Construction of time-dependent dynamical invariants: A new approach

Abstract

We propose a new way to obtain polynomial dynamical invariants of the classical and quantum time-dependent harmonic oscillator from the equations of motion. We also establish relations between linear and quadratic invariants, and discuss how the quadratic invariant can be related to the Ermakov invariant.

Dynamical invariants and quantization of the one-dimensional time-dependent, damped, and driven harmonic oscillator

“…We note that the procedure in [21] This result alone would make us believe that the system is indeed dissipative since it is clear that the allowed classical states would collapse to the zero volume in time. However, if there would be a local transformation to a set of canonical variables, a volume preserved phase-space would emerge.…”

“…We proceed by calculating the first-order dynamical invariants related to (2) with the method proposed by [21]. In this case we define two arbitrary complex functions α (t) and β (t).…”

Dynamical Invariants and Quantization of the One-Dimensional Time-Dependent, Damped, and Driven Harmonic Oscillator

“…We proceed by calculating the first-order dynamical invariants related to Eq. 2a with the method proposed by [21]. In this case, we define two arbitrary complex functions α (t) and β (t).…”

“…Otherwise, the above invariants resemble the case of the oscillator with time-dependent frequency already addressed in the ref. [21].…”

“…In this work, we show how the definition of first-order invariants allows us to approach the quantization of the one-dimension time-dependent, damped, driven harmonic oscillator (TDDDHO). In Section 2, we follow [21] and calculate the linear invariants for the TDDDHO by taking the combinations of the equations of motion. Next, in Section 3, we construct the quadratic invariant and find a Steen-Ermakov-like equation.…”