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Cambridge University Press eBooks

Identical-Particle Systems: Quantum Statistics. The Density Matrix

Ana Marı́a Cetto, Luis de la Peña

Published Dec 18, 2025• 0 citations
Publisher

Abstract

Chapter 15 is devoted to systems of identical particles and the associated quantum statistics. It is shown how the spin (whether integer or half-integer) determines fundamental features of the system’s behavior that cannot be obtained by approximate methods, in particular the exchange effects characteristic of states of two or more identical particles. The spin-symmetry relation is discussed in detail for both the Bose–Einstein and the Fermi–Dirac statistics. Quantum collective phenomena that are a defining feature of condensed matter, including phonons, superconductivity, Bose–Einstein condensates and the quantum Hall effect, are briefly introduced. Since most real quantum systems consist of subsystems in different quantum states that are mixtures of pure states, the chapter includes an introduction to the density matrix, its basic properties and its application to quantum statistics. It concludes with a critical discussion of decoherence and the significance of the collapse of the wave function.

DOI
10.1017/9781009679633.016

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Chapter 15 is devoted to systems of identical particles and the associated quantum statistics. It is shown how the spin (whether integer or half-integer) determines fundamental features of the system’s behavior that cannot be obtained by approximate methods, in particular the exchange effects characteristic of states of two or more identical particles. The spin-symmetry relation is discussed in detail for both the Bose–Einstein and the Fermi–Dirac statistics. Quantum collective phenomena that are a defining feature of condensed matter, including phonons, superconductivity, Bose–Einstein condensates and the quantum Hall effect, are briefly introduced. Since most real quantum systems consist of subsystems in different quantum states that are mixtures of pure states, the chapter includes an introduction to the density matrix, its basic properties and its application to quantum statistics. It concludes with a critical discussion of decoherence and the significance of the collapse of the wave function.
Abstract Two concepts, of fundamental importance to quantum mechanics, will be discussed in this chapter. The first is that of symmetry. Even though the concept of symmetry is familiar in a wide range of natural sciences, the way symmetry constrains the consequences of quantum mechanics is rather subtle and, at the same time, far-reaching. The second is the symmetry property of the states under the exchange of two identical particles. Identical particles with a half-integer spin (fermions) obey the Fermi–Dirac statistics: the wave functions are totally antisymmetric under the exchange of the particles. Identical particles with an integer spin (bosons) are instead subject to the Bose–Einstein statistics. Their wave functions are totally symmetric.
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.
This chapter discusses the properties of systems of identical particles. It presents the wave functions of a system composed of N identical and independent particles. From the properties of the wave functions, the chapter shows that identical particles can be characterized by symmetric as well as antisymmetric wave functions. As a result, it distinguishes two kinds of particles: bosons and fermions. Fermions have half integer spin, whereas bosons have integer spin. Since an arbitrary number of bosons can occupy the same state, atoms with integer spin can abruptly condensate into a single ground state when the temperature of the atoms goes below a certain critical value. In atoms, a limited number of electrons can occupy the same energy level. The Pauli principle prevents the energy states being occupied by an arbitrary number of electrons. Thus, the Pauli principle prevents all atoms having the same size and the same energy.
We define statistics of quantum mechanical particles, obtaining the Bose–Einstein and Fermi–Dirac varieties of indistinguishable particles. After finding the rotation and Lorentz matrices in different Lorentz representations, we describe the spin-statistics theorem, relating fermions with half-integer spin and bosons with integer spin. We explain two simple proofs and say some words on two others. We end by discussing symmetries in more generality, and we discuss the fact that internal symmetries must commute with spacetime ones, due to the Coleman–Mandula theorem.
Synopsis: The connection between a particle’s spin and the statistics it obeys is given. Bose–Einstein and Fermi–Dirac statistics are explained and illustrated with products of wave functions. A new type of particle basis state is described, and two possibilities for the commutation properties of the associated raising (“creation”) and lowering (“annihilation”) ladder operators are examined, giving rise to the Bose–Einstein and Fermi–Dirac cases. The relation between particle occupation number and energy is inferred from detailed balance, and cubical enclosures of particles are studied. Two physics mysteries encountered in the earlier chapters (zweideutigkeit and the vanishing electronic component of specific heat) are explained. The Hartree–Fock theoretical framework for the approximate treatment of many-particle atomic systems is constructed.