Quantization of the Damped Harmonic Oscillator via the Noether Invariants

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.