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Journal of Shanxi University

Exact Solution of the Schrdinger Equation for a Driven Time-dependent Harmonic Oscillator

Wensen Liu

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

In this paper,the exact expressions of the evolution operator and wave function are obtained by means of the combination of the inhomogeneous Bogoliubov transformation(IBT) and the evolution equation associated with a su(1,1)h(4).As a model example,we have got the exact expressions of the evolution operator and Schrdinger wave function of the driven time-dependent harmonic oscillator.These results agree well with those derived in general refrences,but the present method is more simple and definite.what's more,it is easily extended to other su(1,1)h(4) quantum system

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In this paper,the exact expressions of the evolution operator and wave function are obtained by means of the combination of the inhomogeneous Bogoliubov transformation(IBT) and the evolution equation associated with a su(1,1)h(4).As a model example,we have got the exact expressions of the evolution operator and Schrdinger wave function of the driven time-dependent harmonic oscillator.These results agree well with those derived in general refrences,but the present method is more simple and definite.what's more,it is easily extended to other su(1,1)h(4) quantum system
For a harmonic oscillator with time-dependent (positive) mass and frequency, a unitary operator is shown to transform the quantum states of the system to those of a harmonic oscillator system of unit mass and time-dependent frequency, as well as operators. For a driven harmonic oscillator, a unitary transformation which relates the driven system and a system of the same mass and frequency without a driving force is given, as a generalization of previous results, in terms of the solution of classical equation of motion of the driven system. Thus transformations give a simple way of finding exact wavefunctions of a driven harmonic oscillator system, provided the quantum states of the corresponding system of unit mass are given.
In this paper we generalize Wang's approach [J. Phys. A 20, 5041 (1987)] and investigate the algebraic structure of the Schr\"odinger equation associated with a general driven time-dependent oscillator. Using the Lie-algebraic technique we obtain an exact form of the time-evolution operator which, in turn, enables us to derive the propagator of the system readily. Since the propagator is for the most general time-dependent oscillator, results for any special case can be easily deduced from it. These results will be useful for future studies in quantum optics as well as in atomic and molecular physics.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.