A feature of quantum mechanics is that identical particles cannot be labelled. This means that identical particles must be treated as indistinguishable and implies that if we exchange two particles in a many – particle system – e.g. the electrons in a molecule – then the wavefunction describing the system cannot change, apart from a phase factor. Besides some exotic possibilities in strictly two-dimensional systems, this phase factor has only two values: +1 and -1, which lead to symmetric and anti-symmetric wavefunctions, respectively. Particles with symmetric wavefunctions, such as photons, are called bosons, while those with anti-symmetric wavefunctions, such as electrons, are called fermions.
A feature of quantum mechanics is that identical particles cannot be labelled. This means that identical particles must be treated as indistinguishable and implies that if we exchange two particles in a many – particle system – e.g. the electrons in a molecule – then the wavefunction describing the system cannot change, apart from a phase factor. Besides some exotic possibilities in strictly two-dimensional systems, this phase factor has only two values: +1 and -1, which lead to symmetric and anti-symmetric wavefunctions, respectively. Particles with symmetric wavefunctions, such as photons, are called bosons, while those with anti-symmetric wavefunctions, such as electrons, are called fermions.
The degrees of freedom studied so far have been either real or complex variables. These variables commute under multiplication, in the sense that two numbers a, b satisfy ab = ba ; commuting variables are generically called bosonic variables, or bosonic degrees of freedom. Typical of the bosonic case are the degrees of freedom for a collection of quantum mechanical particles. Interactions of fundamental particles are generally mediated by bosonic fields such as the Maxwell electromagnetic field, whereas mass is usually carried by particles that are fermions, the most familiar being the electron. Two key features distinguish fermions from bosons: Fermions obey the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. This is the reason the concept of intensity does not apply to a fermion. A high intensity electric field is a reflection of the presence of a large number of photons, which are bosons, in the same quantum state; for photons, any number of photons can be in the same quantum state. In contrast, an electron is either in a quantum state or it is not; in particular, ignoring spin, an electron exists at a point or there is no electron there. The state function of a multi-bosonic system is totally symmetric in that the exchange of any two bosonic degrees of freedom yields the same state function. In contrast, a multi-fermion system is totally anti-symmetric: the exchange of any two fermion degrees of freedom gives the same state – but with a negative sign .
In this article, we discuss the identity and indistinguishability of quantum systems and the consequent need to introduce an extra postulate in Quantum Mechanics to correctly describe situations involving indistinguishable particles. This is, for electrons, the Pauli Exclusion Principle, or in general, the Symmetrization Postulate. Then, we introduce fermions and bosons and the distributions respectively describing their statistical behaviour in indistinguishable situations. Following that, we discuss the spin-statistics connection, as well as alternative statistics and experimental evidence for all these results, including the use of bunching and antibunching of particles emerging from a beam splitter as a signature for some bosonic or fermionic states.
For a quantum mechanical system of N particles, the Hamiltonian is a sum of the total kinetic energy and the potential energy. In classical physics, it is possible to keep track of individual particles. A system of N identical particles is said to satisfy Bose-Einstein statistics if its wave function is symmetric under the exchange of particles, and these particles are known as bosons. In the nonrelativistic quantum mechanics, this principle must be accepted as an empirical postulate. In the relativistic quantum theory, it can be proven that half-integer-spin particles cannot be bosons and integer-spin particles cannot be fermions. The Pauli exclusion principle plays a critical role in programming physical properties of electrons or more generally fermions.
In this article, we discuss the identity and indistinguishability of quantum systems and the consequent need to introduce an extra postulate in Quantum Mechanics to correctly describe situations involving indistinguishable particles. This is, for electrons, the Pauli Exclusion Principle, or in general, the Symmetrization Postulate. Then, we introduce fermions and bosons and the distributions respectively describing their statistical behaviour in indistinguishable situations. Following that, we discuss the spin-statistics connection, as well as alternative statistics and experimental evidence for all these results, including the use of bunching and antibunching of particles emerging from a beam splitter as a signature for some bosonic or fermionic states.
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 introduces the partition function that involves a sum over independent quantum states of the N-particle system. It outlines the problem of distinguishability, that is, how to distinguish one electron from another when they are replicas of each other and are all the same. It also emphasizes that electrons have no feature or label based on which they can be told apart, and they repeat the same attributes endlessly. The chapter explains that identical particles lead to complications in the quantum mechanical description of the microscopic state of the system as there are no observable changes in the system when any two particles are exchanged. It analyses the uncertainty principle, wherein the position of a particle cannot be accurately specified at each stage, nor can the precise trajectory of a particle be followed.