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American Journal of Physics

Moments of a Wave Packet

Leemon C. Baird

Published Feb 1, 1972• 24 citations
Publisher

Abstract

The properties of a wave packet may be elucidated by a consideration of the statistical moments of the packet. The time evolution of the wave packet has an especially simple description in terms of moments.
DOI
10.1119/1.1986518

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We study some properties of a class of quantum wave packets with power-law spatial tails analitically. Almost all packets of this class have the property that all the moments of the position operator are infinite. We prove that the free evolution of these packets presents an asymptotic decay of the wave packet maximum which is anomalous with respect to the behavior observed in the free evolution of customary quantum wave packets. We also study the temporal evolution of the tails and we show that the dominant term of the asymptotic expansion is conserved.
The motion of wave packets can be easily determined for any Hamiltonian that is quadratic in position and momentum, even if the coefficients of the terms in the Hamiltonian vary with time. The method is based on the existence of an invariant operator, linear in position and momentum, the coefficients of these operators being solutions of the corresponding classical system. This immediately yields a set of very simple wave packets whose evolution is easily determined, and whose magnitude has the same form as the energy eigenfunctions of the harmonic oscillator (i.e., Gaussian or Hermite–Gaussian). This set provides a complete basis for finding the evolution of any state and the exact propagator is readily determined. For the harmonic oscillator, this set includes coherent states, squeezed states, displaced number states, and squeezed number states as special cases. The following important properties hold for every quadratic Hamiltonian, no matter what time dependence is present in the Hamiltonian: (1) The motion of the centroid of any wave packet separates from that of any moments relative to the centroid. (2) Every detail of the evolution of the quantum system can be calculated from the solutions of the corresponding classical system. (3) Wave functions of Gaussian (or Hermite–Gaussian) form will retain that form as they evolve.
We investigate the quantum dynamics of a particle in a one-dimensional parabolic potential subject to both timeperiodic driving and stochastic perturbations modeled as white noise. Using the non-stationary Schrödinger equation and the Green’s function formalism, we derive analytical expressions for the time evolution of the wave function and its statistical moments. It is shown that, due to the quadratic nature of the potential, the wave function retains a Gaussian form, allowing a reduced description in terms of its first and second moments. Under resonant periodic driving, the mean energy exhibits quadratic growth in time. In the presence of white noise, the wave packet undergoes diffusive spreading and loss of coherence. For combined deterministic and stochastic driving, we identify a characteristic decoherence (delocalization) time, which scales inversely with the noise intensity. The results are relevant to quantum transport, driven systems, and tunneling processes in fluctuating environments.