Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System

Abstract

The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.

Quantum features of molecular interactions associated with time-dependent non-central potentials

“…. Substituting equations (75) and (76) with equation (77) into equations (42) and (44), the corresponding solutions are given by…”

“…To attain accurate results when we study molecular systems, it is necessary to introduce an exact Hamiltonian that yields actual time dependence of molecule behaviors. If we consider the convention that time-varying factors have usually been neglected on most studies of dynamical systems, the recent tendency [3,[36][37][38][39][40][41][42][43][44][45][46][47][48] for considering time dependence of physical parameters in this field may open up a new trend in the analysis of molecular interactions.…”