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Quantum Gases
Published in Teunis C. Dorlas, Statistical Mechanics, 2021
In chapter 22 we considered a system of identical, localized particles. These particles can therefore be distinguished according to their position in space. 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. The indistinguishability of particles has an important consequence for the counting of microstates: The microstates of indistinguishable particles are completely determined by the number of particles in each individual particle eigenstate; one says that the statistics of indistinguishable particles is different from that of localized particles.
Overview of Bohmian Mechanics
Published in Xavier Oriols, Jordi Mompart, Applied Bohmian Mechanics, 2019
A standard claim in many quantum mechanics textbooks is that identical particles, for example, two electrons with an antisymmetric wave function, are indistinguishable. It is affirmed that if the particles would have trajectories, they would automatically be distinguishable. In Bohmian mechanics, even with the symmetrization postulate, the adjective “indistinguishable” is inappropriate because one can label one particle’s trajectory r→1[t] and the other r→2[t] and thus distinguish them perfectly at the ontological staff.
Temporal (t > 0) Quantum Mechanics
Published in Francis T. S. Yu, Origin of Temporal (t > 0) Universe, 2019
Pauli exclusive principle [11] states that two identical particles with same quantum state cannot occupy at the same quantum state simultaneously, unless these particles exist with a different half-spin. Quantum entanglement occurs when a pair of particles interacts in such a way that the quantum state of the particles cannot be independently described; even when the particles are separated by a large distance, a quantum state must be described by the pair of particles as a whole. In view of Paul’s principle, we assert that, the atomic model used for his discovery has no coordinate (or position). This is a reasonable assumption since atomic size particles are very small, for which singularity approximation is appropriately used in most of the times.
Gibbs’ paradox according to Gibbs and slightly beyond
Published in Molecular Physics, 2018
In modern texts on statistical mechanics the author is aware of, the concepts of generic and specific phases have not survived. The closest to those would be the distinction between distinguishable and indistinguishable identical particles; that is, particles that are identical but can be distinguished (e.g. because they are allegedly localised in space) or cannot be distinguished, even in principle. A notable difference between the modern concepts of distinguishable and indistinguishable on the one hand and specific and generic on the other hand is that they do not refer to the same thing. The former is a property of particles imagined individually while the latter is a property of states as they are understood in a statistical description of a substance. That point was already made by Gibbs, in a different fashion, in his 1876 paper where he discusses the classical thermodynamics (no statistical mechanics involved at all) of substances in great details [30]. First, among many other things, he derives on purely thermodynamic grounds that the entropy of a monoatomic substance, recast in this paper's notation, reads (cf. Equation (278) in Ref. [30]): where K is a constant. From Equation (11), Gibbs retrieves that for the mixing by diffusion of two different gases the entropy change is kBln 2 per particle while for similar gases it is zero. In passing, this indicates that claim C1 – that classical thermodynamics would be in need of quantum correction – is in fact unsubstantiated for at least Gibbs did not get any inconsistency from applying ‘only’ nineteenth century thermodynamics to the theory of the mixing of gases. Furthermore, Gibbs reflects on what a zero entropy variation means in the case of identical substances [30]: