Does God play dice?

In the posts entitled Bringing some order I, II and III we have described the birth of a new mechanics: quantum mechanics. Quantum mechanics was intended to explain what Newtonian mechanics was unable to describe: the world of the “infinitely” small. As we have been telling throughout this section of the blog, scientists gradually discovered how to explain the world around them. They left the gods behind and decided to be on their own. In the history of (Western) science there were, from the point of view of the writer of this post, two crucial moments: the first one, led by Galileo when he masterfully synthesised what we know today as the scientific method (which has been discussed here y here, for example), and the second one, which coincides with the publication of the Principia Mathematica by Sir Issac Newton, where what we have called Newton’s Programme was formulated.

After the publication of the Principia there was an explosion of studies that developed the knowledge of Nature to unimaginable limits. The methods established by Newton seemed to work, proving the English poet Alexander Pope right, who dedicated to Newton the famous epitaph: Nature, and Nature’s laws lay hid in night; God said, Let Newton be! and all was light.

Indeed, following Newton’s programme, the laws governing mechanical and electromagnetic phenomena were established and a consistent theory of heat (thermodynamics) was developed, among other advances. We also saw how, at the end of the 19th century, when everything seemed to be finished and only two little clouds remained on the horizon of physics (the famous little clouds of Lord Kelvin), they led to two new theories that shattered the world as we know it in a way that was hardly conceivable at the time.

The first little cloud, as we have already told, was the inexplicable constancy of the speed of light, no matter how it was measured, which led to Einstein’s theory of Relativity (first the special one in 1905 and then the general one in 1915). We have been dealing with the second little cloud in a series of posts including the three mentioned above.

As we explained in those posts, it all seemed to start to make sense, especially when Schrödinger discovered his wave equation. Physicists at the time were delighted that they could finally have at hand an accessible mathematical theory that was not as “twisted” as the matrix mechanics proposed by Heisenberg, Born and Jordan. Schrödinger was supposed to bring us back to a continuous world, to a physics where, with a bit of luck, there would be no such frowned-upon quantum jumps. Schrödinger’s elegant theory had only one thing left to explain: what was the physical meaning of the dreaded $latex wave function that appeared in Schrödinger’s equation? This entry will be devoted to explaining what Max Born discovered and what eventually won him the Nobel Prize in Physics in the late year 1954.

But before explaining Born’s discovery and its consequences, it is worth making a small historical digression related to the philosophy of science, which, as we shall see, was not apart from the physics of the early twentieth century. On one side were the physicists who were in favour of quantum mechanics (e.g., Bohr, Heisenberg, Pauli, Dirac) and on the other those who preferred something less revolutionary, more in line with classical ideas (Einstein, Planck, Schrödinger, Wien, etc.). There are two very well documented anecdotes, both by Heisenberg. The first is his discussion with Einstein about matrix mechanics.

In the spring of 1926 Heisenberg was invited to give a talk on his matrix mechanics in Berlin, at the famous physics colloquium, the centre of physics at that time, the home of Einstein or Planck among others. After the talk, Einstein asked Heisenberg to accompany him home to discuss the new mechanics in more detail. In his memoirs (Dialogues on Atomic Physics), Heisenberg gives a good account of this conversation. It is worth quoting the beginning of this conversation:

What you have just told us sounds very strange, –began Einstein. You admit that there are electrons in the atom, and in this you are undoubtedly right. But you want to completely suppress the orbits of the electrons in the atom, even though the electron trajectories are immediately visible in a fog chamber. Can you explain to me a little more precisely the reasons for these curious assumptions?

The orbits of the electrons in the atom cannot be observed, I replied, but from the radiation emitted by the atom in a discharge process, the oscillation frequencies and corresponding amplitudes of the electrons in the atom can be deduced immediately. The knowledge of the whole of the oscillation numbers and amplitudes is also in the above physics something of a substitute for the knowledge of the electron orbitals. And since it is reasonable to admit in a theory only those quantities which can be observed, it seemed natural to me to introduce only these sets as representatives of the electron orbitals.

