TY  - JOUR
A1  - Schuch, Dieter
T1  - Some remarks on analytical solutions for a damped quantum parametric oscillator
T2  - Journal of Physics: Conference Series
N2  - 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.
Y1  - 2019
UR  - http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/82712
UR  - https://nbn-resolving.org/urn:nbn:de:hebis:30:3-827122
SN  - 1742-6596
VL  - 1275
IS  - art. 012033
PB  - IOP Publ.
CY  - Bristol
ER  -