Operator method for quantum anharmonic oscillator (review)

И. Д. Феранчук, L I Komarov, I.V. Nichipor +1 more

Subir Kumar Bose, U. B. Dubey, N. Varma

The present review is devoted to the study of certain aspects of anharmonic, time-dependent and damped oscillator(s) system using different theoretical techniques. A theoretical understanding of these systems is important for application in many problems in physics, mechanics and other fields. We discuss in detail the difficulties in the theoretical analysis of the problem. In particular we discuss here the regular, well-behaved perturbative solution, the large quantum number behaviour of anharmonic oscillator(s) using the technique of coherent states, exact solution of quantum anharmonic oscillators, the electromagnetic radiation emitted by a charged particle executing damped anharmonic oscillator motion using Krylov-Bogoliubov approximation method, use of invariants to obtain solution and coherent states of time-dependent oscillator(s), the derivation of perturbative frequencies of a damped coupled anharmonic oscillators system using suitable canonical transformation in the framework of Hamilton-Jacobi formalism and the quantisation and construction of coherent states of a damped oscillator using time-dependent operators.

Subir Kumar Bose, D. Tripathy

The present review is devoted to the study of the problem of coupled anharmonic oscillators. A perturbative solution is obtained for the system of an undamped and the damped coupled anharmonic oscillators in the coherent state representation. The solution does not contain the vicious secular terms and shows, explicitly, the anharmonic effects of a coupled system. In order to derive the perturbative solution of a damped coupled system a new frame of time, called quasi time τ, has been exploited. The large quantum number behaviour of coupled anharmonic oscillators has been derived using the path integral method.

Alexander V. Turbiner, Juan Carlos del Valle Rosales

Christopher C. Gerry

Jialun Zhang, Raphaël Khan, Sheng Ding +1 more

The exact solution to a quantum forced anharmonic oscillator is derived. The equation of motion is solved by using the integral operators and iteration method. As a result, the time development of the displacement and momentum operators of the forced anharmonic oscillator is obtained. These operators are presented as a Laplace and a subsequent inverse Laplace transform of suitable functionals.