Gaussian wave-packet dynamics: Semiquantal and semiclassical phase-space formalism

Abstract

Gaussian wavepackets are a popular tool for semiclassical analyses of classically chaotic systems. We demonstrate that they are extremely powerful in the semiquantal analysis of such systems, too, where their dynamics can be recast in an extended potential formulation. We develop Gaussian semiquantal dynamics to provide a phase space formalism and construct a propagator with desirable qualities. We qualitatively evaluate the behaviour of these semiquantal equations, and show that they reproduce the quantal behavior better than the standard Gaussian semiclassical dynamics. We also show that these semiclassical equations arise as truncations to semiquantal dynamics non-self-consistent inh. This enables us to introduce an extended semiclassical dynamics that retains the power of the Hamiltonian phase space formulation. Finally, we show how to obtain approximate eigenvalues and eigenfunctions in this formalism, and demonstrate with an example that this works well even for a classically strongly chaotic Hamiltonian.

Generalized quantum Fokker-Planck, diffusion, and Smoluchowski equations with true probability distribution functions

“…If one is interested in the local dynamics around a point (say, at the bottom or top of the potential well), the set of equations get decoupled and it is easy to obtain simple analytic solutions of (55a)-(55e) for x and δx 2 for (54a). The higher order estimates (e.g., fourth order) of the quantum corrections can be obtained from the solutions of the equations of successive higher order derived earlier by Sundaram and Milonni [43] or otherwise [44]. Since the quantum corrections due to the system are calculated by different sets of equations for succesive orders, the measure of accuracy of truncation can be understood easily.…”

Chaotic signatures in the spectrum of a quantum double well

“…This is also in agreement with the study of Pattanyak and Schieve [3] who have shown that, for a double well potential, squeezed coherent states show the most complex behavior for energies close to the separatrix. They argue [9] that the interplay between the classically unstable orbits and the quantum tunneling effects is the origin of this complex behavior. This is in agreement with our observations on the behavior of the energy level statistics in that regime.…”

Quantum phase-space function formulation of reactive flux theory

“…However for a practical calculation we need a recipe for calculation of Q(t). This has been discussed earlier in several contexts 24,25,26,27,39,40 . For the present purpose we summarize it as follows:…”

“…In Appendix A we have derived the equations for quantum corrections upto forth oder 39 . Under very special circumstances, it has been possible to include quantum effects to all orders 27,40 . The present theory thus takes into account of the anharmonicity as an integral part of the treatment.…”