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arXiv:chao-dyn/9609002 · Journal of Physics A Mathematical and General

Eigenstate structures around a hyperbolic point

Stéphane Nonnenmacher, André Voros

Published Jan 7, 1997• 44 citations
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Abstract

Using coherent-state representations of quantum mechanics (Bargmann, Husimi, and stellar representations), we describe analytically the phase-space structure of the general eigenstates corresponding to a one-dimensional bilinear hyperbolic Hamiltonian, H = pq or equivalently . Their semiclassical behaviour is discussed for eigenvalues either near or away from the separatrix energy , especially in the phase-space vicinity of the saddle-point (q,p) = (0,0).

DOI
10.1088/0305-4470/30/1/021

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Using coherent-state representations of quantum mechanics (Bargmann, Husimi, and stellar representations), we describe analytically the phase-space structure of the general eigenstates corresponding to a one-dimensional bilinear hyperbolic Hamiltonian, H = pq or equivalently . Their semiclassical behaviour is discussed for eigenvalues either near or away from the separatrix energy , especially in the phase-space vicinity of the saddle-point (q,p) = (0,0).
We discuss some basic tools for an analysis of one-dimensional quantum systems defined on a cyclic coordinate space. The basic features of the generalized coherent states, the complexifier coherent states, are reviewed. These states are then used to define the corresponding (quasi)densities in phase space. The properties of these generalized Husimi distributions are discussed, in particular their zeros. Furthermore, the use of the complexifier coherent states for a semiclassical analysis is demonstrated by deriving a semiclassical coherent state propagator in phase space.
Just as a coherent state may be considered as a quantum point, its restriction to a factor space of the full Hilbert space can be interpreted as a quantum plane. The overlap of such a factor coherent state with a full pure state is akin to a quantum section. It defines a reduced pure state in the cofactor Hilbert space. Physically, this factorization corresponds to the description of interacting components of a quantum system with many degrees of freedom and the sections could be generated by conceivable partial measurements. The collection of all the Wigner functions corresponding to a full set of parallel quantum sections defines the Husimi–Wigner representation. It occupies an intermediate ground between the drastic suppression of non-classical features, characteristic of Husimi functions, and the daunting complexity of higher dimensional Wigner functions. After analysing these features for simpler states, we exploit this new representation as a probe of numerically computed eigenstates of a chaotic Hamiltonian. Though less regular, the individual two-dimensional Wigner functions resemble those of semiclassically quantized states.
Getting coherent state superpositions to stay put in phase space: QQQ functions and one dimensional integral representations of generator eigenstates
We study quantum mechanics in the phase space associated with the coherent state (CS) manifold of Lie groups. Eigenstates of generators of the group are constructed as one dimensional integral superpositions of CS along their orbits. We distinguish certain privileged orbits where the superposition is in phase. Interestingly, for closed in phase orbits, the geometric phase must be quantized to 2πZ2π\mathbb{Z}2πZ, else the superposition vanishes. This corresponds to exact Bohr-Sommerfeld quantization. The maximum of the Husimi-Kano QQQ quasiprobability distribution is used to diagnose where in phase space the eigenstates of the generators lie. The QQQ function of a generator eigenstate is constant along each orbit. We conjecture that the maximum of the QQQ function corresponds to these privileged in phase orbits. We provide some intuition for this proposition using interference in phase space, and then demonstrate it for canonical CS (H4H_4H4​ oscillator group), spin CS (SU(2)SU(2)SU(2)) and SU(1,1)SU(1,1)SU(1,1) CS, relevant to squeezing.
Husimi functions for eigen states around a hyperbolic point are studied. Husimi functions for eigen states of wave forms with even and odd parity are strictly concerned with the hyperbolic point and the separatrix in classical limit (ℏ→0\hbar{\rightarrow}0ℏ→0), respectively. It is interesting that classical behavior is affected by quantum symmetry.
The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phase space into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schrödinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the Berry-Tabor trace formula, resulting in a summation over non-periodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the The semiclassical quantization of a Hamiltonian system is deeply connected to the structure of its phase space. The generic Hamiltonian system contains a complicated mixture of near-integrable and