A poet once said ‘The whole universe is in a glass of wine’. We will probably never know in what sense he meant that, for poets do not write to be understood. But it is true that if we look at a glass closely enough we see the entire universe. There are the things of physics: the twisting liquid which evaporates depending on the wind and weather, the reflections in the glass, and our imagination adds the atoms. The glass is a distillation of the Earth's rocks, and in its composition we see the secret of the universe's age, and the evolution of the stars. What strange array of chemicals are there in the wine? How did they come to be? There are the ferments, the enzymes, the substrates, and the products. There in wine is found the great generalization: all life is fermentation. Nobody can discover the chemistry of wine without discovering, as did Louis Pasteur, the cause of much disease. How vivid is the claret, pressing its existence into the consciousness that watches it! If our small minds, for some convenience, divide this glass of wine, this universe, into parts – physics, biology, geology, astronomy, psychology, and so on – remember that Nature does not know it! So let us put it all back together, not forgetting ultimately what it is for. Let it give us one more final pleasure: drink it and forget it all!

A failed star

In the previous chapter we have seen how quantum mechanics and the exclusion principle provide the basis for an understanding of all the different types of matter we see around us. What is perhaps more surprising is that quantum mechanics and the exclusion principle also provide the key to understanding stellar evolution and the variety of stars. As a prelude to stars we begin with a planet, Jupiter, which in one sense may be regarded as a star that did not quite make it!

Jupiter is by far the largest planet in our solar system. Some impression of its enormous size can be appreciated from the photo-montage shown in Fig. 10.1.

Prelude: the atom and the nucleus

Now that the atomic basis of matter is taught routinely in schools, it is difficult to imagine the suspicion and hostility towards atoms that existed at the end of the nineteenth century. This seems especially strange since the idea of atoms has been around since the fifth century BC in the writings of the Greek philosophers Leucippus and Democritus. Such distrust of the ‘atomic hypothesis’ is all the more surprising given that Daniel Bernoulli, James Clerk Maxwell and Ludwig Boltzmann had all successfully used an atomic model of gases – with atoms as tiny hard spheres that could move and collide like billiard balls – to explain many thermodynamic properties of gases. Nonetheless, it was only with Einstein's famous 1905 paper on ‘Brownian’ motion – which explained the observed random jiggling motion of grains of pollen floating in water in terms of collisions with water molecules – that almost all of the doubters were silenced and the atomic hypothesis became generally accepted.

As we have seen, the idea of atoms as tiny, hard, indestructible spheres only survived until 1911. It was then that Ernest Rutherford came up with the startling discovery that most of the atom was empty space! From his calculations of the scattering of alpha particles by atoms, Rutherford deduced that almost all the mass of the atom, and all the positive charge, must be concentrated in a tiny sphere much smaller than the apparent size of the atom.

Laser light

Nowadays, everyone has heard of lasers, and laser light displays are a frequent ingredient of modern rock concerts. Laser light has many applications, ranging from astronomy to hydrogen fusion. What is the special feature of laser light that makes it so useful? The answer to this question involves a property of wave motion known as ‘coherence’, with light photons acting together in a special form of quantum mechanical co-operation. This type of quantum co-operation will turn out to be vital for an understanding of the peculiar behaviour of quantum ‘superfluids’. To understand the special nature of laser light, however, we must first explain what is meant by coherence.

Consider the simple wave motion shown in Fig. 7.1. We see that the pattern repeats itself after one wavelength and the frequency of the wave corresponds to the number of wavelengths sent out per second. If this wave is a wave on a string, each point on the string just moves up and down with a certain amplitude: the maximum distance the point can move out to before it starts to come back. Up to now, this is really all we have needed to know about waves. Now consider two waves of the same wavelength but started at slightly different times, as shown in Fig. 7.2.

The double-slit experiment revisited

In this chapter we turn to recent advances in our understanding of the fundamental forces of Nature. As we have said in earlier chapters, the combination of classical electromagnetism, quantum mechanics and relativity provides an astonishingly successful description of electromagnetic forces. The resulting theory is called Quantum ElectroDynamics, or QED for short. For over 50 years physicists searched for similarly successful theories to describe not only the weak forces responsible for natural radioactivity but also the strong forces that hold the nucleus together. It was not until the mid 1970s that real progress was made and these remarkable developments are the subject of this chapter.

Particle physicists now have a unified theory that combines both the electromagnetic and weak forces. The major predictions of this theory have been spectacularly verified by experiments at the CERN high-energy particle physics laboratory in Geneva. We shall describe the theory and these experiments in more detail in this chapter. But particle physicists also believe that they have at last discovered the correct theory of the strong nuclear force.

