The quantum under-, critical- and over-damped driven harmonic oscillators

Yeon, Kyu-Hwang; Kim, Suk-Sung; Moon, Young Min; Hong, Suc-Kyoung; Um, Chung-In; George, Thomas F.

doi:10.1088/0305-4470/34/37/321

Cited by 40 publications

(28 citation statements)

References 9 publications

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“…For instance the time dependent linear potential has been treated through the Lewis and Riesenfeld [34] invariant theory [28,29,36], Feynman's path integrals [37][38][39][40][41], time-space transformation methods [42] and others [43,44]. The quantum oscillator with time-dependent mass and frequency has been dealt through the group-theoretical approach [45], unitary transformations [4], the Lewis and Riesenfeld invariant theory [46][47][48].…”

Section: Introductionmentioning

confidence: 99%

Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

Sandoval-Santana

Ibarra-Sierra

Cardoso

et al. 2016

Journal of Mathematical Physics

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We develop a Lie algebraic approach to systematically calculate the evolution operator of the generalized two-dimensional quadratic Hamiltonian with time-dependent coefficients. Although the development of the Lie algebraic approach presented here is mainly motivated by the two-dimensional quadratic Hamiltonian, it may be applied to investigate the evolution operators of any Hamiltonian having a dynamical algebra with a large number of elements. We illustrate the method by finding the propagator and the Heisenberg picture position and momentum operators for a two-dimensional charge subject to uniform and constant electro-magnetic fields.2

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Section: Introductionmentioning

confidence: 99%

Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

Sandoval-Santana

Ibarra-Sierra

Cardoso

et al. 2016

Journal of Mathematical Physics

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“…In fact, the description of quantized energy for a TDHS, such as (10), is a delicate problem. While some authors [1,3,11,21,22] including ours have investigated quantized energy levels for specific TDHSs, there is another opinion [26,27] that such energy levels do not exist for the case of TDHSs. Hence, explicit demonstrations of the existence of quantized energy levels may be an interesting research topic for further study in the future in this field.…”

Section: Quantized Energymentioning

confidence: 82%

“…Typical examples of TDHSs include a damped harmonic oscillator [1][2][3], a harmonic oscillator driven by a time-varying force [4,5], and a harmonic oscillator with time-dependent parameters such as time-dependent mass [6][7][8][9] and/or frequency [10][11][12]. A powerful method for solving quantum solutions of a TDHS is the invariant operator method [13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Quantum Analysis on Time Behavior of a Lengthening Pendulum

Choi

Song

2016

Physics Research International

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Quantum properties of a lengthening pendulum are studied under the assumption that the length of the string increases at a steady rate. Advanced analysis for various physical problems in several types of quantum states, such as propagators, Wigner distribution functions, energy eigenvalues, probability densities, and dispersions of physical quantities, is carried out using quantum wave functions of the system. In particular, the time behavior of Gaussian-type wave packets is investigated in detail. The probability density for a Gaussian wave packet displaced in the positive at = 0 oscillates back and forth from the center ( = 0). This phenomenon is very similar to the classical motion of the pendulum. As a consequence, we can confirm that there is a correspondence between its quantum and classical behaviors. When we analyze a dynamical system in view of quantum mechanics, the quantum and classical correspondence is very important in order for the associated quantum theory to be valid and viable.

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“…Furthermore, the propagator can be expanded as a complete set of eigenfunctions for the Hamiltonian operator. For a quadratic Lagrangian, (22) can be written in the form [11] K …”

Section: K(x2 T2;xltl) Yx(tl)=xlmentioning

confidence: 99%

Quantization of motion with friction quadratic in velocity

2005

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The motion of a point object through a viscous field is considered. The friction is assumed to depend quadratically on velocity of the particle. The inverse problem of the variational calculus is solved and the Weyl quantization procedure is employed to write a SchrSdinger equation. The solution of this equation shows that the quantum mechanical wave function is oscillatory for small values of the friction. Contrarily, for large values of the friction, the wave function resembles the solution of von Neumann shock problem.

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The quantum under-, critical- and over-damped driven harmonic oscillators

Cited by 40 publications

References 9 publications

Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

Quantum Analysis on Time Behavior of a Lengthening Pendulum

Quantization of motion with friction quadratic in velocity

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