Some remarks on analytical solutions for a damped quantum parametric oscillator

Abstract

The time-dependent Schrödinger equation for quadratic Hamiltonians has Gaussian wave packets as exact solutions. For the parametric oscillator with frequency ω(t), the width of these wave packets must be time-dependent. This time-dependence can be determined by solving a complex nonlinear Riccati equation or an equivalent real nonlinear Ermakov equation. All quantum dynamical properties of the system can easily be constructed from these solutions, e.g., uncertainties of position and momentum, their correlations, ground state energies, etc. In addition, the link to the corresponding classical dynamics is supplied by linearizing the Riccati equation to a complex Newtonian equation, actually representing two equations of the same kind: one for the real and one for the imaginary part. If the solution of one part is known, the missing (linear independent) solution of the other can be obtained via a conservation law for the motion in the complex plane. Knowing these two solutions, the solution of the Ermakov equation can be determined immediately plus the explicit expressions for all the quantum dynamical properties mentioned above. The effect of a dissipative, linear velocity dependent friction force on these systems is discussed.

Dynamical Invariants for Generalized Coherent States via Complex Quantum Hydrodynamics

“…The solutions of the Schrödinger equation strongly depend on the specific time-dependence of ω(t). Few frequencies have analytical solutions that exist, e.g., for ω(t) = 1 a(t+b) (see, e.g., [21,22]). Nevertheless, the dynamical invariant derived in the following exists for any ω(t)!…”

Analysis of Novel Oscillations of Quantized Mechanical Energy in Mass-Accreting Nano-Oscillator Systems

“…Hamiltonians of most actual physical systems such as forced oscillators [1][2][3][4][5], massaccreting oscillators [6][7][8][9][10][11], and damped oscillators [11][12][13][14][15] are a function of time. For this reason, these systems are called time-dependent Hamiltonian systems (TDHSs).…”