Classical and quantum behavior of the harmonic and the quartic oscillators

Abstract

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.

Classical versus quantum evolution for a universe with a positive cosmological constant

“…There are two classes of Hamiltonians that have very special properties regarding the classical and quantum evolution they generate [13]. On the one hand, the quantum equations derived from any harmonic Hamiltonian, which are at most quadratic on the basic variables, do not contain any term.…”

“…When the initial conditions are not known with infinite precision, this uncertainty can be described by an initial probability distribution, and it is then necessary to consider the evolution of such a distribution on the classical phase space. As explicitly shown in [12,13], the evolution of a classical distribution can also be described in terms of its moments. And, interestingly, it turns out that the equations of motion for these classical moments can be obtained from the equations of motion for the quantum moments just by imposing a vanishing value of the Planck constant.…”

“…Note that through this decomposition the wave function that describes the quantum state of the system gets replaced by its infinite set of moments G a,b , which only depend on time. Moments that correspond to a valid wave function must fulfill certain relations due to the Schwarz inequalities, see [13] for a systematic derivation of such relations. The simplest, and probably most important, example is the Heisenberg uncertainty principle that, with this notation, takes the following form:…”

“…In the equations of motion they appear as a power series in 2 . Following the terminology of [13], the first ones will be named distributional effects, whereas the latter ones purely quantum effects. It is important to stress that, if quantum cosmology is to become a testable theory, one needs to discriminate between purely quantum effects and effects that might appear just due to different technical or measurement errors, which would imply a classical probability distribution.…”

“…The problem of time has also been analyzed in [21,22] within this framework. Finally the classical counterpart of the formalism developed in [15,16], for the analysis of the evolution of a classical distribution in terms of its moments, was presented in [13] and applied to the case of a particle on a potential in [23].…”