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American Journal of Physics

Almost-periodic wave packets and wave packets of invariant shape

Detlef Mentrup, Marshall Luban

Published May 12, 2003• 3 citations
Publisher

Abstract

Wave packets of one-dimensional, single-particle quantum systems exhibit a rich diversity of dynamical behavior. We focus on confining potentials where the entire energy spectrum is discrete so that wave packets do not spread indefinitely with time and instead exhibit characteristics of almost-periodic functions. For the harmonic oscillator, for which all wave packets are necessarily strictly periodic, Senitzky has shown that is possible to construct an infinite class of normalized wave packets that maintain a fixed shape while oscillating sinusoidally with time. We present two alternate and instructive methods for deriving this result using coherent states and the displacement operator.

DOI
10.1119/1.1555872

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Wave packets of one-dimensional, single-particle quantum systems exhibit a rich diversity of dynamical behavior. We focus on confining potentials where the entire energy spectrum is discrete so that wave packets do not spread indefinitely with time and instead exhibit characteristics of almost-periodic functions. For the harmonic oscillator, for which all wave packets are necessarily strictly periodic, Senitzky has shown that is possible to construct an infinite class of normalized wave packets that maintain a fixed shape while oscillating sinusoidally with time. We present two alternate and instructive methods for deriving this result using coherent states and the displacement operator.
We discuss nonspreading wave packets in one dimensional Schrödinger equation. We derive general rules for constructing nonspreading wave packets from a general potential V(x,t)\textmd{V}(x,t)V(x,t). The essential ingredients of a nonspreading wave packet, the shape function f(x)f(x)f(x), the motion d(t)d(t)d(t), the phase function ϕ(x,t)ϕ(x,t)ϕ(x,t) are derived. Since the form of the shape of a nonspreading wave packet does not change in time, the shape equation should be time independent. We show that the shape function f(x)f(x)f(x) is the eigenfunction of the time independent Schrödinger equation with an effective potential VeffV_{\textmd{eff}}Veff​ and an energy EeffE_{\textmd{eff}}Eeff​. We derive nonspreading wave packets found by Schrödinger, Senitzky, and Berry and Balazs as examples. We show that most stationary potentials can only support stationary nonspreading wave packets. We show how to construct moving nonspreading wave packets from time dependent potentials, which drive nonspreading wave packets into an arbitrary motion.
We derive, with an invariant operator method and unitary transformation approach, that the Schrödinger equation with a time-dependent linear potential possesses an infinite string of shape-preseving wave-packet states |ϕα,λ( t )⟩ having classical motion. The qualitative properties of the invariant eigenvalue spectrum (discrete or continuous) are described separately for the different values of the frequency ω of a harmonic oscillator. It is also shown that, for a discrete eigenvalue spectrum, the states |ϕα, n ( t )⟩ could be obtained from the coherent state |ϕα,0( t )⟩.
We show that one can obtain analytic solutions of the time-dependent Schrödinger equation that are more complex than the well-known oscillating coherent wave packet. Such Hermite-Gaussian or initially square wave packets exist for a free particle or for one subject to the harmonic oscillator potential. In either case, the Hermite-Gaussian packets retain their nodal structure even after long times. There is a single class of exact solutions for the system with oscillator constant K > 0, K = 0, or K < 0, leading to wave functions for the harmonic oscillator, the free particle, and the inverted oscillator, respectively.
Coherent states play an important role in quantum mechanics because of their unique properties under time evolution. Here we explore this concept for one-dimensional repulsive nonlinear Schr\"odinger equations, which describe weakly interacting Bose-Einstein condensates or light propagation in a nonlinear medium. It is shown that the dynamics of phase-space translations of the ground state of a harmonic potential is quite simple: The center follows a classical trajectory whereas its shape does not vary in time. The parabolic potential is the only one that satisfies this property. We study the time evolution of these nonlinear coherent states under perturbations of their shape or of the confining potential. A rich variety of effects emerges. In particular, in the presence of anharmonicities, we observe that the packet splits into two distinct components. A fraction of the wavepacket is transferred toward incoherent high-energy modes, while the amplitude of oscillation of the remaining coherent component is damped toward the bottom of the well.