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arXiv:quant-ph/0211107 · Europhysics Letters (EPL)

Unitary transformation for the system of a particle in a linear potential

Dae-Yup Song

Published Jun 1, 2003• 12 citations
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Abstract

A unitary operator, which relates the system of a particle in a linear potential with time-dependent parameters to that of a free particle, has been given. This operator, closely related to the one which is responsible for the existence of coherent states for a harmonic oscillator, is used to find a general wave packet described by an Airy function. The kernel (propagator) and a complete set of Hermite-Gaussian–type wave functions are also given.

DOI
10.1209/epl/i2003-00419-1

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A unitary operator, which relates the system of a particle in a linear potential with time-dependent parameters to that of a free particle, has been given. This operator, closely related to the one which is responsible for the existence of coherent states for a harmonic oscillator, is used to find a general wave packet described by an Airy function. The kernel (propagator) and a complete set of Hermite-Gaussian–type wave functions are also given.
The unitary operator that transforms a harmonic oscillator system of time-dependent frequency into that of a simple harmonic oscillator of different timescale is found, with and without an inverse-square potential. It is shown that for both cases, this operator can be used in finding complete sets of wave functions of a generalized harmonic oscillator system from the well-known sets of the simple harmonic oscillator. Exact invariants of the time-dependent systems can also be obtained from the constant Hamiltonians of unit mass and frequency by making use of this unitary transformation. The geometric phases for the wave functions of a generalized harmonic oscillator with an inverse-square potential are given.
The behavior of a particle moving according to the laws of quantum mechanics in a field, which potential changes over time, is studied. The method of unitary transformations for solving the temporal Schrödinger equation is considered. The reduced Hamiltonian of the system of a quantum­-mechanical particle in time­-dependent potential is obtained, as well as the total operator of the evolution of such a system. We find a new version of the unitary transformation, which, compared to the known ones, simplifies solving the problem in analytical form. This transformation associates the linear potential quantum model with time-­dependent parameters and free­particle system. Based on the constructed evolution operator, we consider the wave packet dispersion. It is shown that the wave packet dispersion for this model is the same as that for a free particle. This conclusion is the general result of the behavior of a quantum­mechanical particle moving in the field of a time­dependent linear potential.
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.
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 )⟩.
The Feynman path integral for the generalized harmonic oscillator is reviewed, and it is shown that the path integral can be used to find a complete set of wave functions for the oscillator. Harmonic oscillators with different (time-dependent) parameters can be related through unitary transformations. The existence of generalized coherent states for a simple harmonic oscillator can then be interpreted as the result of a (formal) {\em invariance} under a unitary transformation which relates the same harmonic oscillator. In the path integral formalism, the invariance is reflected in that the kernels do not depend on the choice of classical solutions.
By successive applications of time-dependent rotation, translation and dilatation, it is shown that a charged time-dependent harmonic oscillator in a uniform time-dependent electromagnetic field is equivalent to a free particle. This equivalence is employed to present a simple derivation of the Feynman kernel as well as to obtain a concrete picture of the time evolution of a generic wave packet. A compact formula for an evolving anisotropic Gaussian packet is also derived. A special case of this formula gives a squeezed coherent state for a harmonic oscillator whose frequency is an arbitrary function of time. Finally we discuss an Aharonov-Bohm-Berry-like phase associated with a wave packet whose center traces a closed circuit.