Transformation of the free propagator to the quadratic propagator

B. Bassetti, E. Montaldi, Mario Raciti

José Maria Filardo Bassalo

José Maria Filardo Bassalo

Qiliang Zhao, Pengming Zhang, P. A. Horváthy

The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping “Eisenhart–Duval” lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation, which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to an arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile.

José Maria Filardo Bassalo, Paulo de Tarso Santos Alencar, M. Cattani

Antonio Boulhosa Nassar, Luis C. L. Botelho, José Maria Filardo Bassalo +1 more

Via the Schwinger action principle, we derive a formula for the transformation function (or propagator) of time-dependent systems that can be expressed into a quadratic form. We present also a new type of two-time rotation-translation of coordinates which makes it possible to evaluate the exact transformation function of the three-dimensional time-dependent charged harmonic oscillator in cross time-varying, arbitrary magnetic and laser fields. We demonstrate that such a transformation function can be obtained from that for a free-particle in the new two-time and space coordinate system, whereby the scalar and vector (quadratic) potentials are transformed away from the quantum-mechanical motion. We also show how the quantum-mechanical eikonal theory of this problem arises in a natural way from the Schwinger action principle.