by Manjit Kumar
On Friday, 19 October at the meeting of the German Physical Society, with Rubens and Planck sitting among the audience, it was Ferdinand Kurlbaum who made the formal announcement that Wien’s law was valid only at short wavelengths and failed at the longer wavelengths of the infrared. After Kurlbaum sat down, Planck rose to deliver a short ‘comment’ billed as ‘An Improvement of Wien’s Equation for the Spectrum’. He began by admitting that he had believed ‘Wien’s law must necessarily be true’, and had said so at a previous meeting.44 As he continued, it quickly became clear that Planck was not simply proposing ‘an improvement’, some minor tinkering with Wien’s law, but a completely new law of his own.
After speaking for less than ten minutes, Planck wrote his equation for the blackbody spectrum on the blackboard. Turning around to look at the familiar faces of his colleagues, he told them that this equation ‘as far as I can see at the moment, fits the observational data, published up to now’.45 As he sat down, Planck received polite nods of approval. The muted response was understandable. After all, what Planck had just proposed was another ad hoc formula manufactured to explain the experimental results. There were others who had already put forward equations of their own in the hope of filling the void, should the suspected failure of Wien’s law at long wavelengths be confirmed.
The next day Rubens visited Planck to reassure him. ‘He came to tell me that after the conclusion of the meeting he had that very night checked my formula against the results of his measurements,’ Planck remembered, ‘and found satisfactory concordance at every point.’46 Less than a week later, Rubens and Kurlbaum announced that they had compared their measurements with the predictions of five different formulae and found Planck’s to be much more accurate than any of the others. Paschen too confirmed that Planck’s equation matched his data. Yet despite this rapid corroboration by the experimentalists of the superiority of his formula, Planck was troubled.
He had his formula, but what did it mean? What was the underlying physics? Without an answer, Planck knew that it would, at best, be just an ‘improvement’ on Wien’s law and have ‘merely the standing of a law disclosed by a lucky intuition’ that possessed no more ‘than a formal significance’.47 ‘For this reason, on the very first day when I formulated this law,’ Planck said later, ‘I began to devote myself to the task of investing it with true physical meaning.’48 He could achieve this only by deriving his equation step by step using the principles of physics. Planck knew his destination, but he had to find a way of getting there. He possessed a priceless guide, the equation itself. But what price was he prepared to pay for such a journey?
The next six weeks were, Planck recalled, ‘the most strenuous work of my life’, after which ‘the darkness lifted and an unexpected vista began to appear’.49 On 13 November he wrote to Wien: ‘My new formula is well satisfied; I now have also obtained a theory for it, which I shall present in four weeks at the Physical Society here [in Berlin].’50 Planck said nothing to Wien either of the intense intellectual struggle that had led to his theory or the theory itself. He had strived long and hard during those weeks to reconcile his equation with the two grand theories of nineteenth-century physics: thermodynamics and electromagnetism. He failed.
‘A theoretical interpretation therefore had to be found at any cost,’ he accepted, ‘no matter how high.’51 He ‘was ready to sacrifice every one of my previous convictions about physical laws’.52 Planck no longer cared what it cost him, as long as he could ‘bring about a positive result’.53 For such an emotionally restrained man, who only truly expressed himself freely at the piano, this was highly charged language. Pushed to the limit in the struggle to understand his new formula, Planck was forced into ‘an act of desperation’ that led to the discovery of the quantum.54
As the walls of a blackbody are heated they emit infrared, visible, and ultraviolet radiation into the heart of the cavity. In his search for a theoretically consistent derivation of his law, Planck had to come up with a physical model that reproduced the spectral energy distribution of blackbody radiation. He had already been toying with an idea. It did not matter if the model failed to capture what was really going on; all Planck needed was a way of getting the right mix of frequencies, and therefore wavelengths, of the radiation present inside the cavity. He used the fact that this distribution depends only on the temperature of the blackbody and not on the material from which it is made to conjure up the simplest model he could.
‘Despite the great success that the atomic theory has so far enjoyed,’ Planck wrote in 1882, ‘ultimately it will have to be abandoned in favour of the assumption of continuous matter.’55 Eighteen years later, in the absence of indisputable proof of their existence, he still did not believe in atoms. Planck knew from the theory of electromagnetism that an electric charge oscillating at a certain frequency emits and absorbs radiation only of that frequency. He therefore chose to represent the walls of the blackbody as an enormous array of oscillators. Although each oscillator emits only a single frequency, collectively they emit the entire range of frequencies found within the blackbody.
A pendulum is an oscillator and its frequency is the number of swings per second, a single oscillation being one complete to and fro swing that returns the pendulum to its starting point. Another oscillator is a weight hanging from a spring. Its frequency is the number of times per second the weight bounces up and down after being pulled from its stationary position and released. The physics of such oscillations had long been understood and given the name ‘simple harmonic motion’ by the time Planck used oscillators, as he called them, in his theoretical model.
Planck envisaged his collection of oscillators as massless springs of varying stiffness, so as to reproduce the different frequencies, each with an electric charge attached. Heating the walls of the blackbody provided the energy needed to set the oscillators in motion. Whether an oscillator was active or not would depend only upon the temperature. If it were, then it would emit radiation into, and absorb radiation from, the cavity. In time, if the temperature is held constant, this dynamic give and take of radiation energy between the oscillators and the radiation in the cavity comes into balance and a state of thermal equilibrium is achieved.
Since the spectral energy distribution of blackbody radiation represents how the total energy is shared among the different frequencies, Planck assumed that it was the number of oscillators at each given frequency that determined the allocation. After setting up his hypothetical model, he had to devise a way to share out the available energy among the oscillators. In the weeks following its announcement, Planck discovered the hard way that he could not derive his formula using physics that he had long accepted as dogma. In desperation he turned to the ideas of an Austrian physicist, Ludwig Boltzmann, who was the foremost advocate of the atom. On the road to his blackbody formula, Planck became a convert as he accepted that atoms were more than just a convenient fiction, after years of being openly ‘hostile to the atomic theory’.56
The son of a tax collector, Ludwig Boltzmann was short and stout with an impressive late nineteenth-century beard. Born in Vienna on 20 February 1844, he was, for a while, taught the piano by the composer Anton Bruckner. A better physicist than a pianist, Boltzmann obtained his doctorate from the University of Vienna in 1866. He quickly made his reputation with fundamental contributions to the kinetic theory of gases, so called because its proponents believed that gases were made up of atoms or molecules in a state of continual motion. Later, in 1884, Boltzmann provided the theoretical justification for the discovery by Josef Stefan, his former mentor, that the total energy radiated by a blackbody is proportional to the temperature raised to the fourth power, T4 or T×T×T×T. It meant that doubling the temperature of a blackbody increased the energy it radiated by a factor of sixteen.
Boltzmann was a renowned teacher and, although a theorist, a very capable experimentalist despite being severely shortsighted. Whenever a vacancy arose at one of Europe’s leading universities his nam
e was usually on the list of potential candidates. It was only after he turned down the professorship at Berlin University left vacant by the death of Gustav Kirchhoff that a downgraded version was offered to Planck. By 1900 a much-travelled Boltzmann was at Leipzig University and universally acknowledged as one the great theoreticians. Yet there were many, like Planck, who found his approach to thermodynamics unacceptable.
Boltzmann believed that properties of gases, such as pressure, were the macroscopic manifestations of microscopic phenomena regulated by the laws of mechanics and probability. For those whose believed in atoms, the classical physics of Newton governed the movement of each gas molecule, but using Newtonian laws of motion to determine that of each of the countless molecules of a gas was for all practical purposes impossible. It was the 28-year-old Scottish physicist James Clerk Maxwell who, in 1860, captured the motion of gas molecules without measuring the velocity of a single one. Using statistics and probability, Maxwell worked out the most likely distribution of velocities as the gas molecules underwent incessant collisions with each other and the walls of a container. The introduction of statistics and probability was bold and innovative; it allowed Maxwell to explain many of the observed properties of gases. Thirteen years younger, Boltzmann followed in Maxwell’s footsteps to help shore up the kinetic theory of gases. In the 1870s he went one step further and developed a statistical interpretation of the second law of thermodynamics by linking entropy with disorder.
According to what became known as Boltzmann’s principle, entropy is a measure of the probability of finding a system in a particular state. A well-shuffled pack of playing cards, for example, is a disordered system with high entropy. However, a brand-new deck with cards arranged according to suit and from ace to king is a highly ordered system with low entropy. For Boltzmann the second law of thermodynamics concerns the evolution of a system with a low probability, and therefore low entropy, into a state of higher probability and high entropy. The second law is not an absolute law. It is possible for a system to go from a disordered state to a more ordered one, just as a shuffled pack of cards may, if shuffled again, become ordered. However, the odds against that happening are so astronomical that it would require many times the age of the universe to pass for it to occur.
Planck believed that the second law of thermodynamics was absolute – entropy always increases. In Boltzmann’s statistical interpretation, entropy nearly always increases. There was a world of difference between these two views as far as Planck was concerned. For him to turn to Boltzmann was a renunciation of everything that he held dear as a physicist, but he had no choice in his quest to derive his blackbody formula. ‘Until then I had paid no attention to the relationship between entropy and probability, in which I had little interest since every probability law permits exceptions; and at that time I assumed that the second law of thermodynamics was valid without exceptions.’57
A state of maximum entropy, maximum disorder, is the most probable state for a system. For a blackbody that state is thermal equilibrium – just the situation that Planck faced as he tried to find the most probable distribution of energy among his oscillators. If there are 1000 oscillators in total and ten have a frequency , it is these oscillators that determine the intensity of radiation emitted at that frequency. While the frequency of any one of Planck’s electric oscillators is fixed, the amount of energy it emits and absorbs depends solely upon its amplitude, the size of its oscillation. A pendulum completing five swings in five seconds has a frequency of one oscillation per second. However, if it swings through a wide arc the pendulum has more energy than if it traces out a smaller one. The frequency remains unchanged because the length of the pendulum fixes it, but the extra energy allows it to travel faster through a wide arc. The pendulum therefore completes the same number of oscillations in the same time as an identical pendulum swinging through a narrower arc.
Applying Boltzmann’s techniques, Planck discovered that he could derive his formula for the distribution of blackbody radiation only if the oscillators absorbed and emitted packets of energy that were proportional to their frequency of oscillation. It was the ‘most essential point of the whole calculation’, said Planck, to consider the energy at each frequency as being composed of a number of equal, indivisible ‘energy elements’ that he later called quanta.58
Guided by his formula, Planck had been forced into slicing up energy (E) into hv-sized chunks, where v is the frequency of the oscillator and h is a constant. E=hv would become one of the most famous equations in the whole of science. If, for example, the frequency was 20 and h was 2, then each quantum of energy would have a magnitude of 20×2=40. If the total energy available at this frequency were 3600, then there would be 3600/40=90 quanta to be distributed among the ten oscillators of that frequency. Planck learnt from Boltzmann how to determine the most probable distribution of these quanta among the oscillators.
He found that his oscillators could only have energies: o, h, 2h, 3h, 4h…all the way up to nh, where n is a whole number. This corresponded to either absorbing or emitting a whole number of ‘energy elements’ or ‘quanta’ of size h. It was like a bank cashier able to receive and dispense money only in denominations of £1, £2, £5, £10, £20 and £50. Since Planck’s oscillators cannot have any other energy, the amplitude of their oscillations is constrained. The strange implications of this are manifest if scaled up to the everyday world of a spring with a weight attached.
