Up to now we have limited ourselves to states of a single particle. In the present chapter we shall introduce the description of two-particle states. Once this case is understood, it will be easy to generalize to any number of particles. States of two (or more) particles lead to very rich configurations called entangled states. A remarkable feature is that two entangled quantum particles, even at arbitrarily large spatial separations, continue to form a single entity and no classical probabilistic model is able to reproduce the correlation between particles. In the first section we shall present the essential mathematical formalism, that of the tensor product. This will permit us in Section 6.2 to describe quantum mixtures using the state operator formalism. Section 6.3 is devoted to the study of important physical consequences like the Bell inequalities and interference experiments involving entangled states, which will lead us to a deeper understanding of quantum physics. Finally, in the last section we shall briefly review applications to measurement theory and quantum information theory. The latter is undergoing rapid development at present and has applications to quantum computing, cryptography, and teleportation.

The tensor product of two vector spaces

Definition and properties of the tensor product

We wish to construct the space of states of two physical systems which we assume initially to be completely independent. Let ℌN1 and ℌM2 be the spaces of states of the two systems, of dimension N and M, respectively.

In this appendix we shall describe in more detail how the experiment of Brune et al. mentioned in Section 6.4.1 provided evidence of the phenomenon of decoherence in an entirely controlled manner. In addition to its intrinsic interest, this experiment is a prime example of actual experiments which allow the fundamentals of quantum mechanics to be tested with a precision undreamed of by its founders, and the study of this experiment constitutes a beautiful exercise in quantum physics. It will also allow us to give a small sample of the current ideas on the notion of measurement in quantum mechanics. We shall first return to the interference experiment of Section 1.4.4, this time discussing it within the framework of an elementary theory of measurement. Then we shall examine the realization of Ramsey fringes using Rydberg atoms, and show how the interaction of these atoms with an electromagnetic field progressively blurs these fringes when we try to answer the “which of the two trajectories?” question. Finally, we shall show how the use of a pair of atoms allows decoherence to be tested.

An elementary model of measurement

Let us return to the discussion of the Young's slit experiment with the trajectories labeled as in Fig. 1.13, enlarging on it with the introduction of a mathematical formulation. Let c1(x) [c2(x)] be the complex probability amplitude for an atom to be localized at a point x on the screen after having passed through slit 1 [2]. The (arbitrary) normalization is fixed by |c11(x)|2 = |c2(x)|2 = 1.

This book has grown out of a course given at the University of Nice over many years for advanced undergraduates and graduate students in physics. The first ten chapters correspond to a basic course in quantum mechanics for advanced undergraduates, and the last four could serve to complement a graduate course in, for example, atomic physics. The book contains about 130 exercises of varying length and difficulty, most of which have actually been used in homework or exams.

This book should be interesting not only to students in physics and engineering, but also to a wider group of physicists: graduate students, researchers, and secondary school teachers who wish to update their knowledge of quantum physics. It discusses recent developments not covered in the classic texts such as entangled states, quantum cryptography and quantum computing, decoherence, interactions of a laser with a two level atom, quantum fluctuations of the electromagnetic field, laser manipulation of atoms, and so on, and it also includes a concise discussion of the current ideas about measurement in quantum mechanics as an appendix.

The organization of this book differs greatly from that of the classic texts, which typically begin with the Schrödinger equation and then proceed to study its solution in various situations. That approach makes it necessary to introduce the basic principles of quantum mechanics in a relatively complicated situation, and they end up being obscured by calculations which are often rather complex. Instead, I have striven to present the fundamentals of quantum mechanics using the simplest examples, and the Schrödinger equation appears only in Chapter 9.

This chapter is devoted to an introduction to atomic physics which will be mainly concerned with one-electron atoms. After a brief discussion of the perturbation and variational methods in Section 14.1, in Section 14.2 we study the fine and hyperfine structure of the energy levels as well as the effect of a magnetic field on these levels. In Section 14.3 we examine the coupling of an atom to an electromagnetic field and important applications of this coupling such as the photoelectric effect and the rate of spontaneous emission. In Section 14.4 we give a brief introduction to a subject which has been expanding enormously in the last twenty years, the laser manipulation of atoms, and we discuss Doppler cooling and magneto-optical traps. Finally, Section 14.5 is devoted to a short discussion of two-electron atoms.

Approximation methods

Generalities

In classical physics it is only in exceptional cases that it is possible to solve the Newton or Maxwell equations analytically given the initial conditions at time t = t0 and, in the first case, the forces, or, in the second, the sources of electromagnetic field. In general, it is necessary to resort to an approximation method such as numerical integration of the equations, the perturbation method, or something else. The situation is no different in quantum physics: only in exceptional cases do we know how to “solve the Schrödinger equation” exactly, that is, how to obtain the time evolution of the state vector |ϕ(t〉 as a function of its value |ϕ(t0〉 at initial time t = t0.

The superposition principle is a founding principle of quantum mechanics; we have already made use of it in interpreting the Young's slit experiment. Quantum mechanics is a linear theory, and so it is natural that vector spaces play an important role in it. We shall see that a physical state is represented mathematically by a vector in a space whose characteristics we shall define; this is called the space of states. A second founding principle, which can also be deduced from the Young's slit experiment, is the existence of probability amplitudes. These probability amplitudes will be represented mathematically by scalar products defined on the space of states. In the theory of waves, the use of complex numbers is just a convenience, but in quantum mechanics the probability amplitudes are fundamentally complex numbers – the scalar product will a priori be a complex number. Physical properties like momentum, position, energy, and so on will be represented by operators acting in the space of states. In this chapter we shall introduce the essential properties of Hilbert spaces, that is, vector spaces on which a positive-definite scalar product is defined, and we shall limit ourselves to the case of finite dimension. This restriction will be lifted later on, because the space of states is in general of infinite dimension. The mathematical theory of Hilbert spaces of infinite dimension is much more complicated than that of spaces of finite dimension, and we shall put off studying them until Chapter 7. The reader familiar with vector spaces of finite dimension and operators in such spaces can proceed directly to Chapter 3 after reviewing the notation.

