We apply the Noether symmetry condition to the time-dependent conformal Lagrangian L=A(t)12gijx˙ix˙j−KV(x) with focus on the damped harmonic oscillator in Euclidean space, where (A(t)=eγt and V(x) is the oscillator potential. The Noether symmetries are generated by the homothetic algebra of the Euclidean metric and yield a complete classification of invariants. A key result is a family of conserved quantities parameterized by a constant D≠2, related to the potential multiplier K via the resonance condition K=−2γ2D/(D−2)2. This family includes the Bateman invariant as the special case D=−4. We demonstrate that the Ray-Reid method for constructing invariants via an auxiliary variable corresponds precisely to the choices D=1 and D=4 within our symmetry-based approach. Furthermore, each invariant in the family leads to a distinct quantum Hamiltonian via canonical quantization, resulting in different time-dependent effective masses and decoherence rates. The parameter D thus controls the quantum evolution of the damped oscillator, offering a unified framework that generalizes the Kanai–Caldirola and Bateman models and provides new experimentally testable predictions.
We apply the Noether symmetry condition to the time-dependent conformal Lagrangian L=A(t)12gijx˙ix˙j−KV(x) with focus on the damped harmonic oscillator in Euclidean space, where (A(t)=eγt and V(x) is the oscillator potential. The Noether symmetries are generated by the homothetic algebra of the Euclidean metric and yield a complete classification of invariants. A key result is a family of conserved quantities parameterized by a constant D≠2, related to the potential multiplier K via the resonance condition K=−2γ2D/(D−2)2. This family includes the Bateman invariant as the special case D=−4. We demonstrate that the Ray-Reid method for constructing invariants via an auxiliary variable corresponds precisely to the choices D=1 and D=4 within our symmetry-based approach. Furthermore, each invariant in the family leads to a distinct quantum Hamiltonian via canonical quantization, resulting in different time-dependent effective masses and decoherence rates. The parameter D thus controls the quantum evolution of the damped oscillator, offering a unified framework that generalizes the Kanai–Caldirola and Bateman models and provides new experimentally testable predictions.
For the non-conservative Caldirola–Kanai system, describing a quantum damped harmonic oscillator, a couple of constant-of-motion operators generating the Heisenberg–Weyl algebra can be found. The inclusion of the standard time evolution generator (which is not a symmetry) as a symmetry in this algebra, in a unitary manner, requires a non-trivial extension of this basic algebra and hence of the physical system itself. Surprisingly, this extension leads directly to the so-called Bateman dual system, which now includes a new particle acting as an energy reservoir. In addition, the Caldirola–Kanai dissipative system can be retrieved by imposing constraints. The algebra of symmetries of the dual system is presented, as well as a quantization that implies, in particular, a first-order Schrödinger equation. As opposed to other approaches, where it is claimed that the spectrum of the Bateman Hamiltonian is complex and discrete, we obtain that it is real and continuous, with infinite degeneracy in all regimes.
For the Caldirola-Kanai system, describing a quantum damped harmonic oscillator, a couple of constant-of-motion operators generating the Heisenberg algebra can be found. The inclusion of the standard time evolution symmetry in this algebra for damped systems, in a unitary manner, requires a non-trivial extension of this basic algebra and hence the physical system itself. Surprisingly, this extension leads directly to the so-called Bateman's dual system, which now includes a new particle acting as an energy reservoir. The group of symmetries of the dual system is presented, as well as a quantization that implies, in particular, a first-order Schrödinger equation. The usual second-order equation and the inclusion of the original Caldirola-Kanai model in Bateman's system are also discussed.
In this paper, a new approach for constructing Lagrangians for driven and undriven linearly damped systems is proposed, by introducing a redefined time coordinate and an associated coordinate transformation to ensure that the resulting Lagrangian satisfies the Helmholtz conditions. The approach is applied to canonically quantize the damped harmonic oscillator and although it predicts an energy spectrum that decays at the same rate to previous models, unlike those approaches it recovers the classical critical damping condition, which determines transitions between energy eigenstates, and is therefore consistent with the correspondence principle. It is also demonstrated how to apply the procedure to a driven damped harmonic oscillator.
In this paper, an alternative approach for constructing Lagrangians for driven and undriven linearly damped systems is proposed, by introducing a redefined time coordinate and an associated coordinate transformation to ensure that the resulting Lagrangian satisfies the Helmholtz conditions. The approach is applied to canonically quantize the damped harmonic oscillator and although it predicts an energy spectrum that decays at the same rate to previous models, unlike those approaches it recovers the classical critical damping condition, which determines transitions between energy eigenstates, and is therefore consistent with the correspondence principle. It is also demonstrated how to apply the procedure to a driven damped harmonic oscillator.
We present an approach to the problem of quantization of the damped harmonic oscillator. To start with, we adopt the standard method of doubling the degrees of freedom of the system (Bateman form) and then, by introducing some new parameters, we get a generalized coupled set of equations from the Bateman form. Using the corresponding time-independent Lagrangian, quantum effects on a pair of Bateman oscillators embedded in an ambient noncommutative space (Moyal plane) are analyzed by using both path integral and canonical quantization schemes within the framework of the Hilbert--Schmidt operator formulation. Our method is distinct from those existing in the literature and where the ambient space was taken to be commutative. Our quantization shows that we end up again with a Bateman system except that the damping factor undergoes renormalization. Strikingly, the corresponding expression shows that the renormalized damping factor can be nonzero even if ``bare'' one is zero to begin with. In other words, noncommutativity can act as a source of dissipation. Conversely, the noncommutative parameter $\ensuremath{\theta}$, taken to be a free one now, can be fine tuned to get a vanishing renormalized damping factor. This indicates in some sense a ``duality'' between dissipation and noncommutativity. Our results match the existing results in the commutative limit.
For the quantum Caldirola-Kanai Hamiltonian, describing a quantum damped harmonic oscillator, a couple of constant of motion operators generating the Heisenberg algebra can be found. The inclusion in this algebra, in a unitary manner, of the standard time evolution generator , which is not a constant of motion, requires a non-trivial extension of this basic algebra and the physical system itself, which now includes a new dual particle. This enlarged algebra, when exponentiated, leads to a group, named the Bateman group, which admits unitary representations with support in the Hilbert space of functions satisfying the Schrodinger equation associated with the quantum Bateman Hamiltonian, either as a second order differential operator as well as a first order one. The classical Bateman Hamiltonian describes a dual system of a damped (losing energy) particle and a dual (gaining energy) particle. The classical Bateman system has a solution submanifold containing the trajectories of the original system as a submanifold. When restricted to this submanifold, the Bateman dual classical Hamiltonian leads to the Caldirola-Kanai Hamiltonian for a single damped particle. This construction can also be done at the quantum level, and the Caldirola-Kanai Hamiltonian operator can be derived from the Bateman Hamiltonian operator when appropriate constraints are imposed.