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.
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.
The postulates 1 to 6 of quantum mechanics as stated in Sections 3.7 and 7.2 apply to multi-particle systems provided that each of the particles is distinguishable from the others. For example, the nucleus and the electron in a hydrogen-like atom are readily distinguishable by their differing masses and charges. When a system contains two or more identical particles, however, postulates 1 to 6 are not sufficient to predict the properties of the system. These postulates must be augmented by an additional postulate. This chapter introduces this new postulate and discusses its consequences. Permutations of identical particles Particles are identical if they cannot be distinguished one from another by any intrinsic property, such as mass, charge, or spin. There does not exist, in fact and in principle, any experimental procedure which can identify any one of the particles. In classical mechanics, even though all particles in the system may have the same intrinsic properties, each may be identified, at least in principle, by its precise trajectory as governed by Newton's laws of motion. This identification is not possible in quantum theory because each particle does not possess a trajectory; instead, the wave function gives the probability density for finding the particle at each point in space. When a particle is found to be in some small region, there is no way of determining either theoretically or experimentally which particle it is.
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.
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 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.
Symmetries and wave functions An important difference between the classical and quantum perspectives is their different criteria of distinguishability. Identical particles are classically distinguishable when separated in phase space. On the other hand, identical particles are always quantum mechanically indistinguishable for the purpose of counting distinct microstates. But these concepts and these distinctions do not tell the whole story of how we count the microstates and determine the multiplicity of a quantized system. There are actually two different ways of counting the accessible microstates of a quantized system of identical, and so indistinguishable, particles. While these two ways were discovered in the years 1924–1926 independently of Erwin Schrödinger’s (1887–1961) invention of wave mechanics in 1926, their most convincing explanation is in terms of particle wave functions. The following two paragraphs may be helpful to those familiar with the basic features of wave mechanics. A system of identical particles has, as one might expect, a probability density that is symmetric under particle exchange, that is, the probability density is invariant under the exchange of two identical particles. But here wave mechanics surprises the classical physicist. A system wave function may either keep the same sign or change signs under particle exchange. In particular, a system wave function may be either symmetric or antisymmetric under particle exchange.
In quantum mechanics the situation is different because the uncertainty principle does not allow us to know the initial positions and momenta of the particles with desired accuracy. The individual particle is described by a wave packet. Since the wave packet is localized and spreading with time, it is impossible to distinguish two identical particles as their wave functions always overlap at any point in space. Quantum theory describes any object at energies below its excitation threshold with four degrees of freedom. The wave functions of all the states of a given system of identical particles must have the same symmetry; otherwise, the wave function of a state formally a superposition of states of different symmetry would be neither symmetrical nor antisymmetrical. Hence, for a system of identical particles, at high energies and low concentration, probable number of particles per quantum state will be very small. Wave functions of the particles are well separated.