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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum gas in an external field: exact grand canonical expressions and numerical treatment

P. N. Vorontsov‐Velyaminov, R. I. Gorbunov, Sergei D. Ivanov

Published Jan 25, 2000• 0 citations
Publisher

Abstract

Previously developed approach [1 is extended to systems of noninteracting quantum particles in an external magnetic field. Feynman type series for grand potential, number of particles, internal energy and magnetization are derived. Numerical calculations at constant N for harmonically trapped systems of fermions and bosons in presence of the homogeneous magnetic field are carried out with the aid of both Feynman-type series and series over single particle states. Particular attention is paid to low temperature behavior of fermionic systems.

DOI
10.1117/12.375409

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Previously developed approach [1 is extended to systems of noninteracting quantum particles in an external magnetic field. Feynman type series for grand potential, number of particles, internal energy and magnetization are derived. Numerical calculations at constant N for harmonically trapped systems of fermions and bosons in presence of the homogeneous magnetic field are carried out with the aid of both Feynman-type series and series over single particle states. Particular attention is paid to low temperature behavior of fermionic systems.
Feynman-type series for grand potential , number of particles, internal energy, magnetization and susceptibility are derived for systems of noninteracting quantum particles in an external magnetic field. Numerical calculations at constant N for harmonically trapped systems of fermions and bosons in the presence of a homogeneous magnetic field are carried out with the aid of both Feynman-type series and series over single-particle states. N -convergence to universal curves for chemical potential and energy is presented. At low temperatures magnetization of the Fermi gas exhibits a pronounced quantum behaviour.
An exact Feynman-type presentation of the grand canonical partition function and averages as series over cycles for a system of noninteracting identical particles with a spin in an arbitrary external field is derived, and a numerical procedure for obtaining $\ensuremath{\mu}(\ensuremath{\beta})$ and other dependencies at constant N is developed. It is shown that the same series can be obtained also from the conventional form of the grand potential $\ensuremath{\Omega}$ (i.e., a sum over single-particle energy states). Numerical calculations at constant N are carried out for quantum gas of bosons and fermions in three-dimensional harmonic field and in the P\"oschl-Teller potential.
A system of a finite number of harmonically trapped fermions in one dimension, in the presence of a static magnetic field, is studied within the framework of the static fluctuation approximation, for different repulsive and attractive potential strengths. Specifically, the thermodynamic properties of the system (the chemical potential, total energy, heat capacity, and entropy), as well as its magnetic properties (the magnetization and susceptibility), are calculated. It is observed that the system remains in an ordered phase for a small number of particles N, even at high temperatures T. Disorder sets in for large N, even at low T. The effect of the potential strength on the heat capacity is particularly tangible in the region bordering the quantum and classical regimes. The effect of the temperature (representing disorder) is directly opposite to that of the magnetic field (representing order), as expected on basic physical grounds. These features are consistent with experimental results.
The statistical properties of non-interacting bosons and fermions confined in trapping potentials are most easily obtained when the system may exchange energy and particles with a large reservoir (grand-canonical ensemble). There are circumstances, however, where the system under consideration may be considered as being isolated (micro-canonical ensemble). This paper first reviews results relating to micro-canonical ensembles. Some of them were obtained a long time ago, particularly by Khinchin in 1950. Others were obtained only recently, often motivated by experimental results relating to atomic confinement. A number of formulas are reported for the first time in the present paper. Formulas applicable to the case where the system may exchange energy but not particles with a reservoir (canonical ensemble) are derived from the micro-canonical ensemble expressions. The differences between the three ensembles tend to vanish in the so-called Thermodynamics limit, that is, when the number of particles and the volume go to infinity while the particle number density remains constant. But we are mostly interested in systems of moderate size, often referred to as being mesoscopic, where the grand-canonical formalism is not applicable. The mathematical results rest primarily on the enumeration of partitions of numbers.
This chapter investigates a quantum gas in the grand-canonical ensemble with the grand canonical Hamiltonian ensemble. The path integral formalism pioneered by Feynman has proven to be an extremely useful tool for understanding and handling quantum mechanics, quantum field theory, and statistical mechanics. The path integral approach is a powerful tool that considers nonperturbative calculations for the investigation of many-particle systems. It is worth noting that while the path integral approach seems to be quite cumbersome in quantum mechanics, it does provide a powerful tool in quantum field theory. The entire aforementioned derivation is calculated in order to justify the evaluation of the path integral at equal times. The chapter examines the example of an atomic Fermi gas by considering an electron gas. The Fermi liquid concept is the deviation of the electron gas from the ideal gas behavior. It utilizes the so-called diagram technique to calculate some quantities.