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Bulletin of the American Physical Society

Semiclassical approximation for the average level density of a system of identical interacting particles

H.Michael Sommermann, H.A. Weidenbueller

Published Oct 1, 1993• 0 citations
OpenAlex

Abstract

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.

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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.
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 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.
Semiclassical periodic orbit theory is used in many branches of physics. However, most applications of the theory have been to systems that involve only single-particle dynamics. In this paper, we develop a semiclassical formalism to describe the density-of-states for two noninteracting particles. This includes accounting properly for particle exchange symmetry. As specific examples, we study two identical particles in a disk and in a cardioid. In each case, we demonstrate that the semiclassical formalism correctly reproduces the quantum densities.
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.
The Wigner–Weyl quantum-to-classical correspondence rule is nonunique with respect to coordinate choice. This ambiguity can be exploited to improve the accuracy of semiclassical approximations. For instance, the well-known Langer modification was recently derived by applying a coordinate transformation to the radial Schrödinger equation prior to using the Wigner–Weyl rule—albeit only by presuming exact quantum solutions for all nonradial degrees of freedom [J. J. Morehead, J. Math. Phys. 36, 5431 (1995)]. In this paper, the full classical Hamiltonian is derived in all degrees of freedom, using a (hyper)spherical coordinate Wigner–Weyl correspondence with a Langer-like modification of polar angles. For central force Hamiltonians, the new result is radially equivalent to that of Langer, and to the standard Cartesian form. The new correspondence is superior with respect to all angular momentum operators however, in that the resultant semiclassical eigenvalues are exact—a desirable goal, evidently achieved here for the first time.