Equivalence of a harmonic oscillator to a free particle and Eisenhart lift

Dhasmana, Shailesh; Sen, Abhijit; Silagadze, Z. K.

doi:10.48550/arxiv.2106.09523

Cited by 3 publications

(3 citation statements)

References 47 publications

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“…Alternative ways to relate free and harmonically trapped motions are studied, e.g., in [51][52][53][54]. Motions with the Mathieu profile were considered also in [55].…”

Section: Discussionmentioning

confidence: 99%

Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems

2021

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The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping “Eisenhart–Duval” lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation, which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to an arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile.

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“…Alternative ways to relate free and harmonically trapped motions are studied, e.g., in [51][52][53][54]. Motions with the Mathieu profile were considered also in [55].…”

Section: Discussionmentioning

confidence: 99%

Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems

2021

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“…It seems that the most direct approach is based on the observation (see [36]) that the transformation (2.8) to the solution ψ(x, t) of the Schrödinger equation with the Hamiltonian (2.1). It is worth to notice that we can go even further and relate φ to the solution φ of the free particle by means of the the so-called Niederer transformation, see [37] (for more recent details of this issue see [38])…”

Section: Preliminariesmentioning

confidence: 99%

Dynamics of entropy and information of time-dependent quantum systems: exact results

Andrzejewski¹

2021

Preprint

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Dynamical aspects of information-theoretic and entropic measures of quantum systems are studied. First, we show that for the time-dependent harmonic oscillator, as well as for the charged particle in certain time-varying electromagnetic fields, the increase of the entropy and dynamics of the Fisher information can be directly described and related. To illustrate these results we have considered several examples for which all the relations take the elementary form. Moreover, we show that the integrals of (geodesic) motion associated with some conformal Killing vectors lead to the Ermakov-Lewis invariants for the considered electromagnetic fields. Next, we explicitly work out the dynamics of the entanglement entropy of the oscillators coupled by a continuous time-dependent parameter; in particular, the case when the final value of the entanglement entropy stabilizes. Finally, we study in some detail the behavior of quantum quenches (in the presence of the critical points) for the case of mutually non-interacting non-relativistic fermions in a harmonic trap.

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“…In the previous paper [25], the present authors considered a cosmology of a homogeneous and isotropic space that contains a gravitating minimally coupled scalar field, and extended the corresponding minisuperspace description with an additional degree of freedom by a method of the Eisenhart-Duval lift [26][27][28][29][30][31][32][33][34][35]. The Eisenhart-Duval lift is one of the classical methods in a Hamiltonian dynamical system, which allows for a geometric description of the system even in the presence of the potential term.…”

Section: Introductionmentioning

confidence: 99%

Third quantization for scalar and spinor wave functions of the Universe in an extended minisuperspace

Kan,

Aoyama,

Hasegawa

et al. 2021

Preprint

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We consider the third quantization in quantum cosmology of a minisuperspace extended by the Eisenhart-Duval lift. We study the third quantization based on both Klein-Gordon type and Dirac-type equations in the extended minisuperspace. Spontaneous creation of "universes" is investigated upon the quantization of a simple model. We find that the quantization of the Dirac-type wave function reveals that the number density of universes is expressed by the Fermi-Dirac distribution. We also calculate the entanglement entropy of the multi-universe system.

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Equivalence of a harmonic oscillator to a free particle and Eisenhart lift

Cited by 3 publications

References 47 publications

Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems

Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems

Dynamics of entropy and information of time-dependent quantum systems: exact results

Third quantization for scalar and spinor wave functions of the Universe in an extended minisuperspace

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