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Modern Physics Letters A

GENERALIZED FERMI PARTITION FUNCTION OBTAINED VIA COHERENT STATE REPRESENTATION OF THE DENSITY OPERATOR

Hong-Yi Fan, HUI ZOU, Yue Fan

Published Nov 20, 1999• 4 citations
Publisher

Abstract

We generalize the well-known partition function of Fermi harmonic oscillator (H =ω(f+f + ½))[Formula: see text] to [Formula: see text] for Fermi system described by the n-mode fermionic Hamiltonian [Formula: see text]. This result is derived via constructing the fermionic coherent state representation of the density operator [Formula: see text] in which [Formula: see text] is identified as exp {-β Γ Π}.

DOI
10.1142/s0217732399002571

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We generalize the well-known partition function of Fermi harmonic oscillator (H =ω(f+f + ½))[Formula: see text] to [Formula: see text] for Fermi system described by the n-mode fermionic Hamiltonian [Formula: see text]. This result is derived via constructing the fermionic coherent state representation of the density operator [Formula: see text] in which [Formula: see text] is identified as exp {-β Γ Π}.
An alternative method for evaluating the canonical density matrix (x mod exp(- beta H) mod x') for a one-dimensional harmonic oscillator is proposed. The method uses the coherent states, i.e. the eigenstates of the annihilation operator a, as the basis of the expansion of exp(- beta H)) mod x') in place of the Hermite functions which are used in the conventional treatment. It is shown that the use of the coherent states makes it much easier to evaluate the density matrix than the other methods. As a direct extension of the method, the density matrix for a three-dimensional harmonic oscillator placed in a uniform magnetic field is recalculated in the creation-annihilation operator formalism. Some other uses of the coherent states in operator problems, such as the Wigner distribution function, are also discussed. Two alternative derivations of the Gaussian integral formula are also given, which plays an important role in the present problem as well as in various other problems in physics.
We introduce the generalized fermionic Wigner operator with an s parameter. Based on its remarkable properties, we establish one-to-one mapping between fermion operators and their s-parametrized pseudo-classical correspondence, which may involve fermionic Weyl pseudo-classical correspondence, P-representation and Q-representation in a unified way. Furthermore, starting with the projector of the fermionic coherent state, we obtain the s-ordered operator expansion formula of fermionic density operators, which includes normally ordered, antinormally ordered and Weyl ordered product of operators for different values of s. Applications in calculating some Fermi operators' s-ordered expansions are presented.
Based on the density operator's coherent state representation [Phys. Lett.A252, 281 (1999)], we extend the well-known Bose distribution's partition function Ξ [Formula: see text] for ideal photon gas [Formula: see text] to that of an assembly of photon gas interacting with nonlinear medium described by the Hamiltonian [Formula: see text], where A=(a1,a2,…,an), Ω is a symmetric matrix, and we report that the corresponding partition function is [Formula: see text] As a result, the generalized Bose distribution and thermodynamic functions are derived.
The path integral technique is used to derive a possible expression for the density operator of the fermionic harmonic oscillator. In terms of the Grassmann variables, the fermionic density operator can be written as: ρF(β)=c∗(β)c(β)±c∗(β)c(β)e−βωρ_F (β)=c^* (β)c(β) \pm c^*(β)c(β)e^{-βω}ρF​(β)=c∗(β)c(β)±c∗(β)c(β)e−βω, where +(-) means that the sum over all antiperiodic (periodic) orbits. Our density operator is then used to obtain the usual fermionic partition function which describes the fermionic oscillator in thermal equilibrium. Also, according to the periodic orbit c(β)=c(0)c(β)=c(0)c(β)=c(0), the graded fermionic partition function is obtained.