SO(2,1)-invariant systems and the Berry phase

Cerveró, José M.; Lejarreta, J. D.

doi:10.1088/0305-4470/22/14/001

“…In this adiabatic limit, the invariant I reduces to the adiabatic invariant of the classical DHO [22]. It is quite striking to note that the results obtained herein are similar to the case of the well known and extensively studied [16][17][18][21][22] generalized harmonic oscillator (GHO)…”

Section: Resultssupporting

confidence: 73%

“…the DHO and the GHO, where the latter is studied in detail in [18]) can be established through a redefinition of the parameters of the GHO,…”

Section: Resultsmentioning

confidence: 99%

Igor' Ekhiel'evich Dzyaloshinskii (on his seventieth birthday)

Абрикосов¹,

Andreev²,

Brazovskii³

et al. 2001

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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.

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“…The adiabatic approximation can be obtained by ignoring terms with two or more time derivatives in equation (2.6) and taking for σ (t) the adiabatic solution [15,18,21],…”

Section: Resultsmentioning

confidence: 99%

The Effect of Superthermal Electrons on Mach Probe Diagnostics

Shikama

¹

,

Kado

²

,

Kajita

³

et al. 2004

Jpn. J. Appl. Phys.

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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.

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“…The invariants method [3] is very simple due to the relationship between the eigenstates of the invariant operator and the solutions to the Schrödinger equation by means of the phases. Exploiting the invariant operator theory several authors, for instance [4][5][6][7][8][9][10][11][12][13][14], have studied extensively in the literature two models. One of them is the time-dependent generalized harmonic oscillator with the symmetry of the SU(1, 1) dynamical group, the other is the two-level system possessing an SU(2) symmetry.…”

Section: Introductionmentioning

confidence: 99%

Non-Unitary Transformation of Quantum Time-Dependent Non-Hermitian Systems

Maamache

¹

2017

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Abstract. We provide a new perspective on non-Hermitian evolution in quantum mechanics by emphasizing the same method as in the Hermitian quantum evolution. We first give a precise description of the non unitary transformation and the associated evolution, and collecting the basic results around it and postulating the norm preserving. This cautionary postulate imposing that the time evolution of a non Hermitian quantum system preserves the inner products between the associated states must not be read naively. We also give an example showing that the solutions of time-dependent non Hermitian Hamiltonian systems given by a linear combination of SU(1, 1) and SU(2) are obtained thanks to time-dependent non-unitary transformation.

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SO(2,1)-invariant systems and the Berry phase

Cited by 41 publications

References 6 publications

Igor' Ekhiel'evich Dzyaloshinskii (on his seventieth birthday)

Igor' Ekhiel'evich Dzyaloshinskii (on his seventieth birthday)

The Effect of Superthermal Electrons on Mach Probe Diagnostics

Non-Unitary Transformation of Quantum Time-Dependent Non-Hermitian Systems

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