There is a common theme in mathematics and physics which relates to visualising a theoretical structure. Now by 'visualising' we don't mean necessarily a geometrical model although often geometrical intuition will provide the right model. For example we try to understand the idea of convergence of a sequence by plotting points. We try to understand a group, not by the basic definition with four axioms, but rather by thinking of a group acting on some structure. Of course what provides the best model for one person may be quite different to the best model for another. Also it is often useful to be able to change models in different situations to gain the best intuitive feel. We want to examine here the different models for the atom provided by wave mechanics and by quantum mechanics. These two theories are equivalent but provide such different models to build ones intuition that it is not surprising that a fair argument ensued as to which model provided the best way to visualise an atom.

The model provided by Bohr in 1913 had great appeal, and for that matter it still does. Perhaps it is just the model that we would like to see for in one sense it provides a pleasing unity in the universe between the very large and the very small. That everything should be composed of atoms which are themselves tiny models of a solar system is just the way that we might feel the world should be. The extremely small would be just like the extremely large and so there would be a pleasing simplicity to the world. Max Born expressed that view in 1923:-

It is a model which transfers concepts which we understand well, such as the concept of particles orbiting a central body, to the atom, so it allows us to "understand" what an atom looks like. The difficulty is that, pleasing as this model might be, it suffers from the problem that we have taken concepts we understand and transferred them to a situation where they capture some aspects of that situation but in other respects lead to false conclusions.A remarkable and alluring result of Bohr's atomic theory is the demonstration that the atom is a small planetary system ... The thought that the laws of the macrocosmos in the small reflect the terrestrial world obviously exercises a great magic on mankind's mind, indeed its form is rooted in the superstition(which is as old as the history of thought)that the destiny of men could be read from the stars. The astrological mysticism has disappeared from science, but what remains is the endeavour towards the knowledge of the unity of the laws of the world.

Let us think for a moment about light. Is it a wave, or does it consist of tiny corpuscles? Both waves and corpuscles are concepts which are well understood but although both models capture some aspects of the nature of light, neither seems correct. For example, in 1850 Foucault showed that light travels slower in water than in air. This was in accordance with what the wave theory of light predicted, but contradicted what the corpuscular theory predicted. So the obvious deduction is that light must be a wave. However this deduction assumes that one or other of these two models is correct. All that Foucault had shown is that the simple corpuscular model is not an accurate model to predict all the properties of light. In his doctoral thesis written in 1923, de Broglie proposed his wave/particle duality theory in which particles, even electrons, could also behave like waves. Paul Langevin, de Broglie's thesis supervisor, couldn't decide whether this was a stroke of genius or a completely crazy idea. He sent the thesis to Einstein who wrote:-

With Einstein's backing, de Broglie's model was soon accepted. This model for light provided Schrödinger with the intuition to devise the wave mechanics model of the atom. Schrödinger wrote in 1926:-I believe that it involves more than a mere analogy.

The theory by Heisenberg to which Schrödinger refers is quantum mechanics which he put forward in 1925. Heisenberg invented these concepts by focusing attention on a set of quantised probability amplitudes. These amplitudes formed a non-commutative algebra and only later did Max Born and Pascual Jordan recognise this non-commutative algebra to be a matrix algebra. Even when he published his own work, however, Heisenberg had feared that it lacked an intuitive model. He wrote that his theory labours:-My theory was inspired by L de Broglie ... and by short but incomplete remarks by A Einstein. ... No genetic relation whatever with Heisenberg is known to me. I knew of his theory, of course, but felt discouraged not to say repelled, by the methods of transcendental algebra, which appeared very difficult to me and by the lack of visualizability.

Here was the problem in a nutshell - electrons did not behave in a way which matched our space-time intuition. However, this does not mean that some models will not be more intuitive than others. If Schrödinger found his own theory more intuitive than Heisenberg's theory, then the reverse was true for Heisenberg who wrote to Pauli on 6 June 1926:-... under the disadvantage that there can be no directly intuitive geometrical interpretation because the motion of electrons cannot be described in terms of the familiar concepts of space and time.

Pauli tended to agree with Heisenberg and referred to Schrödinger's theory as "local Zurich superstitions". In his reply to Schrödinger, Pauli wrote:-The more I reflect on the physical portion of Schrödinger's theory the more disgusting I find it. What Schrödinger writes on the visualizability of his theory, I consider trash. The greatest result of his theory is the calculation of the matrix elements.

Schrödinger did not want a scientific argument to become personal, so he replied to Pauli:-Don't take it as an unfriendliness to you but look on the expression as my objective conviction that quantum phenomena naturally display aspects that cannot be expressed by the concepts of continuum physics. But don't think that this conviction makes life easy for me. I have already tormented myself because of it and will have to do so even more.

Schrödinger wrote to Wilhelm Wien on 25 August:-... we are all nice people, and re interested only in the facts and not in whether it finally comes out the way oneself or the other fellow supposed. If outsiders, all the same, find us capricious, we know that such capriciousness serves science better than uniformity.

In a paper he published in September 1926 Heisenberg again repeats his claim that one cannot transfer physical intuition since:-I believe that Born overlooks ... that it would depend on the taste of the observer which he now wishes to regard as real, the particle or the guiding field. There is certainly no criterion for reality if one does not want to say: the real is only the complex of sense impressions, all the rest are only pictures.

Heisenberg wrote in... the electron and the atom do not possess any degree of physical reality as objects of daily experience. ... Investigation of the type of physical reality which is proper to electrons and atoms is precisely the subject of atomic physics and thus also of quantum mechanics.

Schrödinger is reported by Heisenberg to have said to Bohr:-It will scarcely be possible to reproduce how passionately the discussion was carried on from both sides.

Bohr's reply was interesting. He pointed out that it is taking words which refer to well understood concepts of everyday experience and applying them to the atom, which is outside our everyday experience, which causes the problems:-You surely must understand, Bohr, that the whole idea of quantum jumps necessarily leads to nonsense. ... the electron jumps from this orbit to another one and thereby radiates. Does this transition occur gradually or suddenly? If it occurs gradually, then the electron must gradually change its rotation frequency and energy. Its not comprehensible how this can give sharp frequencies for spectral lines. If the transition occurs suddenly, in a jump so to speak, ... one must ask how the electron moves in a jump. Why doesn't it emit a continuous spectrum? And what laws determine its motion in this jump?

Bohr repeated these arguments in print in 1927:-Yes, in what you say you are completely right. But that doesn't prove that there are no quantum jumps. It only proves that we can't visualise them, that means that the pictorial concepts we use to describe the events of everyday life and the experiments of the old physics do not suffice also to represent the process of a quantum jump. That is not surprising when one considers that the processes with which we are concerned here cannot be the subject of direct experience ... and our concepts do not apply to them.

Bohr was able to provide some improvements in the way that we visualise atoms by showing how the uncertainty principle itself related to the duality between waves and particles.Indeed we find ourselves here on the very path taken by Einstein of adapting our modes of perception borrowed from the senses to the gradually deepening knowledge of the laws of Nature. The difficulties met with on this path originate above all in the fact that, so to say, every word in the language refers to ordinary perception.

**Article by:** *J J O'Connor* and *E F Robertson*

**MacTutor History of Mathematics**

[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Wave_matrix.html]