But you do not seriously believe, –said Einstein –that only observable quantities can be accepted in a physical theory.

I thought, –I replied in surprise –that you had established this very thought as the basis of your theory of relativity! […]

I may have used this kind of philosophy, –Einstein replied –but it is nevertheless a contradiction in terms.

What follows is a philosophical discussion of the first order to which I will refer at the end of this post and which the interested reader can read in Heisenberg’s autobiography. What is extremely interesting is the end of this talk, at least as Heisenberg remembers it. Einstein, who was trying at all costs to understand the idea underlying Heisenberg’s quantum mechanics, asked him:

Why do you believe so strongly in your theory, if so many central problems are still totally unclear?

To which Heisenberg answered:

I believe, as you do, that the simplicity of the laws of nature has an objective character, that it is not just an economy of thought. When nature leads us to mathematical forms of great simplicity and beauty, to forms, I say, which have hitherto been unattained by anyone, one cannot help believing that they are ‘truth’, that is, that they represent an authentic feature of nature. It may be that these forms deal, moreover, with our relation to nature, that there is also an element of economy of thought in them. But since these forms would never have been arrived at spontaneously, since they were given to us primarily by nature, they also belong to reality itself, not only to our thinking about reality. You may reproach me for using an aesthetic criterion of truth in speaking of simplicity and beauty. But I must confess that for me a very great force of conviction emanates from the simplicity and beauty of the mathematical scheme which is here suggested to us by nature.

One cannot help but be struck by this passionate defence that mathematical beauty must be one of the pillars on which science is built. No doubt a mathematician would be delighted, but according to Heisenberg there was a much more important reason: such simplicity was closely linked to the scientific method and he made this clear to Einstein:

The simplicity of the mathematical scheme also has the consequence that it is possible to devise many experiments whose results can be calculated in advance with great accuracy in accordance with the theory. If the experiments are subsequently carried out and arrive at the predicted result, it can be concluded with certainty that the theory represents nature in this field.

After reading this reasoning by Heisenberg, it is hard to disagree with him.

The second story has to do with Schrödinger’s wave mechanics. Heisenberg himself told that in summer 1926 Schrödinger was invited by Sommerfeld to give a lecture on wave mechanics in Munich, so he decided to spend a few days at his parents’ house (who lived there) in order to attend Sommerfeld’s lectures and to discuss the subject with Schrödinger. Heisenberg recounts this in his memoirs:

Schrödinger first explained the mathematical principles of wave mechanics, using the model of the hydrogen atom, and we were all excited to see that a problem which Wolfgang Pauli had only been able to solve in a very complicated way with the help of the methods of quantum mechanics could now be dealt with elegantly and simply with conventional mathematical methods. But Schrödinger spoke at the end also of his interpretation of wave mechanics, to which I could not agree. In the ensuing discussion, I raised my objections; in particular, I pointed out that even Planck’s radiation law could not be understood with Schrödinger’s conception. But this criticism of mine was to no avail. Wilhelm Wien replied sharply that he understood, on the one hand, my regret that quantum mechanics was now finished and that it was no longer necessary to speak of such nonsense as quanta jumps and the like; but he hoped, on the other hand, that the difficulties I had pointed out would certainly be solved by Schrödinger within a short time. Schrödinger was not so sure in his answer, but he too thought that it was only a question of time before he could clarify the exact meaning of the problems I had raised. None of the others were impressed by my arguments. Even Sommerfeld, who always treated me with affection, could not escape the convincing force of Schrödinger’s mathematics.

