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Theory for matrix elements of one-body transition operators in the quantum chaotic domain of interacting particle systems

V. K. B. Kota, R. Sahu

Published Dec 9, 2010• 0 citations
Publisher
DOI
10.1142/9789814287395_0040

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Theory for matrix elements of one-body transition operators in the quantum chaotic domain of interacting particle systems

V. K. B. Kota, R. Sahu

Demonstrating the equivalence between the recent theory of Flambaum and collaborators which is based on smoothed strength functions, with the much earlier formulation due to French and collaborators which is based on embedded random matrix ensembles and smoothed transition strength densities, we derive a theory for matrix elements of one-body transition operators in the quantum chaotic domain of isolated finite interacting particle systems with a mean-field and a chaos generating two-body interaction (V). The role of the bivariate correlation coefficient (zeta) arising out of the noncommutability of V and the transition operator (in the theory of Flambaum et al., zeta=0) is tested in numerical embedded ensemble calculations with a one- plus two-body Hamiltonian generating order-chaos transitions.

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There is a newly emerging understanding that in the chaotic domain of isolated finite interacting many particle systems smoothed densities define the statistical description of these systems and these densities follow from embedded (two-body) random matrix ensembles and their various deformations. These ensembles predict that the smoothed form of matrix elements of a transition operator between the chaotic eigenstates weighted by the densities at the two ends (i.e. the bivariate strength density) will be a bivariate Gaussian with the bivariate correlation coefficient arising out of the non-commutability of the hamiltonian and the transition operator involved. The ensemble theory extends to systems with a mean-field and a chaos generating two-body interaction (as in nuclei, atoms and diffusive quantum dots). These developments in many-body quantum chaos are described with special reference to one-body transition operators.

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Interacting many-particle systems with a mean-field one-body part plus a chaos generating random two-body interaction having strength lambda exhibit Poisson to Gaussian orthogonal ensemble and Breit-Wigner (BW) to Gaussian transitions in level fluctuations and strength functions with transition points marked by lambda = lambda c and lambda = lambda F, respectively; lambda F >> lambda c. For these systems a theory for the matrix elements of one-body transition operators is available, as valid in the Gaussian domain, with lambda > lambda F, in terms of orbital occupation numbers, level densities, and an integral involving a bivariate Gaussian in the initial and final energies. Here we show that, using a bivariate-t distribution, the theory extends below from the Gaussian regime to the BW regime up to lambda = lambda c. This is well tested in numerical calculations for 6 spinless fermions in 12 single-particle states.

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Dynamics of Quantum Systems with Interacting Particles

Elisheva Elbaz

•Apr 1, 2016•0 citations
Demonstrating the equivalence between the recent theory of Flambaum and collaborators which is based on smoothed strength functions, with the much earlier formulation due to French and collaborators which is based on embedded random matrix ensembles and smoothed transition strength densities, we derive a theory for matrix elements of one-body transition operators in the quantum chaotic domain of isolated finite interacting particle systems with a mean-field and a chaos generating two-body interaction (V). The role of the bivariate correlation coefficient (zeta) arising out of the noncommutability of V and the transition operator (in the theory of Flambaum et al., zeta=0) is tested in numerical embedded ensemble calculations with a one- plus two-body Hamiltonian generating order-chaos transitions.
There is a newly emerging understanding that in the chaotic domain of isolated finite interacting many particle systems smoothed densities define the statistical description of these systems and these densities follow from embedded (two-body) random matrix ensembles and their various deformations. These ensembles predict that the smoothed form of matrix elements of a transition operator between the chaotic eigenstates weighted by the densities at the two ends (i.e. the bivariate strength density) will be a bivariate Gaussian with the bivariate correlation coefficient arising out of the non-commutability of the hamiltonian and the transition operator involved. The ensemble theory extends to systems with a mean-field and a chaos generating two-body interaction (as in nuclei, atoms and diffusive quantum dots). These developments in many-body quantum chaos are described with special reference to one-body transition operators.
Interacting many-particle systems with a mean-field one-body part plus a chaos generating random two-body interaction having strength lambda exhibit Poisson to Gaussian orthogonal ensemble and Breit-Wigner (BW) to Gaussian transitions in level fluctuations and strength functions with transition points marked by lambda = lambda c and lambda = lambda F, respectively; lambda F >> lambda c. For these systems a theory for the matrix elements of one-body transition operators is available, as valid in the Gaussian domain, with lambda > lambda F, in terms of orbital occupation numbers, level densities, and an integral involving a bivariate Gaussian in the initial and final energies. Here we show that, using a bivariate-t distribution, the theory extends below from the Gaussian regime to the BW regime up to lambda = lambda c. This is well tested in numerical calculations for 6 spinless fermions in 12 single-particle states.