Parametric dependent Hamiltonians, wavefunctions, random-matrix-theory, and quantal-classical correspondence

Abstract

We study a classically chaotic system which is described by a Hamiltonian $H(Q,P;x)$ where $(Q,P)$ are the canonical coordinates of a particle in a 2D well, and $x$ is a parameter. By changing $x$ we can deform the `shape' of the well. The quantum-eigenstates of the system are $|n(x)>$. We analyze numerically how the parametric kernel $P(n|m)=|<n(x)|m(x0)>{|}^2$ evolves as a function of $x-x0$. This kernel, regarded as a function of $n-m$, characterizes the shape of the wavefunctions, and it also can be interpreted as the local density of states (LDOS). The kernel $P(n|m)$ has a well defined classical limit, and the study addresses the issue of quantum-classical correspondence (QCC). We distinguish between restricted QCC and detailed QCC. Both the perturbative and the non-perturbative regimes are explored. The limitations of the random-matrix-theory (RMT) approach are demonstrated.

7 pages, 5 figures, long detailed version

References in corpus (7)

• Chaos and energy spreading for time-Dependent Hamiltonians, and the various Regimes in the theory of Quantum Dissipation
• Quantum dissipation due to the interaction with chaotic degrees-of-freedom and the correspondence principle
• Quantum-Classical Correspondence in Energy Space: Two Interacting Spin-Particles
• Unification of perturbation theory, RMT and semiclassical considerations in the study of parametrically-dependent eigenstates
• Wavepacket dynamics in energy space, RMT and quantum-classical correspondence
• Semiclassical properties of eigenfunctions and occupation number distribution for a model of two interacting particles
• Parametric Evolution for a Deformed Cavity

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