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Europhysics Letters (EPL)

Semi-Classical Approximation for the Average Level Density of a System of Identical Interacting Particles

H.Michael Sommermann, Hans A. Weidenmüller

Published Jul 10, 1993• 15 citations
Publisher

Abstract

For a system of identical particles (we take fermions as an example), we derive the analogue of the Weyl formula for the average level density. This is done by introducing the projection operator onto states in Hilbert space which have the symmetry imposed by particle identity. The semi-classical result is obtained by using the Wigner representation for this operator and by letting ℏ → 0. Numerical examples for few degrees of freedom show in general good agreement with exact diagonalization even near the ground state.

DOI
10.1209/0295-5075/23/2/001

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For a system of identical particles (we take fermions as an example), we derive the analogue of the Weyl formula for the average level density. This is done by introducing the projection operator onto states in Hilbert space which have the symmetry imposed by particle identity. The semi-classical result is obtained by using the Wigner representation for this operator and by letting ℏ → 0. Numerical examples for few degrees of freedom show in general good agreement with exact diagonalization even near the ground state.
We derive the analogue of the Weyl formula for the average level density, in the case of a system of identical particles (taking fermions as an example). This is accomplished by introducing the projection operator onto states in Hilbert space which have the symmetry imposed by particle identity. The semiclassical result is obtained by using the Wigner representation for this operator and by letting h {r_arrow} 0. Numerical examples for few degrees of freedom show in general good agreement with exact diagonalization even near the ground state of the system.
We present a novel analytical approach for the calculation of the mean density of states in many-body systems consisting of confined indistinguishable and independent particles. Our method makes explicit the intrinsic geometry inherent in the symmetrization postulate. In the spirit of the usual Weyl expansion for the smooth part of the density of states in confined single-particle systems, our results take the form of a sum over clusters of particles moving freely around manifolds in configuration space invariant under permutations. In our approach the emergence of the fermionic ground state is a consequence of a delicate cancellation effect of cluster contributions. As an asymptotic expansion, our approximation gives increasingly better results for large excitation energies, and we show that it coincides with the Bethe estimate in the appropriate region. Moreover, our construction gives the correct high-energy asymptotics expected from general considerations. Our expansion in cluster zones is naturally incorporated for systems of interacting particles, opening an alternative road to address the interplay between symmetry, confinement and interactions in many-body systems of identical bosonic or fermionic particles.
We investigate indistinguishability of identical bosons and fermions undergoing arbitrary particle-number-preserving evolutions of their visible degrees of freedom. For the projective indistinguishability measure, defined by the projection of the visible state onto the symmetric/anti-symmetric subspace, we derive an equivalent expression in terms of the dynamically invariant internal state. We further generalize the textbook symmetrization/anti-symmetrization framework for bosons and fermions to arbitrary partial distinguishability by deriving an explicit reconstruction formula for the multiparticle visible state in terms of the indistinguishability function encoding the dynamical invariants. We give complete characterization of the class-functions of indistinguishability by projective measures on generalized symmetries. Finally, we reveal a strikingly counterintuitive effect: introducing additional particle label states can increase the multiparticle indistinguishability of identical particles. The effect originates from the cancellation of collective multiparticle phases.
It is well known from the work of Berezin [24] (Commun. Math. Phys. 40:153–174, 1975) in 1975 that the quantization problem of a classical mechanical system is closely related with coherent states. In particular coherent states help to understand the limiting behaviour of a quantum system when the Planck constant ℏ\hbar ℏ becomes negligible in macroscopic units. This problem is called the semi-classical limit problem. In this chapter we discuss properties of quantum systems when the configuration space is the Euclidean space Rn\mathbb R^nRn. So that in the Hamiltonian formalism, the phase space is Rn×Rn\mathbb R^n\times \mathbb R^nRn×Rn with its canonical symplectic form σ\sigma σ. The quantization problem has many solutions, so we choose a convient one, introduced by Weyl [259] (The Classical Groups, 1997) and Wigner [261] (Group Theory and Its Applications to Quantum Mechanics of Atomic Spectra, 1959). We study the symmetries of Weyl quantization, the operationnal calculus and applications to propagation of observables. We show that Wick quantization is a natural bridge between Weyl quantization and coherent states. Applications are given to the semi-classical limit after introducing an efficient modern tool: the semi-classical measures. We illustrate the general results proved in this chapter by explicit computations for the harmonic oscillator. More applications will be given in the following chapters, in particular concerning propagators and trace formulas for a large class of quantum systems.
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.