Classical and quantum behavior of the harmonic and the quartic oscillators

In a previous paper a formalism to analyze the dynamical evolution of classical and quantum probability distributions in terms of their moments was presented. Here the application of this formalism to the system of a particle moving on a potential is considered in order to derive physical implications about the classical limit of a quantum system. The complete set of harmonic potentials is considered, which includes the particle under a uniform force, as well as the harmonic and the inverse harmonic oscillators. In addition, as an example of anharmonic system, the pure quartic oscillator is analyzed. Classical and quantum moments corresponding to stationary states of these systems are analytically obtained without solving any differential equation. Finally, dynamical states are also considered in order to study the differences between their classical and quantum evolution.

Figure 1

The squared Euclidean distance on phase space between orbits corresponding to consecutive orders ${\mathrm{\Delta }}_n≔[q_n(t)-q_{n-1}(t){]}^2+[p_n(t)-p_{n-1}(t){]}^2$ is shown in a logarithmic plot for $n=2,\dots ,8$. The distance between the second-order and the classical point trajectory (${\mathrm{\Delta }}_2$) corresponds to the black (thickest) line. For the distance corresponding to higher orders (${\mathrm{\Delta }}_n$), the following colors have been used: brown ($n=3$), green ($n=4$), red ($n=5$), blue ($n=6$), purple ($n=7$), and orange ($n=8$); the thickness of the lines being decreasing with the order. The estimated numerical error of these solutions is around $10^{-16}$, thus higher orders are almost numerical error during the first two periods. Note that, at any time, we get a very rapid and strong convergence with the considered order.

Figure 2

In this figure the evolution of the position $q$ over its maximum value $q_{\max }$ (black continuous thick line) with respect to to the time (measured in terms of the period $T$) is shown. The rest of the lines correspond to some moments $G^{a,b}$ rescaled by a factor for illustrational purposes. More precisely, the red (long-dashed) line corresponds to $50G^{0,2}$, the green (dot-dashed) line to $10^3{G}^{0,4}$, the blue (dotted) line to $10G^{1,1}$, and the gray (continuous thin) line to $10^2{G}^{2,1}$. The behavior of the moments is oscillatory, with an increasing amplitude.

Figure 3

In these plots the evolution of $q$, $p$, and $(q^2+p^2)$ (divided by their maximum values) is shown in combination with the operators ${\delta }_1$ and ${\delta }_2$ acting on them. The black (thinnest) line represents the evolutions of the quantity we are considering, for instance in the upper plot $q/q_{\max }$, the blue (thickest) line represents the distributional effects, in the mentioned plot ${\delta }_{1}q$, whereas the red line stands for the purely quantum effects, in the considered graphic ${\delta }_{2}q$.

Figure 4

In this figure the initial value of the momentum is $p(t_0)=100$. Note that enhancement factors, by which differences between solutions are multiplied, differ from the previous case.