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

Abstract

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. *

Alternative approach to the quantization of the damped harmonic oscillator

“…Observe that, up to a phase factor, these are the same wavefunctions as found in Ref. [35] for K = m/2α (where m is the mass of the oscillator). In Fig.…”

“…Equation ( 26) may be solved as described in Ref. [35], and the references therein, for initial condition c m (0) = δ m,n , where |n, 0 is the initial state of our system. This is achieved by solving…”

“…As before, in the critically damped case ξ = 0, so following Ref. [35] we expand around ξ = 0 to obtain…”

“…However, such an approach is not often practical, for example when characterizing superconducting quantum circuits, which rely on experimentally determined estimations of the damping conditions [29][30][31]. By providing a closed description of the system, the BCK approach therefore remains a highly favored approach to quantize the damped harmonic oscillatoras evidenced by continued analysis of the approach [19,20,[32][33][34][35][36]-and solutions to H BCK have been thoroughly investigated [10,37,38].…”

“…Second, in Ref. [35] the authors claim that Eq. ( 2) describes a doubled system of both a damped harmonic oscillator and an amplified harmonic oscillator [Eq.…”