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Physical Review A

Classical dynamics for a class of SU(1,1) Hamiltonians

Zbigniew Wojciech Gortel, Łukasz A. Turski

Published Apr 1, 1991• 9 citations
Publisher

Abstract

Following a recent revival of interest in the properties of SU(1,1) Hamiltonians, we discuss the properties of classical-limit dynamics for a class of such models. Using a coherent-states technique, we analyze the classical energy function determining the motion and discuss the existence of stable and unstable stationary solutions. Appropriate phase diagrams are also presented. Comparison with previous works on similar models is made.

DOI
10.1103/physreva.43.3221

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Following a recent revival of interest in the properties of SU(1,1) Hamiltonians, we discuss the properties of classical-limit dynamics for a class of such models. Using a coherent-states technique, we analyze the classical energy function determining the motion and discuss the existence of stable and unstable stationary solutions. Appropriate phase diagrams are also presented. Comparison with previous works on similar models is made.
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We introduce a classical limit of the dynamics of quantum spin systems based on coherent states of SU(N)\mathrm{SU}(N)SU(N), where NNN is the dimension of the local Hilbert space. This approach, which generalizes the well-known Landau-Lifshitz dynamics from SU(2) to SU(N)\mathrm{SU}(N)SU(N), provides a better approximation to the exact quantum dynamics for a large class of realistic spin Hamiltonians, including $S\ensuremath{\ge}1$ systems with large single-ion anisotropy and weakly coupled multispin units, such as dimers or trimers. We illustrate this idea by comparing the spin structure factors of a single-ion S=1S=1S=1 model that are obtained with the SU(2) and SU(3) classical spin dynamics against the exact solution.
We analyze the semiclassical limit of the stationary Schr\"odinger equation in the coherent-state representation simultaneously for the groups W1{W}_{1}W1​, SU(2), and SU(1,1). A simple expression for the first two orders for the wave function and the associated semiclassical quantization rule is obtained if a definite choice for the classical Hamiltonian and expansion parameter is made. The behavior of the modulus of the wave function, which is a distribution function in a curved phase space, is studied for the three groups. The results are applied to the quantum triaxial rotor.
We discuss the classical and quantum mechanical evolution of systems described by a Hamiltonian that is a function of a solvable one, both classically and quantum mechanically. The case in which the solvable Hamiltonian corresponds to the harmonic oscillator is emphasized. We show that, in spite of the similarities at the classical level, the quantum evolution is very different. In particular, this difference is important in constructing coherent states, which is impossible in most cases. The class of Hamiltonians we consider is interesting due to its pedagogical value and its applicability to some open research problems in quantum optics and quantum gravity.