Abstract The discussion in previous chapters has centred on the use of trajectorybased information to add quantum corrections to the classical picture, in a mainly time-independent picture. This chapter takes an explicitly timedependent standpoint by examining the relevance of various types of wavepacket to classical mechanics and vice versa; Littlejohn (1986) gives a much fuller mathematical account. The intuitive connection is obvious, in the sense that a unit normalized wavepacket corresponds most nearly to a single quantum mechanical particle, but the associated uncertainty relations imply a closer connection with a classical ensemble. Analysis of this connection is one of the main concerns of the first half of the chapter.
The time it takes a particle to tunnel through the asymmetric Eckart barrier potential is investigated using Gaussian wavepackets, where the barrier serves as a model for the potential along a chemical reaction coordinate. We have previously shown that the, in principle experimentally measurable, tunneling flight time, which determines the time taken by the transmitted particle to traverse the barrier, vanishes for symmetric potentials like the Eckart and square barrier [ Petersen , J. ; Pollak , E. J. Phys. Chem. Lett. 2017 , 9 , 4017 ]. Here we show that the same result is obtained for the asymmetric Eckart barrier potential, and therefore, the zero tunneling flight time seems to be a general result for one-dimensional time-independent potentials. The wavepacket dynamics is simulated using both an exact quantum mechanical method and a classical Wigner prescription. The excellent agreement between the two methods shows that quantum coherences are not important in pure one-dimensional tunneling and reinforces the conclusion that the tunneling flight time vanishes.
The dynamics of a quantum mechanical particle in a time-independent potential are found to contain many interesting phenomena. These are direct consequences of the (typical) existence of more than one time scale governing the problem. This gives rise to full revivals of initial wavepackets, fractional revivals (multiple wavepackets appearing at fractions of the revival time) and the striking quantum carpets. A variety of analytic techniques are used to consider the interference that gives rise to these phenomena while skirting calculations involving cross-terms. Novel results include a new theorem on the weighting coefficients
am that govern fractional revivals, a demonstration that
Ψcl, the function that governs the distribution and features of these fractional revivals, really does behave classically, a treatment of the wavepacket dephasing in the infinite square well by means of the Poisson summation formula, and a correct analysis of the spatial distribution of intermode traces. Also, this work presents a coherent treatment of these phenomena, which before now did not exist.