Richard Feynman and nanotechnology

In 1959, in an after dinner speech at a meeting of the American Physical Society in Pasadena, Richard Feynman set out a remarkable vision of the future in a talk entitled ‘There's plenty of room at the bottom.’ The talk was subtitled ‘an invitation to enter a new field of physics’ and marked the beginning of what is now known as ‘nanotechnology’. Nanotechnology is concerned with the manipulation of matter at the scale of a nanometre -a thousand millionth of a metre. Atoms are typically a few tenths of a nanometre in size. Feynman emphasized that such an endeavour does not need new physics:

In his talk Feynman offered two prizes of $1000 each – one ‘to the first guy who makes an operating electric motor which is only 1/64 inch cube’, and a second ‘to the first guy who can take the information on the page of a book and put it on an area 1/25000 smaller’ He had to pay out on both prizes – the first less than a year later, to Bill McLellan, a Caltech alumnus.

Barrier penetration

One of the most startling consequences of de Broglie's wave hypothesis and Schrödinger's equation was the discovery that quantum objects can ‘tunnel’ through potential energy barriers that classical particles are forbidden to penetrate. To gain some idea of what we mean by an energy barrier, let us go back to our roller coaster and look at a larger section of track, as shown in Fig. 5.1. If we start the carriage from rest, high up on the left, at A, and ignore any small frictional energy losses, we know from the conservation of energy that we shall arrive on the other side at the same height we started from, at C. As we went over the little hill B, at the bottom of the valley, the car slowed down as some of our kinetic energy was changed to potential energy in climbing the hill, but because we started much higher up, we had plenty of energy to spare to get us over the top. However, if we started the carriage from rest at A, we do not have enough energy to climb over the hill D and get to E. This is an example of an ‘energy barrier’, and we can say that the region from C to E is ‘classically forbidden’.

What is remarkable about quantum ‘particles’ is that they do not behave like these classical objects.

Science and experiment

Science is a special kind of explanation of the things we see around us. It starts with a problem and curiosity. Something strikes the scientist as odd. It doesn't fit in with the usual explanation. Maybe harder thinking or more careful observation will resolve the problem. If it remains a puzzle, it stimulates the scientist's imagination. Perhaps a completely new way of looking at things is needed? Scientists are perpetually trying to find better explanations – better in the sense that any new explanation must not only explain the new puzzle, but also be consistent with all of the previous explanations that still work well. The hallmark of any scientific explanation or ‘theory’ is that it must be able to make successful predictions. In other words, any decent theory must be able to say what will happen in any given set of circumstances. Thus, any new theory will only become generally accepted by the scientific community if it is able not only to explain the observations that scientists have already made, but also to foretell the results of new, as yet unperformed, experiments. This rigorous testing of new scientific ideas is the key feature that distinguishes science from other fields of intellectual endeavour -such as history or even economics – or from a pseudoscience such as astrology.

Rutherford's nuclear atom

Before quantum mechanics came along, classical physics was unable to account for either the size or the stability of atoms. Experiments initiated in 1911 by the famous New Zealand physicist, Ernest Rutherford, had shown that nearly all the mass and all of the positive charge of an atom are concentrated in a tiny central core that Rutherford called the ‘nucleus’. Most of the atom is empty space! A table of the relative sizes of atoms, nuclei and other quantum and classical objects is given in appendix 1. Rutherford had already won a Nobel Prize earlier, in 1908, for his work on radioactivity. Radioactivity is now known to be due to the ‘decay’ of a nucleus of certain unstable chemical elements: some radiation is given off – in the form of alpha, beta or gamma rays – and a nucleus of a different element is left behind (see Fig. 4.1). As you can imagine, it took physicists some time to disentangle what was going on, and it was Rutherford who showed that the positively charged, heavy, penetrating alpha rays were, in fact, helium atoms which had lost two electrons. Beta rays, on the other hand, were identified as electrons, and gamma rays as high energy photons. At that time, any work involving the different chemical elements was regarded as the province of chemists, and Rutherford was somewhat put out at winning the chemistry Nobel Prize.

Chapters 5–7 have shown us why and how states play such an essential role in quantum theory. States are distinguished through their labeling, which consists of a set of quantum numbers specific to the state. In general, each degree of freedom gives rise to a quantum number, where, in the present context, the phrase “degrees of freedom” refers to the number of spatial coordinates characterizing a given quantum system. The 1-D systems employed in the two chapters illustrating the postulates were thus single-degree-of-freedom systems: all the eigenstates encountered there were labeled by a single quantum number (n for bound states and k for continuum states) related to the energy of the system.

Having studied systems with a single degree of freedom, it might seem that our next step would be to consider systems with more than one degree of freedom, for example, two 1-D particles or a particle in 3-D subject to a potential, etc. Among the goals of such studies would be the assigning of labels to the eigenstates. If the assignment of quantum numbers were based solely on spatial degrees of freedom, then the preceding endeavor would indeed be a way to proceed with our development of quantum theory. However, spatial degrees of freedom are not the only source of the labels which specify eigenstates. An additional source is the sets of eigenvalues of the operators that commute with Ĥ and with each other. Such operators are the main subject of this chapter.