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Physics Letters B

TDHF equations of motion for algebraic hamiltonians in a coherent state representation

D.H. Feng, Robert Gilmore

Published Mar 1, 1980• 33 citations
Publisher
DOI
10.1016/0370-2693(80)90940-5

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arXiv:1107.0856•Jul 5, 2011•1 citation
We investigate the validity of the time-dependent Hartree-Fock (TDHF) approximation for a finite fermion model system. With the help of coherent states we derive equations of motion for the one-particle density, which are of classical Hamiltonian form and can be interpreted in terms of generalized coordinates and momenta. The corresponding trajectories in phase space show transitions between regular and chaotic regimes depending on the parameters of the model. A chaotic behavior indicates a failure of the TDHF approximation because the exact solutions are always regular due to the linearity of the Schr\"odinger equation.
The Wigner problem, i.e. the investigation of general quantum mechanical commutation relations consistent with the Heisenberg evolution equations of a given shape, is studied. We follow a recently proposed generalization of this idea within which the classical analogy is postulated only for the shape of the time evolution equations and not for a Hamiltonian itself. This links our investigation to the problem of alternative Hamiltonians of classical mechanics and to canonically inequivalent phase-space descriptions of physical systems governed by the same Newton equations of motion. Warned that the time evolution and the other symmetry generators may be given ambiguously even in the formalism of classical mechanics, we do not a priori assume the shape of their quantum analogues. Instead we only require that the set of basic algebraic relations, which quantum mechanical observables are to obey, has a Lie algebra structure. Such a requirement appears to be sufficient to find solutions for simple oscillator-like dynamics. New algebras of quantum mechanical observables are not constructed as a linear envelope of the Heisenberg algebra, and their representations reflect physical results unexpected in the framework of the canonical approach. We illustrate our approach in detail for the example of the one-dimensional harmonic oscillator using the representation of the generalized coherent states.
Spectral properties of the Hamiltonian function which characterizes a trapped ion are investigated. In order to study semiclassical dynamics of trapped ions, coherent state orbits are introduced as sub-manifolds of the quantum state space, with the Kähler structure induced by the transition probability. The time dependent variational principle (TDVP) is applied on coherent states' orbits. The Hamilton equations of motion on Kähler manifolds of the type of classical phase spaces naturally arise. The associated classical Hamiltonian is obtained from the expected values on symplectic coherent states of the quantum Hamiltonian. Spectral information is thus coded within the phase portrait. We deal with the bosonic realization of the Lie algebra of the SU(1,1) group, which we particularize for the case of an ion confined in a combined, Paul and Penning trap. This formalism can be applied to Hamiltonians which are nonlinear in the infinitesimal generators of a dynamical symmetry group, such as the case of ions confined in electrodynamic traps. Discrete quasienergy spectra are obtained and the corresponding quasienergy states are explicitly realized as coherent states parameterized by the stable solutions of the corresponding classical equations of motion. A correspondence between quantum and classical stability domains is thus established, using the Husimi representation.