Quantization of the damped harmonic oscillator revisited

Baldiotti, M. C.; Fresneda, R.; Гитман, Д. М.

doi:10.1016/j.physleta.2011.03.009

Cited by 39 publications

(45 citation statements)

References 68 publications

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“…The simplest approach is to write a two-body Hamiltonian describing one damped and one amplified oscillator, with conserved total energy [4]. But the canonical variables of this system [4] are not the positions and momenta of the two oscillators, so again there is no straightforward way of imposing canonical commutation relations on each oscillator [1,2,[5][6][7][8][9]. This doubling of the degrees of freedom has also been applied to an infinite set of oscillators to discuss dissipation of quantum fields [10].…”

Section: Introductionmentioning

confidence: 99%

“…where θ(t) is the step function. The exponential factors are necessary in the Green functions (8) and (9) for their Fourier transforms to exist, and the infinitesimal number 0 + gives the familiar pole prescriptions in the frequency domain, with G r (ω) analytic in the upper-half complex ω-plane and G a (ω) analytic in the lower-half plane. The general solution (6) is constructed with the difference…”

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confidence: 99%

“…In addition to the form (6), we also require the general solution of the reservoir equation (5) in a form that allows imposition of conditions in the infinite past or future. We choose the infinite past t → −∞, and the required form of the general solution of (5) is obtained by using the retarded Green function (8), yielding…”

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confidence: 99%

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Quantum dynamics of the damped harmonic oscillator

Philbin

2012

New J. Phys.

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View the article online for updates and enhancements. Abstract. The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a damped oscillator is determined by an arbitrary effective susceptibility that obeys the Kramers-Kronig relations. This approach offers an alternative description of nano-mechanical oscillators and opto-mechanical systems.

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Section: Introductionmentioning

confidence: 99%

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Quantum dynamics of the damped harmonic oscillator

Philbin

2012

New J. Phys.

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“…A similar but different expression, (ω 2 /ω − )e −γt/m (n + 1/2), has been derived as an energy expectation value of the DHO[18][19][20][21]. This expression, however, behaves in a strange manner such that it diverges in the critical damping limit ω − → 0.…”

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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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“…The Euler-Lagrange equation corresponds to the damped harmonic oscillator whose quantization has been treated in [9] via the extended Dirac formalism for singular non autonomous system. With the Faddeev-Jackiw approach we start from the transformed Lagrangian one form…”

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Faddeev–Jackiw quantization of non-autonomous singular systems

2016

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We extend the quantizationà la Faddeev-Jackiw for non-autonomous singular systems. This leads to a generalization of the Schrödinger equation for those systems. The method is exemplified by the quantization of the damped harmonic oscillator and the relativistic particle in an external electromagnetic field. PACS numbers:The quantization of constrained systems is almost as old as the beginning of quantum mechanics. It was Dirac [1] who elaborated a Hamiltonian approach with a categorization of constraints and the introduction of the socalled Dirac brackets. Later, Faddeev and Jackiw [2] suggested an alternative and generally simpler method based on a symplectic structure. Recently, we have proposed a third approach for classically soluble constrained systems where the brackets between the constants of integration are computed. This method does neither require Dirac formalism nor the symplectic method of FaddeevJackiw [3]. All three approaches were developed for autonomous constrained systems only. The quantization of non-autonomous singular systems has turned to be non trivial [4]. Gitman and Tyutin [5], via notably the introduction of a conjugate momentum of time, could extend Dirac approach and brackets for those systems. In the present work, our aim is to generalize the Faddeev-Jackiw symplectic approach to non-autonomous constrained systems. This leads to a generalization of the Schrödinger equation which encompasses non-autonomous singular systems. The quantization of a relativistic particle in an electromagnetic field is solved by this method, constituting an original derivation of the Dirac equation.Consider a non autonomous Lagrangian one-form as inwhere ξ j are the N -component phase-space coordinates with j = 1, ..., N , where a j and H are time-dependent. The Euler-Lagrange equations lead tȯfor an invertible antisymmetric matrix f ij . The dot denotes differentiation with respect to t. Eq. (2) can not be derived from Hamilton equations through brackets when ∂a j /∂t = 0 (see Eq. (17)). Canonical quantization seems compromised in this case. To solve this problem via the Faddeev-Jackiw approach we first introduce a time parameter τ such that time is promoted to a dynamically variable t = t(τ ) and ξ i = ξ i (τ ). This leads to a Lagrangian L τ , given by L τ dτ = Ldt, so that the action remains the same. L τ has a gauge invariance due to the arbitrariness of the parameter τ . We thus define a new LagrangianL τ = L τ + ω (t ′ − 1) that implements the gauge constraint t ′ = 1 (prime denotes differentiation with respect to τ ) via ω(τ ) a Lagrange multiplier seen as a new variable. Dropping a total time derivative term, one ends up with the following Lagrangian one-formwhere H(ξ, t, ω) = ω defines a new Hamiltonian. The Euler-Lagrange equations with L lead to Eq (2) and the equations ω ′ = 0 and t ′ = 1. Unlike the initial Lagrangian (1), L is autonomous with respect to τ and Eq.(4) is precisely of the form studied by Faddeev-Jackiw [2]with now ζ i the N +2-component phase space coordinates defined by ζ i =...

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Quantization of the damped harmonic oscillator revisited

Cited by 39 publications

References 68 publications

Quantum dynamics of the damped harmonic oscillator

Quantum dynamics of the damped harmonic oscillator

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

Faddeev–Jackiw quantization of non-autonomous singular systems

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