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COMPARATIVE ANALYSIS OF CLASSICAL AND QUANTUM REPRESENTATIONS IN TEACHING PHYSICS

J. Abdullaev

Published May 21, 2023• 0 citations
Publisher

Abstract

The article provides a comparative analysis of classical and quantum statistics. The general properties and differences between the statistics of Maxwell and the statistics of Bose-Einstein, Fermi-Dirac from the methodological point of view are revealed
DOI
10.5281/zenodo.7954919

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Some fundamental changes have been introduced into statistical theory since the publication of a paper by Bose in 1924. The characteristic features of the new statistics, including that of Bose and Einstein, and the later one of Fermi and Dirac, are pointed out in this lecture and compared with the classical statistics. A number of recent applications of the new ideas are described, including the work of Thomas and Fermi on the distribution of electrons in atoms, and the work of Sommerfeld and others on the electron theory of metals. An account is also given of some recent results obtained by the author and H. J. Woods on the distribution of electrons in a metal.
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.
Beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles, a unified quantum statistics, which may interpolate smoothly between Bose-Einstein and Fermi-Dirac statistics, is proposed. The commutation relations for such particle creation and annihilation operators are derived, and statistical partition function and thermodynamical properties of an ideal gas of the particles are investigated. Besides the Bose-Einstein condensation, another kind of particle condensation may appear.