If the weight oscillates with an amplitude of 1cm, then it has an energy of 1 (ignoring the units of measuring energy). If the weight is pulled down to 2cm and allowed to oscillate, its frequency remains the same as before. However its energy, which is proportional to the square of the amplitude, is now 4. If the restriction on Planck’s oscillators applied to the weight, then between 1cm and 2cm it can oscillate only with amplitudes of 1.42cm and 1.73cm, because they have energies of 2 and 3.59 It cannot, for example, oscillate with an amplitude of 1.5cm because the associated energy would be 2.25. A quantum of energy is indivisible. An oscillator cannot receive a fraction of a quantum of energy; it must be all or nothing. This ran counter to the physics of the day. It placed no restrictions on the size of oscillation and therefore on how much energy an oscillator can emit or absorb in a single transaction – it could have any amount.
In his desperation Planck had discovered something so remarkable and unexpected that he failed to grasp its significance. It is not possible for his oscillators to absorb or emit energy continuously like water from a tap. Instead they can only gain and lose energy discontinuously, in small, indivisible units of E=h, where is the frequency with which the oscillator vibrates that exactly matches the frequency of the radiation it can absorb or emit.
The reason why large-scale oscillators are not seen to behave like Planck’s atomic-sized ones is because h is equal to 0.000000000000000000000000006626 erg seconds or 6.626 divided by one thousand trillion trillion. According to Planck’s formula, there could be no smaller step than h in the increase or decrease of energy, but the infinitesimal size of h makes quantum effects invisible in the world of the everyday when it comes to pendulums, children’s swings and vibrating weights.
Planck’s oscillators forced him to slice and dice radiation energy so as to feed them the correct bite-sized chunks of h. He did not believe that the energy of radiation was really chopped up into quanta. It was just the way his oscillators could receive and emit energy. The problem for Planck was that Boltzmann’s procedure for slicing energy required that at the end the slices be made ever thinner until mathematically their thickness was zero and they vanished, with the whole being restored. To reunite a sliced-up quantity in such a fashion was a mathematical technique at the very heart of calculus. Unfortunately for Planck, if he did the same his formula vanished too. He was stuck with quanta, but was unconcerned. He had his formula; the rest could be sorted out later.
‘Gentlemen!’ said Planck as he faced the members of the German Physical Society seated in the room a
t Berlin University’s Physics Institute. He could see Rubens, Lummer and Pringsheim among them as he began his lecture, ‘Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum’, On the Theory of the Energy Distribution Law of the Normal Spectrum. It was just after 5pm on Friday, 14 December 1900. ‘Several weeks ago I had the honour of directing your attention to a new equation that seemed suitable to me for expressing the law of the distribution of radiating energy over all areas of the normal spectrum.’60 Planck now presented the physics behind that new equation as he derived it.
At the end of the meeting his colleagues roundly congratulated him. Just as Planck regarded the introduction of the quantum, a packet of energy, as a ‘purely formal assumption’ to which he ‘really did not give much thought’, so did everyone else that day. What was important to them was that Planck had succeeded in providing a physical justification for the formula he had presented in October. To be sure, his idea of chopping up energy into quanta for the oscillators was rather strange, but it would be ironed out in time. All believed that it was nothing more than the usual theorist’s sleight of hand, a neat mathematical trick on the path to getting the right answer. It had no true physical significance. What continued to impress his colleagues was the accuracy of his new radiation law. Nobody really took much notice of the quantum of energy, including Planck himself.
Early one morning Planck left home with his seven-year-old son, Erwin. Father and son were headed to nearby Grunewald Forest. Walking there was a favourite pastime of Planck’s and he enjoyed taking his son along. Erwin later recalled that as the pair walked and talked, his father told him: ‘Today I have made a discovery as important as that of Newton.’61 When he recounted the tale years later, Erwin could not remember exactly when the walk took place. It was probably some time before the December lecture. Was it possible that Planck understood the full implications of the quantum after all? Or was he just trying to convey to his young son something of the importance of his new radiation law? Neither. He was simply expressing his joy at discovering not one but two new fundamental constants: k, which he called Boltzmann’s constant, and h, which he called the quantum of action but which physicists would call Planck’s constant. They were fixed and eternal, two of nature’s absolutes.62