Systematic Treatment of General Time-Dependent Harmonic Oscillator in Classical and Quantum Mechanics

Kanasugi, Hirotaka; Okada, Hidehiko

doi:10.1143/ptp/93.5.949

Cited by 19 publications

(18 citation statements)

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“…To calculate Ldt we note that the trajectories of a particle in a classical timedependent harmonic oscillator can be mapped onto the ones in a time-independent oscillator, see, e.g., Refs. [21][22][23]. Indeed, the classical time-dependent oscillator…”

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confidence: 99%

Two cold atoms in a time‐dependent harmonic trap in one dimension

2016

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We analyze the dynamics of two atoms with a short-ranged pair interaction in a one-dimensional harmonic trap with time-dependent frequency. Our analysis is focused on two representative cases: (i) a sudden change of the trapping frequency from one value to another, and (ii) a periodic trapping frequency. In case (i), the dynamics of the interacting and the corresponding non-interacting systems turn out to be similar. In the second case, however, the interacting system can behave quite differently, especially close to parametric resonance. For instance, in the regions where such resonance occurs we find that the interaction can significantly reduce the rate of energy increase. The implications for applications of our findings to cool or heat the system are also discussed.

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confidence: 99%

Two cold atoms in a time‐dependent harmonic trap in one dimension

2016

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“…where the function ψ n (t, x n ) is the wave function of a harmonic oscillator with timedependent mass and frequency, which can be written in terms of the wave function of a harmonic oscillator with constant mass and frequency [31][32][33][34]. The general solution of ψ n (t, x n ) can then be expanded in the basis of number eigenstates of the invariant representation, ψ N,n , as…”

Section: Small Perturbations and Backreactionmentioning

confidence: 99%

“…A symmetric 3-metric has only 6 independent components, so it turns out that M is isomorphic to R 6 . Thus, we can make the following choice for the coordinates 33…”

Section: Historial Reviewmentioning

confidence: 99%

“…For this, we shall consider the multiverse as a really closed system, so no external influence is expected to modify its state. Therefore, it seems appropriate to describe the state of the multiverse in a representation that is invariant under the evolution of the third quantised Hamiltonian, which is an extension of the invariant representation used in quantum mechanics [31][32][33][79][80][81][82][83][84][85][86].…”

Section: Boundary Conditions and The Creation Of The Universes In Pairsmentioning

confidence: 99%

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Quantum Cosmology with Third Quantisation

Robles-Pérez

2021

Universe

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We reviewed the canonical quantisation of the geometry of the spacetime in the cases of a simply and a non-simply connected manifold. In the former, we analysed the information contained in the solutions of the Wheeler–DeWitt equation and showed their interpretation in terms of the customary boundary conditions that are typically imposed on the semiclassical wave functions. In particular, we reviewed three different paradigms for the quantum creation of a homogeneous and isotropic universe. For the quantisation of a non-simply connected manifold, the best framework is the third quantisation formalism, in which the wave function of the universe is seen as a field that propagates in the space of Riemannian 3-geometries, which turns out to be isomorphic to a (part of a) 1 + 5 Minkowski spacetime. Thus, the quantisation of the wave function follows the customary formalism of a quantum field theory. A general review of the formalism is given, and the creation of the universes is analysed, including their initial expansion and the appearance of matter after inflation. These features are presented in more detail in the case of a homogeneous and isotropic universe. The main conclusion in both cases is that the most natural way in which the universes should be created is in entangled universe–antiuniverse pairs.

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“…where m(t), γ(t) and ω(t) can be deterministic or random functions of time. The case m(t), γ(t) and ω(t) are deterministic functions has been studied by many authors (see for examples [38,39,40,41, and references therein]). On the other hand, it is possible that the mass, damping coefficient and frequency can be fluctuating functions of time.…”

Section: Introductionmentioning

confidence: 99%

Ornstein–Uhlenbeck process with fluctuating damping

Eab

Lim

2018

Physica A: Statistical Mechanics and its Applications

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This paper studies Langevin equation with random damping due to multiplicative noise and its solution. Two types of multiplicative noise, namely the dichotomous noise and fractional Gaussian noise are considered. Their solutions are obtained explicitly, with the expressions of the mean and covariance determined explicitly. Properties of the mean and covariance of the Ornstein-Uhlenbeck process with random damping, in particular the asymptotic behavior, are studied. The effect of the multiplicative noise on the stability property of the resulting processes is investigated.

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Systematic Treatment of General Time-Dependent Harmonic Oscillator in Classical and Quantum Mechanics

Cited by 19 publications

References 1 publication

Two cold atoms in a time‐dependent harmonic trap in one dimension

Two cold atoms in a time‐dependent harmonic trap in one dimension

Quantum Cosmology with Third Quantisation

Ornstein–Uhlenbeck process with fluctuating damping

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