The behavior of light and its interaction with matter is certainly one of the most interesting phenomena of physics. It has puzzled thinkers and scientists in ancient Greece (Archimedes) and through the Middle Ages, at least since the times of Robert Grosseteste (1168–1235), right up to the theories of Galileo Galilei and Isaac Newton. The classical electromagnetic theory of light (see Tab. 13.1) was beautifully established in the second half of the nineteenth century with the Maxwell equations. Quantum optics is nothing other than the quantum theory of light. It finds its roots in Planck's discovery (1900) that the assumption of energy quantization for a harmonic oscillator allowed us to solve the black-body problem (see Subsec. 1.5.1). The successive early steps that led to the quantization of the electromagnetic field include Einstein's explanation of the photoelectric effect with the introduction of the quantum of light (see Subsec. 1.2.1) and the interpretation of atomic spectra (see Subsec. 1.5.4).

Quantum optics is usually not covered by conventional textbooks on quantum mechanics. We include its discussion here for two main reasons: first, because it is a fundamental and advanced application of the basic framework of quantum mechanics, and, second, because in the last 30 years quantum optics has played (and is still playing) a major role in the advancement and understanding of quantum mechanics.

In the first two chapters we have examined the basic principles – superposition (p. 18), complementarity (p. 19), quantization (p. 44), statistical algorithm (p. 57), and correspondence (p. 72) (see also Subsec. 2.3.4) – and the basic entities, observables and states, of quantum mechanics, as well as the main differences with respect to classical mechanics. While what we have discussed so far is rather a static picture of observables and states, in this chapter we shall deal with quantum dynamics, i.e. with the time evolution of quantum-mechanical systems.

Historically, after Bohr had provided a quantized description of the atom (see Subsec. 1.5.4), Einstein showed the quantized nature of photons (see Subsec. 1.2.1), and de Broglie hypothized the wave-like nature of matter (see Subsec. 1.5.5), the first building block of quantum mechanics was provided by the commutation relations, proposed by Heisenberg in 1925, whose consequence is represented by the uncertainty relation (see Subsec. 2.2.7 and Sec. 2.3). This was the subject of the previous chapters. The dynamical part of the theory was proposed by Schrödinger in 1926, and is known as the Schrödinger equation. It is also known as wave mechanics (see Subsec. 1.5.7). In this chapter we shall show that Heisenberg's and Schrödinger's formulations are only two different aspects of the same theory. We shall also come back to this point in Sec. 8.1.1. Here, first we shall derive the fundamental equation which rules quantum dynamics (Sec. 3.1), and, in Sec. 3.2, we shall summarize the main properties of the Schrödinger equation.

Why yet another book on quantum mechanics? Quantum mechanics was born in the first quarter of the twentieth century and has received an enormous number of theoretical and experimental confirmations over the years. It is considered to be the fundamental physical paradigm, and has a wide range of applications, from cosmology to chemistry, and from biology to information sciences. It is one of the greatest intellectual achievements of the past century. As an effect of its invention, the very concept of physical reality was changed, and “observation,” “measurement,” “prediction,” and “state of the system” acquired a new and deeper meaning.

Probability was not unknown in physics: it was introduced by Boltzmann in order to control the behavior of a system with a very large number of particles. It was the missing concept in order to understand the thermodynamics of macroscopic bodies, but the structure of the physical laws remained still deterministic. The introduction of probability was needed as a consequence of our lack of knowledge of the initial conditions of the system and of our inability to solve an enormous number of coupled non-linear differential equations.

In quantum mechanics, the tune is different: if we have 106 radioactive atoms no intrinsic unknown variables decide which of them will decay first. What we observe experimentally seems to be an irreducible random process. The original explanation of this phenomenon in quantum mechanics was rather unexpected.

In classical mechanics, two (or more) systems that do not interact – that do not exert any kind of force on each other – are completely separated (see Sec. 1.1), i.e. experiments performed locally on one of them cannot in any way influence experiments performed locally on the other. Quantum mechanics, on the other hand, admits that there can be a form of interdependence even in absence of physical interaction. This type of correlation goes under the name of entanglement (see Def. 5.1: p. 183). We already know that entanglement means non-factorizability of the state vector describing a system with at least two different degrees of freedom. In this chapter, we shall see that this may be the case also for systems that are very far away (space-like distant) from each other and we shall learn many interesting consequences of this state of affairs. Recent developments show that one may entangle even particles that have never interacted with each other.

