![]() We perform a rather complete computational analysis of the system's behavior inside the instability regions (lacunae), where the energy of the oscillator increases exponentially, as well as in the stability regions, and in particular in the vicinity of the (in)stability borders. In the case of the starting pure stationary eigenstate the evolution is exactly the same as for the classical microcanonical ensemble of initial conditions of the same starting energy. Using this, we derive the explicit exact formula for the evolution of the expectation value of the energy starting from an arbitrary normalizable initial state. We derive an explicit analytic formula for the quantum propagator in terms of the classical propagator. As a basic paradigm of such a Floquet system we consider the case of the harmonic oscillation of the oscillator frequency, which is convenient to handle theoretically and computationally, while keeping the general features. \bar is approximately 3N divided by the energy derivative of \ln \Omega? That all seems wrong.We study theoretically and computationally the behavior of the classical and quantum parametrically periodically driven linear oscillator. Where M is an integer that describes the extent of excitations in the system andĪ) Find density of states, Ω(M, N) corresponding to the total energyī) Using microcanonical ensemble, show that the internal energy For the 3N-oscillator system, given that the total energy is given as followsĮ = M\hbar \omega (3/2) N\hbar \omega Where \hbar \omega / 2 is the ground state energy of the oscillator. \epsilon = \hbar \omega/2 3\hbar \omega/2 . From quantum mechanics, the allowed energies of a 1D oscillator with angular frequency ω is given by Assume that every atom constitutes an independent oscillator and all oscillators are characterized by the angular frequency ω. This is relevant in that the atoms in a solid are sitting around their equilibrium positions. A system consists of 3N (N > 1) independent, identical, but distinguishable one-dimensional oscillators. ![]()
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