Alternative approach to the quantization of the damped harmonic oscillator

Abstract

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.

Smooth transformations and ruling out closed orbits in planar systems

“…The limit cycle of the Van der Pol oscillator, 5 (see pp. [67][68][69][70][71][72][73][74][75][76][77][78][79][80][81][82] which winds around the origin, necessarily crosses both of these curves, on which the solutions ( 54) and ( 55) do not exist, confirming Theorem 2. See Fig.…”

“…If so, then this dualism may help toward quantization 77 of systems which are not purely Hamiltonian, which is a topic of ongoing research with theoretical and practical significance. [78][79][80][81] A geometric approach to unifying Hamiltonian and gradient dynamics is taken by Esen et al 82 To analyze the saddle-type Hamiltonians discussed here, it may prove useful to introduce hyperbolic actionangle coordinates in the spirit of Waalkens et al 83 (see pp. [25][26].…”