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Scientific Reports

The generalized harmonic potential theorem in the presence of a time-varying magnetic field

Meng-Yun Lai, Xiao‐Yin Pan

Published Oct 17, 2016• 7 citations
Publisher

Abstract

We investigate the evolution of the many-body wave function of a quantum system with time-varying effective mass, confined by a harmonic potential with time-varying frequency in the presence of a uniform time-varying magnetic field, and perturbed by a time-dependent uniform electric field. It is found that the wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schrödinger equation with the latter being translated by a time-dependent value that satisfies the classical driven equation of motion. In other words, we generalize the harmonic potential theorem to the case when the effective mass, harmonic potential, and the external uniform magnetic field with arbitrary orientation are all time-varying. The results reduce to various special cases obtained in the literature, particulary to that of the harmonic potential theorem wave function when the effective mass and frequency are both static and the external magnetic field is absent.

DOI
10.1038/srep35412

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We investigate the evolution of the many-body wave function of a quantum system with time-varying effective mass, confined by a harmonic potential with time-varying frequency in the presence of a uniform time-varying magnetic field, and perturbed by a time-dependent uniform electric field. It is found that the wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schrödinger equation with the latter being translated by a time-dependent value that satisfies the classical driven equation of motion. In other words, we generalize the harmonic potential theorem to the case when the effective mass, harmonic potential, and the external uniform magnetic field with arbitrary orientation are all time-varying. The results reduce to various special cases obtained in the literature, particulary to that of the harmonic potential theorem wave function when the effective mass and frequency are both static and the external magnetic field is absent.
We investigate the many-body wave function of a quantum system with time-dependent effective mass, confined by a harmonic potential with time-dependent frequency, and perturbed by a time-dependent spatially homogeneous electric field. It is found that the wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schrödinger equation with the latter being translated by a time-dependent value that satisfies the classical driven equation of motion. The wave function reduces to that of the harmonic potential theorem wave function when both the effective mass and frequency are static. An example of application is also given.
The many-body wave function of a system of interacting particles confined by a time-dependent harmonic potential and perturbed by a time-dependent spatially homogeneous electric field is derived via the Feynman path-integral method. The wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schrödinger equation with the latter being translated by a time-dependent value that satisfies the classical driven equation of motion. The wave function reduces to that of the Harmonic Potential Theorem wave function for the case of the time-independent harmonic confining potential.
When a system of particles in a bound state is placed in an external field the quantum state of the system depends on the manner in which the field is produced. The two limiting cases of interest are the adiabatic and impulse limits. Since the system does not approach a stationary state in either limit, in general, it is necessary to consider solutions of the time dependent Schrödinger equation. An example of this approach is provided by the problem of a harmonic oscillator in a uniform, time varying electric field for which an exact solution is easy to obtain using the Magnus expansion for the evolution operator. This operator contains the position and momentum in a particularly simple way that allows it to be interpreted in terms of position and momentum shifts produced by the field. The wavefunction of the oscillator is obtained in the adiabatic and impulse limits assuming the field to be switched on in an exponential fashion. Results obtained for these two limiting cases are then shown to be independent of the manner in which the field reaches its final steady value.
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.
The dynamics of time-dependent coupled oscillator model for the charged particle motion subjected to a time-dependent external magnetic field is investigated. We used canonical transformation approach for the classical treatment of the system, whereas unitary transformation approach is used when managing the system in the framework of quantum mechanics. For both approaches, the original system is transformed to a much more simple system that is the sum of two independent harmonic oscillators which have time-dependent frequencies. We therefore easily identified the wave functions in the transformed system with the help of invariant operator of the system. The full wave functions in the original system is derived from the inverse unitary transformation of the wave functions associated to the transformed system.