The starting point of the following analysis is necessarily represented by the pioneering paper of Einstein, Podolsky, and Rosen, who inaugurated an incredible series of theoretical and experimental investigations right up to the present day that have confirmed the predictions of quantum mechanics. The “irony” of this history is that Einstein, Podolsky, and Rosen, EPR for short, aimed to present a definitive proof of the inconsistency of quantum mechanics relative to classical physical theory. In fact, they initiated one of the most prolific fields of modern science.

In classical physics, the question whether there are subsets of identical particles is not particularly interesting. After all, whether the particles are identical or not, we can always track each of them at any time. As we shall see, in quantum mechanics identical particles are indistinguishable because we cannot track trajectories, and this fact induces some important symmetry properties on the total wave function describing the system of particles under consideration. Generally speaking, a symmetry is an effect of an invariance, that is, the system does not change if one effects certain transformations (see Ch. 8). In this case, the system is not modified under exchange of two identical particles. Moreover, such a symmetry yields a most interesting connection between spin and statistics.

After a short introduction to the significance of statistics in quantum mechanics (in Sec. 7.1), we define the permutation operator and discuss the symmetry properties of the wave function of N identical particles (Sec. 7.2). In Sec. 7.3 we show that quantum particles, differently from classical particles (that obey the Maxwell–Boltzmann statistics), may be subject to two different statistics, namely the Fermi–Dirac statistics for fermions and the Bose–Einstein statistics for bosons. In Sec. 7.4 we treat the so-called exchange interaction and, finally, in Sec. 7.5 we consider some applications of the two statistics.

Statistics and quantum mechanics

Identical particles are those particles whose Hamiltonian is symmetric under exchange of one particle with the other.

Introduction

As is well known, the concept of the atom appears for the first time as a hypothesis about the structure of matter by Democritus (c. 460 BC–c. 370 BC). Democritus also proposed a rudimentary mechanics: Atoms moved in straight lines and through their collisions bodies were formed and destroyed. It is also interesting to recall that later Epicurus (341 BC–270 BC) and Lucretius (c. 94 BC–c. 49 BC) introduced a random, reasonless, deviation (clinamen) from the straight line in order to account for the contingency of our world. Though atomism remained a speculative theory for more than 2000 years, in modern ages it was still alive among many scientists and philosophers (Newton, Locke, Spinoza, etc.).

At the beginning of the nineteenth century John Dalton (1766–1844) introduced the concept of the atom in a scientific framework in order to explain some chemical phenomena. He was one of the earliest scientists to work on the structure of matter. He supposed that atoms were the smallest units of an element that enter into chemical combinations, and that a chemical element was composed entirely of one type of atom. Then, compounds contain atoms of two or more different elements, where the relative number of atoms of each element in a particular compound is always the same. Experimentally, he observed that, for each gram of hydrogen found in water there were always 8 grams of oxygen.

In this chapter we shall first summarize some conceptual and formal features of classical mechanics (Sec. 1.1). Modern physics started with the works of Galileo Galilei and Isaac Newton from which classical mechanics, one of the most beautiful and solid intellectual buildings of the human history, came out. The architecture of classical mechanics was developed between the end of the eighteenth century and the second half of the nineteenth century, and its present form is largely due to Lagrange, Jacobi, and Hamilton. As we shall see in this chapter, classical mechanics is built upon the requirement of determinism, a rather complex assumption which is far from being obvious. In Sec. 1.2 we shall present the two main conceptual features of quantum mechanics on the basis of an ideal interferometry experiment: the superposition principle and the principle of complementarity. In Sec. 1.3 a first formal treatment of quantum-mechanical states is developed: quantum states are represented by vectors in a space that turns out to be a Hilbert space. In Sec. 1.4 the significance of probability for quantum mechanics is explained briefly: we will show that probability is not just an ingredient of quantum mechanics, but is rather an intrinsic feature of the theory. Furthermore, we shall see that quantum probability is not ruled by Kolmogorov axioms of classical probability. Finally, we discuss the main evidences which have historically revealed the necessity of a departure from classical mechanics.

A symmetry is an equivalence of different physical situations, and is therefore strictly associated with the concept of invariance. One of the main aims of physics is the search in nature for symmetries and, therefore, for observables that are invariant under certain classes of transformation. In this context, it becomes particularly interesting to look for the conditions under which a certain symmetry is eventually broken. We have already discussed some symmetries of quantum mechanics: time translations (Ch. 3), space translations (Ch. 2) and rotations (Ch. 6). Another example of invariance is represented by the indistinguishability of identical particles, which we discussed in the previous chapter.

Symmetries are induced by physical transformations that generate some invariant properties (see Sec. 8.1). In Sec. 8.2, the continuous transformations induced by rotations and space translations are analyzed, whereas in Sec. 8.3, space reflection and time reversal (discrete symmetries) are discussed. As we shall see in the next chapter, time reversibility is lost in the measurement process. The rigorous mathematical framework of a theory of transformations and of their invariance is called group theory (the subject of Sec. 8.4).

