Construction of accelerating wavepackets

Abstract

Accelerating wavepackets are obtained by applying the extended Galilean transformation to free-space particle wavepackets, or to time-dependent bound states. The transformed Schrödinger equation contains a potential term, which drives the acceleration. The method reproduces Darwin’s result for an accelerating Gaussian wavepacket in a constant force field, and Schrödinger’s oscillating wavepacket in a harmonic well. We also derive a new accelerating Airy wavepacket in a gravitational field, and give a new derivation of oscillating wavepackets based on the higher harmonic oscillator energy eigenstates. These results illustrate an analytic method of dealing with wavepackets accelerating in force fields.

Introduction

There is a wide variety of uses for particle wavepackets in quantum physics. Where explicit calculations are made, they are usually numerical (for an example of the multi-configuration time-dependent Hartree–Fock method, see [1] and the references therein). Here we wish to put forward a set of relatively simple analytical results for wavepackets in linear and quadratic potentials, which illustrate the application of a general method. Specifically: given any form of free-space wavepacket, we show how to construct a corresponding wavepacket which is accelerating under the influence of a uniform field of force. And, from any energy eigenstate of the harmonic oscillator, we show how to construct a corresponding wavepacket which moves under the influence of a harmonic force. In all cases the resulting accelerating wavepackets satisfy Schrödinger’s time-dependent equation exactly.

The exact energy eigenfunctions of the time-independent Schrödinger equation for a particle in a constant force field and in a harmonic well are widely-known (see for example [2]). However, constructing a localized wavepacket for these potentials by summing or integrating over all the eigenfunctions is more complicated. Schrödinger produced a wavepacket which oscillates (without change of envelope) in the harmonic potential well, the first coherent state [3]. Darwin gave a wavepacket solution of the time-dependent Schrödinger equation for an electron in a homogeneous electric field [4]. More recently, analytic solutions have been given for wavepackets moving in potentials proportional to sech²(x), 1/x² and a delta function [5], [6], [7]. Another coherent state has been found for the quantum bouncer on a spring [8], although this one changes shape periodically during its oscillation.

This note demonstrates how the extended Galilean transformation [9], [10], [11], [12], [13], [14] allows us to find solutions to the time-dependent Schrödinger equation for a particle in a force field from wavepacket solutions for the free particle, or from time-dependent energy eigenstates of the harmonic oscillator. The method is restricted to non-relativistic quantum mechanics [14].

We begin with a brief description of the extended Galilean transformation. For the usual Galilean transformation, x′ = x + ut, t′ = t, u being constant. This is generalized to $x^{\prime }=x+{\int }_0^t{\mathit{dt}}_{1}u(t_1)\text{,}{t}^{\prime }=t$: the frames are in relative motion with a variable velocity u = u(t). The derivatives in the two frames are related by

$${\mathrm{\partial }}_x={\mathrm{\partial }}_{x^{\prime }}\text{,}{\mathrm{\partial }}_t={\mathrm{\partial }}_{t^{\prime }}+u(t){\mathrm{\partial }}_{x^{\prime }}\text{.}$$

Suppose that in the frame (x, t) we have ψ(x, t) as the solution to the time-dependent Schrödinger equation:

$$i\hslash {\mathrm{\partial }}_t\psi =\left(-\frac{{\hslash }^2}{2m}{\mathrm{\partial }}_x^2+V(x)\right)\psi \text{.}$$

In a new frame (x′, t′) the Schrödinger equation becomes

$$i\hslash {\mathrm{\partial }}_{t^{\prime }}\psi =\left(-\frac{{\hslash }^2}{2m}{\mathrm{\partial }}_{x^{\prime }}^2-i\hslash u{\mathrm{\partial }}_{x^{\prime }}+V(x)\right)\psi \text{.}$$

We now choose a wavefunction ψ′(x′, t′) in the primed frame such that

$$\psi (x\text{,}t)={\psi }^{\prime }\left(x^{\prime }\text{,}{t}^{\prime }\right)\exp \left[{\mathit{if}}^{\prime }\left(x^{\prime }\text{,}{t}^{\prime }\right)\right]\text{.}$$

(The probability densities are the same at corresponding space–time points in the two frames, provided f′ is real.) Then

$$\begin{array}{cc}\hfill & {\mathrm{\partial }}_{x^{\prime }}\psi =({\mathrm{\partial }}_{x^{\prime }}{\psi }^{\prime }+i{\psi }^{\prime }{\mathrm{\partial }}_{x^{\prime }}{f}^{\prime })\exp \left[{\mathit{if}}^{\prime }\left(x^{\prime }\text{,}{t}^{\prime }\right)\right]\hfill \\ \hfill & {\mathrm{\partial }}_{t^{\prime }}\psi =({\mathrm{\partial }}_{t^{\prime }}{\psi }^{\prime }+i{\psi }^{\prime }{\mathrm{\partial }}_{t^{\prime }}{f}^{\prime })\exp \left[{\mathit{if}}^{\prime }\left(x^{\prime }\text{,}{t}^{\prime }\right)\right]\text{.}\hfill \end{array}$$

Applying (4), (5) to (3), and factoring out exp[if′ (x′, t′)], we obtain a new Schrödinger equation for ψ′(x′, t′) in the primed frame. Now f′(x′, t′) is an arbitrary phase, and by choosing a suitable function for f′, we can construct a solution ψ′(x′, t′) of the transformed Schrödinger equation with a new potential. We will apply this method to two special cases of interest: a wavepacket accelerating in a gravitational field, and an oscillating wave packet in a harmonic well.