Alternative approach to the quantization of the 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.

Figure 1

$|{\psi }_n{(x,t)|}^2$ for Eq. (20) for $n=0,1$ at (a) $t=0$ and (b) $t=250$. All quantities are in natural units with $\hslash =1$.

Figure 2

For an initial state of $|0,0\rangle$, we plot $|c_m{\left(t\right)|}^2$ for $m=0,2,4,6$ in (a) the underdamped case, (b) the critically damped case, and (c) the overdamped case. All quantities are in natural units with $\hslash =1$.

Figure 3

For an initial state of $|2,0\rangle$, we plot $|c_m{\left(t\right)|}^2$ for $m=0,2,4,6$ in (a) the underdamped case, (b) the critically damped case, and (c) the overdamped case. All quantities are in natural units with $\hslash =1$.