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Journal of Physics A Mathematical and Theoretical

Time evolution of a time-dependent harmonic oscillator in a static magnetic field

Wensen Liu, Ji‐Suo Wang

Published Jan 17, 2007• 8 citations
Publisher

Abstract

The analytic expression of the time evolution wavefunction of the two-dimensional harmonic oscillator with time-dependent mass and frequency in a static magnetic field is obtained using an operator-algebraic method slightly different from the usual Lie algebraic technique. The evolution operator of the one-dimensional harmonic oscillator with time-dependent mass and frequency is established first by forming an operator differential equation with the su (1, 1) Lie algebra, which is deduced from the time-dependent linear unitary transformation for boson operators ( a , a †), and then by comparing this operator equation with the time evolution equation of the one-dimensional oscillator.

DOI
10.1088/1751-8113/40/5/013

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Comment on “Wave functions of a time-dependent harmonic oscillator in a static magnetic field”

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We show that the procedure used by Ferreira et al. [Phys. Rev. A 66, 024103 (2002)] is not correct for the following reasons: (i) the invariant $I(t)$ they derived does not satisfy the Liouville--Von Neuman equation. (ii) They found that the eigenvalues of $I(t)$ are time dependent which should not be the case according to the Lewis-Riesenfeld theory. We give a correct procedure to find the solution of the system they considered, i.e., the Schr\"odinger equation for a two-dimensional harmonic oscillator with time-dependent mass and frequency in the presence of a static magnetic field.

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Wave functions of a time-dependent harmonic oscillator in a static magnetic field

C. A. S. Ferreira, P. T. S. Alencar, José Maria Filardo Bassalo

In this paper, we solve for the exact wave functions of a two-dimensional harmonic oscillator under the influence of a static magnetic field. This is done through the use of the so-called Ermakov invariants, similarity transformed so as to obtain a time-independent Schr\"odinger equation. In the same manner, time-dependent eigenvalues are also computed. It is shown that previous results for the invariants are particular cases for the ones found in this work.

Physical Review A•Aug 13, 2002•19 citations
The analytic expression of the time evolution wavefunction of the two-dimensional harmonic oscillator with time-dependent mass and frequency in a static magnetic field is obtained using an operator-algebraic method slightly different from the usual Lie algebraic technique. The evolution operator of the one-dimensional harmonic oscillator with time-dependent mass and frequency is established first by forming an operator differential equation with the su (1, 1) Lie algebra, which is deduced from the time-dependent linear unitary transformation for boson operators ( a , a †), and then by comparing this operator equation with the time evolution equation of the one-dimensional oscillator.
The time-dependent Schrödinger equation describing a generalized two-dimensional quantum parametric oscillator in the presence of time-variable external fields is solved using the evolution operator method. For this, the evolution operator is found as a product of exponential operators through the Wei–Norman Lie algebraic approach. Then, the propagator and time-evolution of eigenstates and coherent states are derived explicitly in terms of solutions to the corresponding system of coupled classical equations of motion. In addition, using the evolution operator formalism, we construct linear and quadratic quantum dynamical invariants that provide connection of the present results with those obtained via the Malkin–Man’ko–Trifonov and the Lewis–Riesenfeld approaches. Finally, as an exactly solvable model, we introduce a Cauchy–Euler type quantum oscillator with increasing mass and decreasing frequency in time-dependent magnetic and electric fields. Based on the explicit results for the uncertainties and expectations, squeezing properties of the wave packets and their trajectories in the two-dimensional configuration space are discussed according to the influence of the time-variable parameters and external fields.
A charged harmonic oscillator in a magnetic field, Landau problems, and an oscillator in a noncommutative space share the same mathematical structure in their Hamiltonians. We have considered a two-dimensional anisotropic harmonic oscillator with arbitrarily time-dependent parameters (effective mass and frequencies), placed in an arbitrarily time-dependent magnetic field. A class of quadratic invariant operators (in the sense of Lewis and Riesenfeld) have been constructed. The invariant operators (Î) have been reduced to a simplified representative form by a linear canonical transformation [the group Sp(4,R)]. An orthonormal basis of the Hilbert space consisting of the eigenvectors of Î is obtained. In order to obtain the solutions of the time-dependent Schrödinger equation corresponding to the system, both the geometric and dynamical phase-factors are constructed. A generalized Peres–Horodecki separability criterion (Simon’s criterion) for the ground state corresponding to our system has been demonstrated.
The invariants for the time-dependent one-dimensional harmonic oscillator and the time-dependent two-dimensional harmonic oscillator in a static magnetic field are derived from the real representation of the exact solution to the equation of motion.Mathematically,the orthogonal functions invariant is the angular momentum of an isotropic time-dependent two-dimensional harmonic oscillator.Based on the invariants obtained here,the wave function for time-dependent two-dimensional harmonic oscillator in a static magnetic field in cylindrical coordinate is simply derived and the dynamical and geometrical phases are easily got by expressing the wave function as the superpositions of the wave functions of time-dependent two-dimensional harmonic oscillator in rectangular coordinate.
We show that the procedure used by Ferreira et al. [Phys. Rev. A 66, 024103 (2002)] is not correct for the following reasons: (i) the invariant I(t)I(t)I(t) they derived does not satisfy the Liouville--Von Neuman equation. (ii) They found that the eigenvalues of I(t)I(t)I(t) are time dependent which should not be the case according to the Lewis-Riesenfeld theory. We give a correct procedure to find the solution of the system they considered, i.e., the Schr\"odinger equation for a two-dimensional harmonic oscillator with time-dependent mass and frequency in the presence of a static magnetic field.