In the following years, Einstein published several important quantum papers, laying the foundations of a quantum theory of the specific heats of solids. However, the next major advance came in 1913 with Danish physicist Niels Bohr’s atomic model. It had long been known that atoms emit radiation only at certain frequencies, called lines because of their appearance in spectra. These spectral lines, which had been arranged purely empirically in regular series, were a great mystery. Everyone assumed that each line must be generated by an oscillatory process of the same frequency in the atoms, but no satisfactory model could be constructed.
Bohr found a quite different explanation. In a famous experiment, the New Zealander Ernest Rutherford had recently shown that the positive charge in atoms (balanced by the negative charge of the electrons) was concentrated in a tiny nucleus. This discovery was itself very surprising and is illustrated by a well-known analogy. If the space of an atom – the region in which the electrons move – is imagined as being the size of a cathedral, the nucleus is the size of a flea. Bohr supposed that an atom was something like the solar system, with the nucleus the ‘Sun’ and the electrons ‘planets’.
However, he made a seemingly outrageous ad hoc assumption. Using the electrostatic force for the known charges of the electron and positive nucleus, he calculated the electron orbits in Newtonian mechanics for the hydrogen atom, which has only one electron. Each such orbit has a definite angular momentum. Bohr suggested that only orbits for which this angular momentum is some exact multiple of Planck’s constant, i.e. 0, h, 2h, ..., can occur in nature. These orbits also have definite energies, now called energy levels. He made the further equally outrageous conjecture that radiation in spectral lines arises when an electron ‘jumps’ (for some unexplained reason) from an orbit with higher energy to one with lower energy. He suggested that the difference £ of these energies is converted into radiation with frequency ω, determined by the relation E = hco found by Einstein for the ‘lump of energy’ associated with radiation of frequency co. Thus, according to Bohr’s theory, an atom emits a light quantum (photon) of a well-defined energy by jumping from one orbit to another.
For hydrogen atoms, it was easy to calculate the energy levels and hence the frequencies of their radiation. Subject to certain further conditions, Bohr’s theory had an immediate success. His hotchpotch of Newtonian theory and strange quantum elements had hardly explained the enigmatic spectral lines, but it did predict their frequencies extraordinarily well, and there could be no doubting that he had found at least some part of a great truth.
During the next decade the Bohr model was applied to more and more atoms, often but not always with success. It was clearly ad hoc. The need for an entirely new theory of atomic and optical phenomena based on consistent quantum principles became ever more transparent, and was keenly felt. Finally, in 1925/6 a complete quantum mechanics was formulated – by Werner Heisenberg in 1925 and Erwin Schrödinger in 1926 (and called, respectively, matrix mechanics and wave mechanics). At first, it seemed that they had discovered two entirely different schemes that miraculously gave the same results, but quite soon Schrödinger established their equivalence.
Heisenberg’s scheme, or picture, is based on abstract algebra and is often regarded as giving a truer picture. In the form in which quantum theory currently exists, it is more flexible and general. Unfortunately, it is rooted in abstract algebra, making it very difficult to describe in intuitive terms. I shall therefore use the Schrödinger picture. Luckily, this will not detract from what I want to say. In fact, one of the main ideas I want to develop is that the Schrödinger picture is actually more fundamental than the Heisenberg picture, and is the only one that can be used to describe the universe quantum-mechanically. Many physicists will be sceptical about this, but perhaps this is because they study phenomena in an environment and do not consider how local physics might arise from the behaviour of the universe as a whole.
Schrödinger’s work developed out of yet another revolutionary idea, put forward by the Frenchman Louis de Broglie in 1924. It finally overthrew the dualistic picture of particles and fields that had crystallized at the end of the nineteenth century. Einstein had already shown that the electromagnetic field possessed not only wave but also particle attributes. De Broglie wondered whether, since light can behave both as wave and particle, might not electrons do the same? Together with its position, the most fundamental property of a particle of mass m and velocity v is its momentum, mv. De Broglie assumed that particles are invariably associated with waves with wavelength λ related by Planck’s constant to their momentum: λ = h/mv.
He applied this idea to Bohr’s model. At each energy level, the electron has a definite momentum and hence a wavelength. We can imagine moving round an orbit, watching the wave oscillations. In general, if we start from a wave crest, the wave will not have returned to a crest after one circuit. De Broglie showed that crest-to-crest matching, or resonance, would happen only for the orbits with quantized angular momentum that figured so prominently in the Bohr model.
Although he had not, strictly, made any new discovery, his proposal was suggestive. It restored a semblance of unity to the world – both electrons and the electromagnetic field exhibited wave and particle properties. De Broglie’s thesis was sent to Einstein, who was impressed and drew attention to its promise. Schrödinger got the hint, and, as they say, the rest is history. During the winter of 1925/6 and the following months he created wave mechanics. This will be the subject of the following chapters.
In 1927 de Broglie’s conjecture was brilliantly confirmed for electrons first in an experiment by the Englishman George Thomson, and then in a particularly famous experiment by the Americans Clinton Davisson and Lester Germer. These experiments paralleled those made about a decade and a half earlier by the German physicist Max von Laue, in which he had directed X-rays onto crystals and observed very characteristic diffraction patterns, from which the structure of the crystals could be deduced. The patterns were explained in terms of the interaction of waves with the regular lattice of the atoms forming the crystals. They demonstrated graphically the wave-like behaviour of the electromagnetic field (X-rays are, of course, electromagnetic waves, like light, but with much higher frequency and shorter wavelength). In the 1927 experiments, electrons were directed onto crystals, and diffraction patterns identical in nature to those produced by X-rays were seen. Thus, the particle nature of electrons was observed long before their wave nature was suspected. With light it was the other way round – wave interference was observed a century before Einstein suspected that light could have a particle aspect too.
Although it was now clear that both light and electrons exhibited wave-particle duality, there were important differences between them. A brief description of the picture as it now appears will help. All particles are associated with fields, and can be described as excitations of those fields. To get some idea of what this means, we can liken the particles to water waves, which are excitations of undisturbed water. However, the analogy is only partial. The classic example of a particle associated with a wave is the photon, which is an excitation of the Maxwell field. Fields and associated particles of different kinds exist. There are fields described by a single number at each point, called scalar fields, and vector fields, which are described by three numbers. Scalar fields represent a simple intensity, while the vector fields – such as Maxwell’s field – are a kind of ‘directed’ intensity. In general relativity we also encountered tensors. Mathematically, scalar, vector and tensor fields belong to one family and obey the same kind of rule under rotations of the coordinate system. In particular, after one rotation they return to the values they had before. However, in 1927 yet another sensational quantum discovery was made, this one by Dirac. He found a quite different family of fields, called spinor fields, which are associated with electrons and protons (as well as many other particles). In their case, one rotation of the coordinate system brings them back to minus the value they had before, and two rotations are needed to restore their original value. Dirac found spinors by trying to make the newly discovered quantum principles compatible with relativity, and achieved a spectacular success even though it was subsequently found that his arguments were not totally compelling. However, the main point is that electrons are associated with a spinor field, photons with a vector field.