Heisenberg, Hayfever, and Heligoland

OR: Why the popular version of quantum mechanics is not the best
The first complete, self-consistent description of quantum mechanics, was developed largely by Werner Heisenberg, in 1925. At the beginning of that year, the understanding of the quantum world was confused and muddled. Heisenberg later described the situation in quantum physics at that time as a “peculiar mixture of incomprehensible mumbo jumbo and empirical success.” Nobody had the faintest idea how to construct a coherent theory to clear up the mess, until Heisenberg came along.
Heisenberg had completed his PhD, at the University of Munich, in 1923, when he was just 21. He was one of the first physicists to be brought up on quantum theory, and after a few months working with Niels Bohr in Copenhagen, in 1924 Heisenberg became Max Born’s assistant in Göttingen. The key to the breakthrough he achieved was based on an idea that he picked up almost immediately on his arrival in Göttingen. The important point is that all the observable features of atoms and electrons deal with two states, and the transition of the atom (or electron, or whatever) from one state to another. We have no picture of what is going on during the transition itself, and images involving things like orbits are just tacked on from our classical image of the behaviour of objects like planets. Heisenberg deliberately abandoned the classical picture of particles and orbits, and took a long, hard look at the mathematics that describes the associations between pairs of quantum states, without asking himself how the quantum entity gets from state A to state B.
Like many physicists at the time, Heisenberg was puzzling over the nature of electron orbits, the way electrons “jump” between orbits, and how this jumping produces the lines seen in atomic spectra. He was bogged down in a morass of mathematics when, late in May 1925, he was struck by an attack of hay fever so severe that he had to ask his professor, Max Born for a fortnight’s leave of absence to recover. He was granted a two-week break, and on 7 June went straight to the rocky island of Heligoland, far from any sources of pollen, to recover (as his birthday was on 5 December 1901, he was still only 23 in the spring and summer of 1925).
Heligoland is a tiny island, less than a square mile in area and rising only about 60 meters above the sea, located in the corner of the North Sea known as the German Bight. Because of its location, ownership of the island changed many times until 1714, when it was taken over by Denmark. In 1807, Heligoland was captured by the British during the Napoleonic Wars, and they held on to it until 1890, when it was swapped with Germany for the African island of Zanzibar. When Heisenberg arrived there, after a three-hour journey by ship from Cuxhaven, at the mouth of the Elbe, it was a fading seaside spa resort. “I must have looked quite a sight,” he tells us, “with my swollen face; in any case, my landlady took one look at me, concluded that I had been in a fight and promised to nurse me through the aftereffects.” But no nursing was required, as the clean air quickly restored him to full fitness, and in between long walks and long swims, with no distractions “I made much swifter progress than I would have done in Göttingen.”
In his autobiographical memoir Physics and Beyond (Harper & Row, New York, 1971), he described his feelings as everything began to fall into place, and at 3 a.m. one night he:

could no longer doubt the mathematical consistency and coherence of the kind of quantum mechanics to which my calculations pointed. At first, I was deeply alarmed. I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures nature had so generously spread out before me.

There were some very peculiar features about the mathematical relationships that Heisenberg had discovered. Because he was describing relationships between two states, Heisenberg had not been able to work with ordinary numbers, but had to use arrays of numbers, which he laid out as tables, which contained information about both states associated with a transition. Among other things, Heisenberg found that these tables did not commute. When two of the arrays were multiplied together, the answer you got depended on the order in which the multiplication was carried out — A x B was not the same as B x A.
I’ve been writing about quantum physics for more than forty years, and in all that time I’ve never been able to come up with a better analogy for these mathematical entities than that of a chess board with pieces arranged on it. A chess board is a two-dimensional array of 64 squares, and each square can be identified by a letter-number combination, starting with a1 and proceeding through a2, a3 and so on all the way up to h8. The “state” of a chess game can be described by an additional letter to tell you which squares are occupied by which pieces – for example, Qc7 would mean that there is a queen on the square c7 (for simplicity, I’ll ignore the difference between black and white pieces). Heisenberg used arrays of numbers not unlike this to describe the quantum state of a system, and worked out the rules for describing the way quantum system interact to change their states – in effect, multiplying the arrays of numbers together, and performing other mathematical manipulations.
Back in Göttingen, Born realised immediately what Heisenberg had discovered. Unlike Heisenberg, Born already knew about a then-obscure branch of pure mathematics dealing with entities known as matrices. He had studied them more than twenty years before; but the one thing that sticks in the mind of anyone who has ever studied matrices is that they do not commute!
In the summer of 1925, working with Pascual Jordan, Born translated Heisenberg’s mathematical insight into the formal language of matrices, and Born, Heisenberg and Jordan together published a full account of the work, in what became known as the “three-man paper”. The equations of Newtonian (classical) mechanics were replaced by similar equations involving matrices, and many of the fundamental concepts of classical mechanics — such as the conservation of energy — emerged naturally from the new equations. Matrix mechanics was seen to include Newtonian mechanics within itself, in much the same way that the equations of the general theory of relativity include the Newtonian description of gravity as a special case.
Unfortunately, few people appreciated the significance of this work. The mathematics were not so much difficult as unfamiliar, and it was not seized upon with the cries of delight that, with hindsight, you might expect. The one exception was in Cambridge, where Paul Dirac picked up the idea and developed it further almost before the ink was dry on the three-man paper. Dirac also found, independently of the Göttingen group, that the equations of matrix mechanics have the same structure as the equations of classical mechanics, with Newtonian mechanics included within them as a special case. Indeed, Dirac’s formulation (quantum algebra) went even further than matrix mechanics, and included matrix mechanics within itself as a special case.
Some mathematicians appreciated the importance of this work, but most physicists were unhappy about its abstract, theoretical nature. They liked the idea of particles in orbits, and were baffled by a theory which deliberately did away with any physical picture of what was going on inside atoms. So when, just a year later, Erwin Schrödinger came up with a version of quantum mechanics based on the familiarity of waves they did seize upon it with delight, and that, not matrix mechanics, became the standard way for physicists to think about the quantum world. This is, perhaps, unfortunate, because the one thing that is now absolutely clear about the quantum world is that it is not like the everyday world, and that although images like waves and orbits may be appealing, and comforting, they do not actually describe quantum reality.

Partly based on material from my book Erwin Schrödinger and the Quantum Revolution.