Symmetries of the quantum damped harmonic oscillator

Guerrero, Julio Becerra; López-Ruiz, Francisco F.; Aldaya, V.; Cossío, Francisco González de

doi:10.1088/1751-8113/45/47/475303

Cited by 13 publications

(25 citation statements)

References 38 publications

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“…It thus turns out that L B , in actuality, describes a doubled system consisting of the uncoupled DHO and AHO, not the DHO itself. The quantization of this system has been studied until recently with various interesting ideas [5][6][7][8][9][10][11][12][13][14][15]. However, in the quantization procedure, (x ± y)/ √ 2, rather than x and y, are treated as fundamental variables, and therefore it is quite doubtful whether the DHO itself is correctly quantized in this approach.…”

Section: Introductionmentioning

confidence: 99%

Quantization of the damped harmonic oscillator based on a modified Bateman Lagrangian

Deguchi

Fujiwara

2020

Phys. Rev. A

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An approach to quantization of the damped harmonic oscillator (DHO) is developed on the basis of a modified Bateman Lagrangian (MBL); thereby some quantum mechanical aspects of the DHO are clarified. We treat the energy operator for the DHO, in addition to the Hamiltonian operator that is determined from the MBL and corresponds to the total energy of the system. It is demonstrated that the energy eigenvalues of the DHO exponentially decrease with time and that transitions between the energy eigenstates occur in accordance with the Schrödinger equation. Also, it is pointed out that a new critical parameter discriminates different behaviours of transition probabilities. *

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Section: Introductionmentioning

confidence: 99%

Quantization of the damped harmonic oscillator based on a modified Bateman Lagrangian

Deguchi

Fujiwara

2020

Phys. Rev. A

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“…Recently, quantization of the Bateman model has been studied in connection with a noncommutative space[15]. For other recent studies concerning quantization of the Bateman model, see, e.g., Refs [16,17]…”

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confidence: 99%

Two quantization approaches to the Bateman oscillator model

2019

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We consider two quantization approaches to the Bateman oscillator model. One is Feshbach-Tikochinsky's quantization approach reformulated concisely without invoking the SU (1, 1) Lie algebra, and the other is the imaginaryscaling quantization approach developed originally for the Pais-Uhlenbeck oscillator model. The latter approach overcomes the problem of unboundedbelow energy spectrum that is encountered in the former approach. In both the approaches, the positive-definiteness of the squared-norms of the Hamiltonian eigenvectors is ensured. Unlike Feshbach-Tikochinsky's quantization approach, the imaginary-scaling quantization approach allows to have stable states in addition to decaying and growing states.

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“…The canonical momenta (16) and (17) are p 1 = mq 1 + c 2 q 2 and p 2 = mq 2 + c 2 q 1 , and the Hamiltonian is (45). From Noether theorem there are four quantities, from (33) and (34) the energies E 1 and E 2 , and from (35) and (36) the momenta P 1 and P 2 , which are related to the Hamiltonian (28) and to the generator of translations (47). The conserved generator of SO(1, 1), (29)…”

Section: Free Motionmentioning

confidence: 99%

“…From (35) and (36) we get the angular momenta J 1 = µ( r 1 ×˙ r 1 ) and J 2 = µ( r 2 ×˙ r 2 ), which satisfy d J1 dt = −cµ( r 1 ×˙ r 2 ) and d J2 dt = cµ( r 2 ×˙ r 1 ). Thus, the angular momentum J = 1 2 J 1 + J 2 and the conserved generator of rotations…”

Section: Central Forcesmentioning

confidence: 99%

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On the Lagrangian description of dissipative systems

Martínez-Pérez

Ramı́rez

2018

Journal of Mathematical Physics

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We consider the Lagrangian formulation with duplicated variables of dissipative mechanical systems. The application of Noether theorem leads to physical observable quantities which are not conserved, like energy and angular momentum, and conserved quantities like the Hamiltonian, that generate symmetry transformations and do not correspond to observables. We show that there are simple relations among the equations satisfied by these two types of quantities. In the case of the damped harmonic oscillator, from the quantities obtained by Noether theorem follows the algebra of Feshbach and Tikochinsky. Further, if we consider the whole dynamics, the degrees of freedom separate into a physical and an unphysical sector. We analyze several cases, with linear and nonlinear dissipative forces; the physical consistency of the solutions is ensured observing that the unphysical sector has always the trivial solution.PACS numbers: 04.20.Fy,12.60.Jv * Electronic address: nephtalieliceo@hotmail.com † Electronic address: cramirez@fcfm.buap.mx 2 Hamilton variational principle. In this approach the variation of the action is done with boundary conditions only at the initial time, independently for each of both variables, and at the final time these variables must coincide. A similar development for classical and quantum mechanics was given by means of an extension of the Closed Time Path formalism to classical mechanics by Polonyi [40,41], who in [42] considers the issue of breaking of time reversal symmetry.The main interest in the study of phenomenological dissipative systems is on their quantum description. Classically, the doubled variable formalism allows to write the equations of motion and after that the additional variables are somehow discarded. In fact, these variables are considered as an artifice which takes account of the dissipative external influence, the whole system being isolated. However, from its construction, the nonconservative Lagrangian has not the standard form due to the time reversed characteristics of the additional sector, i.e. the kinetic term is not positive definite and the potential appears with an unstable term. Thus, an interpretation of its outcome as a whole is not obvious. On the other side, in a quantum theory every interacting degree of freedom in general contributes to the probabilities, spectra and mean values, as they form part of the operator algebra. Thus, it would be desirable to consider the classical theory taking into account the whole dynamics. Moreover, a general knowledge of the relevant quantities in the theory, as delivered e.g. by Noether theorem, is necessary for the definition of the Hilbert space. Actually, in the doubled variable approach, Noether theorem has been applied considering the conservation laws of the conservative part, and these laws are violated due to the dissipative terms [41,43]. Furthermore, Noether theorem has been applied in similar approaches to the symmetries of the whole doubled variables action in [44,45], and for time dependent lagrangians in [46]. ...

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Symmetries of the quantum damped harmonic oscillator

Cited by 13 publications

References 38 publications

Quantization of the damped harmonic oscillator based on a modified Bateman Lagrangian

Quantization of the damped harmonic oscillator based on a modified Bateman Lagrangian

Two quantization approaches to the Bateman oscillator model

On the Lagrangian description of dissipative systems

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