Watching electrons

We have seen that quantum mechanics does not allow us the comfort of being able to visualize the motion of a quantum particle. In a normal game of billiards we can imagine the paths taken by the individual balls (Fig. 1.1). Figure 2.1 shows the physicist George Gamow's attempt to give some impression of how the same game might look if played with quantum particles. Besides illustrating that the notion of a path is no longer valid in quantum mechanics, this cartoon also illustrates another significant difference between the quantum and classical worlds: the exact position of the white ball is not known. Uncertainty has entered physics and replaced the determinism of Newtonian mechanics.

By the nineteenth century, physicists had been able to explain vast amounts of experimental observations on objects as different as planets and billiard balls. If an observation differed from the predictions of classical physics, they looked for something they had overlooked to explain the deviation. In 1864, physicists' confidence in the whole edifice of classical physics seemed to be spectacularly verified by an analysis of some irregularities in the orbit of Uranus. These were attributed to the existence of a then undiscovered planet – the subsequent discovery of Neptune was a triumph for Newtonian physics.

Poets say science takes away from the beauty of the stars – mere globs of gas atoms. Nothing is ‘mere’. I too can see the stars on a desert night, and feel them. But do I see less or more? The vastness of the heavens stretches my imagination – stuck on this carousel, my little eye can catch one-million-year- old light … Or see them [the stars] with the greater eye of Palomar, rushing all apart from some common starting point when they were perhaps all together. What is the pattern, or the meaning, or the why? It does not do harm to the mystery to know a little about it. For far more marvellous is the truth than any artists of the past imagined! Why do the poets of the present not speak of it?

Finally, may I add that the main purpose of my teaching has not been to prepare you for some examination – it was not even to prepare you to serve industry or the military. I wanted most to give you some appreciation of the wonderful world and the physicist's way of looking at it, which, I believe, is a major part of the true culture of modern times. (There are probably professors of other subjects who would object, but I believe they are completely wrong.) Perhaps you will not only have some appreciation of this culture; it is even possible that you may want to join in the greatest adventure that the human mind has ever begun.

De Broglie's matter waves

The early struggles of physicists towards a quantum theory were mostly concerned with attempts to understand the nature of light. The traditional picture of light as a wave motion had been challenged by Planck and Einstein. They had shown that certain experimental results that were impossible to understand in terms of a wave picture could be easily explained if light was thought of as a stream of particles, now called photons. William Bragg, who, with his son, won the 1915 Nobel Prize for studies of crystal structure using X-rays, summarized this dilemma for physics by exclaiming in despair that he was teaching the corpuscular theory of light on Mondays, Wednesdays and Fridays, and the undulatory theory on Tuesdays, Thursdays and Saturdays! Physicists were still wrestling with these apparently contradictory properties of light when, in 1924, Prince Louis de Broglie (pronounced ‘de Broy’) suggested that all matter, even objects that we usually think of as particles – such as electrons – should also display wavelike behaviour! This revolutionary idea was completely unexpected and, what is more, was included by de Broglie in his Ph. D. thesis. Like most people, physicists are generally reluctant to accept any wild new idea, especially if there is not a shred of evidence to support it.

The popularization of science is now an established business for booksellers and publishers. The traditional formula for a ‘trade’ book on popular science is a text of about 100 000 words or so (around 200 pages) with relatively few diagrams or pictures. The target audience is the educated reader with a general interest in science. At the other end of the scale we have popular science reference books such as encyclopedias and atlases. Our target audience lies in between these two extremes. We wish to write a book that will not only interest the ‘educated reader’ as above but, more importantly, capture the interest and imagination of young people. We believe that it is vitally important to give young people a glimpse of the excitement of physics so that they may be motivated to take up the challenge themselves. Nowadays, there are many more alternatives for young people and there is a general perception that science and mathematics are ‘hard’ subjects. It is certainly true that understanding in these subjects does not come without effort and real mastery may indeed take years. So we cannot promise instant gratification. What we can promise is that the study of science and mathematics will provide a gateway to a deeper understanding of a fascinating universe – our universe, a quantum universe. And paradoxically, as our world becomes more and more dependent on science and technology, it has also become increasingly technologically fragile in that fewer people understand the technology on which we all depend.

This appendix is intended for readers with some knowledge of calculus – mainly how to differentiate and integrate sines and cosines. We consider the problem of finding the quantum probability amplitudes for a particle ‘in a box’. This involves solving the Schrödinger equation for a particle of some energy E, confined to a certain region by an electrical potential V:

We consider the case where the electron can only move in one space dimension, rather than in three dimensions as in real life. In spite of the apparent artificiality of this situation, this example illustrates many of the features that occur in more realistic problems. A straightforward three-dimensional generalization of this example can be used to illustrate many features of the quantum physics not only of electrons in a metal, but also of neutrons and protons in the nucleus.

The one-dimensional box potential is shown in Fig. 4.8. Outside the box the potential is assumed to be infinitely high so that there is no probability of finding the particle in these regions. The particle is constrained to be within the box, i.e. the quantum amplitude can only be non-zero between x = 0 and x = L, the two ends of the box. Inside this region, the particle can move freely: in other words, the quantum amplitude for the particle in the box is determined by the Schrödinger equation with no potential term:

To see what form of solution is allowed, we first ‘tidy up’ the equation by introducing the new variable k:

In terms of k, the Schrödinger equation for the particle in the box now reads: which shows the mathematical structure more clearly.

Max Born and quantum probabilities

In this chapter, we will take a short break from our survey of successful applications of quantum mechanics to take a closer look at the foundations of this great edifice of modern physics. In a sense, we are now disobeying Feynman's warning about the results of the double slit experiment of chapter 1. We shall now ask Feynman's ‘forbidden’ question ‘But how can it be like that?’ Nobody disputes that quantum mechanics has been magnificently successful, enabling us to make correct quantitative calculations of atomic and nuclear effects. But there is a great divergence of opinion about the implications of quantum mechanics for the nature of matter, and indeed, of reality itself. To avoid becoming embroiled in a purely philosophical mire, we will focus on two famous paradoxes. The first of these is the ‘Einstein–Podolsky-Rosen’ paradox, named after its originators Albert Einstein, Boris Podolsky and Nathan Rosen, and usually abbreviated by the initials ‘EPR’. The second paradox is named after ‘Schrödinger's cat’.

Electron spin and Pauli's exclusion principle

Over a century ago, a Russian chemist, Dimitri Mendeleev, invented a teaching-aid for students struggling with inorganic chemistry. He realized that the properties of the 63 elements then known were repeated ‘periodically’ as their atomic weight increased. In other words, elements with similar chemical properties were not close together in mass, but instead were found at regular intervals as the mass increased. For example, lithium has a nucleus with three protons and is an alkali metal – a highly reactive soft silvery substance that forms alkaline oxides and hydroxides. The next alkali metal is sodium with eleven protons, then potassium with nineteen, and so on, all with increasing mass. Mendeleev was able to group all the elements into distinct families, and his scheme became known as the ‘periodic table’ of the elements. All good theories should be able to make predictions and this was no exception. Because of the regularities he had observed, Mendeleev realized that his list of elements must be incomplete. He therefore left gaps in his table corresponding to as yet undiscovered elements, and had the satisfaction of seeing gallium, scandium and germanium discovered during his lifetime. Nevertheless, the explanation of these periodicities remained a mystery for over 50 years until the Austrian physicist Wolfgang Pauli put forward his famous ‘exclusion principle’.

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.