After his visit to Munich, Schrödinger went to Copenhagen at Bohr’s invitation (probably as a result of a letter from Heisenberg), where he was subjected to scrupulous interrogation by Bohr. It was so intense that Schrödinger fell ill. This is still recounted by Heisenberg, who witnessed it:

The discussions between Bohr and Schrödinger began already at the Copenhagen station, and lasted every day from early morning until late at night. Schrödinger lived with the Bohrs, so that, for external reasons, there was hardly any occasion to interrupt the dialogue. Bohr was always unusually respectful and affable in his dealings with others. On this occasion, however, he behaved, in my opinion, like an inveterate fanatic, unwilling to make any concessions to his counterpart or to allow the slightest lack of clarity. It is practically impossible to reproduce the passion of the discussions on both sides and to express the deep-seated conviction that was equally palpable in Bohr’s and Schrödinger’s reasonings. […]

The discussion went on for many hours day and night, but no agreement could be reached. A few days later, Schrödinger fell ill, perhaps as a result of the enormous effort; he had to stay in bed with a cold and fever. Mrs Bohr looked after him and brought him tea and cakes, but Niels Bohr would sit on the edge of the bed and return to the subject with Schrödinger: “You must understand that…”. No real understanding could be reached then, because neither side could offer a complete and finished interpretation of quantum mechanics.

Schrödinger himself recalls his stay in Copenhagen in a letter to Wien on 21 October 1926: “Bohr’s approach to atomic problems is really striking. He is completely convinced that any understanding in the usual sense of the word is impossible. Consequently, the conversation almost immediately turns to philosophical questions, and soon one does not know whether one really holds the position he is attacking or whether one should attack the position he is defending”.

But the straw that broke the camel’s back was not the realisation that the two theories were mathematically equivalent (as we mentioned in previous posts), but two discoveries that overturned classical physics’ understanding of the world around us. The first of these was due to Max Born, who came up with the physical meaning of the wave function of the Schrödinger equation, which was a definite bombshell in the foundations of all physics at the time.

Although the physicists of the time were confident that wave mechanics would solve the problem of quantum physics, they gradually realised that there was a difficult problem to solve: that of quantum jumps. How to explain the emission of photons that was so clearly seen in atomic spectra? Schrödinger’s theory did not provide an answer to the discontinuities of which quantum jumps were a clear example. Schrödinger himself, despairing at Bohr’s many arguments about the impossibility of eliminating quantum jumps, told him: “If for all that we must confine ourselves to this damned acrobatics of the ‘quanta’, I regret that I have devoted part of my time to quantum theory”.

It was at that time (mid-1926) that Max Born realised that one way to understand what was going on was to study collisions between particles, something similar to what Rutherford did when he discovered, almost by accident, the atomic nucleus.

In Schrödinger’s view, particles were nothing more than wave packets. A wave packet can move through space without dispersing, so why not assume that particles were in fact nothing more than these wave packets concentrated in a small region of space? Going further, he tried to convince his colleagues that the wave function describing his equation could actually represent electrons.

We won’t go into the details, but his intuition turned out to be false. When Born used the Schrödinger equation to study collisions, and used an idea originating from Einstein himself in one of his attempts to explain wave-particle duality for photons, he discovered something very different: during a collision, the wave representing the particle (in his case the electron) diffused. The animation below shows schematically the difference between what should happen if, indeed, the particles were nothing more than a wave packet that cannot be scattered (what Schrödinger suggested) and what happened according to Born, that the packet after the collision was scattered. The animation below shows how the wave packet after the collision with another particle (the red dot) maintains its shape (which is what Schrödinger proposed) and above what Born discovered, that after the collision the packet dispersed until it practically disappeared.

In his Nobel Prize acceptance lecture, Born tells us in some detail:

A beam of electrons from infinity, represented by an incident wave of known intensity (i.e. \(|\Psi|^2\)) hits an obstacle, say a heavy atom. In the same way that a water wave produced by a ship causes secondary circular waves when it hits a pole, the incident electron wave is partially transformed into a secondary spherical wave whose oscillation amplitude \(\Psi\) varies in different directions. The square of the amplitude of this wave at a large distance from the scattering centre determines the relative scattering probability as a function of direction. Furthermore, if the scattering atom itself is able to exist in different stationary states, then the Schrödinger wave equation automatically gives the probability of excitation of these states, the electron scatters with energy loss, i.e. inelastically, as it is called. In this way it was possible to obtain a theoretical basis for the assumptions of Bohr’s theory that had been confirmed experimentally by Franck and Hertz.

