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Journal of Physics A Mathematical and General

The generalized harmonic oscillator and the infinite square well with a moving boundary

Juan D Lejarreta

Published Jan 1, 1999• 21 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/32/25/314

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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.
In this paper a systematic formalism for dealing with non-relativistic time-dependent quantum Hamiltonians is presented. The starting point is the well-known Lewis and Riesenfeld idea which involves the construction of an invariant operator I ( x , t ) which defines both the dynamics of the physical system and the canonical formalism that has to be used in order to obtain a consistent theoretical framework. In order to exhibit the full power of the formalism we discuss two examples: the generalized harmonic oscillator and the infinite square well with a moving boundary. As the first example has already been analyzed by the present authors from other different points of view, we are able to compare the results of the canonical formalism with these other approaches and, as it was expected, we obtain identical descriptions of this physical system. After this, we turn to the case of the square well with a moving boundary. The main surprise is that in order to obtain consistency with the formalism an effective interaction appears which seems to be due to the time dependence of the boundary. Also consistency with the principle of minimal coupling and gauge invariance is obtained just by using this canonical operator formalism. Finally some interesting physical applications are suggested and discussed.
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.
We investigate the quantum dynamics of a particle subjected to a linear potential using the Lewis--Riesenfeld invariant operator method. Starting from the time-dependent Schrödinger equation associated with a constant external force, we construct the most general Hermitian quadratic invariant and derive the corresponding coupled differential equations for its time-dependent coefficients. By means of an appropriate sequence of unitary transformations, the invariant operator is reduced to the form of a harmonic oscillator Hamiltonian. This reduction enables a clear classification of the system according to the sign of the conserved quantity ω2. Particular attention is devoted to the physically relevant case ω2 >0, which yields a discrete eigenspectrum. Explicit analytical expressions for the invariant coefficients, the displacement parameters, and the transformed wave functions are obtained. The resulting formalism provides an exact quantum description of a particle under a constant force and establishes a direct connection between invariant theory and harmonic oscillator quantization.
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.
In this paper we use the algebraic and the invariant method to study the time-dependent damped harmonic oscillator from classical and quantum points of view. The solution of the classical equation of motion and the wavefunction solving the time-dependent Schrödinger equation are found explicitly. We show that the original time-dependent quantum-mechanical problem is completely related to the well known time-independent harmonic oscillator. In addition, we elucidate the intimate connection between the damped harmonic oscillator (DHO) and the generalized harmonic oscillator (GHO). More importantly, the evolution of the states of the DHO cannot be cyclic, in contradistinction with the states of the GHO. Explicit expressions for both the dynamical and the geometric angles and phases are deduced in the adiabatic limit. The coherent states describing the invariant-angle variables of the classical DHO are constructed; they allow us to recover the classical evolution and invariant from the quantum evolution.