Quantum Dynamics Near a Classical Separatrix
The Schrodinger equation with a slowly time-dependent, double-well potential is investigated both analytically and numerically in a range of parameters for which the equivalent classical system exhibits deterministic chaos. We consider the case where the number of quantum states with energy smaller than the classical separatrix energy is very large, and we study the quantum evolution in the vicinity of the separatrix. The WKB approximation is used to find the instantaneous eigenfunctions and the equation for the quantum energies. For energy values near the separatrix it is found that the minimum separation vanishes only logarithmically with Planck's constant. Thus, we argue that quantum behavior can be observed in a system with a double-well potential, e.g. in a modified Penning trap. Via the rotating basis representation the dynamical problem is reduced to the integration of a finite system of ordinary differential equations. The coefficients of this system are derived from the WKB function and energies. By asymptotic analysis the spread of quantum numbers in the final state is found to be exponentially small in the adiabatic parameter epsilon. This contrasts with the classical situation where the equivalent spread decays linearly with epsilon. Hence, for a physical system with a double-well potential, quantum effects can also be detected by measuring the quantum number-spreading induced by processes that involve crossing of the classical separatrix. Finally, the system of differential equations is integrated numerically. This numerical work shows that the asymptotic analysis is valid provided epsilon is smaller than the inverse of the number of quantum states with energy smaller than the separatrix energy.
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- Physics: General; Mathematics