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Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fields

The classical mechanics for bose-fermi systems

R. Casalbuoni

Published Jun 1, 1976• 464 citations
Publisher
DOI
10.1007/bf02729860

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Physical Science International Journal•Jan 10, 2015•0 citations
I show how the Fermi and Bose pressures in quantum systems, identified in standard discussions through the use of thermodynamic analogies, can be derived directly in terms of the flow of momentum across a surface by using the quantum mechanical stress tensor. In this approach, analogous to classical kinetic theory, pressure is naturally defined locally, a point which is obvious in terms of the stress-tensor but is hidden in the usual thermodynamic approach. The two approaches are connected by an interesting application of boundary perturbation theory for quantum systems. The treatment leads to a simple interpretation of the pressure in Fermi and Bose systems in terms of the momentum flow encoded in the wave functions. I apply the methods to several problems, investigating the properties of quasi continuous systems, relations for Fermi and Bose pressures, shape-dependent effects and anisotropies, and the treatment of particles in external fields, and note several interesting problems for graduate courses in statistical mechanics that arise naturally in the context of these examples.
I show how the pressure in Fermi and Bose systems, identified in standard discussions of quantum statistical mechanics by the use of thermodynamic analogies, can be derived directly in terms of the flux of momentum across a surface by using the quantum mechanical stress tensor. In this approach, which is analogous to classical kinetic theory, the pressure is naturally defined locally. The approach leads to a simple interpretation of the pressure in terms of the momentum flow encoded in the wave functions. The stress-tensor and thermodynamic approaches are related by an interesting application of boundary perturbation theory for quantum systems. I investigate the properties of quasi-continuous systems, the relations for Fermi and Bose pressures, shape-dependent effects and anisotropies, and the treatment of particles in external fields, and note several interesting problems for graduate courses in statistical mechanics.
In this work, we focus our study on some systems in statistical mechanics by using a recent developpment of the fractional classical and quantum mechanics. At first, we present the thermodynamical properties of N classical and quantum oscillators in 3-dimensional space. In each case, the Hamiltonian, describing the system, exhibits fractional powers. Secondly, in the context of the fractional quantum mechanics, we calculate the thermodynamical properties for the systems: the trapped ideal Bose and Fermi gases. For the trapped ideal Bose system, Bose- Einstein condensation and the thermodynamics properties are discussed and plotted with respect to the fractional powers and the temperature. It is shown that the trapped ideal Bose gas exhibits a phase transition at a temperature TC for which the specific heat presents a discontinuity depending on the fractional powers. All calculations are performed in 3-dimensional space.