If the difference between Newtonian and Einsteinian physics is great, quantum mechanics seems separated from both by a chasm. Most accounts of it, however, do not question the framework, essentially absolute space and time, in which it was formulated. They describe how very small systems – mostly atoms and molecules – behave in an external framework. This may make quantum mechanics appear more baffling than need be.
If quantum mechanics is universally true and applies not only to atoms and molecules but also to apples, the Moon, the stars and ultimately the universe, then we ought to consider quantum cosmology. What does the quantum mechanics of the universe look like? It cannot be formulated in an external framework. Like classical physics, quantum cosmology needs a description without a framework. We shall see that many apparent differences between classical and quantum mechanics then appear in a different light. What remains is one huge difference. We shall soon begin to get to grips with it.
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.