Quantum transformations and symmetries

Active and passive transformations

We have seen that time evolution may be represented either in the Schrödinger picture (state vectors evolve and observables are kept constant) or in the Heisenberg picture (states vectors remain constant and observables evolve) (see Subsec. 3.6.1 and Sec. 5.3).

In classical mechanics, as we know, all observables commute with each other and therefore belong to the same “commuting set.” This means that – in principle – we can measure all observables simultaneously. Then, the state manifests itself as a collection of properties (of values of observables) (see Sec. 1.1), and as such it is observable. On the other hand, quantum-mechanical observables may not commute (see Subsec. 2.1.5). As a consequence, the state is not just a collection of observable values (see also Sec. 2.3) so that we may ask ourselves whether it is observable or not, and therefore raise the following question: What is the nature of the wave function? Is it simply a mathematical tool, i.e. does it only represent the sum of all that we can know about a system? Or does it have an ontological status? Since the early days of quantum mechanics, this question has been very popular. As a first attempt, de Broglie tried to interpret the wave function as describing, in classical terms, a wave field that envelopes a classical particle (see also Subsec. 16.3.2). This theory should predict the existence of empty waves, but several experiments have shown negative results (see also the end of Sec. 9.6). More recently, many physicists have tried to propose some methods in order to measure the wave function. Here we shortly report on some of these proposals and discuss to what extent the quantum state can be measured.

In this chapter we move from the atomic to the molecular problem, i.e. we try to understand how quantum mechanics, besides explaining the structure and stability of a single atom, is able to account for the existence and the dynamics of molecules, the smallest building blocks of matter that determine and maintain the chemical properties of macroscopic substances. Molecular quantum physics is the field that deals with these types of problems and constitutes one of the most beautiful and successful applications of the basic principles of quantum mechanics. It is also of enormous relevance, because it lays the foundations of a consistent theory of condensed matter.

Our aim here is only to present the basic framework of molecular physics and to give a flavor of its power. A systematic treatment of the subject would go far beyond the scope of this book. In order to reach this goal, we take as our reference molecule the hydrogen molecular ion, which has the advantage of being simple enough to allow exact calculations. While it embodies all the main ingredients of any molecule, it avoids the difficulties due to multielectronic effects (see Sec. 11.6).

In Sec. 12.1 we present the molecule. In Sec. 12.2 we discuss the powerful Born– Oppenheimer approximation that allows us to deal with molecules with an arbitrary number of nuclei and electrons, a problem that would be otherwise unsolvable.

In this chapter we shall mainly present the basic formalism that was initially developed by Heisenberg, also known as matrix mechanics (see Subsec. 1.5.7). We will first introduce in Sec. 2.1 the concept of quantum observables. Then, the problem of discrete and continuous spectra will be discussed and the basic non-commutability of quantum-mechanical observables will be deduced. While in Sec. 2.1 we discuss observables on a general formal level, in Sec. 2.2 some basic quantum-mechanical observables will be defined, and then different representations discussed and commutation relations derived. In Sec. 2.3 a basic uncertainty relation is derived. In the same section the relationship between uncertainty, superposition, and complementarity will be discussed. Finally, in Sec. 2.4 complete subsets of commuting observables will be shown to be Boolean subalgebras pertaining to a quantum algebra which is not Boolean.

BASIC FEATURES OF QUANTUM OBSERVABLES

This section is devoted to a general and formal exposition of quantum observables. In Subsec. 2.1.1 we shall learn how one can mathematically represent quantum observables as Hermitian operators. In Subsec. 2.1.2 we shall see how to change a basis, while in Subsec. 2.1.3 we shall find the relationship between eigenvalues of the observables and probabilities and learn how to calculate mean values. In Subsec. 2.1.4 we shall deal with an operator diagonalization. Finally, in Subsec. 2.1.5, the basic non-commutability of quantum observables will be presented by means of an example.

PLAYING WITH MATLAB

There is no better way to learn than playing. After all, that is how children learn. In this appendix, we are going to provide the basic guidelines for “playing the quantum computing game” with the help of the MATLAB environment.

COMPLEX NUMBERS AND MATRICES

We began this book by saying that complex numbers are fundamental for both quantum mechanics and quantum computing, so we are going to familiarize ourselves with the way they are dealt with in MATLAB.

To begin with, we need to declare complex number variables. This is easy: a complex has a real part and an imaginary part, both double. The imaginary part is declared by using the “i” or “j” character.