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Communications in Theoretical Physics

Exact Time-Dependent Wave Functions of a Confined Time-Dependent Harmonic Oscillator with Two Moving Boundaries

Chi‐Fai Lo

Published Nov 1, 2009• 2 citations
Publisher

Abstract

By applying the standard analytical techniques of solving partial differential equations, we have obtained the exact solution in terms of the Fourier sine series to the time-dependent Schrödinger equation describing a quantum one-dimensional harmonic oscillator of time-dependent frequency confined in an infinite square well with the two walls moving along some parametric trajectories. Based upon the orthonormal basis of quasi-stationary wave functions, the exact propagator of the system has also been analytically derived. Special cases like (i) a confined free particle, (ii) a confined time-independent harmonic oscillator, and (iii) an aging oscillator are examined, and the corresponding time-dependent wave functions are explicitly determined. Besides, the approach has been extended to solve the case of a confined generalized time-dependent harmonic oscillator for some parametric moving boundaries as well.

DOI
10.1088/0253-6102/52/5/11

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By applying the standard analytical techniques of solving partial differential equations, we have obtained the exact solution in terms of the Fourier sine series to the time-dependent Schrödinger equation describing a quantum one-dimensional harmonic oscillator of time-dependent frequency confined in an infinite square well with the two walls moving along some parametric trajectories. Based upon the orthonormal basis of quasi-stationary wave functions, the exact propagator of the system has also been analytically derived. Special cases like (i) a confined free particle, (ii) a confined time-independent harmonic oscillator, and (iii) an aging oscillator are examined, and the corresponding time-dependent wave functions are explicitly determined. Besides, the approach has been extended to solve the case of a confined generalized time-dependent harmonic oscillator for some parametric moving boundaries as well.
The problem of a harmonic oscillator of time-dependent frequency confined in a one-dimensional infinite square well with a moving wall is studied.It is shown that the exact solution and the Lewis invariant operator of the system can be obtained by performing two consecutive gauge transformations on the time-dependent Schrdinger equation.On the basis of the exact solution the Berry phases for the system are calculated by using the geometric concepts such as the geometric distance and geometric length of the curve.
Abstract The problem of a harmonic oscillator of time dependent frequency confined in a one dimensional infinite square well with a moving wall is studied.It is shown that the exact solution and the Lewis invariant operator of the system can be obtained by performing two consecutive gauge transformations on the time dependent Schrdinger equation.On the basis of the exact solution the Berry phases for the system are calculated by using the geometric concepts such as the geometric distance and geometric length of the curve.
For the one-dimensional generalized harmonic oscillator we obtain in this paper its wavefunctions in closed form by means of two independent methods which are based respectively in (i) the algebraic properties of the dynamical symmetry of the system and (ii) the construction of an invariant operator. The total equivalence of these two formulations is shown and quantal properties are described in terms of a classical solution of the equations of motion. Two possible reductions for the system exist: the static harmonic oscillator and the free particle. In the latter case the quantum system becomes a Fermi oscillator or equivalently it can describe a free particle in a well with one moving boundary which in turn follows certain classical rules. The time-dependent boundary conditions in the well play the role of an effective interaction acting on the particle. The formalism is shown to be compatible with the gauge principle of minimal coupling and several different gauges are constructed and analysed.
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.
The exact solutions to the time-dependent Schrodinger equation for a harmonic oscillator with time-dependent mass and frequency were derived in a general form. The quantum mechanical propagator was calculated by the Feynman path integral method, while the wave function was derived from the spectral representation of the obtained propagator. It was shown that the propagator and the wave function depended on the s solution of a classical oscillator, in which the amplitude and phase satisfied the auxiliary equations. To demonstrate the derivation of the solution from our auxiliary equations, exponential and periodic functions of mass with constant frequency were imposed to evaluate the propagator and wave function for the Caldirola-Kanai and pulsating mass oscillators, respectively.
We solve the time-dependent Schrödinger equation describing a generalized harmonic oscillator in the coordinate representation with constant mass but time-dependent stiffness and correlation parameters. Exact solutions are obtained in terms of confluent hypergeometric functions or parabolic cylinder functions. The dynamics of mean energy is studied in several different regimes, including the special case of imaginary correlation parameter, corresponding to the non-Hermitian Ahmed–Swanson model. In this case, real time-dependent mean values of the Hamiltonian are obtained by using dual wave functions, evolving according to the time-dependent Schrödinger equations with the given Hamiltonian and its Hermitially conjugated partner. The adiabatic evolution is considered as one of examples.