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Journal of Physics A Mathematical and General

State reconstruction of wave packets moving in time-dependent potentials and the existence of Wronskian pairs

Ulf Leonhardt, T. Kiss, P. J. Bardroff

Published Jan 1, 1999• 1 citation
Publisher

Abstract

We determine a class of time-dependent potentials that support the state reconstruction of one-dimensional wave packets. For this we extend the projection method (Leonhardt U and Raymer M G 1996 Phys. Rev. Lett. 76 1985) to explicitly time-dependent situations. We require the existence of Wronskian pairs (regular and irregular wavefunctions) and analyse the consequences on the possible potentials.

DOI
10.1088/0305-4470/32/2/014

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We determine a class of time-dependent potentials that support the state reconstruction of one-dimensional wave packets. For this we extend the projection method (Leonhardt U and Raymer M G 1996 Phys. Rev. Lett. 76 1985) to explicitly time-dependent situations. We require the existence of Wronskian pairs (regular and irregular wavefunctions) and analyse the consequences on the possible potentials.
We show how to reconstruct the quantum state of one-dimensional wave packets in the continuous part of the spectrum. We assume that the position probability distribution is measured for a sufficiently long time interval and that the potential is known, but otherwise arbitrary. Our paper fills the gap between the tomographic state reconstruction based on free evolution [J. Bertrand and P. Bertrand, Found. Phys. 17, 397 (1987)] and the state determination of moving bound states [U. Leonhardt and M. G. Raymer, Phys. Rev. Lett. 76, 1985 (1996)]. Our result may be relevant with respect to the state determination of atomic beams [Ch. Kurtsiefer, T. Pfau, and J. Mlynek, Nature (London) 386, 150 (1997)] moving in arbitrary potentials.
We show how to infer the quantum state of a wave packet from position probability distributions measured during the packet's motion in an arbitrary potential. We assume a nonrelativistic one-dimensional or radial wave packet. Temporal Fourier transformation and spatial sampling with respect to a newly found set of functions project the density-matrix elements out of the probability distributions. The sampling functions are derivatives of products of regular and irregular wave functions. We note that the ability to infer quantum states in this way depends on the structure of the Schr\"odinger equation.
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 propose a simple and straightforward method based on Wronskians for the calculation of bound-state energies and wavefunctions of one-dimensional quantum-mechanical problems. We explicitly discuss the asymptotic behaviour of the wavefunction and show that the allowed energies make the divergent part vanish. As illustrative examples we consider an exactly solvable model, the Gaussian potential well, and a two-well potential proposed earlier for the interpretation of the infrared spectrum of ammonia.
We propose a simple and straightforward method based on Wronskians for the calculation of bound-state energies and wavefunctions of one-dimensional quantum-mechanical problems. We explicitly discuss the asymptotic behaviour of the wavefunction and show that the allowed energies make the divergent part vanish. As illustrative examples we consider an exactly solvable model,the Gaussian potential well, and a two-well potential proposed earlier for the interpretation of the infrared spectrum of ammonia.
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.