Quantum mechanics, like relativity, is a pillar of contemporary physics, with remarkable, puzzling consequences. In these chapters some very simple ‘games’ – distributing prizes of different kinds – are used to explain the implications for ‘identity’. Classical particles have ‘Maxwell‐Boltzmann’ statistics, according to which swapping properties makes a difference; similarly for people, as it makes a difference if we swap bank accounts! Quantum particles with ‘Bose‐Einstein’ statistics (called bosons) or ‘Fermi‐Dirac’ statistics (fermions) are like dollars in bank accounts: if we swap $1 it makes no difference. Chapter 17 shows how Schrödinger used the prize games to argue that – like the dollars – bosons and fermions are not ‘identifiable things’. Chapter 18 extends the models to other possible quantum particles, arguing that quantum mechanics also restricts identity by restricting what states particles can possess together – for instance, Fermi's ‘exclusion principle’, that fermions must be in different states.
Quantum mechanics, like relativity, is a pillar of contemporary physics, with remarkable, puzzling consequences. In these chapters some very simple ‘games’ – distributing prizes of different kinds – are used to explain the implications for ‘identity’. Classical particles have ‘Maxwell‐Boltzmann’ statistics, according to which swapping properties makes a difference; similarly for people, as it makes a difference if we swap bank accounts! Quantum particles with ‘Bose‐Einstein’ statistics (called bosons) or ‘Fermi‐Dirac’ statistics (fermions) are like dollars in bank accounts: if we swap $1 it makes no difference. Chapter 17 shows how Schrödinger used the prize games to argue that – like the dollars – bosons and fermions are not ‘identifiable things’. Chapter 18 extends the models to other possible quantum particles, arguing that quantum mechanics also restricts identity by restricting what states particles can possess together – for instance, Fermi's ‘exclusion principle’, that fermions must be in different states.
Quantum mechanics, like relativity, is a pillar of contemporary physics, with remarkable, puzzling consequences. In these chapters some very simple ‘games’ – distributing prizes of different kinds – are used to explain the implications for ‘identity’. Classical particles have ‘Maxwell‐Boltzmann’ statistics, according to which swapping properties makes a difference; similarly for people, as it makes a difference if we swap bank accounts! Quantum particles with ‘Bose‐Einstein’ statistics (called bosons) or ‘Fermi‐Dirac’ statistics (fermions) are like dollars in bank accounts: if we swap $1 it makes no difference. Chapter 17 shows how Schrödinger used the prize games to argue that – like the dollars – bosons and fermions are not ‘identifiable things’. Chapter 18 extends the models to other possible quantum particles, arguing that quantum mechanics also restricts identity by restricting what states particles can possess together – for instance, Fermi's ‘exclusion principle’, that fermions must be in different states.
When identical particles are not localized they can no longer be distinguished . This situation occurs in the case of a gas since the particles of a gas are unconstrained in their motion except by the walls of the container. It also occurs when the particles are so close together that their wave functions overlap, that is when they are free to exchange positions. It should be stressed that this is a typical quantum-mechanical phenomenon, formulated as QM 3. Classical particles are always distinguishable because one can in principle follow each particle along its trajectory. According to quantum mechanics this is impossible even in principle. According to principle QM 3, there are in fact two different kinds of statistics depending on the type of particle: Bose-Einstein statistics for bosons and Fermi-Dirac statistics for fermions . This is related to the fact that fermions satisfy the Pauli exclusion principle ; there can be no more than one particle in each eigenstate.
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.
In the previous chapters, the focus was on systems of particles following the law of classical mechanics. Of course, this was mainly done within the framework of classical mechanics but a quantum description of individual (discretised) states was also used, while assuming distinguishability of the particles (for instance, particles located at the nodes of a lattice (spin system, quantum harmonic oscillators,) ). The present chapter as well as Chapters 7 and 8 are dedicated to the study of statistical physics in a fully quantum framework. The first obstacle encountered by a quantum approach is the impossibility to define a quantum phase space (in which the ergodic hypothesis would hold for instance). This problem is inherent to quantum mechanics: because of Heisenberg’s uncertainty principle, one cannot define both the momentum and the position of a particle. Besides, statistical physics considers systems made of a large number of particles, a priori in mutual interaction. In classical physics, model situations for which the interactions between the constituents of the system are neglected can be studied. In quantum mechanics, the indistinguishability of identical particles makes the study of the problem in the absence of interaction already difficult: even for non-interacting particles, the symmetrisation (or antisymmetrisation) of the wave function of https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_25.tif "/> identical particles induces correlations which have non-trivial thermodynamical effects such as the existence of a “Fermi sea” for fermions (see Section 8.1 ) or the Bose–Einstein condensation phenomenon for bosons (see Section 7.4 ). The present chapter addresses these two points consecutively.
It is widely accepted that the papers by Bose in 1924 and by Einstein in 1924-1925 definitely opened the path to
the quantum concept of ‘indistinguishability’. Since then, distinction in physics between ‘identity’ and
‘indistinguishability’ of particles has come to be required.
However, even before the rise of quantum physics, Maxwell, Boltzmann and Gibbs, among others, had to tackle
the difficulties and paradoxes that appear when counting identical particles. Later, after Sackur’s and Tetrode’s
first quantum derivation of the canonical partition function of an ideal gas, much confusion arose around its
dependence on the number of molecules in the system. Many other physicists, such as Planck, Lorentz or
Schrodinger took part in those debates. In 1921, Ehrenfest and Trkal devoted a whole paper to this subject.
Clearly, in the days before Bose’s seminal work, embroiling and misunderstanding were commonplace.
In considering the emergence of the distinction between ‘identical’ and ‘indistinguishable’ particles, we will go
through the so-called statistical route to quantum mechanics. The first developments in statistical mechanics
became immediately confused and entangled with the irruption of quantum physics. In the first quarter of the
twentieth century the mutual influence between both disciplines was profuse.
We will pay special attention to the story of the so-called ‘Gibbs Paradox.’ We will argue that what is now often
known by this name in some textbooks of statistical mechanics did not exist as such in those foundational days.
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.