Quantum physics is now one hundred years old, and this description of physical phenomena, which has transformed our vision of the world, has never been found at fault, which is exceptional for a scientific theory. Its predictions have always been verified by experiment with impressive accuracy. The basic concepts of quantum physics such as probability amplitudes and linear superpositions of states, which seem so strange to our intuition when encountered for the first time, remain fundamental. However, during the last few decades an important evolution has occurred. The spectacular progress made in observational techniques and methods of manipulating atoms now makes it possible to perform experiments so delicate that they were once considered as only “thought experiments” by the founders of quantum mechanics. The existence of “nonseparable” quantum correlations, which forms the basis of the Einstein–Podolsky–Rosen “paradox” and which violates the famous Bell inequalities, has been confirmed experimentally with high precision. “Entangled” states of two systems which manifest such quantum correlations are now better understood and even used in practical applications such as quantum cryptography. The entanglement of a measuring device with its environment reveals an interesting new pathway to better understanding of the measurement process.

In parallel with these conceptual advances, our everyday world is being invaded by devices which function on the basis of quantum phenomena. The laser sources used to read compact disks, in ophthalmology, and in optical telecommunications are based on light amplification by atomic systems with population inversion. Nuclear magnetic resonance is widely used in hospitals to obtain ever more detailed images of the organs of the human body.

Up to now we have mainly studied bound states, except for the brief mention of one-dimensional scattering in Section 9.4. However, essential information on interactions between particles, atoms, molecules, etc., as well as on the structure of composite objects, can be obtained from scattering experiments. Bound states – when they exist, which is not always the case – give only partial information on such interactions, whereas it is nearly always possible to perform scattering experiments. In this chapter we shall limit ourselves to potential scattering, which can be used to describe elastic collisions of two particles of masses m1 and m2. Indeed, in the center-of-mass frame the problem is reduced to that of a particle of mass m = (m1m2)/(m1 + m2) in a potential (Exercise 8.5.6).

In Sections 12.1 and 12.2 we develop the elementary formalism of elastic scattering theory with emphasis on the low-energy limit, which plays an extremely important role in practice. In Section 12.3 we generalize the formalism to the inelastic case; more precisely, we examine the effect of inelastic channels on elastic scattering. Finally, Section 12.4 is devoted to some more formal aspects of scattering theory.

The cross section and scattering amplitude

The differential and total cross sections

A scattering experiment is shown schematically in Fig. 12.1. A beam of particles of mass m1 and well-defined momentum moving along the z axis collides with a target composed of particles of mass m2. To simplify the discussion, we assume that m1 ≪ m2 and we neglect the recoil of the target in the collision. In general, it is necessary to go from the laboratory frame to the center-of-mass frame via a simple kinematic transformation (Exercise 8.5.6).

In this chapter we examine some simple applications of quantum mechanics in situations where it is possible to model quantum systems accurately by restricting ourselves to a space of states of finite dimension. If each energy level, including degenerate ones, is counted once, the dimension of ℌ is equal to the number of levels, and this is why we use the term system with a finite number of levels. The first two examples (Section 5.1) are taken from quantum chemistry and allow us to study a stationary situation where the Hamiltonian is time-independent. But the most important point in this chapter is the introduction of time dependence, which will be implemented by coupling a two-level system to an external periodic classical field. This will be illustrated by three examples of great practical importance: nuclear magnetic resonance (Section 5.2), the ammonia molecule (Section 5.3), and the two-level atom (Section 5.4).

Elementary quantum chemistry

The ethylene molecule

The ethylene molecule C2H4 will serve as an introduction to the subject. The “skeleton” of this molecule is formed by the so-called σ bonds, pairs of σ electrons of opposite spin common to two carbon atoms or to a carbon and a hydrogen atom, thus forming the (C2H4)++ ion (Fig. 5.1). The remaining two electrons, called π electrons, are mobile – they can jump from one carbon atom to another. It is said that they are delocalized. The separate treatment of the π and σ electrons is, of course, an approximation, but one that plays an important role in the theory of chemical bonding. Let us begin by putting the first π electron in place.

The derivation of the Fermi Golden Rule in Section 9.6.3 is limited to sufficiently short times t ≪ τ2, and the exponential decay law (9.171) cannot be justified using only the arguments of that section. A method due to Wigner and Weisskopf permits this law to be justified for long times with the help of another approximation scheme. Let us consider the following situation. A state of an isolated system a of energy Ea decays to a continuum of states b of energy Eb. Examples of such a situation are the de-excitation of an excited state of an atom, a molecule, a nucleus, and so on with the emission of a photon, or the decay of an elementary particle. The states of energy Ea and Eb are the eigenstates of a Hamiltonian H(0):

and a time-independent perturbation W is responsible for the transition a → b; in the case of spontaneous photon emission, W is given by (14.58). The states a and b are not stationary states of the total time-independent Hamiltonian H = H(0) + W. We can assume that the diagonal matrix elements of W are zero: Waa = Wbb = 0 and we use |ψ(t)〉 to denote the state vector of the system, the initial state being |ψ(t = 0)〉 = |a〉.