The End of Time: The Next Revolution in Physics

CHAPTER 12

The Discovery of Quantum Mechanics

About a hundred years ago, a dualistic picture of the world took shape. The electron had just been discovered, and it was believed that two quite different kinds of thing existed: charged particles and the electromagnetic field. Particles were pictured as little billiard balls, possessing always definite positions and velocities, whereas electromagnetic fields permeated space and behaved like waves. Waves interfere, and recognition of this had led Thomas Young to the wave theory of light (Figure 22).

By the end of the nineteenth century, the evidence for the wave theory of light was very strong. However, it was precisely the failure of light, as electromagnetic radiation, to behave in all respects in a continuous wavelike manner that led first Max Planck in 1900 and then Einstein in 1905 to the revolutionary proposals that eventually spawned quantum mechanics. A problem had arisen in the theory of ovens, in which radiation is in thermal equilibrium with the oven walls at some temperature. Boltzmann’s statistical methods, which had worked so well for gases, suggested that this could not happen, and that to heat an oven an infinite amount of energy would be needed. The point is that radiation can have any wavelength, so radiation with infinitely many different wavelengths should be present in the oven. At the same time, the statistical arguments suggested that, on average, the same finite amount of energy should be associated with the radiation when in equilibrium. Therefore there would be an infinite amount of energy in the oven – clearly an impossibility. Baking ovens broke the laws of physics! Planck was driven to assume that energy is transferred between the oven walls and the radiation not continuously but in ‘lumps’, or ‘quanta’.

Accordingly, he introduced a new constant of nature, the quantum of action, now called Planck’s constant, because the same kind of quantity appears in the principle of least action. Until Planck’s work, it had been universally assumed that all physical quantities vary continuously. But in the quantum world, action is always ‘quantized’: any action ever measured has one of the values 0, ½h, h, ¾h, 2h, .... Here h is Planck’s constant. (The fact that half-integer values of h, i.e. ½h, ¾h, ..., can occur in nature was established long after Planck’s original discovery. By then it was too late to take half the original quantity as the basic unit.) The value of h is tiny.

Most people are familiar with the speed of light, which goes seven times round the world in a second or to the Moon and back in two and a half seconds. The smallness of Planck’s constant is less well known. Comparison with the number of atoms in a pea brings it home. Angular momentum is an action and can be increased only in ‘jerks’ that are multiples of h. Suppose we thread a pea on a string 30cm long and swing it in a circle once a second. Then the pea’s action is about 10³² times h. As we saw, the atoms in a pea, represented as dots a millimetre apart, would comfortably cover the British Isles to a depth of a kilometre. The number 10³², represented in the same way, would fill the Earth – not once but a hundred times. Double the speed of rotation, and you will have put the same number of action quanta into the pea’s angular momentum. It is hardly surprising that you do not notice the individual ‘jerks’ of the hs as they are added.

When people explain how our normal experiences give no inkling of relativity and quantum mechanics, the great speed of light and the tiny action quantum are often invoked. Relativity was discovered so late because all normal speeds are so small compared with light’s. Similarly, quantum mechanics was not discovered earlier because all normal actions are huge compared with h. This is true, but in a sense it is also misleading. For physicists at least, relativity is completely comprehensible. The mismatch between the relativistic world and its non-relativistic appearance to us is entirely explained by the speed of light. In contrast, the mere smallness of Planck’s constant does not fully explain the classical appearance of the quantum world. There is a mystery. It is, I believe, intimately tied up with the nature of time. But we must first learn more about the quantum.

Einstein went further than Planck in embracing discreteness. His 1905 paper, written several months before the relativity paper, is extraordinarily prescient and a wonderful demonstration of his ability to draw far-reaching conclusions from general principles. He showed that in some respects radiation behaved as if it consisted of particles. In a bold move, he then suggested that ‘the energy of a beam of light emanating from a certain point is not distributed continuously in an ever increasing volume but is made up of a finite number of indivisible quanta of energy that are absorbed or emitted only as wholes’. Einstein called the putative particles light quanta (much later they were called photons). In a particularly beautiful argument, Einstein showed that their energy E must be the radiation frequency ω times Planck’s constant: E = hω. This has become one of the most fundamental equations in physics, just as significant as the famous E = mc².

The idea of light quanta was very daring, since a great many phenomena, above all the diffraction, refraction, reflection and dispersion of light, had all been perfectly explained during the nineteenth century in terms of the wave hypothesis and associated interference effects. However, Einstein pointed out that the intensity distributions measured in optical experiments were invariably averages accumulated over finite times and could therefore be the outcome of innumerable ‘hits’ of individual light quanta. Then Maxwell’s theory would correctly describe only the averaged distributions, not the behaviour of the individual quanta. Einstein showed that other phenomena not belonging to the classical successes of the wave theory could be explained better by the quantum idea. He explained and predicted effects in ovens, the generation of cathode rays by ultraviolet radiation (the photoelectric effect), and photoluminescence, all of which defied classical explanation. It was for his quantum paper, not relativity, that Einstein was awarded the 1921 Nobel Prize for Physics.

The great mystery was how light could consist of particles yet exhibit wave behaviour. It was clear to Einstein that there must be some statistical connection between the positions of the conjectured light quanta and the continuous intensities of Maxwell’s theory. Perhaps it could arise through significantly more complicated classical wave equations that described particles as stable, concentrated ‘knots’ of field intensity. Maxwell’s equations would then be only approximate manifestations of this deeper theory. Throughout his life, Einstein hankered after an explanation of quantum effects through classical fields defined in a space-time framework. In this respect he was surprisingly conservative, and he famously rejected the much simpler statistical interpretation provided for his discoveries by the creation of quantum mechanics in the 1920s.

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.

Both electrons and photons can, depending on the circumstances, exhibit wave or particle behaviour. Otherwise they behave very differently. Many photons can be present simultaneously in the same state (a state being a characteristic set of properties of particles, such as position and direction of motion), but for electrons this is impossible – there can be at most one in any given state. The two kinds of particle have different statistical behaviour, so-called Fermi-Dirac statistics for electrons and Bose-Einstein statistics for photons. In fact, there are now known to be many different particles, each with an associated field. They satisfy either Fermi-Dirac statistics, and are thus called fermions, or Bose-Einstein statistics, in which case they are called bosons. In addition, nearly all particles have an antiparticle. An antiparticle is identical to the original particle in some respects, but opposite to it in others; in particular, a particle and its antiparticle always have opposite charges.

In many ways, the story of fundamental physics during the last seventy years has been the discovery of particles and the understanding of the manner in which they interact. All particles that have so far been discovered – there is a whole ‘zoo’ of them – are either spinor or vector particles. Ironically, particles corresponding to the simplest scalar fields have not yet been discovered, though it is confidently believed that they will be soon, mainly on the grounds of indirect but rather persuasive theoretical arguments. Currently, an immense amount of work is being done in the attempt to unify the two broad categories of particles – fermions and bosons – by means of an idea called supersymmetry. In the last two or three years, there has been another great surge of excitement in the field of superstring theory. This combines the idea of supersymmetry with the idea that the complete ‘zoo’ of particles known at present are simply different manifestations of the vibrations of a string, much as a violin string can vibrate at its different harmonics. This is the dream of the theory of everything (TOE). Some readers may be familiar with these ideas, originally embodied in the acronym GUT – grand unified theory. This was the aim of physicists who wished to describe within a single, unified theoretical framework all the forces of nature except gravity (long recognized as especially difficult to include). More recent, and more ambitious since it aims to include gravity, is the quest for the big TOE.

I am not going to make any attempt to discuss this work, nor will I try to explain the connection between a particle and its associated field. If a theory of everything is found, it may well change the framework of physics. We may find ourselves in a quite new arena and have to change our ideas about space and time yet again. However, as of now I believe we can glimpse the outlines of an arena large enough to accommodate not only the present ‘zoo’ but also whatever entities some putative theory of everything will come up with. The arena I have in mind is vast and timeless. I see it not as a rival to the theory of everything, but as a general framework in which such a theory can be formulated.

Now it is time to talk about the ideas that Schrödinger introduced in the winter of 1925/6. That was when the door was opened onto the vast arena.