Classical propagators of quadratic quantum systems

Kyu Hwang Yeon, H. J. Kim, Chung‐In Um +2 more

Albino Hernández-Galeana, J. M. Rivera-Rebolledo, José Luis López-Bonilla +1 more

Olga V. Man’ko

The “classical propagator” for quadratic quantum systems is presented and its relation to the quantum propagator is shown. New formalism of quantum mechanics, based on the probability representation of the quantum state, is applied to the problem of an ion in a Paul trap, for which the classical propagator is written in the explicit form.

Kyu Hwang Yeon, Chung‐In Um, Thomas F. George

Solutions of the Schr\"odinger equation and propagators for the general quadratic and the linear Hamiltonian system of canonical variables whose coefficients are arbitrary and time dependent are obtained when the corresponding classical solution is oscillatory, linear, or monotonic. All of them are given by the coefficients of the Hamiltonian and the classical solution of the system under the conditions for which the classical solution exists.

Vishesh Kaushik, Navin Khaneja

Asish Kumar Dhara, S. V. Lawande

A class of time-dependent classical Lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view. Quantum mechanics is introduced through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out an explicit path integration, that the propagator for such a time-dependent system is related to the propagator of an associated time-independent problem. The expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of Lewis and Reisenfeld (1969) is discussed. Explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach.