Equivalence between free and harmonically trapped quantum particles

arises as an approximation to Maxwell’s equations [3]. In this optical context the parameter α = 4π/λ is proportional to the (stationary, monochromatic beam’s) wave number or inversely proportional to the light’s wavelength λ.

The Schrödinger equation for a free particle in one spatial dimension x is given by

where the parameter α = 2M/ contains the particle mass M and Planck’s constant h, namely = h/(2π). It is obviously equivalent to eq. ( 1) above. Since the wave functions for free particles and those subjected to harmonic potentials factorize with respect to their spatial coordinates we will only discuss the onedimensional case.

It has recently been realized that a simple coordinate transformation [2] maps paraxial beams onto twodimensional harmonic oscillator wave functions. Clearly, the same coordinate transformation can be used for the mapping of a free onto a harmonically trapped particle; in a more general setting this was noted before [4,5]. This case has added meaning since a non-adiabatic physical transition constitutes an experimentally implementable sequence of changing environmental conditions. It turns out that modelling such a transition using the standard eigenstate-projection or quantum propagator techniques is more cumbersome. State projection typically leads to * Electronic address: O.Steuernagel@herts.ac.uk infinite sums over eigenfunctions which have to be truncated and are difficult to simplify, propagator techniques involve non-trivial integrals. The results reported here may therefore not only be of fundamental but also of technical interest.

It is noteworthy that the discontinuities of the mapping of the time-coordinate in eq. ( 3) maps from a Schrödinger equation with a continuous to another with a discrete energy spectrum.

The coordinate transformations [4] x(ξ, τ

applied to the wave function mapping

where

yield the solution ψ(ξ, τ ) for the Schrödinger equation

of a harmonic oscillator with mass M , spring constant k and resonance frequency ω = k/M . Although this transformation maps t ∈ [-∞, ∞] onto τ ∈ (-π/2, π/2), compare with the Gouy-phase of optics [2], the periodicity arising through the use of trigonometric functions meaningfully represents the oscillator’s motion for all times τ .

In beam optics the confocal parameter b parameterizes the strength of the beam’s focussing and the curvature of its hyperbolic flow lines [2]. Here, it serves as a rescaling parameter of the transverse coordinate transformation and thus allows us to stretch or squeeze the width of the wave packet we want to map.

In order to preserve the wave function normalization we have to determine the spatial coordinate stretching at the mapping time t 0 = τ 0 = 0. This yields the normalization factor N = (b ω) 1/4 which the wave function has to be multiplied with. The specific choice

yields the natural mapping x = ξ and N = 1 which we will use from now on.

The inverse to the coordinate transformations (3) are

and

and go together with the wave function multiplication φ = ψf , inverting eq. ( 4).

The concatenated coordinate transformations from an initial harmonic trapping potential with spring constant k and wave function ψ(ξ, τ ; k), via the free particlecase φ, to a final harmonic potential with spring constant K and wave function Ψ(ξ, τ ; k, K) is given by

and

.

Here Ω = K/M , and Ψ(ξ, τ ; k, K) solves Schrödinger eq. ( 6) with k substituted by K.

As expected for a map from one harmonic oscillator to another, the inverse of the coordinate transformations (10) and ( 11) are given by the same functional expressions with the quantities pertaining to one potential swapped with those of the other (k ↔ K and ω ↔ Ω). FIG. 1: Probability density P (x, t) ∼ |φ0(x, t; 0, p0) + φ0(x, t; 0, -p0)| 2 of a free quantum particle in an equal superposition of two Gaussian states as described by eq. ( 13)), with = 1, M = 1, σ0 = 3/2, x0 = 0, and opposing momenta p0 = 4. The motion of the two half waves leads to interference at the origin. 2: Probability density P (x, t) of an initially free quantum particle which at time t = τ = 0 has the same state as the free particle displayed in Fig. 1 and suddenly gets trapped by a potential with spring constant k = 5. This plot illustrates that the trapped particle is in a superposition state of two squeezed coherent states.

A well-known textbook example is the freely evolving Gaussian wave-packet with initial position spread σ 0

where

Here x 0 parameterizes spatial, and p 0 = M v 0 momentum displacement of the wave functions. If either of these two quantities is non-zero the mapping onto a harmonically trapped state results in a state with oscillating centerof-mass. In general, although a wave function of Gaussian shape, this freely evolving wave packet will also not ‘fit’

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