At last a plausible explanation for the mysterious quantum jumps inside Bohr’s atom. But Born was aware of what he was doing: he was bringing indeterminacy into physics as no one had ever done before. So much so that in his original article “Zur Quantenmechanik der Stossvorgänge” (On the Quantum Mechanics of Collision Processes) in Zeitschrift für Physik, received by the journal’s editorial office on 25 June 1926, he wrote:

Thus, Schrödinger’s quantum mechanics gives a fairly definite answer to the question of the collision effect, but it is not a causal description. No answer is given to the question “what is the state after the collision?”, but only to the question “how probable is a specific result of the collision?” […].

Here is where the whole problem of determinism arises. From the point of view of our quantum mechanics, there is no magnitude which in each case causally determines the result of the collision, but neither experimentally at the moment do we have any reason to believe that there are any internal properties of the atom which condition a definite result for the collision. Should we expect to discover such properties later (…) and determine them in individual cases? Or should we believe that the agreement of theory and experiment is a pre-established harmony founded on the non-existence of such conditions? I myself am inclined to renounce determinism in the world of atoms. But that is a philosophical question for which physical arguments alone are not decisive.

Born’s approach was so revolutionary that it took some time for the scientific community to accept it. Even his colleague Heisenberg did not like the idea… but that was about to change, when shortly afterwards Heisenberg himself discovered the uncertainty principle that bears his name and which came to strike the death blow to determinism in physics, but we will deal with that story in a future entry.

As a postscript to this entry, it is worth reproducing the fragments of Einstein’s letter to Born dated 4 December 1926, probably some time after he had read his daring conclusion:

Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the ‘old one’. I, at any rate, am convinced that He is not playing at dice.

From then on Einstein always insisted, when talking about quantum mechanics, that God does not play dice. Nothing changed his mind. Throughout his life he always maintained that quantum mechanics was incomplete. Thus, in another letter to Born (with whom he was close friends), dated almost 20 years later, on 7 September 1944, he wrote:

We have turned out to be antipodes in relation to our scientific expectations. You believe in a God who plays dice, and I believe in absolute law and order in a world that exists objectively, and which I am foolishly speculatively trying to understand […]. Even the initial great success of quantum theory does not make me believe in a fundamental crapshoot, although I am aware that our young colleagues interpret this as a symptom of old age. No doubt the day will come when we will see whose instinctive attitude was correct.

In the end, Einstein “lost” his bet (another interesting one to tell in the future), as it seems that a dice game is indeed needed to explain and predict the phenomena of the microscopic world. Nevertheless, his sentence God does not play dice has remained forever in our cultural heritage.

Learn more

[1] Werner Heisenberg, Diálogos sobre la física atómica. La Editorial Católica. Madrid. 1975. Also available in the Universal Collection of Círculo de Lectores in the volume entitled “Física cuántica (Werner Heisenberg, Niels Bohr, Erwin Schrödinger)” Círculo de Lectores. Barcelona. 1996.

[2] David Lindley, Incertidumbre. Einstein, Heisenberg, Bohr y la lucha por la esencia de la ciencia, Ariel, 2008.

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

Final comments

The English translation of the original German letter from Einstein to Bohr is due to Born’s daughter Irene Born, and is taken from The Born-Einstein Letters. Correspondence between Albert Einstein and Max and Hedwig Born from 1916 to 1955 with commentaries by Max Born, Macmillan Press, London and Basingstoke, 1971.

Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the ‘old one’. I, at any rate, am convinced that He is not playing at dice.

In German Einstein does not refer to the ‘old man’ but to ‘Jacob’. David Lindley in [2] argues that it refers to the biblical character, although in the translation Irene Born opts for ‘the old man’ which could be understood as Jacob or as God, which is in agreement with the later sentence God is not playing at dice.

The featured image is a collage of Raphael’s painting in the Sistine Chapel, dice and an observing Einstein.

The animation shown here does not correspond to the Schrödinger equation but to sound waves, but the effect to which Born referred can be seen: the scattering of the particles. The animated file is taken from this blog.