# Parametric dependent Hamiltonians, wave functions, random matrix theory, and quantal-classical correspondence Academic Article

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### abstract

• We study a classically chaotic system that is described by a Hamiltonian $\mathcal{H}(Q,P;x),$ where $(Q,P)$ are the canonical coordinates of a particle in a two-dimensional 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 $P(n|m)=|〈n(x)|{m(x}_{0})〉{|}^{2}$ evolves as a function of $\ensuremath{\delta}x\ensuremath{\equiv}(x\ensuremath{-}{x}_{0}).$ This kernel, regarded as a function of $n\ensuremath{-}m,$ characterizes the shape of the wave functions, and it also can be interpreted as the local density of states. The kernel $P(n|m)$ has a well-defined classical limit, and the study addresses the issue of quantum-classical correspondence. Both the perturbative and the nonperturbative regimes are explored. The limitations of the random matrix theory approach are demonstrated.

### publication date

• January 1, 2001