Quantization of time-dependent non-central singular potential systems in three dimensions by using the Nikiforov-Uvarov method

Abstract

Quantum solutions of a time-dependent Hamiltonian for the motion of a time-varying mass subjected to time-dependent singular potentials in three dimensions are investigated. A time-dependent inverse quadratic potential and a Coulomb-like potential are considered as the components of the singular potential of the system. Because the Hamiltonian is a function of time, special techniques for deriving quantum solutions of the system are necessary. A quadratic invariant operator is introduced, and its eigenstates are derived using the Nikiforov-Uvarov method together with a unitary transformation method. The Nikiforov-Uvarov method enables us to solve the eigenvalue equations of the invariant operator, which are second-order linear diffierential equations, by reducing the original equation to a hypergeometric type. According to the invariant operator theory, the wave functions of the system are represented in terms of the eigenstates obtained in such a way. The difference of the wave functions from the eigenstates of the invariant operator is that the wave functions have time-dependent phases while the eigenstates do not. By determining the phases of the wave functions via the help of the Schrödinger equation, we identify the full wave functions of the system and address their physical implications.

Quantization of time-dependent singular potential systems: Non-central potential in three dimensions

“…(2) reduces to that of the system given in Ref. 15. Hence, our system that will be treated from now on is a generalization of the system that has been managed in Ref.…”

“…Hence, our system that will be treated from now on is a generalization of the system that has been managed in Ref. 15. The Schrödinger equation for this Hamiltonian is given by If we think the time-dependence of the Hamiltonian, it may be impossible to derive exact quantum solutions of the system relying only on the conventional separation of variables method.…”

“…You can see an example of such particular cases from Ref. 15, which corresponds to the case chosen the parameters as µ(t) = µ 0 /(1 + kt) and…”

“…(B1) of Ref. 15: If we think that the expression in the square root of this equation should be a square of a polynomial, it is possible to find the allowed functions for each k as…”

“…On the other hand, the Hamiltonian in Ref. 15 is just a specific example of the one that will be treated here. Starting from a general quantum description of the time-dependent non-central potential system, quantum features of the system based on Schrödinger solutions will be investigated.…”