On wave properties of identical particles

F. J. Bloore, S. J. Swarbrick

Arnaud Ralko, Trong Tuong Truong

Z. Ficek

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.

Gennaro Auletta, Mauro Fortunato, Giorgio Parisi A

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.

Yoshio Ohnuki, S. Kamefuchi