Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats.

In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics.

The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.

This chapter begins with a brief outline of some of the key motivations for considering a quantum theory: the early attempts to determine the spectral distribution of energy density of black bodies; stability of atoms and molecules; specific heats of solids; interference and diffraction of light beams; polarization of photons. The experimental foundations of wave mechanics are then presented in detail, but in a logical order quite different from its historical development: photo-emission of electrons by metallic surfaces, X- and γ-ray scattering from gases, liquids and solids, interference experiments, atomic spectra and the Bohr hypotheses, the experiment of Franck and Hertz, the Bragg experiment, diffraction of electrons by a crystal of nickel (Davisson and Germer), and measurements of position and velocity of an electron.

The need for a quantum theory

In the second half of the nineteenth century it seemed that the laws of classical mechanics, developed by the genius of Newton, Lagrange, Hamilton, Jacobi and Poincaré, the Maxwell theory of electromagnetic phenomena and the laws of classical statistical mechanics could account for all known physical phenomena. Still, it became gradually clear, after several decades of experimental and theoretical work, that one has to formulate a new kind of mechanics, which reduces to classical mechanics in a suitable limit, and makes it possible to obtain a consistent description of phenomena that cannot be understood within the classical framework. It is now appropriate to present a brief outline of this new class of phenomena, the systematic investigation of which is the object of the following sections and of chapters 4 and 14.

The density matrix ρ is defined and its basic properties are derived in detail: ρ is a Hermitian non-negative operator with unit trace, its eigenvalues are non-negative and lie in the closed interval [0, 1]; if ρ is a projector, it projects onto a one-dimensional sub-space, the trace of ρ2 is ≤ 1, the equality holding only if ρ is a projector. A necessary and sufficient condition for ρ to be a projector is that all vectors in its definition are identical up to a phase. Pure cases are sets that can be described by a state vector, unlike mixtures. The density matrix of a mixture is not a projection operator, and this may be viewed as its distinguishing feature with respect to pure cases. The mean value of an observable can be reexpressed with the help of the density matrix, and a Schrödinger equation for density matrices holds. Last, but not least, pure cases cannot unitarily evolve into mixtures.

The applications of the formalism presented in this chapter deal with orientation of spin-½ particles, polarization of pairs of photons, thermal-equilibrium states. In the second part, quantum entanglement is presented, with emphasis on the existence of state vectors for entangled pairs of particles that belong to the tensor product of Hilbert spaces H1 ⊗ H2, but are not of the simple product form ψ1 ⊗ ψ2. The only correct interpretation of such states has a statistical nature, associated to series of repeated measurements performed by two observers on the constituents of the entangled pair. Hidden-variable theories, with the associated Bell inequalities, are introduced, and a way of using entangled quantum states is described which makes it possible to transfer the polarization state of a photon to another photon. This property provides the foundation of the modern investigations of quantum teleportation.

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’.