in the eternal pantheon of physics.

  1 Such extremely low temperatures were now available to physicists since 1908, following the Nobel Prize–winning research of the Dutch physicist H. Kammerling Onnes on liquefying and cooling helium. Einstein and Onnes were well acquainted, having met at the First Solvay Congress, and interacted frequently during Einstein’s regular visits to Leiden.

  2 At very high energies there are processes in which massive particles, such as an electron and a positron, annihilate one another and disappear, but such processes are not relevant here.

  3 It is a good approximation to neglect the interaction energy between atoms, based on the assumption that the gas is dilute enough that most of the time atoms are well separated, so that their mutual interaction is weak.

  4 The gas can be kept at a fixed temperature as its density (and hence its pressure) increases simply by making the container out of a heat-conducting material and putting it in contact with a large “bath” that will add or subtract the necessary heat.

  5 This state of an ordinary gas-liquid is referred to as “phase-separated,” and the gas is termed “saturated,” hence Einstein’s use of the term in this new context.

  6 Quantum mavens might worry that having a particle at rest in a finite volume violates the uncertainty principle. They would be right; “zero energy” here means an essentially infinitesimal energy, much less than the natural thermal energy scale kT, determined by the macroscopic size of the container for the gas.

  7 Superconductivity is a phenomenon whereby, at low temperatures, a solid such as aluminum conducts electricity with no resistance and hence no dissipation of energy; superfluidity is a similar effect in liquids, whereby they lose their viscosity and flow without dissipating energy.

  8 The required temperature in these experiments was 170 billionths of a degree above absolute zero, by far the coldest temperature ever created by man at that time.

  CHAPTER 26

  THE ROYAL MARRIAGE: E = mc2 = hυ

  I said to myself that classical physics wasn’t sufficient, that all of the ancient edifice … was shaken and that it was necessary to reconstruct the edifice; but I didn’t think it was necessary to change fundamental notions, I thought that it was by the introduction of new things, completely unknown, that one would take into account quanta…. For example by a synthesis between waves and particles … it was more or less the point of view of Einstein.

  —LOUIS DE BROGLIE

  “A younger brother of the de Broglie known to us has made a very interesting attempt to interpret the Bohr-Sommerfeld quantization rules in his dissertation. I believe that it is the first feeble ray of light to illuminate this, the worst of our physical riddles. I have also discovered something that supports his construction.” So Einstein wrote to Lorentz in December of 1924, just as he was completing his masterpiece on the quantum ideal gas. In the midst of his cogitations on the meaning of Bose statistics, during the summer of 1924, a second bolt from the blue singed his mail slot. This revelation came in the form of a doctoral thesis, sent to him by his old friend Paul Langevin, the French physicist, with the somewhat skeptical assessment, “[the thesis] is a bit strange, but after all, Bohr also was a bit strange, so see if it is worth something all the same.” Just as with Bose, Einstein was willing to look at this young man’s ideas with an open mind and recognize the crucial insight they represented. He wrote back to Langevin with a warm and eloquent endorsement: “Louis de Broglie’s work has greatly impressed me. He has lifted a corner of the great veil. In my work I [have recently] obtained results that seem to confirm his. If you see him please tell him how much esteem and sympathy I have for him.”

  Louis de Broglie, like Bose, had venerated Einstein from his earliest exposure to modern physics, but, unlike Bose, he had good reason to expect that Einstein might someday take his ideas seriously. Louis’ older brother, Maurice, was one of the most distinguished experimental physicists in France and had been one of two scientific secretaries at the First Solvay Congress of 1911 (hence Einstein’s allusion to the “de Broglie known to us” in his letter to Lorentz). Moreover, the de Broglies were an eminent noble family of France; Maurice and Louis counted among their predecessors ministers, generals, and famous literary figures. Maurice himself held the title of Duke de Broglie, whereas Louis had the inherited rank of Prince of the Holy Roman Empire, awarded to all the direct descendants of his ancestor Duke Victor-François in reward for his martial feats, by order of the emperor Francis I.

  Louis Victor Pierre Raymond de Broglie was born on August 15, 1892, the youngest of the five offspring of his father Victor de Broglie. His brother, Maurice, was the second child, seventeen years older than Louis and first son (hence duke). He served as a father figure and mentor to Louis, particularly after the death of Victor in 1906; “at every stage of my life and career,” Louis wrote to his brother, “I found you near me as guide and support.” Maurice had begun a career as a naval officer in 1895 but, against the wishes of his family, abandoned this path in 1904 and devoted himself to science, taking the rather unusual step of installing a private laboratory in the family town house on the tony Rue Chateaubriand, adjacent to the Arc de Triomphe. He would become an enormously influential figure in French physics, successor to Langevin at the College de France, and nominated for the Nobel Prize multiple times, including in the same year, 1925, when his younger brother was first nominated.1

  Louis’ sister and close companion Pauline described the young prince glowingly:

  this little brother had become a charming child, slender, svelte, with a small laughing face, eyes shining with mischief, curled like a poodle…. His gaiety filled the house. He talked all the time, even at the dinner table, where the most severe injunctions to silence could not make him hold his tongue…. He had a prodigious memory … and seemed to have a particular taste for … political history…. He improvised speeches inspired by accounts of the newspapers and could recite unerringly complete lists of the Ministers of the Third Republic…. A great future as a statesman was predicted for Louis.

  After finishing his secondary education in 1909, graduating from the elite Lycée Janson de Sailly, and obtaini
ng his licence in historical studies, he was encouraged to “continue and prepare the diploma in history.” However, this early passion was dying out, and confusion about his future set in. “I could see that to do that it was necessary to go often to the library, make a large bibliography and such things. That didn’t appeal to me too much, and that was the beginning of the year of my ‘moral crisis.’ ” He studied law for a year while hesitating “between several intellectual directions … and [I] wasn’t very much in agreement with my brother, who would have preferred that I do either the Ecole Polytechnique or study diplomacy…. I wanted to do neither, and yet I didn’t want to follow him [into physics], which increased the crisis a bit.” Finally he committed himself to a course in advanced mathematics, and by the end of the year, according to his brother, “The hesitations are over, he has crossed the Rubicon, and the course of his thoughts has turned towards physics and more particularly theoretical physics.”

  The precise timing of this crisis and conversion is not clear, but it overlapped closely with Maurice’s service as secretary at the First Solvay Congress, which afforded Louis a unique window into the ferment of the new atomic theory. De Broglie recalled: “I began to think about quanta from the moment that my brother gave me the notes of the Solvay Congress of 1911, probably at the beginning of 1912,” and “with the ardor of my age I became enthusiastically interested in the problems that had been treated and I promised myself to devote all my efforts to achieve an understanding of the mysterious quanta, which Max Planck had introduced ten years earlier into theoretical physics, but whose deep significance had not yet been grasped.” Two years later, in 1913, he graduated with his licence de science, having performed brilliantly on his exams. “His enthusiasm was returning,” Maurice recalled, “with the certainty of being at last on the right track.”