Revivals and classical -motion bases of quantum wave packets
This thesis explores the boundary between classical and quantum mechanics by studying wave packets, coherent superpositions of the stationary states of a quantum system. Such wave packets travel as localized entities along the trajectories predicted by classical mechanics for small windows of time before they spread out and decay away. Our investigations focus on two central issues—the revivals of the shape and classical motion of these wave packets that occur long after their initial decay, and the classical-motion bases that describe the quantum wavefunction in terms of constitutive objects that move classically.
We study the infinite square-well potential, a simple model of complete confinement in a one-dimensional interval. The quantum motion seen in this potential is compared with classical models of a particle bouncing between two walls and of a wave traveling along a stretched string with both ends secured. We uncover a remarkable wave-motion basis, with which the wavefunction at any moment in time can be decomposed into a sum of distinct wave propagations of the initial quantum wavefunction in the classical wave equation. These results are extended to the finite square-well potential and we show how the wave-motion basis can be reconciled with the seemingly disparate theory of revivals for highly excited quantum wave packets.
We explore the commonalities of the quantum revivals seen in a wide variety of systems by developing a mathematical formalism called phase-difference equations. These equations connect physical models for revivals with the subsequent prediction of revival times in a general way and offer a comprehensive “calculus” for understanding revival phenomena. We apply this calculus to several examples to demonstrate its power and versatility.
Using a recently developed semiclassical basis for quantum states, we explore the radial wave packets of the hydrogen atom. Viewed in the semiclassical basis, the revivals of these wave packets are shown to arise from constructive interference among groups of classical particles on neighboring tracks of a Lagrange manifold in phase space, in a way that parallels the interference among discrete stationary states found in the standard view of revivals.