681 research outputs found
Fermi and Bose pressures in statistical mechanics
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.Comment: RevTeX4, 18 pages. Submitted to American Journal of Physic
Fractional operators and special functions. II. Legendre functions
Most of the special functions of mathematical physics are connected with the
representation of Lie groups. The action of elements of the associated Lie
algebras as linear differential operators gives relations among the functions
in a class, for example, their differential recurrence relations. In this
paper, we apply the fractional generalizations of these operators
developed in an earlier paper in the context of Lie theory to the group SO(2,1)
and its conformal extension. The fractional relations give a variety of
interesting relations for the associated Legendre functions. We show that the
two-variable fractional operator relations lead directly to integral relations
among the Legendre functions and to one- and two-variable integral
representations for those functions. Some of the relations reduce to known
fractional integrals for the Legendre functions when reduced to one variable.
The results enlarge the understanding of many properties of the associated
Legendre functions on the basis of the underlying group structure.Comment: 26 pages, Latex2e, reference correcte
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