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arXiv:1008.3929

Equivalence between free and harmonically trapped quantum particles

Ole Steuernagel

Published Aug 23, 2010• 0 citations
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Abstract

It is shown that general solutions of the free-particle Schroedinger equation can be mapped onto solutions of the Schroedinger equation for the harmonic oscillator. This is done in such a way that the time evolution of a free particle subjected to a sudden transition to a harmonic potential can be described by a simple coordinate transformation applied at the transition time. This procedure is computationally more efficient than either state-projection or propagator techniques. A concatenation of the map and its inverse allows us to map from one harmonic oscillator to another with a different spring constant.

DOI
10.48550/arxiv.1008.3929

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It is shown that general solutions of the free-particle Schroedinger equation can be mapped onto solutions of the Schroedinger equation for the harmonic oscillator. This is done in such a way that the time evolution of a free particle subjected to a sudden transition to a harmonic potential can be described by a simple coordinate transformation applied at the transition time. This procedure is computationally more efficient than either state-projection or propagator techniques. A concatenation of the map and its inverse allows us to map from one harmonic oscillator to another with a different spring constant.
In quantum physics the free particle and the harmonically trapped particle are arguably the most important systems a physicist needs to know about. It is little known that, mathematically, they are one and the same. This knowledge helps us to understand either from the viewpoint of the other. Here we show that all general time-dependent solutions of the free-particle Schrödinger equation can be mapped to solutions of the Schrödinger equation for harmonic potentials, both the trapping oscillator and the inverted “oscillator”. This map is fully invertible and therefore induces an isomorphism between both types of system, they are equivalent. A composition of the map and its inverse allows us to map from one harmonic oscillator to another with a different spring constant and different center position. The map is independent of the state of the system, consisting only of a coordinate transformation and multiplication by a form factor, and can be chosen such that the state is identical in both systems at one point in time. This transition point in time can be chosen freely, the wave function of the particle evolving in time in one system before the transition point can therefore be linked up smoothly with the wave function for the other system and its future evolution after the transition point. Such a cut-and-paste procedure allows us to describe the instantaneous changes of the environment a particle finds itself in. Transitions from free to trapped systems, between harmonic traps of different spring constants or center positions, or, from harmonic binding to repulsive harmonic potentials are straightforwardly modelled. This includes some time-dependent harmonic potentials. The mappings introduced here are computationally more efficient than either state-projection or harmonic oscillator propagator techniques conventionally employed when describing instantaneous (non-adiabatic) changes of a quantum particle’s environment.
Abstract In the context of quantum dynamics there exists a coordinate transformation which maps the free particle to the harmonic oscillator. Here we extend this result by reformulating it as a unitary operation followed by a time coordinate transformation. We demonstrate that an equivalent transformation can be performed for classical systems in the context of Koopman–von Neumann dynamics. We further extend this mapping both to dissipative evolutions as well as for a quantum–classical hybrid, and show that this mapping imparts an identical time-dependent scaling on the dissipation parameters for both types of dynamics. The derived classical procedure presents a number of opportunities to import squeezing dependent quantum procedures (such as Hamiltonian amplification) into the classical regime.
We construct the coherent states of a free particle by implementing a coordinate transformation in the extended configuration space, which establishes a correspondence between the solutions of the Schrödinger equation for a harmonic oscillator and those for a free particle. Using this framework, we derive analytical expressions for the coherent states of a free particle, avoiding the complexities of non-normalizable fiducial states, integrals of motion, or group-theoretic approaches. Our method provides a systematic way to characterize these states at any instant while ensuring that they satisfy the Robertson-Schrödinger uncertainty relation.
Physical consequences of the theorem of equivalence among harmonic oscillators of different frequencies, including a free particle as a special case, are explored. A new pictorial way of understanding time-dependent coherent states and squeezed states are presented and contrasted with the standard view. Three-dimensional spherical analogues of squeezed coherent states are constructed. It is also shown as a corollary of the theorem that a free wave packet which coincides initially with any one of the eigenfunctions of a harmonic oscillator maintains its shape, that is, the probability density of position does not change apart from a time-dependent scaling of spatial coordinates. Most of the results are derived without use of any sophisticated techniques but with the knowledge of free wave packets alone.